af u1 l1 expressions and properties 1-28-09

SolvingOne-VariableEquations

AIIFAlgebra II Foundations

Teacher Manual

Table of Contents

Lesson Page Lesson 1: Expressions and Properties ...............................................................................................................1

Lesson 2: Solving Equations by Using Properties .........................................................................................28

Lesson 3: Order of Operations..........................................................................................................................45

Lesson 4: Solving Equations Using Order of Operations .............................................................................71

Lesson 5: Solving One-Variable Inequalities..................................................................................................92

Lesson 6: Solving Absolute Value Equations and Inequalities..................................................................122

Lesson 7: Ratios, Proportions, and Percent of Change ...............................................................................142

Assessments ......................................................................................................................................................166

CREDITS Author: Dennis Goyette and Danny Jones Contributors: Robert Balfanz, Dorothy Barry, Leonard Bequiraj, Stan Bogart, Robert Bosco, Carlos Burke, Lorenzo

Hayward, Vicki Hill, Winnie Horan, Donald Johnson, Kay Johnson, Karen Kelleher, Kwan Lange, Dennis Leahy, Song-Yi Lee, Hsin-Jung Lin, Guy Lucas, Ira Lunsk, Sandra McLean, Hemant Mishra, Glenn Moore, Linda Muskauski, Tracy Morrison, Jennifer Prescott, Gerald Porter, Steve Rigefsky, Ken Rucker, Stephanie Sawyer, Dawne Spangler, Fred Vincent, Maria Waltemeyer, Teddy Wieland

Graphic Design: Gregg M. Howell © Copyright 2009, The Johns Hopkins University, on behalf of the Center for Social Organization of Schools. All Rights Reserved. CENTER FOR SOCIAL ORGANIZATION OF SCHOOLS Johns Hopkins University 3003 N. Charles Street—Suite 200 Baltimore, MD 21218 410-516-8800 410-516-8890 fax All rights reserved. Student assessments, Cutout objects, and transparencies may be duplicated for classroom use only; the number is not to exceed the number of students in each class. No other part of this document may be reproduced, in any form or by any means, without permission in writing from the publisher. Transition to Advanced Mathematics contains Internet website IP (Internet Protocol) addresses. At the time this manual was printed, the website addresses were checked for both validity and content as it relates to the manual’s corresponding topic. The Johns Hopkins University, and its licensors is not responsible for any changes in content, IP addresses changes, pop advertisements, or redirects. It is further recommended that teachers confirm the validity of the listed addresses if they intend to share any address with students.

Solving One-Variable Equations Planning Document

Page i

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Planning Document: Solving One-Variable Equations Overview Solving One-Variable Equations is based on solving one-variable equations. The one-variable lessons include:

• Expressions and properties • Solving equations using properties • Order of operations (PEMDAS) • Solving equations using the order of operations • Solving one-variable inequalities • Solving absolute value equations and inequalities • Ratios, proportions, and percents

The number of total suggested days for the unit is 12 to 14.5. Adjustments may be needed based on student performance during the unit and amount of time available until the end of the semester.

Vocabulary

Integer Coefficient Constant Term Like terms Mathematical properties Numerical expressions Algebraic expressions Zero pair Additive inverse property Identity property of addition Multiplicative identity property Multiplicative property of zero Multiplicative property of negative one Reciprocals (inverses) Multiplicative inverse property Numerator Denominator Commutative property of addition Associative property of addition Commutative property of multiplication Associative property of multiplication Distributive property Variable Unknown Solution Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Substitution Property of Equality Symmetric Property of Equality Input Output Operations PEMDAS GEMDAS Order of operations Doing Undoing Solve Solving OR AND Less than Less than or equal to Greater than Greater than or equal to Inequality Compound Compound inequality Addition Property of Inequality Subtraction Property of Inequality

Page ii

AIIFVocabulary (Continued)

Multiplication Property of Inequality Division Property of Inequality Absolute Absolute Value Absolute value inequality Ratio Proportion Proportional Percent of change Percent Change in value Original value Scaled

Material List

Student Journal Setting the Stage Transparencies Dry-erase boards Markers and erasers Overhead tiles Student tiles

Tile pad Equal tile pad Graphing calculators Calculator view screen Blank transparencies Lesson specific transparencies Overhead projector

Construction paper Poster paper and markers Graphic organizer Scissors Centimeter rulers Inequality pad

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Solving One-Variable Equations Lesson 1: Expressions and Properties Page 1

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Lesson 1: Expressions and Properties Objectives • Students will be able to represent expressions with symbols, tiles, words, scenarios, and drawings. • Students will be able to represent equations with symbols, tiles, words, scenarios, and drawings. • Students will be able to represent and write equivalent expressions. • Students will be able to use and determine mathematical properties and rules related to expressions and

equations. Essential Question • How can words, symbols, tiles, drawings, and scenarios represent mathematical expressions and

equations? Tools • Student Journal • Setting the Stage transparency • Overhead projector • Dry–erase boards, markers, erasers • Overhead Tiles • Student Tiles • Tile Pad (Located at end of unit.) • Equal Tile Pad (Located at end of unit.) • Poster Paper and Markers • Calculator

Warm Up • Problems of the Day Number of Days • 2 days (A suggestion is to complete activities 1 and 2 on the first day. On the second day, complete

Activity 3, Practice Exercises, and Lesson Quiz.) Notes • Prior to teaching, you will need to prepare transparencies from the master hard copies supplied in this

manual. • At the end of each lesson in Algebra II Foundations there are Practice Exercises, Outcome Sentences, and a

small quiz. The authors suggest that teachers use these tools as needed and as time allows.

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Teacher Reference Lesson 1 Introduce students to this lesson by stating something such as, “One of the main goals of mathematics is to help describe the world around us. This is done by describing patterns related to numbers and by generalizing those patterns into expressions, equations, and functions. The basic foundation of these expressions, equations, and functions are mathematical properties. It is because these properties always hold true that scientists, business people, engineers, doctors, mathematicians, and almost anyone else in a career field can consistently calculate accurately and appropriately. It would be very strange if an engineer on the East Coast solved the same equation as an engineer on the West Coast and obtained a different answer. If people using math in the world around us could obtain different answers for the same calculations, we wouldn’t be very confident in bridges, buildings, investments, surgeries, or anything that relied on mathematical calculations. Because we can be confident that there are many solid consistent truths (properties) in mathematics, let’s investigate them as our foundation for this course. In this lesson, we will study these mathematical properties with various representations so that each of you can understand the properties in depth. We will represent the properties with numbers, variables, tiles, drawings, and words, both written and spoken.” Vocabulary The following is a list of the main vocabulary and rules developed or discussed in this lesson. You may add other words to this list as you find the need. Integer Coefficient Constant Term Like terms Mathematical

properties Numerical expressions

Algebraic expressions

Zero pair Additive inverse property

Identity property of addition

Rules for adding integers

Rules for adding rational numbers in fraction form

Rules for multiplying integers

Rules for dividing integers

Multiplicative identity property

Multiplicative property of zero

Multiplicative property of negative one

Reciprocals (inverses)

Multiplicative inverse property

Numerator Denominator Rule for multiplying rational numbers in fraction form

Commutative property of addition

Associative property of addition

Commutative property of multiplication

Associative property of multiplication

Distributive property

Variable unknown

The authors suggest that the teacher establishes a method to deal with the many terms in this lesson as well as future lessons. Here are three suggestions that the teacher could use separately or in combination.

• Create a word wall that you add to as new words are introduced. • Have students complete a vocabulary organizer for each new word. A sample blank vocabulary

organizer is supplied at the end of the unit. • Assign students to keep a journal of all the new and important mathematical terms and ideas.


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Teacher Reference Pre-Activity Before students begin Activity 1, it is highly suggested that you have students explore the tiles supplied in the materials tub. Tiles will often be used to represent properties, expressions, and equations and would be helpful if students are familiar with them. There is an overhead version of the tiles to model at the overhead projector as needed. The following is one method you could use to allow the students to explore the tiles. Give each group a set of tiles and have the students explore the various tiles for a few minutes. In their groups, have the students either categorize the tiles by shape, size, and color or have students describe the different characteristics of the tiles. After groups have completed categorizing or describing the tiles, have students share at the overhead projector. Then through a series of guiding questions and tile displays on the overhead, have students come to the following conclusions and agreements. Note: Some teachers will have students mark the tiles with symbols such as (+1), (–1), (+x), etc.

• The small blue square can represent 1 positive unit (+1). • By flipping the small blue tile it becomes a small red square that can represent 1

negative unit (–1). • There is also a rectangle that is broken into five units. You may want to remove this

from the set so as not to confuse the students with the x or y variable. • The short light blue rectangle can represent positive x units (x). • By flipping the short light blue rectangle it becomes a short red rectangle that can

represent negative x units (–x). • The medium light blue square can represent positive x-squared (x2). • By flipping the medium light blue square it becomes a medium red square that can

represent negative x-squared (–x2). • The long purple rectangle can represent positive y units (y). • By flipping the long purple rectangle it becomes a long red rectangle that can

represent negative y units (–y). • The large purple square can represent positive y-squared (y2). • By flipping the large purple square it becomes a large red square that can represent

negative y-squared (–y2). • The large green rectangle can represent a rectangle with dimension x by y. (xy

square units). • By flipping the large green rectangle it becomes a large red rectangle that can

represent a negative xy (–xy). • You will also want students to realize the relationships between the tiles in terms of

dimensions. For example, the small blue tile is one unit in length and one unit in width, representing 1 square unit. The short light blue rectangle is x units in length and one unit in width, representing x square units. The long purple rectangle is y units in length and one unit in width, representings y square units. All three of these objects have the same width but different lengths.

The students do not need to master all the details of the tiles at this time. Students need to become familiar with them in order to use them to represent various numbers and expressions in this lesson.

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Setting the Stage Teacher Reference The goal for this Setting the Stage is to help students understand that there are multiple methods to represent an idea, concept, or fact. Eventually, students will link this multiple representation idea to algebraic facts and concepts with various methods such as symbols, words, graphs, numbers, tables, and drawings. Ask the class if they have ever played charades. Display the Setting the Stage transparency. If a good portion of the class has not played charades, have student volunteers who have played the game, explain and model the game. Have a student volunteer record the rules at the overhead or board, as students explain the game. Lead a discussion with the class about how charades uses hand and body gestures to represent words or word phrases. Have different student volunteers come to the front of the class and demonstrate how they might represent a specific word. You can pick the word or use some from the following list: run, sleep, confused, angry, proud, soccer. Make sure the students don’t know the word first, but have to guess the word based on the volunteer demonstrating. Then, have the class get into groups of four. You could have students get into groups by passing out expressions in sets of four. The students would have to find four people that have equivalent expressions. Give the class two to three minutes to create hand and body gestures that could be used in a game of mathematics charades. Example math words might include: variable, unknown, adding, subtracting, multiplying, and dividing. Have each group demonstrate one or more of its mathematical words. Lead a class discussion that bridges the idea that in math there will be various ways to represent ideas: numbers, symbols, graphs, drawings, words, scenarios, etc.


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Setting the Stage Transparency

C H A R A D E S

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Teacher Reference Activity 1 Before looking at specific mathematical properties in Activity 2, have students investigate in this activity the basic concepts of integers and how to represent numerical and algebraic expressions with multiple representations. Guide the class through the first page on integers and have the students record answers to Exercises 1 through 3. Make sure students understand how to represent integers with symbols and tiles. Now have students investigate the representation of a numerical expression with words, scenarios, symbols, tiles, and drawings. Have the class continue working in groups from the Setting the Stage activity. Give the class about two minutes to write down a list of numbers and mathematical symbols and combine them in meaningful ways using prior knowledge. Make sure to tell the class to represent their combination multiple ways. Model an example, using the numbers 4 and 6 and the operation of addition(+):

Symbolically: 4 + 6 Written Words: four plus six or the sum of four and six Verbal Words: "Four plus six" Tiles: (Display two groups of tiles, one containing 4 units and the other 6 units, then combine the two groups) Scenario: I had four video games and I bought six more video games.

Walk around the classroom, checking with students and asking questions to help students correctly represent a numerical expression. Now model an algebraic expression such as 2x – 3 with multiple representations.

Symbolically: 2x – 3 Written Words: Three less than the product of two and some number. Verbal Words: "Two times x minus three" Tiles: (Display two positive x tiles and three negative unit tiles) Scenario: The cost of two identical pairs of pants minus three dollars.

Now have each group create its own algebraic expression and represent the expression with multiple representations. Bring the class back together and have each group share. Have the class agree on a definition and explanation of the terms numerical expression and algebraic expression. Tell the class that the many representations they shared were great examples of expressions and that they need to be able to represent expressions in different ways to fully understand and appreciate the power of mathematics. To help the students create the definitions, here are a few things to keep in mind.

• The term numerical means involving numbers and the term expression means phrase. This means that a numerical expression is a mathematical phrase involving only numbers and one or more operational symbols with or without grouping symbols.

• Algebraic expressions are expressions containing variables. • A single number or variable with no operation can also be considered an expression. Note: Different

textbooks define numerical and algebraic expressions differently. You may want to refer to the Algebra and Algebra II textbooks that your school uses as you help students create a definition.

Have the class work on Exercises 4 through 16 in their groups. Bring the class together and have each group share one of the exercises with the class. The vocabulary terms coefficient, constant, term, and like terms are introduced in Exercises 14 and 15.


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Activity 1 SJ Page 1 In this course, you will develop skills and concepts that lead to an understanding of algebraic concepts, such as unknowns, algebraic expressions and equations, graphing algebraic functions, and applying algebra to represent real-world situations. Before we jump into these concepts, let’s make a few links to prior understanding. Remember, integers are the positive and negative natural numbers including zero. You can represent integers on a number line by placing points above each number as shown below. 1. What do the arrows on the number line represent? Sample response: The arrows at each end represent

that these numbers continue infinitely in each direction. Representing integers in relation to natural numbers and whole numbers with a Venn-Diagram may look like the figure to the right. 2. Place 10 other integers on the Venn-Diagram in

the appropriate location. Answers will vary. Look at Student Journals to determine appropriate responses.

There are many ways to represent an individual integer. We generally represent an integer with a positive or negative number, but we can also represent them with words, scenarios, graphs, and even tiles. Explore the following example of the integer negative 7. Number: –7 Tiles: Words: Negative seven Scenario: “It’s minus seven degrees Fahrenheit outside.” Drawing or Graph: 3. Pick a different integer other than a negative seven and represent it with the following methods.

Answers will vary a. Words: b. Number: c. Scenario: d. Tiles: e. Drawing or Graph:

–8 –7 –6 –5 –4 –3 –2 –1 0 1 2

0 1 2 3 4–1 –2 –3 –4

integers –1

–2–3

–5–4

whole numbers 0

1 2 3 4 5 natural numbers

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SJ Page 2 4. Use some, or all, of the following numbers and operations to create a numerical expression and

represent it with symbols, words, tiles, and a scenario. 7, –9, 3, +, –, •, ÷, (, ) Answers will vary. A sample response

might be: 7 + (–9); “Seven plus a negative nine”; “I had seven dollars in my checking account and then I wrote a check for $9.”

5. Write an algebraic expression, words, and a scenario for the tile representation below. Sample response: 2x + 4; “Two x tiles plus four unit tiles or two unknowns

plus four”; “Four more minutes than twice the amount of unknown minutes.” 6. Write an algebraic expression for the statement below and describe the unknown quantities. Then,

represent the scenario with tiles. "$15 more than Tanya."

Answers will vary. A sample response might be: If T represents the amount of money Tanya has, then the expression would be T + 15.

7. Write an expression and a scenario for the following statement. 20% of 80 Sample response: 0.2(80); Twenty percent of an eighty-dollar coat. 8. Translate “The quotient of x and seven” into an algebraic expression. The algebraic expression is x / 7. 9. Translate “Thirty yards less than the length” into an algebraic expression. L – 30. 10. A 24-ounce jar of apple juice is poured into two different sized containers. Write the algebraic

expression for the amount of juice poured into the smaller container, in terms of the larger container, if g ounces were poured into the larger container.

24 – g.

Numerical Expression – Algebraic Expression –


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11. Write an algebraic expression, words, and a scenario, for the tile representation below. SJ Page 3

Sample response: 4x – 6; "Four unknowns minus six"; "Six less than four times the amount of total yards in the football game."

12. Write an algebraic expression for the scenario below and describe the unknown quantities. Include a

tile representation. Three years older than Susan.

Answers will vary. A sample response might be: Let S be Susan’s age, then the expression would be S 3+ .

13. Write any algebraic expression of your choice, and then represent it with multiple methods. Answers will vary. 14. For expressions, the parts that are added are called terms. For example, 6x and 9 are the terms for the

expression 6x + 9. What are the terms for each of the expressions below? a. 3x + 5 b. 8 + (–5) c. 5x + 9x 3x, 5 8, –5 5x, 9x 15. In a term, the number multiplied by a variable is called a coefficient. A term that is just a number is

called a constant. For example, 6 is the coefficient of the term 6x and 9 is the constant in the expression 6x + 9. Determine the coefficients and constants for each of the expressions below.

a. 3x + 5 b. 8 + (–5) c. –7x + 9x 3 is the coefficient 8 is the constant –7 is the coefficient 5 is the constant –5 is the constant 9 is the coefficient 16. If two terms are the same except for their coefficients, the terms are said to be like terms. For example,

8x and –9x are like terms and 8x and 5y are not like terms. Two constants are also like terms. Determine which terms below are like terms.

a. 3x b. 8 c. 5x d. 9y e. –10x f. –4y g. –9 h. 1 i. x j. y k. 0.5x l. 0.3 3x, 5x, –10x, x, and 0.5x are like terms 8, –9, 1, and 0.3 are like terms because they are all constants 9y, –4y, and y are like terms

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Teacher Reference Activity 2 In this activity, students will learn and review some universal mathematics truths. We will officially refer to most of these truths as mathematical properties. The goal will be for students to review them quickly, to represent the properties with multiple representations, and to understand how unknowns (letters or variables) can help us write a generic statement that would represent the property. Tell students, “We know that adding a positive one and a negative one results in a value of zero. We will refer to 1 + (–1) as a zero pair.” Use tiles to represent this zero pair on the overhead. Then have students represent 5+(–5) with tiles at their desks and ask them to describe why this also represents zero. Students should be able to show you that there are five sets of zero pairs each representing zero. Now, place a positive x tile and negative x tile on the overhead and ask students why this also represents a zero pair. Have students represent 4x + (–4x) with tiles and ask them to describe why this represents zero. Have students read the additive inverse property then ask students how the concept of zero pairs helps them understand this property. Make sure students understand that the variables “a” and “–a” represent all possible values and their opposites. Have students complete Exercises 1 and 2. The goal of Exercises 3 through 22 is to help students become familiar with some of the basic mathematical properties. (Note: The addition, subtraction, multiplication, and division properties of equality will be introduced in Lesson 2 where students will begin to solve equations.) Because there are so many properties in this lesson, divide the class into expert groups and assign each group a different set of exercises to study. If you have 6 groups of 4, divide the exercises as follows: Group 1-Exercises 3 through 7; Group 2-Exercises 8 through 13; Group 3-Exercise 14 through 17; and Group 4: Exercises 18 through 22. Have each group complete its assigned exercises. You may want each expert group to create a mini-presentation or some method (acrostic or memory technique) that can help all the students remember their property or rule. You will need to stop by the group that is assigned Exercises 18 through 22 to model the distributive property (see suggestions below). After the expert groups have presented their exercises and properties have them regroup so that each new group has at least one person from each of the original groups. Now have the new groups complete Activity 3. Suggestions for Modeling Distributive Property Have students parallel you with their own example as you model one similar to the examples below. (Note: It may be convenient for you and the students to use dry-erase boards as they parallel you.) Example 1: Money Demonstrate to students that calculating the price of three candy bars for $1.05 each is the same as determining the sum of 3 times $1 and 3 times $0.05. It is the same as 3 one-dollar bills and 3 nickels. ( ) ( ) ( )= +3 $1.05 3 $1.00 3 $0.05 Example 2: First method with tiles Model the distributive property for the expression ( )( )+ −3 3 2 using tiles and symbols at the overhead, while

students parallel you with ( )− +4 3 4 . Three sets of a group of 3 unit tiles and –2 unit tiles is the same as three sets of 3 unit tiles and 3 sets of –2 unit tiles.

( )( ) ( ) ( )+ − = + −3 3 2 3 3 3 2


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Model the reverse of the distributive property for the class with the expression ( ) ( )− +3 2 3 5 , while the students parallel you with ( ) ( )+ −2 3 2 5 . ( ) ( ) ( )− + = − +3 2 3 5 3 2 5 Now model the distributive property for the expression ( )+2 2x , while students parallel you with ( )+3 2 4x . Note: You may want to model the reverse of the distributive property with variables as well. ( ) ( )+ = +2 2 2 2 2x x Have a student volunteer model the distributive property with 3(2x – 3) using tiles and symbols. You may need to help the student change the subtraction to adding the opposite first. Explain that there are cases where tiles will not be a good model for the distributive property. Knowing how to manipulate the symbols will be important with expressions like ( )−12 7 5x because a student wouldn’t have enough tiles and ( )−0.5 0.2 3x because a student wouldn’t have partial tiles. Example 3: Second method with tiles This method lends itself to multiplying polynomials and factoring. It is similar to the second example, but deals with an area model related to length times width. You can represent ( )( )+ −3 3 2 as a rectangular area

model with a width of 3 and a length of ( )+ −3 2 . Building a row at a time would look like: By spreading the last representation out a little it should become clear that there are 3 sets of 3 and 3 sets of –2. This method could also be used with x and y–values.

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Activity 2 SJ Page 4 Additive Inverse Property

For any number a, a + (–a) = 0. 1. Think of four other ways to represent the additive inverse property by completing each exercise below. a. Words: A number plus its opposite is zero. b. Tiles: c. Symbols: a + (–a) = 0

d. Scenario: A debt of $20 added to a credit of $20 leaves a zero balance. We also know that adding zero to any number does not change the value of the number. This is called the identity property of addition. 2. Think of four different ways to represent the identity property of addition by completing each exercise

below. a. Words: Any number plus zero is that number. b. Tiles: c. Symbols: a + 0 = a

d. Scenario: The offense was at the 15 yard line and made a gain of zero yards to be at the 15 yard line again.

3. In the past, you’ve learned certain rules for adding integers. In your group, experiment with the

following calculations. Use tiles and the concept of zero pairs as needed, then write the rule(s) in your own words. Complete more examples if needed.

a. –8 + 5 b. –8 + (–5) c. 8 + 5 d. 8 + (–5)

e. Rule(s) for adding any integer to another integer: Answers will vary. Textbook definitions

usually contain the words absolute value. Because students have not reviewed absolute value yet in this course they might write something like: To add two integers with the same sign, ignore the signs and add the numbers then write the common sign. To add two integers with different signs, ignore the signs, subtract the numbers, and write the sign of the number that was larger without signs. A more textbook type answer would be: To add two integers with the same sign, add the absolute values of each number then write the common sign. To add two integers with different signs, determine the difference of the absolute values of each number then write the sign of the number with the greatest absolute value.

4. Use tiles to add the following like terms, then explain if your rule for adding integers also works for

adding like terms.

a. –6x + 4x b. –6x + (–4x) c. 6x + 4x d. 6x + (–4x)

e. Do rules for adding integers work for adding like terms? Explain. Students should find that the rules are basically the same. At this point students have not reviewed the distributive property, so students may develop more of an intuitive understanding with tiles instead of a formal distributive method of something like –6x + 4x = (–6 + 4)x = –2x.


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SJ Page 5 5. A statement you might hear a mathematician say is, “Subtracting is the same as adding the opposite.”

With symbols, the mathematician’s statement might look like a – b = a + (–b). Complete each example below with a calculator to examine each statement, and then write your conclusion as to whether you agree or disagree with this statement.

a. 7 – 15 b. 7 + (–15) c. 23 – 17 d. 23 + (–17) e. –7 – (–6) f. –7 + 6 g. Conclusion: Students should discover that adding the opposite is the same as subtracting. 6. How might you use tiles to represent the idea that subtracting is the same as adding the opposite? Be

prepared to share how your group represents this idea with tiles. Answers will vary. 7. Study the following properties and determine a way to show the property using tiles. For each

property, draw a sketch of what you did with the tiles. a. Commutative Property of Addition: You can add numbers in any order. For example,

4 + 5 = 5 + 4 or a + b = b + a. Answers will vary. b. Associative Property of Addition: The sum of three or more numbers does not depend on how

they are grouped. For example, 3 + (4 + 5) = (3 + 4) + 5 or a + (b + c) = (a + b) + c. Answers will vary.

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SJ Page 6 8. Sometimes people forget what to do with the sign when multiplying or dividing integers.

a. Complete the following calculations with a calculator, then write a rule for the sign of the product when multiplying integers.

–9(6) –9(–6) 9(6) 9(–6)

Rule for Multiplying Integers: Answers will vary. Sample response: The product of two positive integers is positive. The

product of two negative integers is positive. The product of a positive integer and a negative integer is negative.

b. Complete the following calculations with a calculator, then write a rule for the sign of the

quotient when dividing integers.

72/9 –72/9 –72/(–9) –72/(–9) c. Rule for Dividing Integers: Answers will vary. Sample response: The quotient of two positive integers is positive. The

quotient of two negative integers is positive. The quotient of a positive integer and a negative integer is negative.

9. What value can you multiply any number by and the result is the same number? Show an example

with numbers, then show another example with variables. This is called the Multiplicative Identity Property.

Sample response: You can multiply any number by one and get the same number. 5(1) = 5 a (1) = a 10. What value can you multiply any number by and the result is always zero? Show an example with

numbers, then show another example using a variable. This is called the Multiplicative Property of Zero.

Sample response: Multiplying any number by zero will give a value of zero. 5(0) = 0 a (0) = 0 11. What value can you multiply any number by and the result is the opposite number? Show an example

with numbers , then show another example using a variable. This is called the Multiplicative Property of Negative One.

Sample response: Multiplying any number by negative one will give the opposite number. 5(–1) = –5 a (–1) = –a

You may want students to also understand the property as –5(–1) = 5 or –a (–1) = a.


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SJ Page 7 12. The concept of multiplication can relate to multiple groups of objects. For example, the product of 3 and

4 can be thought of as 3 groups of 4 objects or 4 groups of 3 objects. See the figures below. Describe how this concept matches the Commutative Property of Multiplication, which states that you

can multiply numbers in any order. For example, 5(7) = 7(5) or ab = ba. Answers will vary. 13. Many calculators accept the use of parenthesis ( ) when entering numeric expressions. Use your

calculator to show that the statements below are true. a. 3 •(4 • 5) has the same value as (3 • 4)• 5 b. 0.5• (4 • 0.6) has the same value as (0.5 • 4) •0.6

The property that you just modeled is the Associative Property of Multiplication: The product of three or more numbers does not depend on how they are grouped. For example, 3 (4 • 5) = (3 • 4) 5 or a(bc) = (ab)c.

14. Study the method of adding the fractions 1/2 and 3/5 shown below and use the method to add the

fractions 1/8 and 3/7.

( )( )

( )( )+ = + = + =

1 5 3 21 3 5 6 112 5 2 5 5 2 10 10 10

Sample response: ( )( )

( )( )

1 7 3 81 3 7 24 318 7 8 7 7 8 56 56 56

+ = + = + =

a. Write a real-world situation that represents adding your two fractions. Answers will vary. b. If you had to generalize the method you used to add two fractions, how might you do that if the

fractions were ab

and cd

?

Answers may vary. Sample response: a c ad bc ad bcb d bd bd bd

++ = + =

( ) =3 4 12

3 Groups of 4 4 Groups of 3

( ) =4 3 12

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SJ Page 8 15. The method for multiplying rational numbers in fraction form is to multiply the numerators and

multiply the denominators. Two examples are given below, one is with specific numbers and the other is a general example with variables to represent all numbers.

( )( )

⎛ ⎞ = =⎜ ⎟⎝ ⎠

2 42 4 83 5 3 5 15

⎛ ⎞ =⎜ ⎟⎝ ⎠

a c acb d bd

, when ≠ 0b and ≠ 0d

Use the rule above to determine when each condition below could be true. Then, give one example for

each condition.

a. A positive rational number times a positive rational number gives a product that is smaller than either positive rational number. Sample response: If you multiply any two rational numbers that each have a value between 0 and 1 the product will result in a smaller number. For

example, the result of multiplying 12

and 34

gives a product of 38

, which is smaller than either

12

or 34

. You may need to have students compute with the equivalent decimal values such as

0.5 ( 0.75) is equal to 0.375. b. The product of two positive rational numbers is larger than either rational number. Sample

response: If you multiply any two rational numbers that each have a value greater than 1 the

product will result in a larger number. For example, the product of 32

and 105

is 3010

or 3, which

is greater than either 32

or 105

. You may need to have students compute with the equivalent

decimal values such as 1.5 ( 2) is equal to 3. c. The product of two positive rational numbers gives a value that is between each rational

number. Sample response: If you multiply a rational number that is between 0 and 1 with another rational number that is greater than one, the product will result in a value that is

between each rational number. For example, the product of 12

and 32

is 34

. You may need to

have students compute with the equivalent decimal values such as 0.5 ( 1.5) is equal to 0.75. 16. We know that multiplying reciprocals, also known as multiplying inverses, gives a value of 1. For

example, ⎛ ⎞ = =⎜ ⎟⎝ ⎠

5 6 30 16 5 30

. This mathematical truth is also called the Multiplicative Inverse Property.

Show that this property is true by multiplying 4/5 and 5/4. • =4 5 15 4

a. If you had to generalize the method you used to multiply two reciprocals, how might you do

that if the fractions were b/c and c/b? Answers will vary. You should point out to students that these fractions only work if b and c

are not equal to zero. b. How might you use your calculator to show that multiplying two reciprocals gives a value of 1? Answers will vary.


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SJ Page 9 17. You have probably heard a mathematics teacher say, “To divide by a fraction, multiply by the

reciprocal.” Using the dry–erase board, your calculator, or any other tool, prepare a short presentation on how you could show this fact to be true. You may want to include an example with variables to

show how to divide ab

by cd

.

Answers will vary. 18. Represent the Distributive Property with tiles and symbols for the following expressions. Draw a

sketch of each representation. a. ( )− +2 3 5x Tile Symbol ( ) ( ) ( )2 3x 5 2 3x 2 5− + = − + b. ( )−3 7x Tile Symbol ( ) ( ) ( )3 x 7 3 x 3 7− = + − c. ( ) ( )+ −3 2 3 2x Tile Symbol ( ) ( ) ( )3 2x 3 2 3 2x 2+ − = − d. ( ) ( )− +4 4 4x Tile Symbol ( ) ( ) ( )4 x 4 4 4 x 4− + = − +

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SJ Page 10 19. You can represent the Distributive Property of multiplication over addition with the symbols in an

equation as shown below. ( )+ = +a b c ab ac

a. Write a definition of the Distributive Property of multiplication over addition in your own words.

Answers will vary. b. Give an example of the Distributive Property of multiplication over addition with numbers

and/or variables. Answers will vary. 20. Show how you might represent the Distributive Property of multiplication over subtraction with

symbols in an equation. ( )a b c ab ac− = − a. Write a definition of the Distributive Property of multiplication over subtraction in your own

words. Answers will vary. b. Give an example of the Distributive Property of multiplication over subtraction with numbers

and/or variables. Answers will vary.


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SJ Page 11 21. The term Distributive Property is related to the words distribute, distribution, and distributor. Study

the following definitions, and then explain how these definitions are similar to, or different from, the mathematical term, Distributive Property.

• Distribute: The action of sharing something among a number of recipients.

For example, the political candidates will distribute the fliers throughout the neighborhood.

• Distribution: The way in which something is shared in a group or spread over an area. For example, changes to the wilderness affected the distribution of its wildlife.

• Distributor: The agent or company that supplies goods to stores and other businesses. For example, the wholesale distributor shipped the computer monitors to all the stores in the region.

Explanation: Answers will vary.

22. The Distributive Property can be used to multiply calculations in your head. For example, if you

bought six cans of soda that cost $0.98 each, you could represent $0.98 with $1.00 – $0.02 then multiply by 6 to determine the total cost of $5.88.

6(1.00 – 0.02) = 6.00 – 0.12 = 5.88

a. Use this method to determine the total price of purchasing eight jars of salsa that cost $0.96 each.

Describe how you completed the calculation in your head.

Sample response: I determined the products of 8 and 1 as well as 8 and 0.04 then subtracted to obtain $7.68.

8(1.00 – 0.04) = 8.00 – 0.32 = 7.68

b. Use this method to determine the total price of purchasing seven cans of chili that cost $1.07 each. Describe how you completed the calculation in your head.

Sample response: I determined the products of 7 and 1 as well as 7 and 0.07 then added to obtain $7.49.

7(1.00 + 0.07) = 7.00 – 0.49= 7.49

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Teacher Reference Activity 3 Continuing from Activity 2, make sure to reassign the expert groups so that each new group has one expert from each group in Activity 2. In this activity, students will investigate equivalent expressions. Lead a class discussion about the term equivalent. Ask guiding questions such as “What does it mean to be equivalent?” Give an example, for instance, “Four quarters make one dollar, so four quarters are equivalent to one dollar.” Ask for student volunteers to give other examples of equivalence. The students’ examples could be mathematical or non-mathematical. The key concept is that the students understand the term equivalent (equivalence). Have a student volunteer write the students’ examples on the board or on the overhead projector. Divide each group into pairs to work on Exercises 1 through 6. After completing these exercises, have each pair compare results with their group. Obtain a couple of examples for each exercise and lead a class discussion on the similarities and differences between equivalent expressions. The key concept is that students understand that there are many ways to create equivalent expressions. Students will now investigate equations by combining expressions. Lead a discussion about the symbol that mathematicians use to show equivalent expressions and the mathematical term that mathematicians use to describe two equivalent expressions. The class should refer to the terms equal sign and equation. Discuss the meaning of an equation. Ask guiding questions such as, “What makes an equation an equation?” “Why do we write equations?” or “What are we trying to do when we write an equation?” Ask student volunteers to model equations using symbols, tiles, written words, spoken words, scenarios, and drawings. The important concept is that students understand that the equal sign between two expressions represents equivalence as do the words “equal” or “is.” In general, any derivative of the verb “to be” can, under the right circumstance, represent equal or equivalent. Model writing equations from words using the application examples below, while the class follows along using dry–erase boards. You could have a student volunteer to do the modeling with assistance from the class as needed.

• “Together Kevin and Tanya have $42.” Write an equation for the amount of money that Kevin and Tanya have together. Sample answer: k + t = 42.

• Model the percent problem, “What percent of 30 is 15?” with an expression while the class models

“What percent of 80 is 20?” on dry–erase boards. Ask the students what important words in the problem helped them write their equation.

NOTE: You may also model the equation with two–variable tiles and the equal tile pad. This scaffolds, for the students, the use of the equal tile pad. Tell students that each portion of the tile pad represents an expression and that the equal sign between each pad makes the double tile pad an equation. A transparency master is available for the equal tile pad. Now have the student groups complete Exercises 7 through 9. Review these exercises as needed by having volunteers present. An equal tile pad is available at the end of the unit for students. You may want students to cut the pad out and put it in their binder.


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Activity 3 SJ Page 12 In this activity, you will investigate writing equivalent expressions. An example of equivalent expressions is 3 + 7 and 2(5). There are various ways you can show that these two expressions are equivalent. Study the three different representations below. Numerically: Each expression has a value of 10 so they are equivalent: 3 + 7 = 10 and 2(5) = 10 Scenario: A woman that had three one-dollar bills in her purse and added another seven one-dollar bills to her purse would have the same value as a person who had two five-dollar bills in his or her wallet. Tiles: 3 unit tiles added to 7 unit tiles is the same as 2 sets of 5 unit tiles. 1. Create two equivalent numerical expressions. Each numerical expression should have different

numbers and different operations. Answers may vary. A sample response might be: 3 + 4 and 7 • 1. 2. Write an equivalent numerical expression, using multiplication or division, for the expression 12 – 36. Answers may vary. A sample response might be: –2 • 12. It is also possible to create equivalent algebraic expressions. For example, the expression 2x + 3 is equivalent to the expression 5x – 3x + 1 + 2. With tiles this might look like. . . 3. Write an equivalent algebraic expression for 6x + 3x – 8 – 9. Use tiles if needed. Answers may vary. A sample response might be: 9x – 17. 4. Draw an equivalent tile representation for the expression 3x – x + 4 – 2. Answers may vary. A sample response might be:

Equivalent Expressions

Two expressions that have the same value.

is the same as

is the same value as

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SJ Page 13 5. Write, as a scenario, an equivalent numerical expression for, “I had 20 video arcade tokens and then

used 10 of the tokens playing video arcade games.” Answers may vary. A sample response might be: “I had 5 video arcade tokens and then I bought five

more.” 6. Write, as a scenario or real–world situation, an equivalent expression for, “One hundred fifty fewer cell

minutes than the available total.” Answers may vary. A sample response might be: “The first week I had my new cell phone I used ninety

minutes of the total available. The second week I used sixty minutes of the remaining available minutes in my cell plan.”

7. Write an equation for the following tile representation.

The equation is 5x = 5. 8. For the question, “What is 40% of 90?”, what are the important words that will be used to write an

equation to represent the question? Explain each use of the important words you chose. The important words in the problem are "What", "is", and "of". "What" will be used to represent the variable, "is" represents the equal sign, and "of" represents multiplication. a. Write the equation which represents the question.

The equation is x = 0.40 • 90. 9. Draw a tile representation of the equation 3x – 4 = 2. Sample tile representation is

=

=


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Practice Exercises SJ Page 14 1. Use some, or all, of the following numbers and operations to create a numerical expression and

represent it with symbols, words, tiles, and a scenario. 6, –7, 12, 1, 15, –9, +, –, •, ÷ Answers will vary. 2. Represent the scenario below with an expression. "The temperature dropped 8°F."

Answers will vary. A sample response might be: Let T represent the temperature before it dropped. Then the current temperature would be T – 8.

3. Write an algebraic expression, words, and a scenario for the following tile representation. Answers may vary. A sample response might be: 2x –x + 5 – 2; “Positive two

x tiles plus one negative x tile plus five unit tiles plus 2 negative unit tiles." “Jacob, a salesman at a wholesale computer store, began the day with two crates of computers and five individual computers. He was able to sell a whole crate of computers and two additional individual computers.”

4. Translate “The product of x and nine” into an algebraic expression and then write an equivalent

algebraic expression.

The algebraic expression is 9x. Equivalent expressions will vary. A sample response might be x(5 + 4). 5. Write an equivalent expression for 24 + 36 using multiplication. Answers will vary. A sample response might be: "6 • 10". 6. Write an equation for, “50 is what percent of 200?” The equation would be 50 = x • 200. 7. Write, in words, the following formula for the perimeter of a rectangle: P = 2l + 2w, where l represents

the length of the rectangle and w represents the width of the rectangle. "The perimeter of a rectangle is the sum of twice the length and twice the width."

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SJ Page 15 8. Determine which property or rule is represented in each equivalent situation below. a. ( ) ( ) ( )5 4 8 5 4 5 8− + = − + Distributive Property b.

Additive Inverse Property (zero pairs) c. 11 55 55 11+ = + Commutative Property of Addition

d. The area of a rectangle can be determined by multiplying the length times the width or by multiplying the width times the length.

Commutative Property of Multiplication

e. An accountant decided to add all the numbers in a list by adding groups of numbers that made a value of 10 first as shown below.

( ) ( ) ( )7 6 5 4 3 5 7 3 6 4 5 510 10 10 30

+ + + + + = + + + + += + + =

Some students may refer to the Associative Property of Addition and/or the Commutative Property of Addition. It is actually a combination of both properties.

f. ( )21 17 21 17− − = − + − Subtraction is the same as adding the opposite.

g. 9 3 9 210 2 10 3

⎛ ⎞÷ = ⎜ ⎟⎝ ⎠

Dividing is the same as multiplying by the reciprocal.

9. Use the distributive property to change the following expressions. Note: You may need to change a

subtracting to adding the opposite. a. ( )+5 7x = 5x + 5(7) b. ( )+2 2x = 2x + 2(2) c. ( )+a b c = ab + ac d. ( )−2 8x = 2x – 2(8) or 2x + 2(–8) e. ( )− −2 5x = –2x – 2(–5) or –2x + (–2)(–5) f. ( )−a b c = ab – ac or ab + a(–c)

=


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Outcome Sentences SJ Page 16 I didn’t know that zero pairs The distributive property now makes sense to me because All of these properties Using tiles really Reviewing expressions I didn’t know that expressions An equivalent expression is For me, converting words to expression and equations is ______________________________________ because

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Teacher Reference Lesson 1 Quiz Answers 1 The area of a triangle is A = (1/2)bh. Sample answer might be: If the base is 5 inches and the height is

10 inches, then the area is A = (1/2)(5)(10) = 25 square inches. 2. Answers will vary. A sample response might be: a. 4( 5 – 3) = 4(5) – 4(3) b. 6(3x + 2) = 6(3x) + 6(2) c. 9(4x) – 9(1) = 9(4x – 1) 3. a. is matched with 6. b. is matched with 4. c. is matched with 5. d. is matched with 2. e. is matched with 1. f. is matched with 3.

10

5


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Lesson 1 Quiz Name: 1. For the statement, “The area of a triangle is half its height times its base,” write an equation that

represents the words and give an example. Also, provide a drawing of your example. 2. Give three examples of the Distributive Property. 3. Match the column on the left with the appropriate answer from the column on the right.

a. 3x + 6 = 6 + 3x 1. A fraction divided by a second fraction is the same as the fraction multiplied by the reciprocal of the second fraction.

b. 7 + (–7) = 0 2. The Associative Property of Addition

c. ( ) =4 415 5

3. Subtracting is the same as adding the opposite.

d. 3 + (6 + 5) = (3 + 6) +5 4. The Additive Inverse Property

e. ÷ = •1 4 1 52 5 2 4

5. Identity Property of Multiplication

f. 3x – 6 = 3x + (–6) 6. The Commutative Property of Addition

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Lesson 2: Solving Equations by Using Properties Objectives • Students will be able to solve one-variable equations using the addition, subtraction, multiplication, and

division properties of equality. • Students will be able to check a solution to an equation by using the substitution property of equality. • Students will represent the addition, subtraction, multiplication, and division properties of equality with

symbolic equations, words, numbers, and tiles. • Students will translate real-world scenarios into equations and then solve and check for the unknown. Essential Questions • How can mathematical properties be used to solve equations and real–world applications?

Tools • Student Journal • Setting the Stage transparency • Overhead projector • Dry–erase boards, markers, erasers • Overhead Tiles • Student Tiles • Tile Pad • Equal Tile Pad • Poster Paper and Markers

Warm Up • Problems of the Day

Number of Days • 1 to 1 ½ days (A suggestion is to complete Activity 1 and as much of Activity 2 as possible on the first day.

On the second day, have students finish Activity 2, Practice Exercises, and Lesson Quiz.) Vocabulary Solution Substitution Substitution Property of Equality Addition Property of Equality Subtraction Property of Equality Division Property of Equality Multiplication Property of Equality Symmetric Property of Equality

Solving One-Variable Equations Lesson 2: Solving Equations by Using Properties Page 29

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Teacher Reference Setting the Stage: Describing Problems and Giving Solutions Give the students a problem, not necessarily related to math, that has at least one solution. A sample set of questions follow to choose from or ask one of your own that you believe will interest students. Before you have students answer, you may, as a class, have to agree on what is the actual problem. • What if you came to school today and didn't have enough money for lunch? How could you solve this

problem? • What if the electric power goes out at your house and the temperature is below freezing? • What if your MP3 player quit working? Have student volunteers share their solutions. Then lead a discussion about what it means for something to be a solution to a problem, any type of problem. It could be a mathematical or non–mathematical problem. Some guiding questions may be: • What does it mean for something to be a solution to a problem? • How might a solution to a mathematics problem be different from a solution to a family problem? • How might a solution to a mathematics problem be similar to a solution to a family problem? • Can you use similar strategies to solve a mathematics problem and a transportation problem? • What strategies can people use to solve problems? Are these strategies similar to how a person might solve

a mathematics problem? • How do you know if your solution to a problem is correct? Divide the class into groups of four. Display the Setting the Stage transparency. Assign each group one of the illustrations on the transparency. On a dry-erase board have each group describe a problem that the illustration might represent. On another dry-erase board have each group describe at least one solution to the problem they chose. For example, for illustration 1, the problem might be that the cloths are dirty and a solution could be to wash them. Have the students display their dry-erase boards. Then have the class complete a walk around the room to analyze each problem and solution. Have the class reflect on the similarities and differences of each of the solutions and strategies used. Tell students that as they progress through this mathematics course they will often solve problems and give solutions. Sometimes this will be for a real-world application problem and sometimes this will be for a specific symbolic equation. Optional Extension: Problem Solving Strategies: As students encounter story problems and real-world problems, it will help if they have a basic strategy or plan to attempt these problems. It was implied with this Setting the Stage that there are two steps: describe the problem and then give solutions. We know that this is probably not enough steps for a real-world mathematics problem, but it was a place to begin a discussion. Many teachers often give multiple steps as well as a graphic organizer, to help students solve problems. An optional approach is to develop (or have students develop) some type of graphic organizer or list of steps to solve a mathematics problem. A suggested graphic organizer is displayed to the right. Please adapt it to your needs or use another strategy that you have found useful.

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What could be the problem? What could be the solution?

1. 2.

3. 4.

5. 6.


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Teacher Reference Activity 1 Continue the discussion of solutions from the Setting the Stage with the class, but lead toward solutions of equations. Ask the students questions such as, “What is a solution to an equation?” or “What does it mean to be a solution to an equation?” Have the students agree on a definition of solution to an equation. A sample definition might be: A solution to an equation is the value or values that make the equation true. Have the students record the definition in their Student Journal. Model for the class, or have a student volunteer model, how to determine if an answer is a solution to an equation by substituting the solution into the equation. Use the equation 6x = 18 with a solution of x = 3, while the students use dry–erase boards to determine if x = 5 is a solution to the equation 4x = 20. Have the class hold their dry–erase boards up and visually inspect their results. Ask the class if x = 5 was a solution to their equation 4x = 20. Now, have students complete Exercises 1 and 2. Then complete a discussion to finalize the concept of solution and its relation to substitution. You may want to model a few real-world examples.

The formula for the perimeter of a rectangle is P = 2l + 2w. If the perimeter of a rectangular fence is 160 feet what are some possible solutions for the length and width?

A width of 10 feet and a length of 70 feet is a solution because substituting 10 in for w

and 70 in for l gives a true equation: 160 = 2(70) + 2(10). Have groups of students complete Exercises 3 through 11. The goal is for students to understand “solution,” not how to complete a set of steps to solve for the variable (this will be developed in later activities and lessons). Therefore, students may determine solutions with guess and check for this activity. After students complete Exercises 3 through 11, have each group share one of the exercises with the class. Let the class know that in the next activity we will find solutions for equations by solving equations using mathematical properties.

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SJ Page 17 Activity 1 1. Give examples of how substitution can be used in the world around us. Give as many examples as you

can. The first example is given. 2. The Substitution Property of Equality states that if a = b, then a can be substituted for b in any

expression or equation and b can be substituted for a in any expression or equation. Give examples of how substitution can be used in mathematics. Give as many examples as you can. The first example is given.

3. The formula for the area of a rectangle is A = lw. Give three solutions for the length, l, and width, w,

that would make an area of 36 square meters. Answers will vary. A sample response might be: l = 1 and w = 36; l = 2 and w = 18; or l = 3 and w = 12 4. What value x is a solution to the equation x + 6 = 10? Explain your answer. If x = 4, the equation becomes 4 + 6 = 10. Because this is a true statement, x = 4 is a solution to the

equation x + 6 = 10.

Solution to an Equation:

Substitution in the World Around Us

Substitute Teacher

Substitution In Mathematics If the side of square is 5 cm then the area is determined by substituting 5 for s in the formula A=s2.


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SJ Page 18 5. What values of x are solutions to the equation x + 6 = 6 + x? Explain your answer. Answers will vary. A sample response might be: “All values of x are solutions because the equation

represents the commutative property of addition.” 6. What values of c are solutions for c + (–c) = 0? Explain your answer. Answers will vary. A sample response might be: “All values of c are solutions because the equation

represents the additive inverse property.” 7. Write an equation that represents the circumference, C, of the circle based on the radius. If the

circumference is 314 square feet, determine the solution for the radius that makes the equation true. The equation for the circumference of the circle is C = 2πr. For 314 = 2πr, a value of r that would make

this true is approximately 50. 8. Write a scenario, or real–world example, and a matching equation that has a solution of t = 5 hours. Answers will vary. Sample response might be: Tyler drove 60 miles per hour on his 300 mile drive. How

long did it take Tyler to drive the 300 miles? Equation is 300 = 60t. 9. Write a real-world example and matching equation that has a solution of p equals $25. Answers will vary. Sample response: Erdeen spent $75 for three blouses. On average how much did

each blouse cost? The equation is 3p=75. 10. The area of a triangle is one-half the base times the height. Write a formula for the area of a triangle and

give three solutions for the height, h, and base, b, that would make an area of 100 square meters. The formula for the area of a triangle is A = (1/2)bh. Answers will vary. Sample response: base of 100

meters and a height of 2 meters; base of 50 meters and height of 4 meters; a base of 20 meters and a height of 10 meters.

11. Write a percent real-world example and matching equation that has a solution of $5. Answers will vary. Sample response: Tara received a 10% discount of $5 off a pair of slacks. What was

the original price of the slacks? The equation is 5 = 0.10p.

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Teacher Reference Activity 2 The goal for this activity is for students to conceptually understand and apply the four main properties of equality: addition property of equality, subtraction property of equality, multiplication property of equality, and division property of equality. While students will solve some equations in this lesson using mathematical properties, we do not expect students to become experts in solving equations yet. There will be further lessons in this unit that develop order of operations so that students solve equations accordingly. For now we want students to become familiar with the concept and mathematical truth that a person can add, subtract, multiply, or divide the same thing to both sides of an equation and both sides remain equal. This will be convenient to solve some simple equations. Guide the students through the addition property of equality. Ask additional questions such as:

• What is the difference between writing a generic symbolic statement such as: “If a = b, then a + c = b + c” and a numeric example such as, “If –3 + 5 = 2, then –3 + 5 +4 = 2 + 4?”

• Why might writing a generic symbolic statement be more convenient for a mathematician than an example with numbers?

• Conceptually understanding a mathematical property is as important as remembering the generic symbols that represent the property. Describe why using tiles helped you conceptually understand the addition property of equality.

You can either have pairs of students work on Exercises 1 through 3, or you can assign different groups to work on different properties and then present. The goal is for all students to understand and be able to use the four properties of equality. You can add these properties to your word wall, have students complete a vocabulary organizer for each property, and/or have students add this to a vocabulary journal. For the remainder of this activity, have students apply their new knowledge to other equations and basic real-world problems. Have each group of four divide into pairs to work on Exercises 4 through 15. Have each pair check the results with the other pair in their group. Walk around the class as students complete these exercises and guide as needed. For Exercise 6 the most important thing students will need to remember is that they must distribute the division of 4 in the first method. This leads to Exercise 8 where students see that properly using the distributive property is important. After the class has completed these exercises, you may want to ask the students why the symmetric property of equality is valuable. Note: The symmetric property of equality comes in handy when a solution has the variable on the right and students would like to change it to the left.


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Equal Tile Pad Transparency

=

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SJ Page 19 Activity 2 The Addition Property of Equality states that if the same number is added to both sides of an equation, then both sides of the equation remain equal. Using symbols, the following statement would be, if a = b, then a + c = b + c. Two examples of this property are shown below. Numbers: If –3 + 5 = 2, then –3 + 5 +4 = 2 + 4 Tiles: Use this information as an example and complete Exercises 1 through 3. 1. In your own words, write a statement that represents the Subtraction Property of Equality. Sample Response: If the same number is subtracted from both sides of an equation both sides of the

equation remain equal. a. Write the statement with symbols. Sample response: If a = b, then a – c = b – c. b. Give an example with numbers. Answers will vary.

Sample response: If 6 = 2 + 4, then 6 – 3 = 2 + 4 – 3. c. Give an example with tiles. Answers will vary. 2. In your own words, write a statement that represents the Multiplication Property of Equality. Sample Response: If the same number is multiplied to both sides of an equation both sides of the

equation remain equal. a. Write the statement with symbols. Sample response: If a = b, then ac = bc. b. Give an example with numbers. Answers will vary. 3. In your own words, write a statement that represents the Division Property of Equality. Sample Response: If the same non-zero number is divided into both sides of an equation both sides of

the equation remain equal. a. Write the statement with symbols. Sample response: If a = b and c ≠0, then a ÷ c = b ÷ c. b. Give an example with numbers. Answers will vary.

Sample response: If 24 = 4 (6), then 24 / 3 = 4 (6) / 3.

= =

= =


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SJ Page 20 4. Will these four properties of equality always work for any type of number, such as integers, rational

numbers, or real numbers? If not, give an example. Sample response: Yes, except for dividing by zero. 5. For each of the equations below, determine which property you could use to solve for the unknown?

a. + = −8 11x b. − =7 9x Subtraction Property of Equality Addition Property of Equality

c. = −7 98x d. = −135x

Division Property of Equality Multiplication Property of Equality

e. Use the mathematical properties of equality to solve the four equations. Show your work and check the solutions.

i. x 8 11

x 8 8 11 8x 19

+ = −+ − = − −

= −

Check 19 8 11

11 11− + = −

− = −

ii. x 7 9x 7 7 9 7

x 16

− =− + = +

=

Check 16 7 9

9 9− =

=

iii. 7 x 987 x 987 7x 14

= −−=

= −

( )

Check 7 14 98

98 98− = −− = −

iv.

( )

x 135

x5 5 135

x 65

= −

⎛ ⎞ = −⎜ ⎟⎝ ⎠

= −

Check 65 13513 13

− = −

− = −

6. For the solved equation below, state which property of equality was used for each step. + =

+ ==

4 8 202 5

3

xx

x

a. The same equation could have been solved in a different way. State which property of equality was used for each step shown below.

4 8 20

4 123

xxx

+ ===

b. Describe why you think either method shown above worked to solve 4x + 8 =20. Answers will vary. Make sure students understand with the first method they would need to

distribute the division through 4x + 8.

Sample response: The Division Property of Equality was used first to change 4x + 8 = 20 to x + 2 = 5 by dividing both sides by 4. Then the Subtraction Property of Equality was used to obtain the solution of x = 3 by subtracting 2 from both sides.

Sample response: The Subtraction Property of Equality was used first to change 4x + 8 = 20 to 4x = 12 by subtracting 8 from both sides. Then the Division Property of Equality was used to obtain the solution of x = 3 by dividing both sides by 4.

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SJ Page 21

7. Solve − =2 35x and state which properties of equality were used for each step in the solution.

Answers will vary. Some students may multiply by 5 first and others may add 2. Sample response: The Multiplication Property of Equality and Distributive Property was used first to obtain the equation x – 10 =15. Then the Addition Property of Equality was used to obtain the solution x = 25.

8. Sam solved the following equation incorrectly. Determine the mistake, explain how to fix it, and then

solve the equation correctly.

( )

8 56

6 8 6 56

8 308 8 30 8

38

x

x

xx

x

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =− + = +

=

( )

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =− + = +

=

x 8 56

x6 8 6 56x 48 30

x 48 48 30 48x 78

9. The area of a rectangle is 96 square inches and the length is 12 inches. What is the width of the

rectangle? Use the rectangle area formula of A = lw to solve for the width, show your work, check your solution. Describe how you used mathematical properties to solve and check this problem. Sample response: The width of the rectangle is 8 inches. I used the Substitution Property of Equality to replace area and length in the formula with 96 and 12, and then I used the Division Property of Equality to solve for the unknown width. I also used the substitution property to check the solution.

Given Information Substitute and Solve Check A lwA 96l 12

==

=

96 12w96 12w12 12

8 w

=

=

=

( )

A lw96 12 896 96

===

10. Write and solve an equation for the following percent problem.

25% of what is $20 Sample response: The equation is 0.25x = 20. Dividing both sides by 0.25 gives a solution of x = $80. 11. The distance formula for traveling at a constant rate is d = rt where d is the distance traveled, r is the

rate, and t is the time of travel. At what rate must you travel to go 300 hundred miles in 6 hours? Sample response: You must travel at a rate of 50 miles per hour to travel 300 miles in 6 hours. 12. The solution to a problem is 21. Write a real-world application and equation that results in the given

solution. Show that your equation has the given solution. Sample response: “Jimmy went to the store to buy some school supplies. He wasn't sure how much

money he had before he went to the store, but he spent $7.50 and had $13.50 left after buying his school supplies.” The equation is x – 7.50 = 13.50. Checking the solution, we get 21.00 – 7.50 = 13.50.

Sample response: Sam did not use the distributive property correctly when he multiplied by 6. He should have multiplied the 8 by 6 also.


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SJ Page 22 13. Write an equation for the scenario below, and then solve the equation. Describe the properties you used

to solve the equation. “Ron, who had $40, had five times the sum of Nick's amount of money and $4. How much money did

Nick have?” Sample response: The equation is 5(x + 4) = 40. Using the distributive property: 5x + 20 = 40.

Using the division property of equality: x + 4 = 8 Using the subtraction property of equality: x = 4.

14. You have learned that you can add, subtract, multiply, or divide the same quantity to both sides of an

equation to solve it. Can you do the same with squaring or taking the square root of both sides of an equation? Look at each example below and determine if you can. If you can, write an explanation.

a. = 5x b. = 8x c. =2 81x d. =2 36x Explanation: Answers will vary. Students should determine that they can take the square root of both sides of an

equation or square both sides of an equation to solve for a variable.

Note: You may need to share with students that there is a positive and negative square root with undoing a square.

15. Study the following two different categories, then create a definition for the Symmetric Property of

Equality. Answers will vary. Sample response: For an equation, the expression on the left and the expression on

the right could switch sides and the equation would still be true.

Examples of the Symmetric Property of Equality

Non-Examples of the Symmetric Property of Equality

= −5 11 6 and − =11 6 5

+ =25 2 27 and = +27 25 2

= 12x and =12 x

=a b and =b a

= −5 11 6 and + =5 6 11

+ =25 2 27 and + =20 7 27

+ = +3 4 4 3

=a b and + = +a c b c

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SJ Page 23 Practice Exercises 1. Before pulling away from the curb, a cab driver charges an initial fee of $2.00,

then an additional $1.50 for each one mile traveled. Determine how far a customer traveled if her fee was $32. Write an equation that represents the fee based on how far, in miles, a customer travels.

The equation for the company's fares is F = 1.50x + 2.00, where F represents the

fare and x represents the miles traveled. The customer traveled 20 miles for a fare of $32.

2. Apex Recycling Company pays $1.35 for every pound of copper. Write an equation for the amount of

money the recycle company pays for copper. The equation for the amount of money the recycle company pays is P = 1.35x, where P represents the

total payment and x represents the pounds of copper. Whenever Anthony remodels a house, he collects the old copper pipes. He sold a large pile of copper to

Apex Recycling Company and they paid him $58.05. How many pounds did Anthony recycle? Anthony recycled 43 pounds of copper. 3. Hannah works in sales for a large company that sells treadmills to fitness centers. She earns $550.00 for

each fitness center that purchases a treadmill from her company plus another $75.00 for each treadmill she sells. Write an equation that represents the amount of money Hannah earns for each fitness center. Determine how many treadmills she must sell if her goal is to earn $1,150.00 per fitness center.

Answers will vary. The equation for the amount of money Hannah earns for each fitness center is M =

75x + 550, where M represents the amount of money earned and x represents the number of treadmills sold. Hanna must sell 8 treadmills to each fitness center.

4. The height of a tree and the thickness (diameter) of its trunk are generally related to the tree’s age.

Assume that the height and thickness of a particular type of tree can be described by the following equations:

h = 0.4a + 1, where h is height in meters and a is age in years. t = 0.4a – 2.5, where t is thickness in centimeters and a is age in years. a. Calculate the age of the tree if it has a height of 30 meters.

The age of a tree that has a height of 30 meters is 72.5 years. b. Calculate the age of a tree if the thickness is 25.5 centimeters. The age of a tree with thickness of 25.5 centimeters is 70 years.


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SJ Page 24 5. Many stores and banks have coin machines where people can pour in their loose change – pennies,

nickels, dimes, and quarters – and receive a receipt to turn in for the equivalent amount in dollar bills. Write an equation representing the amount of money the coin machine will print a receipt for if you pour in p pennies, n nickels, d dimes, and q quarters.

Sample response: The equation to represent the total amount will be T = 0.25q + 0.10d + 0.05n + 0.01p,

where T represents the total amount of money.

Use your equation to solve the following: a. If Kyle pours 7 nickels and 17 dimes into the coin machine, how many quarters does he need to

also pour in to have a total of $11.05? Kyle must have 36 quarters to have a total of $11.05.

b. If Kelly has 30 dimes, no pennies, and 44 quarters, how many nickels does she have if her total is $15.15?

Kelly would have 23 nickels.

6. How can mathematical properties be used to solve equations and real–world applications? Answers will vary. 7. Create a graphic organizer of all the properties and terms that have been developed in Lessons 1 and 2

of this unit so far. Answers will vary.

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SJ Page 25 Outcome Sentences Using properties to solve equations I now understand that The equality property of division is still hard to understand because The four properties of equality make solving equations Substitution now makes sense because


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Teacher Reference Lesson 2 Quiz Answers 1. 3x 6 15 Given

3x 6 6 15 6 Addition Property of Equality3x 93x 9 Division Property of Equality3 3x 3

− = −− + = − +

= −−=

= −

( )

Check3 3 6 15

15 15− − = −

− = −

2. The height of the can of soup is approximately 4 inches.

219.64 (3.14)(1.25) h Given19.64 4.91h 19.64 4.91h Division Pr operty of Equality4.91 4.91

4 h

=≈

=

=

3. The opposite of 5 was added to the equation instead of just adding 5 to create a zero pair.

4x 5 124x 5 5 12 5

4x 174x 174 4

4

− =− + = +

=

=

x4

174

17x4

=

=

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Lesson 2 Quiz Name: 1. Solve the equation − = −3 6 15x . State the properties or rules used. Check your solution. 2. The volume of a cylinder is given by the formula V=πr2h. A large can of mushroom soup has a volume

of 19.64 cubic inches. The radius of the can is approximately 1.25 inches. Determine the height of the can, rounded to the nearest hundredth of an inch. State the properties used to solve your equation. Note: use 3.14 for π.

3. The following equation is solved incorrectly. Determine the mistake, explain how to fix it, and then

solve the equation correctly.

4 5 124 5 ( 5) 12 ( 5)

4 74 74 4

4

xx

xx

− =− + − = + −

=

=

4x 7

474

x

=

=

Solving One-Variable Equations Lesson 3: Order of Operations Page 45

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Lesson 3: Order of Operations Objectives • Students will be able to evaluate an expression using the correct order of operations. • Students will be able to create an expression based on a series of operations. • Students will be able to determine the input needed for an expression to yield a given output. Essential Questions • How can working backward help solve problems in mathematics? Tools • Student Journal • Setting the Stage transparency • Dry–erase boards, markers, erasers • Graphic Organizer • Scissors • Overhead projector • Poster Paper and Markers • Expression Cut Outs

Warm Up • Problems of the Day Number of Days • 2 days (A suggestion is to complete activities 1 and 2 on the first day. On the second day, Activity 3,

Practice Exercises, and Lesson Quiz.) Vocabulary Series of operations input Output operations PEMDAS GEMDAS Order of operations

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Teacher Reference Setting the Stage The goal for this Setting the Stage is for students to link organizing the order of events to prior knowledge. Organizing the order of events will be valuable for completing the order of operations in arithmetic and to solve equations later in this lesson and unit. Display “Transparency 1”. Have one student read the operation example and then have a second student read the homework example. Discuss with the class why the order does or doesn’t matter in the two examples. Sample responses may include that order matters in completing an operation and the order of studying homework generally doesn’t matter. Some guiding questions that you might want to pose to students are:

• Why is the order of events important in one situation and not in another? • How might changing the order of events change the outcome? • What other important missing steps may be needed in the operation example? • When might the order of studying subjects matter?

Display “Transparency 2”. Talk with students about using graphic organizers to record the steps of a task. Ask students why a graphic organizer may be helpful for listing the steps of a task. Students should talk about how a graphic organizer helps see the relationship between steps. To help students think about their thinking and reflect on their own understanding, have each student create a step-by-step graphic organizer with arrows that represent a task with which he or she is familiar. Answers will vary. Have each student share his or her graphic organizer with a partner or with the small group. After the students have shared, you may want to ask some guiding questions such as:

• Would following the steps in your graphic organizer backward make sense? Explain. • Are there certain steps in your task that cannot be undone? • What is unique about steps that can be undone? • What is unique about steps that cannot be undone? • Do you think most mathematical procedures can be undone? If so, give an example.

Let students know that in this lesson they will use arrows to represent the steps of a mathematical task and that they will become proficient with “undoing” the steps by the end of this lesson. You may want to have students reverse their steps and rewrite the steps to undo their task (Note: some tasks cannot be undone).


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Setting the Stage Transparency 1

ORDER MATTERS or DOES IT?

Operation: When a surgeon operates on a patient: • First the room, the tools, the doctor’s

hands, and the incision area need to be sterilized.

• Second, the doctor makes the incision.

• Third, the doctor performs the operation.

• Finally, the doctor closes and seals the incision. Completing Homework: When working on

homework, it is best to do English first, then mathematics, followed by science, and finally history.

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Sometimes, to keep track of the steps to perform a task, people use arrows as a graphic organizer. For example, Gabrielle organized the steps needed to tile her bathroom shower walls as follows.

Pick a task that you have had to complete recently. Create a graphic organizer with arrows that represents the steps needed to complete the task.

Trowel on the Tile Mastic

Mark the Layout Lines

Start Tiling

Cut Tiles to Fit

Drill for Tub or Shower Supply Pipe

Apply the

Grout


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Teacher Reference Activity 1 The goal of this activity is to use the students’ intuitive understanding of order of operations to have them complete the order of operations backward. Most students will be successful with this activity even if they have struggled in the past with the order of operations for arithmetic or for solving equations. This activity will be a good foundation for the next lesson related to solving equations. Before students begin Exercise 1, you will want to quickly discuss series of operations, input, and output as these terms relate to the graphic organizer that they will use in this activity. Give the students a quick calculator challenge and have them draw the corresponding graphic organizer on their dry-erase board. You can model one example while they do another. For example, say to the students, “Everyone pick your favorite number from 1 to 100 and input that into your calculator, now add 4 to your number, now multiply that sum by twelve, now divide that product by 6, now subtract your original number twice. What did you get for the output?” Start with a blank graphic organizer and fill it in with an example as you complete each operation. The ovals represent input and outputs, the arrows represent an operation with a specific value, and the circles represent the ongoing value as each operation is performed. The output will always be 8. You may want to give extra credit for students who can write a description of why this works. The descriptions may vary. Some students may change the statement into an expression and then simplify as shown below.

( ) ( )4 124 2 2 2 8 2 8

6x

x x x x x x+

− − = + − = + − =

Note: Translating between a series of operations and an expression is developed in Activity 3. There is no need to develop the relation between a series of operations and the expression at this time. Encourage students to use calculators for the exercises in this activity. Assign students to work in pairs. You can assign pairs by having each student find another student that has a matching expression. Expression cutouts are available at the end of the activity. The goal would be that each student finds another student that has an expression with the same operations in the same order, the numbers may be different. Have the students complete Exercises 1 and 2 individually before checking with their partners. Make sure to walk around the classroom to work with individuals and pairs. Note: The calculations must occur after each operation. If needed for Exercise 2, help students determine that they need to complete the opposite operations in the opposite order, but try to let them come to this conclusion intuitively. For Exercise 3, you might suggest that one partner answers Exercises 3a and 3b while the other answers Exercises 3c and 3d, then they can check each other’s work before completing Exercises 4 and 5. You may even want a few students to share their work with the class. Continue walking around the classroom to work with pairs. For Exercise 4, have the students determine strategies that will work for determining outputs given inputs and inputs given outputs. You might pick one or two groups to share their strategies. You may want to let students know that whole number inputs and outputs are not the only solutions to the inputs and outputs. There are many rational number and integer inputs and outputs.

Input Output55 +4 59 ●12 708 ÷6 118 –55 63 –55 8

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The concept of undoing is very important for solving equations. You may want students to create a small poster of their response to Exercise 5 and hang posters in the room as a constant reminder of how to undo operations. Use Exercise 6 as an extension to further engage students with the concept of doing and undoing a series of operations in order. Have every student write an operation followed by a number on their dry-erase board. Make sure the students write bold enough and large enough to see the symbols from across the room. Then have the students place their dry-erase boards in order somewhere in the classroom. A chalk tray or heater along the wall works well. Then give an input value and see how quickly the class can determine the output. Then give the students an output to determine the input. A sample of five operations in a row is shown below for an input of 5. It may be interesting to see if the students can create an expression to represent the display of dry-erase boards in order. Having 20 to 30 boards up front is generally fun for students to try to determine the input or output given a value. Other investigations could include:

• switching dry-erase boards around to determine the affect • placing the dry-erase boards in an order that creates the largest number • placing the dry-erase boards in an order that creates the smallest number • placing the dry-erase boards in an order that creates a number as close to zero as possible

Input: 5

( )• −8

÷12

÷3

−6

( )+ −4 Output?


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Activity 1 SJ Page 26 The diagram below represents a series of operations with numbers. Each operation is completed before going to the next operation. For example, if the input is 14, first add 2 to get 16, then divide by 8 to get 2, then subtract 3 to get 1− , then multiply by 10 to get −10 , then subtract 7 to get an output of −17 . 1. Determine the outputs for the following inputs that match the diagram of operations and numbers

above. Check the outputs with your partner. a. Input: 62 Output: ________ 43 b. Input: 22 Output: ________ –7 2. Determine the inputs for the following outputs that match the diagram above. Check your inputs with

your partner. a. Output: 63 Input: ________ 78 b. Output: 13 Input: ________ 38 c. What was different about solving this exercise as compared to Exercise 1? Sample response: We

had to do opposite operations backward. 3. Use the diagram below and experiment with inputs or outputs to complete the exercises.

a. Write one example of a positive integer input that gives a negative integer output. Sample response: A positive integer input of 5 gives a negative output of –77.

b. Write one example of a negative integer output that comes from a negative integer input. Sample response: A negative output of –105 come from a negative integer input of –5.

c. Determine a whole number input that gives a whole number output. Sample response: An input of 35 gives an output of 7.

d. Determine a whole number output that has a whole number input. The input and output values should be different than Part C.

Sample response: An output of 21 needs an input of 40.

Input Output +2 ÷8 –3 ●10 –7

Input Output ÷5 +1 ●2 –15 ●7

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SJ Page 27 4. Describe the strategies that you used in Exercise 3. Will these strategies always work? Explain your

answer. Answers will vary. Sample Response: I just started with one as an input and then I tried two and then

kept going until I found a whole number input that gave a whole number output. (Note: If the equation ( )( )y x 5 1 2 15 7= ÷ + ∗ − ∗ is entered into the graphing calculator, then by viewing the table, it is easy

to see all the inputs and outputs that are whole numbers. It is not expected at this time that the students will know how to change the diagram into an expression or an equation, but you may have some students that would benefit from this calculator strategy. Please consult the graphing calculator manual for your classroom graphing calculator)

5. If doing is completing the operations in order from the input to the output, describe what undoing

would be.

Sample response: Undoing would be completing the opposite operations in the opposite order. It would be going from the output to the input. Note: This concept is the most important concept of the activity.

6. Based on the numbers and operations displayed at the front of class, determine the inputs or outputs as

directed by your teacher. Record your responses below.

Input Output


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Matching Expression Cutouts

3 4x +

2 7x +

3 4x −

4 5x −

63x + 9

4x + 6

3x − 9

4x −

( )6 3x +

( )5 4x− +

( )3 1x− −

( )( )8 3x − −

64

x +

59

x +−

6

6x −

5

7x −−

4 9x + −

7 6x + −

( )4 2x − + −

5 8x − +

54x

25x

32x⎛ ⎞

⎜ ⎟⎝ ⎠

54

x⎛ ⎞⎜ ⎟−⎝ ⎠

2 45

x +

8 79

x + 2 4

6x⎛ ⎞−⎜ ⎟

⎝ ⎠ 4 5

2x⎛ ⎞−⎜ ⎟

⎝ ⎠

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Teacher Reference Activity 2 Have a student volunteer read the first paragraph of this activity. Have the students work in pairs on Exercise 1. The goal is for students to understand that completing the order of operations correctly in the real world is important. Now have each pair team up with another pair to form a group of four. Have the students work through Operation Order, Exercises 2 through 5. Cutouts for this activity are at the end of the activity. Save the cutouts to use in Activity 3. The goal of these exercises is for students to discover which operations can be done in any order and which need to be done in a specific order. This will help students better understand the order of operations. After students have answered Exercise 5, lead a class discussion about PEMDAS. Generally, mathematics teachers emphasize that in arithmetic, multiplication and division are at the same level in the order of operations followed by addition and subtraction at the same level. One method to represent this is. Before students answer Exercise 6, continue the discussion of PEMDAS and specifically focus on grouping as it relates to symbols and operations in expressions which lead to a new acronym GEMDAS. You can have a student volunteer read the paragraph before Exercise 6 to help with the discussion. You may want to ask questions such as:

• Why might multiplication and division be at the same level of order of operations? • What does the E stand for in PEMDAS and how do you complete the opposite of E? • How does a square root symbol represent grouping?

After the discussion, have students complete Exercises 6 through 8. Have each group trade graphic organizers with another group for Exercise 8. Have some groups share their graphic organizer with the rest of the class. You can have the class discuss the similarities and differences of the different graphic organizers. A sample graphic organizer is given in Exercise 7. Optional Activity: To make it more interesting, you could use the following variation to have students complete an alternative Operation Order. Give each group a packet with a set of arrow cutouts from the back of the lesson. Tell the class that each member of the group will take a turn. Each group member should start with any number, positive or negative, from negative ten to positive ten, and write this input value in a blank diagram. The student then selects two individual arrows with operations from the envelope without looking and places them in the order selected on the table. The student should then write what the two arrows say in the diagram provided. The other group members assist the student as needed to determine the output. The student who selected the arrows should then switch the order of the arrows, write the new expression in a second diagram, and then evaluate the new output. The other students should then take their turns through the same process. Each group should cycle through enough of these examples to make conclusions as to when the order matters or doesn’t matter in evaluating two operations. The group then should discuss and complete a list when order matters and when order doesn’t matter. Students should discover that the order doesn’t matter when only adding and subtracting or when only multiplying and dividing, but for most other combinations the order matters.

P E MD

AS


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Activity 2 SJ Page 28 Knowing the order of operations is important so that everyone will calculate the same way. There are four main operations: addition, subtraction, multiplication, and division. Study and answer the following exercise related to money to see how important the order of operations is for real-world mathematics. 1. Camille is an accountant and charges $50.00 for the first time she meets with a new client. She then

charges $25.00 per hour after that to prepare a tax return. She met recently with a new client, and then she worked an additional 2 hours to complete his tax returns. Two different billing agents in her company calculated the total fee in two different ways. Which calculation below is correct? In your own words, explain why one person may have calculated it incorrectly?

a. 50 25 2 $150+ • = b. 50 25 2 $100+ • = Sample response: The method in part a is incorrect and the method in part b is correct. The

person in a added 50 and 25 first and then multiplied by 2. This person doesn’t know that multiplication comes before addition in an expression. Use the problem to transition to the next exercise.

To investigate the order of operations in more detail, let’s complete the activity Operation Order. Cut out the Operation Order arrows at the end of the activity. Place the arrows in a pile. The object of the activity is to determine which operations need to be completed in order and which operations do not. Operation Order 2. Select any number from –10 to +10 and write it in the INPUT section below. Continue to use this input

number for the entire activity. Pick two arrows from the pile that have the addition operation. Write the operations in the two arrow blocks in the diagrams below in the order you chose them. Determine the value of the output and write it in the OUTPUT section.

Answers will vary. A sample input could be –3 followed by adding –4 and then adding 9 for an output of 2.

b. Switch the order of operations and write them in the two arrow blocks below. Using the same input value, determine the output and write it in the OUTPUT section.

Answers will vary. A sample input could be –3 followed by adding 9 and then adding –4 for an output of 2.

c. Were the output values for both conditions the same or different? Explain why they may have

been the same or different. Sample response: The output values were the same. You can add –4 and then add 9 and get the

same answer as adding 9 and then adding –4.

Input Output

Input Output

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SJ Page 29 3. You just investigated completing two addition operations in different order; now continue

investigating the combination of other operations in different orders. Pick arrows from your pile as needed to investigate each exercise below. Make sure to determine the output based on the same input using the operations in different order.

a. Pick two subtraction operations from your pile and determine the two outputs based on

completing the operations in a different order. Answers will vary. b. Pick two multiplication operations from your pile and determine the two outputs based on

completing the operations in a different order. Answers will vary. c. Pick two division operations from your pile and determine the two outputs based on

completing the operations in a different order. Answers will vary.

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output


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SJ Page 30 d. Pick an addition operation and a subtraction operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary. e. Pick a multiplication operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary. f. Pick an addition operation and multiplication operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary. g. Pick an addition operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary.

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

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SJ Page 31 h. Pick a subtraction operation and multiplication operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary. i. Pick a subtraction operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order. Answers will vary. 4. Record your findings for Exercises 2 and 3 in the table below by listing which combinations of

operations where order mattered and which combinations where order did not matter when determining the output value.

Combinations Where Order Doesn’t Matter Combinations Where Order Matters

Input Output

Input Output

Input Output

Input Output


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SJ Page 32 5. You have probably memorized the acronym PEMDAS to help you complete the order of operations,

where P stands for parenthesis (or other grouping symbols), E stands for exponents, M stands for multiplication, D stands for division, A stands for addition, and S stands for subtraction. In your own words, describe how Exercises 2 and 3 relate to the acronym PEMDAS.

Answers will vary. Some teachers manipulate the letters to better represent the actual order because

multiplication and division are at the same level and addition and subtraction are at the same level. For example, here are two examples that some teachers use to better represent the order.

Example 1: P E [MD] [AS] Example 2: Because there are so many different ways to combine the operations with numbers, mathematicians use grouping symbols such as parenthesis to make sure some operations occur before others. This is represented by the P in PEMDAS. Other grouping methods that can be classified under P are brackets and braces, such as 4 [ 3 + 2 ] and –2 { 6 – 3 }. Square roots and, in some cases, division also imply grouping. For example, in the

expression 4 12+ the addition should be completed before the square root, and in the expression 17 52− the

subtraction should be completed before division. Because there are other methods to represent grouping, it may be convenient to remember the order of operations as GEMDAS and not PEMDAS, where G represents grouping. 6. For each of the expressions below, use the correct order of GEMDAS one step at a time, and show the

correct order of operations to obtain the given value. The first has been completed for you

a. ( ) ( )( )

− + = +

= += +=

2 23 9 5 8 3 4 83 4 6412 64 76

b. [ ]÷ + − + ===== −

24 8 6 2 7

1

c. − =•

===

25 46 2

1.75

d. • − + =====

25 4 7 6

21

e. ( )⎡ ⎤− + + =⎢ ⎥⎣ ⎦====

229 7 3 1

50

f. ( )⎡ + + ⎤ + =⎣ ⎦====

2 2 2 2 2 2

22

P E M D

A S

÷ + −÷ + −

+ −−

24 8 6 916 8 6 92 6 98 9

−•−

2 5 46 2

2 5 41 2

2 11 2

• − +

• +• +

+

5 16 7 6

5 9 65 3 615 6

2 2

2

2

[2 3] 1[4 3] 1[7 ] 149 1

+ +

+ +

++

2[2(4) 2] 22[8 2] 22[10] 220 2

+ ++ +

++

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SJ Page 33 7. Work with your group members to create a graphic organizer for the order of operations. Answers will vary. The graphic organizer below is an example of what students may create. The

students who created this graphic organizer wanted to use E and R, because to them R for roots was the opposite of E for exponents.

8. Trade your graphic organizer with another group so that they can test it to solve the following

exercises.

a. 26 3 3

5 6− =

+ b. 2 24 3 5+ =

c. 10 3 8 14− • = − d. ( )( )5 10 2 1 3 6 4− + − =

G

Start with the innermost Group.

( ) [ ] { } | |

Rep

eat s

teps

as

need

ed.

Complete Exponent (power) and Root operations.

ER

Complete Addition or Subtraction going left to right.

+ −AS

Complete Multiplication or Division going left to right.

MD( )( ) × • ÷ /

3 3 2


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Operation Order Cutouts for Activity 2 SJ Page 35

● 5 ●( –6 ) ●( –1 ) ● 2 ● 4 ● 10 ●( –5 )

÷( –4 ) ÷ 3 ÷ 2÷( –1 ) ÷( –8 ) ÷( –5 ) ÷10

+ 3 + ( –4) + 10 + 4 + 6 + 8 + ( –6)

–5 – ( –4) – ( –5) – 6 – 9 – ( –7) – 2

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Teacher Reference Activity 3 Use this activity as a bridge to Lesson 4. The goal is to help students further understand the connection between an expression and the order of operations. Review with the students how they can use their graphic organizer to complete the order of operations correctly. Remind students that in Activity 1 they did a series of operations to determine outputs from inputs. They also determined inputs from outputs by completing the series of inverse operations backward. Tell the class that this activity is somewhat similar to the first activity, and in addition they will write the matching expression. Have pairs of students complete Exercises 1 through 4, then discuss the methods they used to evaluate an expression with a table and symbolically. You may want to ask questions such as:

• What benefit does the table have that the symbolic expression does not? • What advantage does the symbolic expression have that the table does not? • What method would you prefer to use to evaluate an expression?

After Exercise 4, have the class work in groups of four. Have each group add the additional cutout arrows at the end of the activity to the Operation Order cutouts from Activity 2 for this activity. Additional blank arrows have been provided in case you would like to add specific operations and numbers to the students’ piles of arrows. Each group should number their members 1 through 4. Model, with a volunteer group in front of the class, how a group can create and evaluate an expression. Have each member of the volunteer group pick a random arrow from the pile. Then have member #1 come to overhead or board and write the information from the arrow in the table and write the first part of the expression. For example, if member #1 drew “square”, then he/she would write “square” in the table and x2 as the expression. If member #2 drew “●2,” then he/she would write “●2” in the table and place 2 on the expression so it becomes 2x2. If member #3 drew “+5,” then he/she would write “+5” in the table and place +5 on the expression. If #4 drew “÷11,” then he/she would write “÷11” in the table and then divide the expression by 11. The final expression and table should look like the following.

+22 511

x

Now have the volunteer group pick a number between –10 and 10 and use this number as the input to evaluate the expression. The members should evaluate the expression for parts they made. For example, if the input was x = 5, member #1 would square 5 to obtain a result of 25. Then member #2 would multiply 25 by 2 to obtain a result of 50, then member #3 would add 5 to obtain a result of 55, and finally member #4 would divide by 11 to obtain a result of 5. When the students complete the exercises in their groups, they can physically pass one paper or the dry-erase board around to the individual members to complete the tasks. You could also have this volunteer group model how to go from an output to an input. Group members should switch the order of operations. One technique is to have the group member who performed the first operation do the second operation for the next problem; the group member who performed the second operation do the third operation for the next problem; the group member who performed the third operation do the fourth operation for the next problem; the group member who performed the last operation do the first operation for the next problem. Have the groups repeat Exercises 5 and 6 as needed, then bring the class together and have groups share with the rest of the class. Perhaps ask specific groups that had interesting expressions to share.

INPUT: x

Square

●2

+5

÷11

OUTPUT


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Activity 3 SJ Page 37 You’ve investigated using a graphic organizer to complete operations from left to right and right to left. Another method to organize the operations could be vertical. Complete Exercises 1 and 2 using a vertical table. The input is at the top and the operations continue in order going down the table to the output. 1. Determine the outputs given the following inputs. a. Input: 8 Output: –15 b. Input: 20 Output: 15 2. Determine the inputs given the following outputs. a. Output: –10 Input: 10 b. Output: 5 Input: 16 Using a method, such as the table, can help a person keep organized when evaluating an expression given a value of an unknown variable. For example, to evaluate the expression ( )23 4 6 x + + when x = 2, you could use the table or you could evaluate it symbolically.

( )( ) ( )2 2

2

3 2 4 6 6 4 6

10 6100 6106

+ + = + +

= += +=

3. Complete a table for the expression 4 6 122

x + − and then evaluate it for the different x values. Evaluate

it symbolically too. a. x = 3

( )4 3 612 3

2+

− = −

b. x = –3

( )4 3 612 15

2− +

− = −

4. What is your opinion of using a table to evaluate an expression as compared to symbolically? Answers will vary.

INPUT

6+

÷2

–10

●5

OUTPUT

INPUT: x = 2

●3

+4

Square

+6

OUTPUT: 106

INPUT: –3

●4

+6

÷2

–12

OUTPUT: –15

INPUT: 3

●4

+6

÷2

–12

OUTPUT: –3

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SJ Page 38 Operation Order II Within your group make a pile of the Operation Order arrows and add the additional arrows of square and square root to your pile. Decide who will be #1, #2, #3, and #4. Your teacher will model, with a volunteer group at the front of the class, how to create and evaluate an expression with your group. Complete Exercises 5 through 6 with a similar method. Each member of your group will be responsible for one operation for each expression. 5. Have each member of your group randomly pick an arrow from the pile.

a. As a group, in the order that you chose, write the expression and fill in the table.

Answers will vary.

b. Evaluate the expression for x = 15.

Answers will vary.

c. Determine the input if the output was 30. Answers will vary.

6. Have each member of your group randomly pick two arrows from the pile. For this exercise, you will each need to add two operations to the expression and table.

a. As a group, in the order that you chose, write the expression

and fill in the table. Answers will vary.

b. Evaluate the expression for x = –4.

Answers will vary.

c. Determine the input if the output was 100. Answers will vary.

INPUT: x

OUTPUT:

INPUT: x

OUTPUT:


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Additional Operation Cutouts for Activity 3 SJ Page 39

Square Square root

Square Square root

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SJ Page 41 Practice Exercises 1. Determine the output for the following inputs that match the diagram of operations and numbers

above. Check the outputs with your partner. a. Input: 78 Output: ________ 46 b. Input: –10 Output: ________ –9 2. Determine the input for the following outputs that matches the diagram above. Check your inputs with

your partner. a. Output: 11 Input: ________ 22 b. Output: –4 Input: ________ –2

3. Use the expression 26 711

x − to fill in the table.

a. Evaluate the expression for an input of x = 6. 19 b. Evaluate the expression for an input of x = –6. 19 c. Determine the input if the output is 157. ±17 4. Write an expression that matches the table on the right.

( )3 x 54

2+

−

a. Evaluate the expression for an input of x = 3. 8 b. Determine the input if the output is 23. 13

Input Output –7 ●5 –3 ÷8 +2

INPUT: x

square

●6

–7

÷11

OUTPUT:

INPUT: x

+5

●3

÷2

–4

OUTPUT:


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SJ Page 42 5. Describe how you know what order the operations should occur when determining the value of the

expression • −25 4 15

2.

Sample response: By using GEMDAS the first thing to do is complete the operation above the division

sign because it is behaving as a grouping symbol, then square 4 because it has an exponent of 2, then multiply by 5 because multiplying comes after exponents, then because the grouping is done, divide by 2, then subtract 15.

6. Logan completed the following steps to determine the value of the expression ( )5 3 4+ . He made a

mistake along the way. Determine the mistake and describe why Logan may have made the mistake.

( ) ( )5 3 4 8 432

+ ==

Sample response: The mistake was to add 5 and 3 first. Logan may have been just

doing the math from left to right instead of using the order of operations.

7. Evaluate the expression 25 15

3x − for the following x value inputs.

a. x = 0 b. x = 9 –15 120 8. Determine the value of x for the expression ( )25 4 7x + − for the following outputs. a. Output is 238 b. Output is 73 x = 3 or x = –11 x = 0 or x = –8

9. Determine the value of x for the expression −5 86x for the following outputs.

a. Output is 2 b. Output is –1/2 x = 16 x = 1

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SJ Page 43 Outcome Sentences I now understand that the order of operations I would conclude that adding numbers in any order I used doing and undoing operations to I was surprised that the order of operations I would conclude that determining the input when given the output I would like to find out more about Creating a diagram to match an equation is Creating a table to match the equation is How can working backwards help solve problems in mathematics?


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Teacher Reference Lesson 3 Quiz Answers 1. a. –80 b. 160 2. a. 0 b. 28 3. Answers will vary. 4. a. -11 b. –5 c. –11

INPUT: x

●(–2)

+5

÷3

–8

OUTPUT:

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Lesson 3 Quiz Name: The following diagram represents a series of operations with numbers. Each operation is completed before going to the next operation. 1. Determine the output for the following inputs for the diagram above. a. Input: –4 Output: ________ b. Input: 12 Output: ________ 2. Determine the input for the following outputs for the diagram above. a. Output: –20 Input: ________ b. Output: 400 Input: ________ 3. Describe how the acronym GEMDAS can help you evaluate an expression.

4. Use the expression − + −2 5 8 3x to fill in the table.

a. Evaluate the expression for an input of x = 7. b. Evaluate the expression for an input of x = –2. c. Determine the input if the output was 1.

Input Output ●6 +4 ÷2 –6 ●5

INPUT: x

OUTPUT:

Solving One-Variable Equations Lesson 4: Solving Equations Using Order of Operations Page 71

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Lesson 4: Solving Equations Using Order of Operations Objectives • Students will be able to solve one-variable equations symbolically by isolating the variable and completing

the opposite operations in reverse order of arithmetic. o The one-variable equations will include whole, integer, and rational coefficients and solutions.

• Students will be able to solve for the unknown in basic real-world applications related to formulas. • Students will be able to solve one-variable equations with tables and tiles. Essential Questions • How can the order of operations be used to solve equations? • How do you solve equations using opposite operations? • How do you apply what you have learned about solving equations to solving real-world applications

involving formulas? Tools • Student Journal • Setting the Stage transparency • Dry-erase boards, markers, erasers • Student Journal • Overhead projector • Dry–erase boards, markers, erasers • Overhead Tiles • Student Tiles • Tile Pad • Equal Tile Pad • Poster Paper and Markers Warm Up • Problems of the Day Number of Days • 1 to 2 days (A suggestion is to complete Activity 1 and Practice Exercises 1 through 5 on the first day. On

the second day, complete Activity 2, the remaining Practice Exercises, and Lesson Quiz.) Vocabulary Doing Undoing Solve Solving

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Teacher Reference Setting the Stage

The goal for this Setting the Stage is to have the students follow steps for instructions forward and backward. They will link this concept to solving equations and checking solutions. Display the transparency , and tell the class that this represents the instructions for installing a battery pack to a particular electronic device when it is first purchased. The instructions involve placing the battery pack in the device, connecting the battery wire cable connector to the device's module connector, putting on the battery pack cover, and then putting the outer shell cover on the device. After a couple of years of use the battery pack no longer holds a charge and has to be replaced. Have the class work in groups of four to write the directions necessary to remove the old battery pack and then to install a new one. Have each group share the steps they wrote. Have each group write a problem of their choice that requires them to follow steps backward to resolve the problem. It doesn't have to be mathematical in nature. Have each group share their problem with steps for the solution with the class. After each group has shared their steps for solving the battery pack problem and their own problem, place the following equation, solution, and check on the board. Solution Check

2 4 182 4 4 18 4

2 222 222 2

11

xx

xx

x

− =− + = +

=

=

=

( )2 4 18

2 11 4 1822 4 18

18 18

x − =− =− =

=

Then ask the class, “How is the reinstallation with the battery pack process similar to checking your solution to a mathematical problem?” The key concept is that the students learn that following the installation instructions backward to undo the installation is like solving an equation for the variable and that re–installing is like checking your solution.


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Following Steps Forward and Backward

Battery Pack Installation Instructions: 1. Place the battery pack into

the battery pack module slot.

2. Connect the battery pack

wire module connector to the module connector on the device.

3. Screw on the battery pack module enclosure cover. 4. Install the outer shell cover.

Portable Wireless Video Display

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Teacher Reference Activity 1 The goal of this activity is for students to become successful in solving basic one-variable equations. A table method will be used in conjunction with solving equations symbolically. This helps students keep track of the order of operations as well as using the opposite operations. Tell the class that in Lesson 2 they learned how to solve equations using properties. The order of operations to solve the equations didn't matter in Lesson 2 as long as the properties were properly used. For example, the equation 3 6 12x + = , can be solved at least two different ways:

First Method Second Method 3 6 12 Divide both sides by 3.

2 4 Subtract 2 from both sides.2

xx

x

+ = →+ = →

=

3 6 12 Subtract 6 from both sides.3 6 Divide both sides by 3.

2

xxx

+ = →= →=

Now ask the class, “What if we solve the equation 4 6 9x + = by dividing both sides by 4 first? Would the division result in whole numbers or fractions?” Ask the class, “How many of you like to work with fractions?” Chances are not many students, if any, will raise their hand. Now ask the class, “What if we subtracted 6 first? Would the subtraction result in whole numbers or fractions? Which method uses the order of operations backward?” The key concept is that students realize that the second method involves using the order of operations backward, also known as opposite operations, and that the second method tends to yield equations with friendlier numbers. Tell the class they will use both a table method and symbolic method to solve equations using the order of operations backward. Have a student volunteer read the information before Exercise 1 then have the students

answer Exercises 1 through 3. You may also want to model solving an example problem such as 4 2 103

x + =

both with a table and symbolically, while the class solves − =2 4 83

x on their dry-erase boards. Ask the students

what they think the left column represents. You may want to guide them or give them a hint like, “Think order of operations.” Do the same for the right column. The key is for the students to see that the left column represents the doing (for evaluating expressions) and that the right column represents the undoing (for solving equations). Have each group write two questions about something that they still do not understand and write two statements of something that they never knew before. Collect these and then randomly read some or all as needed. Discuss, as a class, the responses to the questions. After you have discussed Exercises 1 through 3, discuss with students the method you would like students to use to check the solution. Most teachers like to have students substitute the solution into the original equation to make sure it makes a true statement after each side of the equation is evaluated. Have students work in pairs to complete Exercises 4 through 9. Walk around the room and help students as needed. Bring the class together and have student volunteers share results with the class. Have a student volunteer model the example problem before Exercise 10 while the rest of the class guides the student as needed. Now have the class work in pairs on Exercise 10 and have a student volunteer(s) share their results with the class. Discuss the advantages and disadvantages of the table technique with the class.


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Talk with students about solving equations that have fractions. Have students read the paragraph before Exercise 11 as part of the discussion. Have the students decide which method they would rather use and complete Exercise 11. Then have individual students use the method of their choice to solve the equations in Exercise 12. Have students partner with someone who used a different method and compare answers as well as the steps to solve the equation. The goal is to have students take ownership of one method to solve equations with fractions. Some students may want to use a calculator to use a third method by changing the fractions to decimals. Have the students use the table method and the symbolic method to solve equations that have decimals in Exercise 13. After students have mastered solving equations by using the opposite operation in reverse order, you may want the students to revisit the graphic organizer they made in Lesson 3 for completing the order of operations. Now, have the students modify the graphic organizer or make a new one that can be used for solving one-variable equations. You could either display their graphic organizers or have them keep them in their notebook to use when they are solving simple one-variable equations.

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SJ Page 44 Activity 1 Today you will use doing and undoing operations with a table to solve one-variable equations. We will sometimes refer to “doing” as going forward and “undoing” as going backward.

The operations in the equation + =3 2 234

x can be represented in a table as shown.

The steps in the left column show how to start with x and work forward to 23. The steps in the right column show how to start with 23 and work backward to x.

This type of table can be used to help solve one-variable equations.

1. Study the left column and the right column. In your own words, describe how they compare?

Answers may vary. Sample response: The left column statements are mathematically opposite to the right column statements. Make sure the students discuss opposite operations.

2. Use the column on the right and work backward to undo the operations and solve for x. Students should determine that x is 30 by multiplying 23 and 4 then subtracting 2 and finally dividing

by 3. The equation can also be solved symbolically as shown below.

( )

3 2 23 Start with the equation.4

3 24 4 23 Multiply both sides by 4.4

3 2 923 2 2 92 2 Subtract 2 from each side.

3 90 3 90 Divide both sides by 3.3 3

30

x

x

xx

xx

x

+ =

+⎛ ⎞ =⎜ ⎟⎝ ⎠

+ =+ − = −

=

=

=

3. Compare the table method and symbolic method to solve the equation. Describe the similarities and

differences in these two methods. Sample response: We see the same operations are performed in the same order to solve it. The only difference is that the operations are performed on both sides of the equation to maintain equivalence.

Divide by 3Multiply by 3

x

Add 2

Divide by 4

Subtract 2

Multiply by 4

23

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SJ Page 45 Solve Exercises 4 through 9 by using both a table and symbolic mathematics. Make sure to check your solutions. 4. Complete the right column in the following table that matches the equation − − = −3 15 78x and use the

table to help you solve for x.

3x 15 783x 15 15 78 15

3x 633x 633 3x 21

− − = −− − + = − +

− = −− −=− −

=

5. Fill in the table so that it matches the equation ( )− = −3 4 5 99x and use it to solve for x.

( )( )

3 4x 5 993 4x 5 99

3 34x 5 33

4x 5 5 33 54x 284x 284 4x 7

− = −

− −=

− = −− + = − +

= −−=

= −

6. Fill in the table so that it matches the equation − + = −6 50 211x and use it to solve for x.

6x 50 211

6x 5011 11( 2)116x 50 22

6x 50 50 22 506x 726x 726 6x 12

− + = −

− +⎛ ⎞ = −⎜ ⎟⎝ ⎠

− + = −− + − = − −

− = −− −=− −

=

Divide by –6 Multiply by –6

x=12

Add 50

Divide by 11

Subtract 50

Multiply by 11

–2

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o

Divide by –3 Multiply by –3

x=21

Subtract 15 Add 15

–78

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Step

s to

und

o

Divide by 4 Multiply by 4

x= –7

Subtract 5

Multiply by 3

Add 5

Divide by 3

–99

Steps to do

Step

s to

und

o

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SJ Page 46

7. In your pair, create a table that matches the equation ⎛ ⎞− =⎜ ⎟⎝ ⎠

3 7 66x and use it to solve for the variable.

Walk around the room while groups are working on the problem. Pick at least one group to present. Give the group a blank transparency and overhead marker to represent the tables and method of solving the equations.

( )

x3 7 66x3 7

663 3x 7 26

x 7 7 2 76

x 96

x6 6 96

x 54

⎛ ⎞− =⎜ ⎟⎝ ⎠⎛ ⎞−⎜ ⎟⎝ ⎠ =

− =

− + = +

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

=

8. In your pair, create a table that matches the equation ( )+=

23 2108

4x

and use it to solve for the

variable. Make sure at least one group shares with the whole class. Provide transparency or poster paper. You may want to spend some time talking about the opposite of squaring (and square roots).

( )

( ) ( )

( )( )

( )( )

2

2

2

2

2

2

3 x 2108

43 x 2

4 4 1084

3 x 2 432

3 x 2 4323 3

x 2 144

x 2 144x 2 12

x 2 2 12 2x 10 or 14

+=

⎛ ⎞+⎜ ⎟ =⎜ ⎟⎝ ⎠

+ =

+=

+ =

+ = ±

+ = ±+ − = ± −

= −

Subtract 2 Add 2

x=10 or –14

Square

Multiply by 3

Square Root

Divide by 3

108

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s to

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o


Multiply by 6 Divide by 6

x=54

Subtract 7

Multiply by 3

Add 7

Divide by 3

6

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Step

s to

und

o


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SJ Page 47 9. With your partner create a table and a matching equation. Trade the table with the other pair in your

group, but don’t show them the equation. Now, create the equation that matches the table from the other pair. As a group, check your work.

Make sure the pairs don’t share the equations before the other pair has a chance to determine the

matching equations. Collect the tables and equations or have the groups post them followed by having students walk around the room analyzing the different tables and matching equations.

A table can also be used to solve for an unknown variable in a formula. For example, the perimeter formula of a rectangle is l= +2 2P w , where P is the perimeter, l is the length, and w is the width. If we know that the perimeter is 325 inches and the width is 45 inches, we can determine the length by substituting the values into the formula, build a matching table, then solve.

= +

= + •= +

2 2325 2 2 45325 2 90

P wlll

325 2 90325 90 2 90 90

235 2235 22 2

117.5

l

ll

l

= +− = + −

=

=

=

l

Length is 117.5 inches. 10. Use this same concept to solve for the unknown radius of a circle. The formula for the area of a circle is

π= 2A r . If the area of a circle is 3,020 square centimeters, what is the approximate length of the radius? How would you check your solution?

The approximate length is 31 centimeters. Sample response: You can check your answer by substituting r = 31 into the formula 3020 = π r 2.

2

2

2

2

3020 r

3020 r

961.8 r

961.8 r31.0 r

=

=

≈

≈≈

π

ππ π

Note: In this case we did not represent the negative square root, because a measurement of negative

centimeters would not make sense.

l

l

w w

Square root Square

r 31≈

Multiply by π Divide by π

3020

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Step

s to

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o Divide by 2Multiply by 2

l

Add 90 Subtract 90

325

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SJ Page 48 There are two different methods you can use to solve equations that contain fractions. One method is to complete the opposite operations with the fractions. The second method is to first multiply both sides of the equation by the least common denominator of all the fractions then complete the opposite operations. Complete Opposite Operations First Multiply by the Least Common Denominator First

3 2 24 5

3 2 2 224 5 5 5

3 124 5

34 4 123 4 3 5

4815165

y

y

y

y

y

y

− =

− + = +

=

⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

=

=

( )

3 2 24 5

3 220 20 24 5

60 40 404 515 8 40

15 8 8 40 815 48

4815165

y

y

y

yy

y

y

y

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

− =− + = +

=

=

=

11. Study the two different methods above, then describe which method you prefer to use and why.

Answers will vary. You may want to have students create a matching table for the first method. 12. Solve the following equations for the variable.

a. − =2 1 211 3

y b. + =5 3 53 4 6z

77y6

= 1z20

=

13. Solve the following equations for the variable. Use a table as needed.

a. 7.5 0.5 1.7x − = − b. 1.25 1.5 7.8x− + = − x = – 0.16 x = 7.44


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Teacher Reference Activity 2 The goal of this activity is to help students understand how to solve a one-variable equation when the variable is on both sides of the equal sign. Place the equation 2 5 6x x− = + on the overhead or board and ask the students how we might solve an equation if the variable is on both sides of the equal sign. Have a student voluntary write the class responses on the board. Keep the responses posted until you have completed the modeling with tiles. Model solving the equation 2 5 6x x− = + with tiles and symbolically. Have the students guide you as they can. 2 5 6x x− = + 2 5 5 6 5x x x x− − + = − + + 11x = Now, model solving other examples with the students. You may want to have a volunteer student solve one equation with tiles at the overhead, while the class solves a different equation at their desks. You can use the following examples to model solving equations with variables on both sides using tiles.

5 6 2 3x x− = + 3 8 4x x+ = − ( )3 4 2 5x x− = − ( )2 3 4 2x x− = − Have the students compare how they solved the equation with tiles to what was recorded on the board. Students should notice that they need to first add or subtract one of the variable terms from both sides of the equal sign so that the variable is to one side of the equal sign. Have the student complete Exercises 1 through 4. Mention to the students that Exercises 2 and 3 are examples of using the tiles for the distributive property which they did in Lesson 2. You may want to review this property with your students. For Exercise 4, tell the students that when the equation has values too large to represent with algebra tiles, that they can solve these equations using symbolic mathematics only.

=

=

=

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SJ Page 49 Activity 2 Tiles can be used to model equations with variables on both sides of the equal sign and solve the equations symbolically. 1. Solve the equation − = +2 5 3x x with tiles and draw a sketch of how you solved it using the tiles. Then,

solve the equation symbolically. 2x 5 x 3− = + 2x x 5 5 x x 3 5− − + = − + + x 8= 2. Solve the equation + = −3( 2) 6x x with tiles and solve it symbolically. Draw a sketch of the tiles and

describe how you solved this equation for x by manipulating the tiles and symbols. Sample Response: I first subtract six units from both sides and one x from each side. Then I divided both sides by two.

3x 6 x 6

3x x 6 6 x x 6 62x 122x 122 2x 6

+ = −− + − = − − −

= −−=

= −

=

=

=

=

=

=


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SJ Page 50 3. Use tiles to solve the following problems. Draw a sketch of the tiles, then solve the problem

symbolically. a. ( ) ( )3 1 2 2x x− = + See student work for tile representation. 3x 3 2x 4

3x 2x 3 3 2x 2x 4 3x 7

− = +− − + = − + +

=

b. ( )4 2 4 5x x− + = + See student work for tile representation. 4x 8 4 x 5

4x 4 x 54x x 4 4 x x 5 4

3x 93x 93 3x 3

− + = +− = +

− − + = − + +=

=

=

What if the numbers are too large in value to use algebra tiles? Use symbolic mathematics and the distributive property. 4. Solve the following problems using symbolic mathematics. a. ( )7 5 3 11x x− = − b. ( ) ( )12 9 6 4 12 6x x− + + = + −

( )7 x 5 3x 117 x 35 3x 11

7 x 3x 35 35 3x 3x 11 354x 244x 244 4x 6

− = −− = −

− − + = − − +=

=

=

( ) ( )12 x 9 6 4 x 12 612x 108 6 4x 48 6

12x 102 4x 4212x 4x 102 102 4x 4x 42 102

16x 14416x 14416 16

x 9

− + + = + −− − + = + −

− − = +− − − + = − + +

− =− =− −

= −

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SJ Page 51 Practice Exercises 1. Use the table method to solve the following equations.

a. − =2 1 115

x

x = 28

b. + =7 5 183

x

x = 7

2. Find the mistakes in the table below for the equation −=

2 9 311y and correct the mistakes.

The two columns of steps are in reverse order and the left column should be the right column and the

right column should be the left column. See students answers for corrections.


x

Subtract 1

Divide by 5

Add 1

Multiply by 5

11

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s to

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o


x

Add 5

Divide by 3

Subtract 5

Multiply by 3

18

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s to

und

o


y

Add 9

Divide by 2

Subtract 9

Multiply by 2

3

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s to

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o


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SJ Page 52

3. The formula for the area of a trapezoid is ( )= +1 212

A b b h . If the area of a trapezoid is 672 square

inches, the height is 24 inches, and length of one of the bases is 30 inches, what is the length of the other base?

1b 26= inches 4. The Pythagorean Theorem states that for a right triangle, the sum of the square of each leg of the

triangle, sides a and b, is equal to the square of the hypotenuse, side c. + =2 2 2a b c What is the length of the unknown leg of a right triangle with a hypotenuse of 20 centimeters and one

leg with a length of 16 centimeters? a = 12 cm

Subtract 30 Add 30

1b = 26

Multiply by ½

Multiply by 24

Divide by ½

Divide by 24

672

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b1

24 in.

30 in.

a

b

c

a

16 cm

20 cm

Steps to do

Square root Square

a = 12

Add 216 Subtract 216

220 Step

s to

und

o

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SJ Page 53 5. Solve the following equations. Use a table as needed.

a. 4 39 17x + = − b. ( )5 3 45x + = c. 13 224

y −=

x = –14 x = 6 y = 101

d. 7 11 138

y −+ = e. ( )6 7 11 13x− + − = f. 12 5x + =

y = 23 x = –11 x = 13 g. 2 16 65x − = h. 22 5 7x− = − x = 9 or x = –9 x = –3

i. 5 10 3x− = − j. 4 8 2x− = − x = 5 x = 6 6. Solve the following equations. Use tiles as needed. a. 3 5 13x x− = − b. 8 2 16x x− = − c. 3 8 3 10x x− = − + x = –4 x = 8 x = 3

d. ( )2 3 4 3x x− = − e. 4 2 5 10 3 4x x x− − = + − f. 6 12 83

x − =

x = 2 x = –2 x = 6 g. 2 3 4x x− − = − h. 3 2 1 2 1 3x x x x+ − − = + − − i. 3 24 8x x− = + x = –7 x =2 x = 16 j. 4 4 23x x− = + k. − + = − +4(3 5) 9 3(3 6) 1x x l. + = −2( 3) 4x x x = –9 x = –2 x = –10


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SJ Page 54 7. Solve the following equations. Round your answer to nearest tenth, if needed, or leave the answer as a

fraction. a. 0.5 1.5 1.3x x− = − b. 5.1 8.2 2 16x x− = − c. 3 6.2 3.8 10x x− = − + x = –0.4 x ≈ –2.5 x ≈ 2.4

d. 3 3 134 2 3

x x− = − e. 4 32 15 2x x− = + f. 6 12 8

3 5x − =

x = – 14/27 x = –30/7 x = 14/5 8. The following equations were solved incorrectly. Determine the mistake and solve the equation

correctly. Show your work. a. 2 5 4 7

2 2 5 7 4 2 7 712 212 22 26

x xx x x x

xx

x

− + = −− − + + = − − +

=

=

=

b. ( )2 4 3 22 4 3 2

2 2 4 2 3 2 2 26

x xx x

x x x xx

+ = −+ = −

− + + = − − +=

Sample responses: a. They subtracted 2x from both sides of the equation instead of adding 2x. The solution should be

x = 2. b. They didn’t distribute the multiplication of 2 correctly on the left side of the equation. The

solution should be x = 10. 9. Substitute the correct values into the formula and solve for the unknown for each exercise.

a. The formula for the area of a triangle is 2bhA = where A represents the area, b represents the

base length, and h represents the height. If the height of a triangle is 21 inches and the area is 325.5 square inches, what is the base length?

The base is 31 inches.

21 in.

b

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SJ Page 55 b. The Pythagorean Theorem is 2 2 2a b c+ = , where a and b represent the leg lengths of a right

triangle and c represents the hypotenuse length. If the one leg is 18 centimeters and the hypotenuse is 34 centimeters what is the approximate length of the other leg?

The other leg is approximately 28.8 centimeters.

c. The volume of a cone is 213

V r hπ= , where r is the radius of the base of the cone and h is the

height of the cone. Determine the radius of a cone that has a volume of 235.5 cubic centimeters and a height of 12 centimeters. Use 3.14 for π .

The radius is approximately 4.3 centimeters. d. The formula ( )1 tA P r= + represents the amount in an interest bearing account, where A is the

amount in the account, P is the original amount placed in the account, r is the annual interest rate, and t is the number of years the account has been in existence. Determine the annual interest rate for an account that began with $2,000.00 two years ago and now has a value of $2,205.00. Round to the nearest hundredth.

The annual interest rate is approximately 0.05 or 5%.

e. The slope of a line containing two points, ( )1 1,x y and ( )2 2,x y is 2 1

2 1

y ymx x

−=

−. Where m is the

slope, 1x is the x-value for the first point, 1y is the y-value for the first point, 2x is the x-value for the second point, and 2y is the y-value for the second point. Determine the y-value for the second point, 2y , if the slope of the line is –3, the x-value for the first point is –1, the y-value for the first point is 7, and the x-value for the second point is –6.

y2 = 22 8. How can the order of operations be used to solve equations? Answers will vary.

h

r

34 cm 18 cm

?


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Outcome Sentences SJ Page 56 The table method for solving equations When using the order of operations to solve equations Tiles are useful to solve equations with variables on both sides because I need more help understanding Tiles are difficult to use when The method I prefer to use to solve equations is ________________________________________________ because

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Teacher Reference Lesson 4 Quiz Answers 1. x = 3. 2.

2(x 2) 6 3(x 2) 32x 4 6 3x 6 3

2x 2 3x 92x 2x 2 9 3x 2x 9 9

7 x

+ − = − −+ − = − −

− = −− − + = − − +

=

3.

( )P 2 2w576 2 192 2ww 96cm

= += +

=

l


x

Add 3

Divide by 5

Subtract 3

Multiply by 5

6

Steps to do St

eps

to u

ndo

=

=

=

=


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Lesson 4 Quiz Name:

1. Create a table that matches the equation + =9 3 65

x and use it to solve for the variable. Also solve the

equation symbolically. 2. Use tiles to solve + − = − −2( 2) 6 3( 2) 3x x . Draw a sketch of your complete tile solution. Solve the

equation symbolically too. 3. The perimeter formula of a rectangle is l= +2 2P w , where P is the perimeter, l is the length, and w is

the width. If the perimeter of a rectangle is 576 centimeters and the length is 192 centimeters, determine the width.

x

Steps to do

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und

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Lesson 5: One-Variable Inequalities Objectives • The students will be able to solve one-variable inequalities. • The students will be able to graph and describe the solution to one-variable inequalities. • The students will be able to solve one-variable inequalities related to real-world applications.

Essential Questions • How are solutions to inequalities described in written and graphed formats? Tools • Student Journal • Setting the Stage transparency • Dry–erase boards, markers, erasers • Overhead projector • Overhead Tiles • Student Tiles • Tile Pad • Inequality Tile Pad • Inequality Tile Pad Transparency • Poster Paper and Markers Warm Up • Problems of the Day Number of Days • 2 days (A suggestion is to complete activities 1 and 2 with Practice Exercises 1 through 4 on the first day.

On the second day, have students complete Activity 3, the remaining Practice Exercises, and Lesson Quiz.) Vocabulary OR ADD Inequality Less than Less than or equal to Greater than Greater than or equal to Compound Compound inequality Addition Property of Inequality Subtraction Property of Inequality Multiplication Property of Inequality Division Property of Inequality

Solving One-Variable Equations Lesson 5: Solving One-Variable Inequalities Page 93

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Teacher Reference Setting the Stage The goal of this Setting the Stage is to help students review and develop their intuitive understanding of inequalities from real-world examples. After this Setting the Stage, you can transition students to the formal term “inequality”, and the symbolic method and number line method to represent inequalities. Place the transparency on the overhead projector and then ask the students if they know any of the requirements, ratings, or conditions to these real-world situations. You may need to supply the students with some of the following information orally or by writing it on the transparency:

Movie Ratings: The MPAA (Motion Picture Association of America) movie ratings were:

• G-General Audiences: All ages admitted. • PG - Parental Guidance Suggested: Some material may not be suitable for children. • PG-13 - Parents Strongly Cautioned: Some material may be inappropriate for children under 13. • R – Restricted: Under 17 requires accompanying parent or adult guardian. • NC-17: No One 17 and Under Admitted.

FBI Age Requirements: Many federal investigative jobs have age requirements, and applicants must be at least 21 years of age and under the age of 37 at the time of appointment.

Space Shuttle Launch Conditions: The countdown for a Shuttle launch will stop if the temperature exceeds 99 degrees for more than 30 consecutive minutes. And in no case may the Space Shuttle launch if the temperature is 35 degrees or colder.

Now, have the class work in pairs on their dry-erase boards. Each pair needs to pick at least one condition and represent it as many ways as possible. Encourage students to use drawings, words, symbols, and graphs to represent the conditions. You may want to allow students to pick another real-world application of their choice that has inequality characteristics. Other possible applications for inequalities that you may want to use are age requirements for various reality TV shows, military age requirements, college requirements for standardized test scores, rental car age requirements, and driver license age requirements. Walk around to look at how students represent each condition or have the pairs hold up their dry-erase boards to show you and the class. Don’t worry if students do not formally use number lines or inequality statements correctly at this point. Mainly, you are helping students link to prior understanding and to share their current thinking about inequalities. You will have a chance, if needed, to formally introduce students to graphing and representing inequalities as you transition to the first activity. Have the students keep the work they did on their dry-erase boards, they will get a chance to work with their representations again in the activity. Option: Put a number line at the front of class and have students stand at different locations to represent various inequalities. For example, you could have all the students stand in such a way that they all need to be in a position on the number line that represents x < 3. Note: You may want to let students know that there are three possible relationships between any two real numbers (=, <, >). Examples: 5 = 5, 5 < 6, and 5 > 4.

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Movie Ratings

Space Shuttle Launch Conditions

FBI Age Requirements

Lost in the Wilderness of Space!: PG-13


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Teacher Reference Activity 1 The goal of this activity is to introduce the students to the more formal methods to represent inequalities, with words, graphs (number lines), and symbols (inequalities). Display the Setting the Stage transparency. Use one of the conditions from the Setting the Stage and guide the class through representing the condition with the formal words, symbols, and number line. For example, you may want to lead the students through representing the age requirements for the FBI.

• A person applying for the FBI must be 21 or older and less than 37 years of age. • Using “a” to represent age, a mathematical inequality would be 21≤ a < 37. • A number line representation could be:

Model your thinking aloud or solicit help from students. You may want to use terms such as endpoints, range, interval, variable, less than, greater than, less than or equal to, and greater than or equal to as you think out loud. You will also want to briefly describe when to fill in the endpoints or leave them hollow.

Now have pairs of students go back to the condition they chose for the Setting the Stage and make any appropriate changes on the dry-erase boards to represent their condition formally with words, symbols (inequality), and a graph (number line). You can walk around the classroom to assess the students and have students share with the class as needed. You may want to ask guiding questions, such as:

• How do the specific characteristics on your graph reflect the real-world concept? • What advantage does the graph have compared to the real world scenario? • What advantage does the symbol representation have compared to the graph?

For Exercise 1, have the students record the representations for their condition. Now, write the term inequality in the center of the board and have the students create a graphic organizer or word web for the term. You can have various students come to the front to add to the organizer. Continue to have student pairs work on Exercise 2. Have student volunteers share their results with the class. If no student gave a fraction as an answer, ask the class, “Can we also place the w at 3/2?” Place w anywhere on the transparency number line so that w is less than 3. Discuss with the class that w can have many possible values when it is described as a number less than 3. Have the student pairs complete Exercise 3 and then share their results with the class. Then have the class complete Exercises 4 through 6 and share their results. After the class has shared results, have them continue with Exercises 7 through 12. While the class is working on the exercises, walk around checking the students understanding of the material and answer questions they may have. If a student pair has any intriguing questions, pose the question to the rest of the class and let other student pairs give answers. After the class has completed the activity, have student volunteers share their results on Exercises 7 through 12. Lead a follow-up discussion with the class on what they have learned about graphing inequalities and writing symbolic and descriptive inequalities. Have a student volunteer list the class responses on the board or on a blank transparency on the overhead projector. Tell the class to record information they find useful.

18 20 22 24 26 28 30 32 34 36 38 40

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SJ Page 57 Activity 1 1. Record the different representations for the condition you chose. Answers will vary. 2. Write the inequality that represents w less than 3. The inequality is w < 3.

a. With your partner draw a number line, then each of you place a w anywhere on the number line, so that w is less than 3. You cannot choose the same place on the number line. Answers will vary. See number lines as needed.

b. Write the placement location of the w on your number line.

Answers will vary. A sample response might be –2.

c. Write your partner's placement location. Answers will vary. A sample response might be 1.

d. Whose value, or location, represents the largest number? Explain why. Now, write an inequality which represents a true statement for all three values, including the 3. Answers will vary. A sample response might be: "My partner placed w at 1. My partner had the larger number because 1 is greater than –2." The inequality for all three values is –2 < 1 < 3.

e. Write a non–integer value which is greater than your w value, but less than 3.

Answers will vary. A sample response might be: –1/2. 3. Erase the variables and then place other variables on your number line according to the conditions (or

criteria) below and write the corresponding inequalities.

a. w is greater than or equal to –2 w ≥ –2

b. x is between –3 and 2, excluding the endpoints –3 and 2 –3< x < 2

c. y is more than 0 but less than or equal to 5 0 < y ≤ 5

d. z is greater than 5 or less than –4 z > 5 or z < –4 Note: This statement cannot be represented with one statement like in part c. You may want to briefly discuss with students the reason.


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SJ Page 58 4. Erase the variables and consider the following new criteria for the variables.

w is between 0 and 1 x is between –1 and 0 y is between 4 and 5 z is between –3 and –2

a. Determine if w x< is always true, sometimes true, or never true. Give examples with numbers

to help determine the answer. Sample response: Never true. For example 0.5 is not less than –0.5.

b. Determine if wy xy> is always true, sometimes true, or never true. Sample response: Always true. For example, 0.5(4.5) is 2.25 which is greater than –0.5(4.5) which is –2.25.

5. a. Write two inequality questions for the variables w, x, y, and z from Exercise 4. A sample

question could be, “Is w x< always true, sometimes true, or never true?” Answers will vary. A sample response might be: “Is y > x always, sometimes, or never true?”

and “Is –1(z) > w?”

b. Trade questions with your partner and answer your partner's inequality questions. Answers will vary.

6. Discuss your inequality questions with another student pair. How were your inequality questions

similar and how were they different? Answers will vary.

7. Below are several pairs of correctly and incorrectly graphed inequalities. For each incorrect graph

describe what is incorrect. Correctly Graphed Incorrectly Graphed a. x ≤ 5 Sample response: The endpoint should be solid. b. x < 5 Sample response: The arrow is pointed in the wrong direction. c. –1 ≤ x Sample response: The endpoint should be solid.

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SJ Page 59 d. –1 < x

Sample response: The arrow is pointed in the wrong direction. e. –4 < x < 6 Sample response: The endpoints should not be solid. f. –4 ≤ x ≤ 6 Sample response: There should be no arrows and the graph should have the numbers between –4

and 6 not outside that range. 8. What are the characteristics of a correctly graphed inequality? Answers will vary. A sample response might be: “The endpoints are solid if the inequality has less than

or equal to.” 9. Write a description for each correctly graphed inequality from Exercise 7. The first one has been

completed for you. a. x ≤ 5: All real numbers less than or equal to a positive five.

b. x < 5: Sample response: x is all real numbers less than five. c. –1 ≤ x: Sample response: x is all real numbers greater than or equal to negative one. d. –1 < x: Sample response: x is all real numbers greater than negative one. e. –4 < x < 6: Sample response: x is all real numbers between negative four and positive six. f. –4 ≤ x ≤ 6: Sample response: x is all real numbers greater than or equal to negative four and less

than or equal to six.

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0 0

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SJ Page 60 You will often see the variable of an inequality on the left, such as the x in the inequality x ≥ 1. A written or verbal description of this inequality would generally be “x is greater than one.” There are, however, other equivalent methods to represent this inequality with symbols. There are also other methods to write or verbally describe this inequality. Below are some examples. The graph of the inequality remains the same for the different symbols, written, or verbal representations. Graph: Symbol methods: x ≥ 1, 1 ≤ x Written or verbal: x is greater than or equal to 1,

1 is less than or equal to x 10. For each written description below of an inequality, draw a graph of the inequality, then write two

different symbol inequalities and another written or verbal description.

a. w is less than or equal to positive three. Symbols: w ≤ 3, 3 ≥ w

Written or verbal: Sample response: Positive three is greater than or equal to w.

b. y is greater than negative five. Symbols: y > –5, –5 < y

Written or verbal: Sample response: Negative five is less than y.

c. An unknown number is at least a negative ten. Symbols: x ≥ –10, –10 ≤ x

Written or verbal: Sample response: Negative ten is less than or equal to some number.

d. An unknown number is at least negative four but no more than positive eight.

Symbols: –4 ≤ x ≤ 8, 8 ≥ x ≥ –4

Written or verbal: Sample response: An unknown number is no more than positive eight but at least negative four.

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0 –5 5

0 –5 5

–4 4 0

–4 4 0

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SJ Page 61 11. Write a symbolic inequality for each real-world example, then draw a graph.

a. Male adult Nile crocodiles usually mature around 10 years with an approximate length of 3 meters. One of the largest recorded adult male Nile crocodiles measured 8.6 meters.

3 ≤ l ≤ 8.6

b. An adult male Nile crocodile may easily exceed 500 pounds and many adults reach 2,200 pounds.

500 < p ≤ 2200

c. The weather report called for low temperatures ranging from –5°F to 5°F. The inequality is –5 ≤ t ≤ 5. The graph is: 12. Write the symbolic inequality and a real-world example to match the graph of the inequality.

The inequality is 92 ≤ a ≤ 100. Answers for the real–world application will vary. A sample response: "To obtain an A in a class, you must have an average between at least 92% and 100% on all your work."

0 10

0 5 –5

90 100

0

800

1600


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Teacher Reference Activity 2 In this activity we will expand on solving one-variable equations to solving basic one-variable inequalities. The goal is for students to apply what they might know about writing and solving one-variable equations to writing and solving one-variable inequalities. Assign different groups one of the exercises between 1 and 4. Have the students write an inequality that matches the scenario and then solve the inequality to answer the question. Make sure the students are prepared to share their answers with the class. Have each group share how they wrote and solved an inequality for each scenario. For the next part of this activity, students will use tiles and an inequality tile pad (located at end of unit) to solve inequalities using the Addition and Subtraction Properties of Inequalities. The class will continue to work in groups of four but as pairs on this activity. Write the inequality x – 3 ≥ 2 on the inequality tile pad transparency and place the appropriate tiles on the pad to represent the inequality. Ask the class what they must do to isolate the variable on the inequality tile pad. Guide the class through adding three positive tiles to both sides of the pad to solve for the variable. As you model with the tiles also record the symbols. The zero pairs can be removed to yield the inequality x ≥ 5. x – 3 ≥ 2 x – 3 + 3 ≥ 2 + 3 x ≥ 5 For Exercise 5, have a student volunteer model a second example, y + 4 < – 3, on the overhead transparency using the overhead tiles while the class works on y + 6 < 4 on their dry–erase boards and the inequality tile pad. The class, and volunteer student, will need to cover up the equal sign for this example. The class, and volunteer student, will also need to orient the inequality tile pad to show < (less than). Lead a class discussion about adding and subtracting the same quantity from both sides of the equal sign and inequality sign. Ask the class, “How is adding or subtracting the same quantity from both sides of the inequality symbol similar to the Addition and Subtraction Properties of Equality?” Have the class agree on the definitions for the Addition and Subtraction Property of Inequality and write the definitions in their journal in the blank box provided. Have the class work in their groups, but as pairs, for Exercises 6 through 8. Have student volunteers share their results with the class. Sample definition for addition property of inequality: For all real numbers a, b, and c, if a > b, then a + c > b + c. The second part of this activity will expand the properties of inequalities to the Multiplication and Division Properties of Inequalities. You will use the Inside–Outside Circles strategy for this part of the discovery activity. Split the class in half. Have half the class form an inner circle facing outward and the other half form an outside circle facing the inner circle of students. Have half of the inner circle students each choose a positive

> =

> =

> =

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number less than 10 and the other half a negative number greater than –10 and place the number on a dry-erase board. Make sure that the inner circle students are arranged as every other one has a positive number and every other one has a negative number. Have half of the outer circle of students write a greater than inequality with numbers only (such as 5 > 2) and the other half write a less than inequality with numbers only (such as –3 < 6). Have the students on the inside pair up with a student on the outside. Tell the class to take the number from the inside student, of the student pair, and multiply both sides of the equality held by the outside student of the student pair. Determine if the inequality statement is true after the multiplication. Tell one of the student circles to rotate to the right by one student to form a new pair. Repeat the process until the students have had enough examples to determine that multiplying by a positive number is okay but that multiplying by a negative number results in a dilemma. You may also want the students to experiment with dividing. Tell the class to go back to their groups of four to discuss and list their findings from the Inside–Outside Circles activity. Have the class answer Exercises 9 through 11 and have student volunteers share their results with the class. Ask the students guiding questions, such as “When did the inequality no longer become a true statement after multiplying or dividing both sides by some value?” Together as a class, write the definitions of the properties for multiplying and dividing both sides of an inequality. Have the class write these definitions in the blank box provided in their student journal. Model solving the inequality 5x – 5 ≥ 6x + 4 by subtracting 6x from both sides so that eventually you will need to multiply both sides by a negative one which will cause the inequality to switch. Model solving 5x – 5 ≥ 6x + 4, while the class solves 8x + 8 < 9x – 7 using the same process as yourself. Lead a discussion with the class on how solving inequalities is similar and different than solving equations. Ask guiding questions like: • What are the similarities and differences in solving one-variable inequalities and one-variable equations? • How did the order of operations assist you in solving one variable inequalities? Have the students continue with Exercises 12 through 17 in their groups or pairs. Walk around the classroom and guide students as needed with the exercises or answer any questions they may have. Ask for student volunteers to share their results with the rest of the class. Now you can introduce students to simple real-world applications for solving inequalities. Have the students work in groups to complete Exercises 18 through 21. Walk around the classroom and give suggestions as needed. If you feel that you need to model solving a real-world application you can use the following problem.

The lead marching band for one of the Marti Gras parades needs a banner that is 5 feet in width. What are the possible lengths of the banner if the border has to be 50 feet or less?

P = 2l + 2w ( )+ ≤

≤2 2 5 50

20l

l Length would need to be less than or equal to 20 feet.


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Activity 2 SJ Page 62

Your teacher will assign you a scenario below. You will write an inequality for the scenario then solve the inequality to answer the question.

1. The maximum safe load for each two-person chair on a ski lift is 425 pounds. The weight of one person

on a chair is 208 pounds. What additional weight can the chair carry?

Sample answer: The chair can hold up to an additional 217 pounds. Solving x + 208 ≤ 425 for x gives x ≤ 217.

2. A cell phone plan allows a maximum of 750 minutes each month. If 147 minutes

have been used this month, what range of minutes is left? Sample answers: There are up to an additional 603 minutes left. Solving

x 147 750+ ≤ for x gives x 603≤ . 3. A family is planning a long trip across the United States that will take at least 36 hours to drive. They

drove for 14 hours on the first day. How many hours do they have left to drive?

Sample answer: There are at least 22 hours left to drive. Solving x 14 36+ ≥ for x gives x 22≥ .

4. Only customers 48 inches tall, or taller, can ride on most roller coasters. If Gabrielle

is only 44.5 inches tall, how much does she have to grow to be able to ride the roller coaster?

Sample answer: Gabrielle has to grow at least 3.5 inches. Solving x 44.5 48+ ≥ for

x gives x 3.5≥ . 5. Create the inequality, y + 6 < 4, on the inequality tile pad using tiles. Solve the inequality with algebra

tiles. Write a description of the solution and graph the solution. Look at students tiles as needed. The symbolic solution to the

inequality is y < –2. The descriptive solution is "All real numbers less than a negative two." The graphed solution is:

Addition Property of Inequality – Subtraction Property of Inequality –

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SJ Page 63 Use the Addition Property of Inequality and the Subtraction Property of Inequality to solve the inequalities in Exercises 6 through 8. Use tiles as needed. 6. Solve and graph the solution to the inequality 4n ≤ 3n +8. Write a descriptive solution to the inequality. The solution to the inequality is n ≤ 8. The descriptive solution is “All real numbers less than or equal

to a positive eight.” 7. Solve and graph the solution to the inequality 14 ≤ n – 8. Write a descriptive solution to the inequality. The solution to the inequality is 22 ≤ n. The descriptive solution is “All real numbers greater than or

equal to a positive twenty–two.” 8. Solve and graph the solution to the inequality 13.5 + 7.2 + 9.8 + p > 40. Write a descriptive solution to

the inequality. The solution to the inequality is p > 9.5. The descriptive solution is “All real numbers greater than

positive nine and one-half.” The following questions are for the Inside–Outside Circles activity. 9. Describe what happened when you multiplied both sides of an inequality by a positive number. Answers will vary. A sample response might be: “When we multiplied both sides of an inequality by a

positive number, the inequality was still a true statement.” 10. Describe what happened when you multiplied both sides of an inequality by a negative number.

Answers will vary. A sample response might be: “When we multiplied both sides of an inequality by a negative number, the inequality was no longer a true statement.”

11. What might we do to make the inequality statement true when multiplying by a negative number? Do you think this would be true when dividing by a negative number?

Answers will vary. A sample response might be: “Flip the inequality so that a greater than becomes a

less than and a less than becomes a greater than.” Yes. It is true for dividing by a negative number.

4–4 0

0 10 –10

0 6 –6

Multiplication Property of Inequality: Division Property of Inequality:


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SJ Page 64 We will now solve inequalities using the Multiplication Property of Inequality and the Division Property of Inequality. Solve and graph the solutions to following inequalities. Be careful when multiplying or dividing both sides of an inequality by a negative number. Write a descriptive solution to the inequality. 12. –6g ≤ 144

The solution to the inequality is g ≥ –24. A descriptive solution is “All real numbers greater than or equal to negative twenty–four.”

13. 75

y≥ −

−

The solution to the inequality is y ≤ 35. A descriptive solution is “All real numbers less than or equal to thirty-five.”

14. 21x < –28

The solution to the inequality is 4x3

−< . A descriptive

solution is “All real numbers less than or equal to negative four thirds.” You did not divide by a negative number so the inequality symbol stays the same.

15. 5 158y

≥ −

The solution to the inequality is y ≥ –24. A descriptive solution is “All real numbers greater than or equal to a negative twenty–four.”

16. 3 334

q−≤ −

The solution to the inequality is q ≥ 44. A descriptive solution is “All real numbers greater than or equal to forty–four.”

15. 2 7 95w + ≤

The solution to the inequality is w ≤ 5. A descriptive solution is “All real numbers less than or equal to five.”

16. 13k – 11 > 7k + 37 The solution to the inequality is k > 8. A descriptive

solution is “All real numbers greater than eight.” 17. 7 + 3t ≤ 2(t + 3) – 2(–1 – t) The solution to the inequality is –1 ≤ t. A descriptive

solution is “All real numbers greater than or equal to a negative one.”

0 24–24

0 44–44

0 5 –5

0 8 –8

0 24–24

0 35–35

0 2–2

0 2 –2

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SJ Page 65 18. The Booster Club needs to make a new school banner to display during championship tournaments.

The length of the banner needs to be 15 feet. What are the possible widths, if the border of the banner can be no more than 42 feet? Write an inequality and then solve it.

P = 2l + 2w ( ) + ≤≤

2 15 2 426

ww

The width would need to be less than or equal to 6 feet.

19. The cheerleaders would like to purchase small pompoms to pass out at the championship game. A case

of 25 pompoms cost $20. What is the maximum number of cases that the cheerleaders can purchase if they can spend no more than $250?

20p ≤ 250 p ≤ 12.50 They would need to purchase less than or equal to 12.50 cases. If it is assumed that partial cases cannot be purchased then they would need to purchase 12 or fewer cases.

20. Girl Scout Cookies™ cost $3.50 per box. If Mercedes wants to serve them at her party but only has a

budget of $20.00 for snacks, how many boxes can she purchase? 3.50p ≤ 20 p ≤ 5.71 She would need to purchase less than 5.71 boxes, which works out to 5 or

less boxes. 21. Your MP3 player holds 8 GB of music. Each song you put on your MP3 averages 0.004 GB. Write an

inequality that will let you know how many songs you can store on your MP3 player. How many songs at most can you store on your MP3 player?

The inequality is 0.004s ≤ 8. Solving the inequality yields s ≤ 2,000 songs.


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Teacher Reference Activity 3 This activity will introduce the class to compound inequalities. The class has already worked with compound inequalities in the Setting the Stage perhaps without knowing they were compound inequalities. Lead a class discussion about inequalities of the form 4.50 ≤ x ≤ 7.50. First talk about how this may represent a real-world concept such as books at the local book fair ranging in price from $4.50 to $7.50. Then ask guiding questions such as:

• Can we write the inequality 4.50 ≤ x ≤ 7.50 into two separate inequalities with the same solution? • Is the inequality 4.50 ≤ x ≤ 7.50 the same as 4.50 ≤ x AND x ≤ 7.50? Why or why not? • How might 4.50 ≤ x ≤ 7.50 be the same as 4.50 ≤ x AND x ≤ 7.50? • What might change in the inequality statement if the price of books all increased by a dollar?

Have student volunteers share their thoughts and reasoning on the questions. Tell the class that the word AND is a conjunction because it joins two inequalities into one and the solution must satisfy both inequalities simultaneously.

You can use the following Venn diagram concept to aid the class in a visual explanation of AND, which means intersection. The set S contains the integers 1 through 15; set E contains the integers {4, 6, 10, 12}; the set D contains the integers {3, 6, 12, 15}. Have the students shade the area in the Venn diagram in Exercise 1 to represent the numbers that are only in E AND D. The numbers 6 and 12 are in both sets. Ask the students questions to help them understand the connection of the term and, and the intersection of the two sets E and D. Example questions could be:

• How does the Venn diagram represent the intersection of

both sets? • What part of the Venn diagram represents the term

“and?” • What would change in set E and D if there were no

numbers in the intersection on the graph?

You may want to model creating a Venn diagram with the sets A {2, 4, 6, 8, 10, 12, 14} and B {3, 6, 9, 12} for the conjunction AND, while students complete Exercise 2. Visually inspect the student’s results on their dry-erase boards. Create other example sets as needed.

Discuss the word "compound" with the class and ask them what it means. It is important for students to understand the term compound as it relates to inequalities. Now, model graphing the following compound inequality and have the class follow along on their dry-erase boards as you model.

–4 < x and x < 2 It may be useful to show students how to use three number lines to graph the compound inequality. The third number line would represent the result of the compound inequality. You might want to use dashed lines projected onto the third number line to show the students the overlap of each inequality. Model how the original two inequalities can be written into a single inequality, –4 < x < 2, using the final results.

1

2

5 7 89

11

13 14

0 4–4

0 4–4

0 4–4

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Model one of the sample exercises below while the class does a different sample exercise on their dry-erase boards. Have the class show you their results and inspect them visually. Sample Exercises –6 ≤ x and x < 12 5 < x and x ≤ 9 –5 ≤ x < 11 –3 < y < 5 –2 ≤ z + 3 < 8 –3 < z + 2 < 7 Have the students work in pairs for Exercises 3 through 8. For Exercise 6, explain or ask the class what the term inclusive means. Walk around the class facilitating and answering any questions or problems the students may come across. After the students have completed Exercises 3 through 8, ask for student volunteers to share their results and explain how they determined which terms were similar. Now tell the class there is another type of compound inequality (also known as a conjunction, or a logical disjunction) related to the word OR. Lead a discussion about a real-world scenario such as discount prices at many restaurants are available to kids 10 and under and adults 55 years of age and older. A matching compound inequality would be n ≤ 10 or n ≥ 55. You can ask guiding questions such as:

• Can we write the inequality n ≤ 10 or n ≥ 55 into one inequality statement? • What might change in the inequality statement if the age of the adult changed to 60? • What might the graph of the compound inequality look like?

You can model a Venn diagram example to aid the class in a visual explanation of OR. Have the students follow along with you using their dry–erase boards. The set S contains the integers 1 through 15; set E contains the integers {2, 4, 6, 8, 10, 14}; and the set D contains the integers {1, 3, 5, 9, 11, 13}. Use a Venn diagram to display these sets. Shade which numbers are in E or D?

Have the students hold up their dry–erase boards and visually check their results. The class should have all the numbers in both sets {1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14}. Ask the students if they understand the connection of the term OR, and the union of the two sets E and D. The key here is for the students to see that we want everything in both sets. We will use this approach when graphing compound inequalities. For the following set of examples, model one set for OR with the students while the students work on another set for OR. Make sure to tell the students that we do not want duplicates. Visually inspect the student’s results on their dry-erase boards. You may want to model creating a Venn diagram with the sets A {2, 4, 6, 8, 10, 12, 14} and B {3, 6, 9, 12} for the conjunction OR, while the students complete Exercise 9. Visually inspect the student’s results on their dry-erase boards. Create other example sets as needed.

Now, model graphing –4 > x or x > 2 while the class models –6 > x or x > 5 for Exercise 10. The third number line would represent the combination of the both inequalities. You may want to use dashed lines to show the students that the graphed inequality is transposed to the solution graph.

S

E 2 4 6 14 8 10

D 1 3 5 13 9 11

7

12

15

4–4

0 4–4

0 4–4

0


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Make sure the students understand that for a logical disjunction inequality, we include both graphs. We could also tell the students that OR represents a choice, hence the inclusion of both inequalities in the solution. If needed, model additional sample compound inequalities while the class does a different compound inequality. Have the class show you their results and inspect them visually. Sample exercises –6 ≥ x or x > 12 5 > x or x ≥ 9 –5 > y or 5 < y –3 > z + 2 or 8 < x + 2 Have the students work in pairs in their groups of four for Exercises 11 and 12. After the pairs have completed these exercises, have them check their answers with the other pair in the group. Ask for student volunteers to share their results and explanations. After the students have shared their results and explanations, have them complete Exercises 13 through 15 using the same process. Again, ask for student volunteers to share and explain their results.

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SJ Page 66 Activity 3 In this activity, you will investigate compound inequalities. The word AND is a conjunction because it joins two inequalities and the solution must satisfy both inequalities simultaneously. To understand the concept of the conjunction AND, complete Exercises 1 and 2 as your teacher directs. 1. The Venn diagram on the right represents the 3 sets S,

E, and D. The set S contains the integers 1 through 15; set E contains the integers {4, 6, 10, 12}; the set D contains the integers {3, 6, 12, 15}. Shade the area in the Venn diagram which represents the numbers that are both in sets E AND D.

2. For the set A { –3, –1, 0, 1 , 3, 5} and B {–4, –3, –2, –1, 0, 1, 2, 3,

4}, draw a Venn diagram of the two sets and shade the area that contains the numbers that are common to both sets.

For Exercises 3 and 4, graph the solution to the given compound inequality. Give a written description of the solution and write the inequalities as a compound inequality. 3. a ≤ 6 AND a ≥ –2 The solution inequality is –2 ≤ a ≤ 6. A written description

of the solution is “All real numbers greater than or equal to negative two and less than or equal to positive six.”

4. –5 < x –4 AND x – 4 < 2 The solution inequality is –1 < x < 6. A written description of the

solution is “All real numbers greater than negative one and less than positive six.”

See students’ diagrams.

0

0

0

0

0

0

0

0

1

2

5 7 89

11

13 14


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SJ Page 67 Now, we will concentrate on writing compound inequalities from various application problems. For Exercises 5 through 8, write a compound inequality for the given situation then graph the inequality. 5. Starting salaries for college graduates range from $30,000, for educational services, to $56,000 for

chemical engineering.

The compound inequality is $30,000 ≤ S < $56,000. NOTE: Each mark on the number line represents $10,000.

6. According to Hooke's Law, in order to displace a spring by a certain amount of centimeters from its

"resting" length, a force F, in Newtons, needs to be applied to the spring. The equation for Hooke's Law is F = kx, where k represents the spring constant of elasticity. If the spring constant was calculated to be 1.68 and forces from 2 to 5 Newtons were applied to the spring, what will be the range of the displacements of the stretched spring? Round your answer to two decimal places.

The compound inequality is 1.19 ≤ x ≤ 2.98. 7. The National Weather Service classifies hurricanes using the Saffir-Simpson Hurricane Scale. It is used

to inform people of potential property damage and flooding caused by the storm's surge, winds, and rain. Wind speed is the major criteria that is used to categorize hurricanes. What is the range of wind speeds for a category 3 hurricane?

Let w represent the wind speed, the inequality would be

111 ≤ w ≤ 130. 8. There are several stages of sleep which affect your heart rate. During the most restful period of sleep,

your heart rate can be reduced by about 20%. During the most restless period of sleep, called REM, your heart rate can increase by about 50%. REM stands for “rapid eye movement,” and it is during REM when your dreams take place. If your normal heart rate while you’re awake is 66 beats per minute, what is the range of your heart rate while you are sleeping? Round your answer to the nearest heart beat.

Let h represent the heart beats per minute. The inequality is 53 ≤ h ≤ 99.

Category Number

Wind Speed

1 74–95 mph 2 96–110 mph 3 111–130 mph 4 131–155mph 5 > 155 mph

0

0

100

50 100

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SJ Page 68 9. For sets A {-3, -1, 0} and B {1, 2, 3, 4}, draw a Venn diagram representing these sets and then shade in

the portion representing A OR B. Use curly brackets to list all the items in sets A OR B. The items in set A OR set B are {–3, –1, 0, 1, 2, 3, 4}. The Venn diagram is For Exercises 10 through 12, graph the solution to the given compound inequality. Give a written description of the solution. 10. –6 > x OR x > 5 The description of the inequality is “All real numbers

less than negative six or greater than five. 11. a ≥ 6 OR a ≤ –2 The description of the inequality is “All real numbers less

than or equal to negative two or all real numbers greater than or equal to six.”

12. –2 ≥ x + 4 OR x + 4 > 9

The description of the solution is “All real numbers less than or equal to negative six or all real numbers greater than five.”

13. Write a compound inequality and description for the graph on

the right, then write an application problem to go with your compound inequality.

The compound inequality is x < –30 OR x > 45. The description of the solution is “All real numbers less

than negative thirty or all real numbers greater than forty–five.” Application answers will vary.

0

0

0

0

0

0

0

0

0 –30 30

0

0

0 0


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SJ Page 69 Now, let’s write compound inequalities from various written and application type problems. For the following exercises, write a compound inequality for the given situation, find and graph its solution. Don’t forget to label the variable if necessary. 14. The product of –5 and a number is greater than 35 or less than 10.

Let n be the number. The compound inequality is 5n 10− < OR 5n 35− > and the solution inequality is n > –2 OR

n < –7. 15. A store is offering a $30.00 mail-in rebate on all digital

cameras costing at most $200.00 or at least $400.00. Let c be the cost of the camera. The solution inequality is

c < 200 OR c ≥ 400.

0

0

0

0

0

0

0–10 10

200 400300

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SJ Page 70 Practice Exercises 1. Graph the solutions to the following inequalities. Provide a written description of your solution.

a. w ≥ –5 All real numbers greater than or equal to negative five.

b. x < 4 All real numbers less than positive four.

c. 80 – 24t > 32 The solution to the inequality is t < 2. The written

description is “All real numbers less than two.”

d. 911

a− >

The solution to the inequality is a < –99. The written description is “All real numbers less than negative ninety–nine.”

e. 4(y + 1) – 3(y – 5) ≥ 3(y – 1) The solution to the inequality is y ≤ 11. The written

description is “All real numbers less than or equal to eleven.”

f. x < 5 AND x ≥ –3 The solution to the inequality is –3 ≤ x < 5. The written

description is “All real numbers less than five but greater than or equal to negative three.”

g. –3 < d – 4 ≤ 1 The solution inequality is 1 < d ≤ 5. The written description is

“All real numbers less than or equal to five but greater than one.”

2. How are solutions to inequalities described in written and graphed formats? Answers will vary.

0

0

0

0

0 –11 11

0 –99 99

0


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SJ Page 71 3. Give a written and symbolic description of the graphed inequality below. “All real numbers greater than or equal to a negative six but

less than a positive six.” –6 ≤ x < 6

4. Different animals can hear different ranges of sound. Sound is measured in hertz which for hearing

means vibrations per second. Sounds that an animal hears are actually caused by the air pressure changing back and forth very quickly. For example, on a piano the vibration that the note “A” above “middle C” makes is actually 440 hertz. This is caused by the string inside the piano vibrating 440 times per second which makes air pressure changes of 440 times per second that your ears sense. Humans can hear sounds as low as the 20 hertz and as high as 20,000 hertz. Bats can hear in the range of 20 to 200,000 hertz

a. Write an equality that represents the range humans hear. 20 ≤ h ≤ 20,000 b. Write an equality that represents range that bats hear. 20 ≤ b ≤ 200,000 c. Use inequalities to represent the range of sounds bats can hear that a human cannot hear.

Let x represent the hertz range that bats can hear and humans cannot. The compound inequalities is 20,000 < x ≤ 200,000.

5. Salmon sharks thrive in water temperatures which range from

41°F to 64°F. Write a compound inequality to represent the temperatures where the salmon shark may not thrive. Graph the compound inequality.

Let T represent the water temperature. The compound inequality is T < 41 OR T > 64.

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0

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SJ Page 72 6. Solve the compound inequality –3h + 4 > 19 OR 7h – 3 > 18 for h. Then graph the solution. h < –5 OR h > 3 The solution graph is: 7. Solve the compound inequality 4t – 3 < 17 OR –8 ≤ 3t –2 t < 5 OR –2 ≤ t The solution graph is: 8. Write a compound inequality for the following graph. State the compound inequality in words. Sample response: The compound inequality is x < –2 OR x ≥ 6. All real numbers less than negative two

or greater than or equal to positive six.

0

0

0

0

0

0

0

0

0


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Outcome Sentences SJ Page 73 Graphing inequalities Writing a description for an inequality The difference between AND and OR for a compound inequality is I know that an AND compound inequality I know that an OR compound inequality Determining an inequality from a graph I am having trouble understanding

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Teacher Reference Lesson 5 Quiz Answers 1a. The solution is x < 5. 1b. The solution is x ≥ –12. 2a. The solution is x > 4 OR x < –4. 2b. The solution is –2 ≤ x < 2. 3a. The inequality is x > –5. 3a. The inequality is –9 < x ≤ 21.

0

0

–6 60

0


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Lesson 5 Quiz Name: 1. Solve and graph the following inequalities. a. 3x – 6 < 9

b. − + ≤3 5 144

x

2. Solve and graph the following compound inequalities. a. –3x + 5 < –7 OR –2x – 3 > 5 b. 6x – 4 < 8 AND 4x – 1 ≥ –9 3. Write an inequality for the following graphs. a. b.

–6 6 0

0

0

0

0

0

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Number Line Transparency


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Inequality Tile Pad Transparency

= <

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Lesson 6: Solving Absolute Value Equations and Inequalities Objectives • Students will be able to solve absolute value equations symbolically. • Students will be able to solve absolute value inequalities symbolically. • Students will be able to graph on a number line the solution to absolute value equations and absolute value

inequalities. • Students will be able to model real-world phenomena with absolute value equations and absolute value

inequalities. • Students will apply absolute value inequalities to tolerance specifications for consumer products. Essential Questions • How can absolute value help model real-world phenomena? • How do we solve absolute value equations and inequalities? Tools • Student Journal • Dry–erase boards, markers, erasers • Centimeter Rulers • Overhead projector • Inequality Tile Pad Transparency • Inequality Tile Pad Warm Up • Problems of the Day Number of Days • 1 ½ days (A suggestion is to complete Activity 1, the first part of Activity 2, and Practice Exercises 1

through 8 on the first day. On the second-half day, have students complete Activity 2, Practice Exercises, and Lesson Quiz.)

Vocabulary Absolute Absolute value Absolute value inequality Compound inequality

Solving One-Variable Equations Lesson 6: Solving Absolute Value Equations and Inequalities Page 123

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Teacher Reference Setting the Stage

The goal for this Setting the Stage is for students to develop a broader and more in-depth understanding of the term and concept of absolute. This Setting the Stage will be a bridge to the term and concept of absolute value in the activities to follow. Ask students what is absolute in their life and have a value level discussion on this topic. Students may discuss this with corresponding statements, such as: • I absolutely need to graduate from high school to go to college. • I absolutely turn down any invitation to do drugs. • My parents have absolute control of our house. • I’m absolutely certain the answer to that problem is x = 25. You may need to familiarize students with the term absolute by sharing some of the following examples of using absolute as an adjective.

Without limit in authority: • The dictator had absolute control of the taxes.

Free from impurities:

• The bar was made of absolute gold.

Certain or actual: • The student told the principal the absolute truth.

Total and complete:

• The country declared an absolute ceasefire.

You may also want to share with students the science term of absolute zero. Scientist record this as 0º Kelvin or –273º Celsius.

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Teacher Reference Activity 1 This activity begins with understanding the definition and concept of absolute value then transitions to investigating solving equations with absolute values. Create a large number line on the wall or floor on which students can stand to represent different numbers between –10 and 10. It is best to have the units on the number line represent a real unit such feet or meters. Have three student volunteers come to the front of the class. Have a student stand in the middle to represent zero on a number line. Have the other two students stand, on each side of the center student, one unit away from zero. Tell the class that the student to the left of zero represents negative one while the student to the right of zero represents positive one. Ask the class, “How far from zero are the other two students?” The class should say that each student is one unit from zero. Now have the students on each side of zero move another unit further away. Ask the class, “How far from zero are the two students now?” The class should say that each student is two units from zero. You could ask the class, “What if the student to the left of zero was ten units from zero, what number would this student represent?” The class should say that the student represents the number negative ten. Now ask, “How far from zero would the student be located?” The class should say 10 units away from zero. Tell the class, “So, a point (or person) on a number line could represent a positive or negative number, but when we talk about the distance that point or person is from zero the value is always positive.” As needed you can have students think about other points on a number line such as –24 and 87. The key concept is that students realize that positive or negative numbers have a signed value but are always a positive number of units away from zero. Tell the class that they have been investigating the concept of absolute value and have the students draw a sketch for Exercise 1. Have the class agree on a definition for absolute value and record it in Exercise 2. The class’ definition should be close to “The absolute value of any number n is its distance from zero on a number line.” Ask students how this concept of absolute value relates to discussion of absolute as an adjective in the Setting the Stage. Show the students the symbol used to represent the absolute value of numbers and then have student volunteers state the absolute value for the following expressions: 6 12− 76− 345 1,000,000−

Answers: 6; 12; 76; 345; 1,000,000 Remind the class that the absolute value symbol behaves as a grouping symbol in the order of operations. Then have students determine the value of the following numerical expressions. You may want to model one while the students practice another.

−5 12 −7 20 − −3 8 6 − −5 9 13 − • +5 2 7 − • −6 5 8 Answers: 7, 13, –6, –20, 3, 38 Have students complete Exercise 3 in pairs.


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Before students work in pairs on Exercise 4, you may want to ask a leading question such as, “At what two points on a number line would two students need to stand if the absolute value of each point is 6?” You may need to show students that this same question can be represented symbolically by asking, “What values of x make the equation = 6x true?” For Exercise 4, students should use this intuitive distance definition of absolute value. Make sure to discuss solutions to Exercise 4 with students, especially why Exercise 4d does not have a solution.

Introduce students to the more formal symbolic definition of absolute value if 0

if 0x x

xx x

≥⎧= ⎨− <⎩

Complete a few examples with students as they parallel you with different expressions. Two examples are given below and additional samples are supplied. ( )− = − − =15 15 15 , because –15 < 0. =9 9 , because 9 > 0.

Additional Samples: 8 −14 −52 125 −8,345,000 Have students answer Exercise 5 in pairs using the definition. Model using the symbolic definition of absolute value to solve equations. An example is given below and samples are provided.

For = 14x , = 14x or − = 14x , which gives = ±14x .

Additional Samples: = 10x = 22x = 1x = 12

x

Note: Some text books offer a different statement to represent solving absolute value equations. For example, If =x n , then x = –n or x = n. You may want to use this statement instead. Have students work in pairs on Exercise 6. Have each pair group with another pair to make a group of four. Have these groups solve more complicated equations that include absolute values by completing Exercises 7 through 11. Remind students that the absolute value symbols behave as grouping symbols. The goal is for students to work together to determine how to solve these types of equations. You may need to walk around and guide some groups as needed. If there is a need to complete an example, try to find a student to model an example as other students complete a different example.

5 4 6x − = and 7 7 21x + = 2 6 8 18x + + = and 3 5 2 16x + + = 4 6 11 6x − + = | and 7 12 13 10x + + = Note: These two have no solution because absolute value

can never equal a negative number. Make sure students check the solutions by substituting the solutions back into the original equation.

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SJ Page 74 Activity 1 1. Draw a sketch of one example of two students demonstrating absolute value in front of class. Answers will vary. 2. Record the class definition of absolute value below. Answers will vary. 3. Determine the value of the expressions below. a. −9 b. 18 c. −1 d. −8 15 9 18 1 7 e. −7 2 f. −4 5 g. −3 6 h. −2 6 8 5 20 –18 4

i. −62

j. −62

k. − •10 5 3 l. −14 182

3 –3 5 2 m. − −8 5 n. − +2 5 5 o. − −8 p. − +7 4 3 15 –8 11 4. Applying what you now understand about absolute value, determine what values of x would make the

following equations true. Describe how to solve these equations. a. 5x = b. 12t = c. 3812g = d. 6k = − x = ±5 t = ±12 g = ±3812 No solution Students should use the concept of distance from zero to determine solutions. Since distance cannot be

negative there is no solution to d.


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SJ Page 75 A symbolic definition of absolute value is:

if 0

if 0x x

xx x

≥⎧= ⎨− <⎩

Examples: ( )− = − − =20 20 20 because –20 < 0. =4 4 because 4 > 0. 5. Determine the value of the following expressions using the symbolic definition of absolute value. a. 3 b. −16 3 3= because 3 > 0 ( )16 16 16− = − − = because –16 < 0 c. −37 d. 225 ( )37 37 37− = − − = because –37 < 0 225 225= because 225 > 0 6. Solve the following equations.

a. = 3x b. = 35z x = ±3 z = ±35

c. = 81t d. = 34

x

t = ±81 3x4

= ±

7. Describe how to solve the absolute value equation below by using GEMDAS. Make sure to describe

each step you used for solving it. Sample response: Steps Description 4 8 16x − = ( )1 4x 8 16 or 4x 8 16− − = − = First undo the absolute value. ( )4x 8 16 or 4x 8 16− − − = − = Then simplify as needed.

4x 8 or 4x 24− = = Then undo the subtraction to both sides. x 2 or x 6= − = Then undo the multiplication to both sides.

( ) ( )4 2 8 16 4 6 8 16

8 8 16 24 8 16and16 16 16 1616 16 16 16

− − = − =

− − = − =

− = == =

Then check the solutions.

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SJ Page 76 8. Solve the following absolute value equations using opposite operations. Check your solutions with the

original absolute value equation.

a. | 2x – 8 | = 6 b. | t + 5| = 3

The solutions are x = 1 and x = 7. The solutions are t = –2 and t = –8.

c. | 3d – 6| – 7 = –4 d. Solve | f + 9| + 13 = 13

The solutions are d = 3 and d = 1. The solution is f = –9. Ask the students to explain why this problem only has one solution.

e. | 15v – 12| + 18 = 15 f. –2| x+ 6| = –8

There are no solutions. The solutions are x = –2 and x = –10. 9. The approximate minimum and maximum number of days of an elephant’s pregnancy can be

modeled with the equation − =630 30d . Determine the minimum and maximum numbers of days an elephant may remain pregnant.

d = 600 or d = 660

The maximum is 660 days and the minimum is 600 days. 10. Find the mistake for the solved equation, 2 6 7 11x + − = , and correct it below.

( )+ − =+ − =

− ===

2 6 7 112 6 7 11

2 1 112 12

6

xx

xxx

( )+ − = −+ − = −

− = −= −= −

2 6 7 112 6 7 11

2 1 112 10

5

xx

xxx

The absolute value was undone before adding 7 to both sides.

The correct solutions are x = 6 and x = –12.


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Teacher Reference Activity 2 In this activity students will investigate absolute value inequalities. Students will begin with a review of absolute value then bridge to a difference concept on the number line and finally investigate the concept of tolerance to develop a working knowledge of solving absolute value inequalities. Have the class continue to work in their groups of four for this activity. Ask the students what they remember about solving absolute value. Have the students discuss absolute value in groups and make a list of facts that came up in their discussion. Have a student volunteer record the group responses on the board or overhead projector, while the students record responses in Exercise 1. The concept of absolute value can be extended beyond the distance from zero on a number line to the concept of the distance between any two points on a number line. Have two volunteers come to the front of class and stand on the number line so that the distance between them is 5 units. Record the two points on the board. Have the students change positions two or three times but keep the condition that the distance between them is 5 units. Record each set of points. Have the students demonstrate at least one example where both points are positive, where one is positive and the other is negative, and finally where both points are negative. Some sample responses might be 1 and 6, –2 and 3, and –8 and –3. Write the matching absolute value expressions on the board, such as: −1 6 , − −2 3 , and ( )− − −8 3 . Then have the students calculate the values to make sure they work. Now, have a new set of volunteers come to the front and model the following absolute value expressions. −2 4 − −5 5 ( )− −9 4 You may want to point out to students the relationship between finding the difference in the two points with subtraction, i.e. the difference between 9 and –4 is 13 which can be determined with ( )− −9 4 . Have three students come up front. Have one student stand at 6 and ask the other two students to stand at the points that represent a distance of 4 units from 6. Students should stand at 2 and 10. Show students how this represents the symbolic absolute value equation − =6 4x . Point out to students the subtraction combined with the absolute value represents finding the difference (distance) between the two points. Have these same students model again. Have one student stand at –2 and then have the other two students stand 5 units from that student. Students should stand at –7 and 3. Show students how this represents

( )− − =2 5x . If needed, have the students reverse the model by giving a different equation, such as − =4 8x

or ( )− − =1 3x . Now, have half the class come to the front and have one student stand at 3 and ask the other students to stand on the number line so that the distance from this student will be less than 4 units. Students should stand between –1 and 7. Show students how this represents the absolute value inequality − <3 4x . You may have too many students for the whole class to stand on number line. Adjust the number of students as needed.

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Have half the class sit down and the other half model the situation of all the students greater than 4 units away from a student standing at three. Students should be standing at values greater than 7 and less than –1. Show students how this represents the absolute inequality − >3 4x . Show students that these two situations created compound inequalities as in Lesson 5. Have students work in pairs to complete Exercise 2. You may want to have a more formal discussion about how the symbolic definition of absolute value can be used to solve similar absolute value inequalities.

if 0

if 0x x

xx x

≥⎧= ⎨− <⎩

Now tell the class, “What if we had | x + 4 |< 5, how might we use the definition to solve for x?’ Have a student volunteer write the class responses on the board. The class should end up with x + 4 < 5 and –1(x + 4) < 5, which simplifies to x < 1 and x > –9, which is the compound inequality –9 < x < 1. Have student volunteers share their results on the board or overhead projector. The graphed solution is: Have a student volunteer model a second example from the samples below while the class works on a different sample from the list. | x – 3 | – 2 > 4 | x – 5 | – 3 > 3 | 2x + 4 | ≤ 5 | 3x + 2 | ≤ 7 Ask the class, "How are absolute value inequalities similar to compound inequalities with AND and OR?" Have a student volunteer record the class responses on the board. The key concepts are that when an absolute value expression is less than or equal to a quantity then it is like an AND compound inequality and when an absolute value expression is greater than or equal to a quantity then it is like an OR compound inequality. Have students complete Exercise 3. Students should be ready to complete an activity on tolerance as it relates to absolute value inequalities and the real world. Ask for a volunteer to read the two paragraphs, and then complete a discussion on how companies have certain tolerance specifications for their product. Divide the class into at least four groups then have the students cycle through the four stations. Set a time limit for students at each station and clearly designate when and how to switch from station to station. There are four simple station options supplied in this lesson for students to apply their knowledge of absolute value inequalities. To make the stations more life-like, the authors suggest that you make at least four stations in which students can interact with real-world applications. For example, you may want to change Station 1 by supplying a set of 10 real pencils and then give conditions for the proper length. Or you may want to replace a station in its entirety with other objects to measure or investigate. For example, you may want to have 10 bags of chips that claim to each have a weight of 12 ounces. Then have the students measure the weight of the bags with an electronic scale from the science department to actually determine the range of weights. If possible you could determine from the company what product tolerance specification they allow.

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Activity 2 SJ Page 77 1. In the last lesson we solved absolute value equations. Within your group, discuss what you remember

about solving absolute value equations and what the absolute value represented. Make a list below of the facts about absolute value that came up during your discussion. • Represents the distance from 0 on a number line. • Distance is always positive. • Absolute value equations usually have two solutions when they have solutions. • You can take the absolute value of a negative quantity, but the absolute value cannot equal a

negative quantity. 2. Solve and graph the solution to the following absolute value inequalities. a. − <2 7x b. 5 10x − ≤ c. 3.5 1.5x − < –5 < x < 9 –5 ≤ x ≤ 15 2 ≤ x ≤ 5 d. 2 8x − < − e. 4 5x − > f. 6 10x − > no solution x ≤ –1 OR x ≥ 9 x < –4 OR x > 16 You can also use the symbolic definition of absolute value to solve absolute value inequalities.

if 0 if 0

x xx

x x≥⎧

= ⎨− <⎩

For example, for − <3 5x the two statements would be − <3 5x AND ( )− − <3 5x . Solving these inequalities gives –2 < x < 8. 3. Solve the following absolute value inequalities using the definition. a. − <6 10x b. 6 10x − > c. 6 10x + ≤ –4 < x < 16 x < –4 OR x > 16 –16 ≤ x ≤ 4

d. 5 7 10x − + < e. 2 6 4x − < f. 2 13 2

x − <

2 < x < 8 1 < x < 5 1 7x6 6

< <

g. 0.7 8.5x − ≤ h. 6 6 8x − − ≥ i. 4 1 6 1x − − ≤ –7.8 ≤ x ≤ 9.2 x ≤ –8 OR x ≥ 20 –1.5 ≤ x ≤ 2

2 9 –5 150–5 105 3 42 5

84 –4 0 12 16 20–8 4 6–4 0–2 2 8 10 12

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SJ Page 78 Companies that manufacture products often set tolerance limits that help determine if the product passes inspection. For example, a company that produces crayons that are 10 centimeters in length might set a tolerance of 0.2 centimeters. This means every crayon created should be within 0.2 centimeters of the 10 centimeter length. This tolerance can be modeled mathematically by the absolute value inequality |x – 10| < 0.2. This represents that crayons only pass inspection if they are longer than 9.8 centimeters and shorter than 10.2 centimeters. Any crayon outside that range would be rejected by inspectors.

Sometimes companies put conditions on using their product. For example, on a can of paint you might read the directions, “Only paint when the temperature is between 50 and 90 degrees Fahrenheit.” This condition can be modeled mathematically by the absolute value inequality |x – 70| < 20. This means that if you paint with temperatures in this range the product should work properly and if you paint in a temperature that is less than or equal to 50 degrees Fahrenheit or greater than or equal to 90 degrees Fahrenheit the paint may not adhere properly or have the quality it should.

Station 1: Pencil Making This station represents a pencil manufacturing company. The company would like to make pencils that meet the conditions of the absolute value inequality |x – 18| < 0.3 for length in centimeters. Write a compound inequality that represents the range of pencils lengths that are acceptable. 17.7 < x < 18.3 Write a compound inequality that represents the range of pencils lengths that do not meet the

conditions. x ≤ 17.7 or x ≥ 18.3 Analyze the sample of pencils shown below. Determine which pencils pass inspection and which do

not.

PassNot Pass

Not Pass

PassPass


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Station 2: Silicone Caulk SJ Page 79 Silicone caulk is used to seal the space between such things as walls and windows or around tubs and showers. Some silicone caulk is specifically for indoor use, some for outdoors, and some can be used for both. There are certain conditions that should occur when caulking. The surface should be clean and dry and the temperature should be between 40ºF and 100ºF. Write an absolute value inequality that represents the range of temperatures that are acceptable to apply silicone caulk. | x –70 | < 30 Write an absolute value inequality that represents the range of temperatures that are not acceptable to apply silicone caulk. | x –70 | ≥ 30 Write a compound inequality that represents the range of temperatures that are acceptable to apply silicone caulk.

40 < x < 100 Write a compound inequality that represents the range of temperatures that are not acceptable to apply silicone caulk.

x ≤ 40 or x ≥ 100 Station 3: Wood Product Manufacturers A wood product manufacturing company makes fiberboard panels which are wood products made from smaller wood fibers bound together into a panel or sheet. High temperature, high pressure, and glue are used to form the panels. These panels can be used for building’s structural sheathing as shown in the figure below. The company sells fiberboard panels with dimensions of 4 feet wide, 8 feet long, and 5/8 inch thick. The company lists the following tolerance specifications. Length/Width ±1/8 inch Thickness ±1/16 inch Write three different absolute value inequalities that represent the actual dimensions that can occur for length, width, and thickness. Sample response: Length: |x – 96 | < 1/8 Width: | x – 48 | < 1/8 Thickness: | x – 5/8 | < 1/16 Explain why you had to change the value so that they represented the same units. Answers will vary. Sample response: The units either need to be feet or inches because x can only be one type of unit at the same time. It can’t represent both inches and feet. For example, I changed 8 feet into 96 inches, so that adding or subtracting 1/8 of an inch would make sense. Write the matching inequality for thickness. Thickness: 9/16 < x < 11/16.

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SJ Page 80 Station 4: Boxes for All Boxes for All is a company that manufactures boxes for packaging products, such as perfume and cologne. The company claims that each box it manufactures will be within 5% of the requested length, width, and height of the box. Jan, a perfume designer, has created a new perfume product that had great sales for the past year. She would like to update the box in which the bottle is packaged. She would like the new box to have a width of 35 millimeters, length of 45 millimeters, and a height of 60 millimeters. Write an absolute value inequality that represents the range of widths the boxes could be and fall within the condition of 5%. | x – 35 | < 0.05 (35) or | x – 35 | < 1.75 Write an absolute value inequality that represents the range of lengths the boxes could be and fall within the condition of 5%. | x – 45 | < 0.05(45) or | x – 45 | < 2.25 Write an absolute value inequality that represents the range of heights the boxes could be and fall within the condition of 5%. | x – 60 | < 60 (0.05) or | x – 60 | < 3 Determine if the cut out for the box on the next page would pass inspection under the conditions given above.

The box should not pass inspection. The height should be between 57 and 63 millimeters and the height is actually 65 millimeters.


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SJ Page 81 Cutout for Box

Top

Botto

m

Leng

th

Wid

th

Height

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SJ Page 83 Practice Exercises 1. Solve the following absolute value equations. a. | 9x – 3 | = 15 b. | 5n + 2 | – 4 = 7 c. | 7z + 6 | + 12 = 9 x = 2 and x = –4/3 n = 9/5 and n = –13/5 No solutions 2. Solve the absolute value inequality | x | < 5. Graph and write a description of the solution. A written description might be: All real numbers less than

a positive five but greater than a negative five. –5 < x < 5 3. Solve and graph the absolute value inequalities. Write a description of the graphed inequality as well.

a. | y | < 4 The written solution is: “All real numbers between –4

and 4 but not including –4 and 4.”; symbolic solution is –4 < y < 4.

b. | t | ≤ 6

The written solution is: “All real numbers between –6 and 6 including –6 and 6.”; symbolic solution is –6 ≤ t ≤ 6.

c. | d | ≥ 1

The written solution is: "All real numbers less than or equal to –1 or greater than or equal to 1."; symbolic solution is d ≤ –1 OR d ≥ 1.

d. –5 > | f | There is no solution.

e. | v | < 4 The written solution is: "All real numbers less than 4

and greater than –4 but not including –4 and 4."; symbolic solution is –4 < v < 4.

f. | g | > 3 The written solution is: "All real numbers less than

–3 or greater than 3 but not including –3 and 3."; symbolic solution is g < –3 OR g > 3.

0

0

0

0

0

0


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SJ Page 84 4. Write the absolute value inequality from the written description and graph the solution. “All real numbers less than a positive nine and greater than a negative nine and including a positive

nine and negative nine.” The absolute inequality is –9 ≤ x ≤ 9. 5. Graph the solution to | z + 5 | < 2. The solution graph is: 6. Graph the solution to | 3w – 4| ≥ 5. The solution graph is: 7. Write the symbolic absolute value inequality and written description for the graphed inequality.

a. The absolute value inequality is| x | ≥ 12. The written description is: "All real numbers less than

or equal to negative twelve or greater than or equal to positive twelve."

b. The absolute value inequality is| x | > 0. The written description is: "All real numbers less than

or greater than zero." 8. Graph and write the solution to the following absolute value inequalities.

a. | z | ≥ 5 The written solution is: "All real numbers less than or equal to –5 or greater than or equal to 5.";

symbolic solution is z ≤ –5 OR z ≥ 5.

0 8 –8

0

0

4 0 –4

0 –3 3

0–3 3

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SJ Page 85 b. | y – 4 | < 5

The written solution is: "All real numbers between –1 and 9."; symbolic solution is –1 < y < 9.

c. | 2t + 3| ≥ 3.

The written solution is: "All real numbers less than or equal to –3 or greater than or equal to 0"; symbolic solution is t ≤ –3 or t ≥ 0.

d. | 4z – 1| – 4 ≤ 3.

The written solution is: "All real numbers less than or equal to 2 and greater than or equal to –3/2"; symbolic solution is –3/2 ≤ z ≤ 2.

e. | 4d – 1| + 4 ≤ 1.

No solutions. Any positive number added to 4 will be greater than 1 and never less than or

equal to 1. 9. Write an inequality, description, and graph its solution for: “The temperature inside the freezer ranges

within 3 degrees of –18°F.” The inequality is | t – (–18) | ≤ 3. The written solution is: "All

real numbers less than or equal to –15 degrees Fahrenheit and greater than or equal to –21 degrees Fahrenheit."

10. Write an inequality and graph its solution for: “A cruise ship is trying to maintain a speed of 22 ± 0.6

knots while traveling to the islands.”

The inequality is | s – 22 | ≤ 0.6. The written solution is: "All real numbers less than or equal to 22.6 knots and greater than or equal to 21.4 knots."

11. Write a compound inequality and an absolute value inequality for the following graph. The compound inequality is x < –2 OR x > 2; | x | > 2

0

0

0

–18

22 21 23

–4 40


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Outcome Sentences SJ Page 86 When solving absolute value equations When solving absolute value inequalities Absolute value inequalities I am having a hard time distinguishing between Absolute value means What do I absolutely know?

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Teacher Reference Lesson 6 Quiz Answers 1. y = –17/3 is not a solution and y = 9 is a solution. 2. z = 15 or z = –3 3. –27 ≤ k ≤ 13 4. |p – 12 | ≤ 0.02 11.98 ≤ p ≤ 12.02 NOTE: Each mark on the number line represents 0.02 inches.

–10 100

12 12.111.9


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Lesson 6 Quiz Name: 1. Determine if y = 17/3 and y = 9 are the solutions to | –3y + 5 | – 6 = 16. 2. Solve | 2z – 12 | – 8 = 10. 3. Solve and graph the following inequality.

+

≤7

54

k

4. Write, solve, and graph an absolute value inequality for: “A tile company makes square tiles that have

a side length of 12 inches with a tolerance of 0.02 inches.”

0

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Lesson 7: Ratios, Proportions, and Percent of Change Objectives • Students will determine percent of change for real-world applications. • Students will determine and write the ratio of various real-world applications. • Students will use proportions to solve for unknowns in real-world applications. Essential Questions • How are ratios used to model real-world phenomenon? • What are the similarities and/or differences between ratios, proportions, and percent of change? Tools • Student Journal • Dry-erase boards, markers, erasers • Graphing calculator • Setting the Stage transparency • Poster Paper and Markers Warm Up • Problems of the Day Number of Days • 2 days (A suggestion is to complete Activity 1 and Activity 2 on the first day. On the second day, complete

Activity 3, Practice Exercises, and Lesson Quiz.) Vocabulary Ratio Proportion Proportional Percent of change Percent Change in value Original value Scaled

Solving One-Variable Equations Lesson 7: Ratios, Proportions, and Percent of Change Page 143

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Teacher Reference Setting the Stage Display the Setting the Stage transparency. Tell students that each object is a different size of the same map of a video game. Ask the class, “What is the area of Object 1 compared to the larger Object 2?” Have a student volunteer record all the different responses on the transparency. Next, ask the class, “How can we write the comparison of the two areas mathematically and what do we call this comparison?” Have a student volunteer record the various responses on the transparency. Lead a class discussion based on the student’s responses that will lead toward the various ways that ratios can be written: 22 to 88, 22:88, 22/88, or it can be simplified to 1 to 4. Ask the students why a ratio like 22/88 loses meaning when simplified to 0.25. The concept for the students to understand is that a ratio is a comparison between two values and simplifying it to a single value “destroys” some of that comparison. Let the class know that ratios can be simplified to lowest terms, but generally not simplified to a single term. Have the class draw a simple object (something to easily calculate the area) in one of the quadrants on their dry-erase boards. Tell the students to draw a new object that is the same shape with a ratio of 1 to 4 in area. Ask for student volunteers to show the class their drawings and to give the areas of the two drawings. Make sure the class agrees with the student’s results that the area of the second drawing is four-times the area of the first drawing.

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Possible Layout of a Mapping Area for a Playing Field in Video Game

Object 1

Object 2


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Teacher Reference Activity 1 In this activity, students continue to work with ratios. You may want to work with students on creating a formal definition of ratio to place on the word wall and in their student journal. A sample definition is: A ratio is an expression comparing two quantities to each other. Some definitions include the term division such as: A ratio is an expression of two quantities by division. Model some simple ratios with the class using the students as examples. For instance, you could ask the class, “How many girls to boys are in class today?” Then have the students write the results on the dry-erase boards and then hold them up for everyone to see. Have student volunteers give their results and another student volunteer record the results on the board or a blank transparency on the overhead projector. Have the class agree on the correct ratio of girls to boys in the class. Other ratios you may want to demonstrate with the class include: left handed students to right handed students or students who like a particular song or dance to those who do not. You may want to also demonstrate the reverse ratios with the students. For example, the ratio boys to girls will be different than girls to boys. This may be a good opportunity to review the reciprocal of a fraction. Have the students work individually on Exercises 1 through 7. For Exercises 5 and 6, have students pair and then exchange and solve the ratio problems they wrote. Have the students work in the same pairs for Exercise 7. Bring the class back together and have student volunteers share their results.

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SJ Page 87 Activity 1 In this activity, you will work with ratios. Ratios are commonly used as an expression to compare two quantities to each other.

For each exercise below, write a ratio, in all 3 formats. 1. Bryan scores 3 points for every 2 points that Jon scores. What is the ratio of the number of points Bryan

scores compared to Jon? The ratio representing Bryan’s points to Jon’s points is 3 to 2, 3:2, and 3/2. 2. In a cooking recipe, you will need one-and-three-quarter cups of water for each cup of rice. What is the

ratio of rice to water?

The ratio of rice to water is 1 to 1.75, 1:1.75, or 1/1.75.

3. On a recent trip, you drove 300 miles and used 12 gallons of gas. Write a ratio comparing the miles driven to the amount of gas used. Simplify your ratio to lowest terms.

The ratio of miles driven to gallons used is 300/12 = 25/1. 4. On the grid-side of a dry-erase board, draw a simple sailboat like the one pictured below. Make sure to

place all the vertices on the intersections of the grid lines. Draw a second sailboat on the same grid that is a different size than the original but the same shape. Use a metric ruler to measure the length of the boats drawn. Write the ratios for the length of the boat of the first sailboat drawn compared to the second sailboat drawn.

Answers will vary. For Exercises 5 and 6, write your own ratio problem. 5. Answers will vary. 6. Answers will vary.


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SJ Page 88 7. For the following drawings, write a ratio comparing the area of object A to the area of object B. Write

the ratio in all 3 formats. a. b.

a. The area of triangle A is 14 square units. The area of triangle B is 33 square units. The ratio of area A to area B is 14 to 33, 14:33, or 14/33.

b. The area of circle A is π494

square units. The area of circle B is 1π square units. The ratio of area

A to area B is 49/4 to 1, 49/4:1, or (49/4)/1.

d

A

B B

A

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Teacher Reference Activity 2 Guide the students in a short discussion on the paragraphs and drawings before Exercise 1 and then have the students complete Exercises 1 through 4 in pairs. The goal of these four exercises is for students to make a conceptual connection to ratios and proportions as well as construct knowledge and understanding of ratios and proportional thinking. If you have students that like to draw or that have the ability to sketch, you could extend these concepts by having students create proportional drawings to display and present. After students have completed Exercise 4, complete a guided discussion on solving for an unknown in a proportion. You may want to model how to solve proportions for the unknown with one example as students solve a different proportion. A few examples are given below. It is suggested that you don’t mention the technique cross products because students sometimes confuse cross products with multiplying fractions. The authors suggest that students multiply by the denominators to clear the fractions. • Draw two sets of similar triangles on the board with a missing side length then solve one while the

students solve the other. x = 16 y = 9.6 • Model one of the example problems below while the students do a different example on their dry-erase

boards. Have student volunteers share results and the class agree on the solution to the problem(s) solved.

125 15x = 3 18

4 x= 30

4 6y

= 3 711d

= 2415 16n =

31 249d

=

x = 4 x = 24 y = 45 d = 33/7 n = 22.5 d = 7.875

• If a survey of 100 students at school results in 75 voting for the new mascot to be a sting ray, what should we expect in a survey of 300 students? We should expect about 225 students voting for the mascot to be a sting ray.

• Model finding the height of a building using similar right triangles and the shadows created by the

building and a telephone pole with a known height. Add the term proportion and its definition to the word wall and have the class complete the definition in Exercise 5. You may prefer to have students create their own definition or you may have a particular way you would like it stated. A sample definition is: A proportion is two ratios equal to each other. You may want students to add this definition to the vocabulary organizer or to their vocabulary journal. Have the students work in groups on Exercises 6 through 10. For Exercises 11 and 12, have students individually create problems then exchange the problems within their group, then have each group discuss and agree on the solutions.

12 ft

x ft

18 ft

24 ft

8 ft

20 ft

y ft

24 ft


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You may want to discuss how to form the proportion before the students do Exercise 13 or you may decide to discuss forming proportions after checking the student’s displayed results. For Exercise 13, you will need to fill enough boxes of pretend fish for your groups. The “fish” box is a small box that contains some uniform small object such as centimeter cubes, chips, or packing “peanuts” each representing a fish. The box will be considered the lake and the objects the fish. The students will capture and count a handful of fish and “tag” them in some way. The fish will be returned to the “lake” and thoroughly mixed with the fish that were not tagged. Then the students complete a second capture and record the number of fish that are tagged and the number of fish that are not tagged. The students will then use proportions to estimate the total number of fish in the lake. The students could use something similar to the following proportion.

=Tagged in lake Tagged in sample

Total in lake Total in sample

Give each group a poster paper. After the groups have gathered the data and determined a solution, have them write their solution on poster paper explaining in detail each step they did. Have each group display their poster paper and then have the students walk around the room to look at each poster. Students can put small notes on the poster of either positive comments or questions. Have the groups look at the comments for their poster. Then have the students actually count the number of total fish in the lake to determine how close their estimate was to the true population. Optional Students generally have an intuitive understanding of proportions. You can ask them to complete the following statements to make a link to their prior understanding before starting this activity and/or at the end of the activity to assess their knowledge.

• 1 foot is to 12 inches as 2 feet is to 24 inches 1 foot is to 12 inches as 3 feet is to

• 1 pound is to 16 ounces as 2 pounds is to 32 ounces 1 pound is to 16 ounces as 5 pounds is to

• 14 days is to 2 weeks as 21 days is to • 9 feet is to 3 yards as 300 feet is to • 30 miles per 1 gallon is the same as 90 miles per • 30 miles per 1 gallon is the same as 15 miles per • If 5 boxes sell for $1.25, then 3 boxes sell for . • If 20 gallons of fuel cost $44.00, then 15 gallons cost . • If $13.50 will buy 2 hotdogs and 1 soda at the baseball game, then $40.50 will buy

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SJ Page 89 Activity 2 Artists that sketch and paint animals use ratios to help make sure the parts of the animals are in proportion. For example, the sketch of the cat below shows that the size of the head compared to the height of the cat is 1 to 3. You may hear artist describe this as, “The head size is one-third the height” or “the height is three times the size of the head.” These ratios are important to artist in order to make animals look lifelike. For example, while the following drawing of a cat may be artistically appealing, the head size compared to the body height is no longer in a ratio that represents a real life cat. In this case the head is three-fourths the height. An artist will use the word proportional to describe an animal sketch that has a harmonious relation of parts to each other or to the whole. A mathematician will use the word proportional to state that two ratios are equal to each other. 1. Study the following drawings of dogs and determine which dogs are proportional to each other in

terms of head size compared to height. Be prepared to show your measurements and represent your findings mathematically. Answers will vary depending on how accurate students measure.

a. b. c. d. ~1 to 4 ~1 to 2.5 ~1 to 4 ~1 to 5 e. f. g. h. ~1 to 3 ~1 to 3.5 ~1 to 5 ~1 to 3.5 i. j. k. l. ~1 to 3.5 ~1 to 4 ~1 to 4 ~1 to 4

= =head size 3 unit 3height 4 unit 4

4 units 3 units

= =head size 1 unit 1height 3 unit 3

1 unit

3 units


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SJ Page 90 2. Study the dog drawings in Exercise 1 and determine which dogs are proportional in terms of head size

compared to length. Answers will vary depending on how accurate students measure. Sample response: Approximately 1 to 4: a, c Approximately 1 to 3.5: b, d, f, g, j Approximately 1 to 4.5: e Approximately 1 to 3: h, i, k, l Artists will often use the ratio of parts in one drawing to make a scaled version of the same drawing. Notice how in each drawing below the ratio of the head size to the height is the same even though the actual measurements are different. These ratios are a proportion because they are equal. Another way that artists use ratios and proportions is to make sure the height and length of the new drawing are the same ratio as the original drawing. If drawings are scaled in one direction more than another, the drawings would not be proportional and the ratios that represent the dimensions would not be equal. For example, the two drawings below have different ratios. 3. Measure the height and length of the following to determine which drawings are proportional. a is proportional to b. a. b. c. d. e.

= = =head size 9 mm 16 mm 1height 27 mm 48 mm 3

9 mm

27 mm

16 mm

48 mm

= =height 26 mm 13length 18 mm 9

26 mm

18 mm

88 mm

36 mm

= =height 36 mm 9length 88 mm 22

≠13 99 22

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SJ Page 91 4. An artist started to draw a scaled version of a turtle. The artist forgot to determine the actual length for

the new drawing. Determine the new length and describe how you determined the new length. Answers will vary.

Sample response: ~5.5 cm 5. Write a definition of proportion. Answers will vary. You may want to put the definition on your word wall, have the students record it

in a vocabulary organizer, and/or have the students record it in a vocabulary journal. 6. Solve the following proportions for the unknown variable.

a. 445 15x = b. 3 8

7 v= c.

123

6 9x =

x = 12 v = 56/3 ≈ 18.7 x = 14/9 ≈ 1.56 7. A survey of the duck and geese population at a local park was conducted by a state employee of the

Department of Fish and Game. The employee counted 24 ducks to 16 geese. A few days later, the same employee counted only the ducks and ended up counting 288 ducks. The employee then estimated the number of geese using his ratio from a few days earlier. How many geese did the employee estimate were at the park?

Approximately 192 geese 8. A long piece of pipe was cut into two pieces, with the lengths of the pieces having a ratio of 3 to 5. If the

pipe was originally 32 meters long, what are the lengths of the two pieces of pipe? 12 meters and 20 meters.


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SJ Page 92 9. Most states have an expressway speed limit of 65 miles per hour. How many feet per second is

equivalent to 65 miles per hour? Note: There are 24 hours in a day, 60 minutes in an hour, 60 seconds in a minute, and 5,280 feet in one mile.

65 miles per hour is 95 and 1/3 feet per second. Students can use proportions to complete each part.

First they can set up proportion to change 65 miles to 343,200 feet. Then they can use proportions to change 1 hour into 60 minutes and 60 minutes into 3600 seconds. Finally, students can determine that 343,200 feet per 3600 seconds is the same as 95.333… feet per second.

10. A building casts a shadow of 130 feet at the same time a 60 foot flagpole casts a shadow of 5 feet. How

tall is the building? The building is 1,560 feet tall. 11. Write your own proportion problem similar to Exercises 6. Answers will vary. 12. Write your own real-world proportion problem similar to Exercises 7 through 10. Answers will vary. 13. When biologists try to estimate the number of fish in a lake, it is not possible to count them all. The

biologists will capture a large sample size of the fish, tag them and release them back in the lake. After enough time has passed for the fish to be thoroughly mixed with the non-captured fish, the biologists then capture another sample of the fish. The biologist will count the number of recaptured fish that are tagged. Using a proportion, the biologists will then estimate how many fish are in the lake. Obtain one of the “fish” boxes from your teacher. Within your group, “capture” a small sample of fish, tag them, “mix” them back into the lake, take a second sample and determine how many fish are in your lake. Write up a detailed report on a piece of poster paper.

Answers will vary.

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Teacher Reference Activity 3 In this activity, students will develop knowledge and understanding about percent of change. Have the groups of students complete Exercises 1 through 3. Have various groups share their responses. Answers for Exercises 1 through 3 will vary. Look for unique responses. If you can, guide the class towards the concept that describing change is relative to the quantity with which the change is being compared. Discuss the following ideas with the class and then model calculating percent of change. Have students record your steps for Exercise 4.

• During a recent shopping trip, you noticed a pair of your favorite sneakers on sale for $80, from its original price of $100. What was the change in price as a ratio from its original price? What was the percent change from the original price to the sale price?

The problem may be simple enough for students to determine that it was 20/100 and 20%. However, we want the students to be able to represent the calculations mathematically. Ask for student volunteers to share their results with the class. Have a student volunteer record the shared results on the board or on a blank transparency on the overhead projector. Have the class agree upon the steps to determine percent of change.

• Ask a student volunteer to use the agreed upon steps for the following problem.

Suppose a certain item at the local grocery store sold for seventy cents a pound last week, you see that it has been marked up to seventy-nine cents a pound. What is the percent of change? Was the percent of change an increase or decrease?

Ask the class “Did the steps that you agreed upon work to solve this problem correctly?” If the steps need to be adjusted based on the solution of the above problem, lead the discussion with the class to revise the steps to solve percent of change problems. Have another student volunteer use any revised steps on the same problem above. A common method to determine percent change is:

Change in Value 100 Percent of ChangeOriginal Value

⎛ ⎞=⎜ ⎟

⎝ ⎠

Have the groups of students complete Exercises 5 through 7. Have a discussion about how the answers for these exercises compared to Exercises 1 through 3. After the discussion, have students complete Exercises 8 through 12 and then have volunteers share results with the class. For Exercise 13, have each student write their own percent of change problem and then exchange their problem with their partner.


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SJ Page 93 Activity 3 Henrique, an online video-gamer, is trying to reach the SILVER level in order to earn the capability of invisibility for the virtual character he is playing. Friday night, before Henrique went to bed, he checked his current level. His level is displayed on the screen to the right. The gray squares represent Henrique’s current level and the unshaded squares represent how far he needs to go before reaching the SILVER level. 1. Relative to the SILVER level how might you describe the

current level of Henrique? Sample response: Henrique is two-sixths, 2/6, of the way to the Silver Level. Students may also refer to

this as 1/3 or approximately 33%. 2. Relative to the SILVER level how might you describe the change that would need to occur for Henrique

to reach the SILVER level? Sample Response: Henrique needs to gain an additional four-sixths of the Silver Level. Students may

also refer to 2/3 or approximately 67%. 3. Relative to Henrique’s current level, how might you describe the change that would need to occur for

Henrique to reach the SILVER level? Sample Response: Henrique would need to triple his current level.

Students may refer to three times or 300%. 4. Record the method to calculate the percent change as your teacher models it. Answers will vary depending on teacher’s method. After playing Saturday, Henrique checked his level points again. See the display to the right. 5. Relative to Friday’s level, how might you describe the

change that occurred to Henrique’s level as a percent? Sample response: Henrique increased his current level by

150%. 6. Relative to the SILVER level, how might you describe the change that occurred since Friday as a

percent? Sample response: Henrique increased his level by three-sixths of the Silver Level. Students may refer to ½ or 50%.

7. What percent change would need to occur for Henrique to reach the SILVER level after Saturday? Sample response: Either approximately 16.7% if relative to the Silver level or 20% relative to

Saturday’s level.

Henrique’s Current Level

Silver Level

Henrique’s Current Level

Silver Level

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SJ Page 94 Use your knowledge of percent change to determine the unknown values for the following exercises. 8. Inflation has caused the price of your favorite meal at your favorite restaurant to go up. Originally, the

cost of the meal was $14.50. The new price of the meal is $15.35. Determine the percent of change in the cost of the meal, to the nearest whole percent. State whether the percent change is an increase or a decrease.

The percent increase is 6%. 9. A pair of jeans normally cost $55. The same pair of jeans is now 27% off. What is the

price of the jeans? The cost of the jeans is $40.15. 10. The National Football League’s, NFL™ , football field is 120 yards long (counting the end zones). The

Canadian Football League, CFL, is 25% longer. How long is the CFL’s football field? The CFL’s football field is 150 yards long. 11. In 2002 there were 12.1 million Federal employees. In 2006, there were 14.6 million Federal employees.

What was the percent change in Federal employees from 2002 to 2006?

The percent change was an approximate 20.66% increase. 12. A dining room table had a price increase of 15% to $550.00 from its original cost, due to an increase in

demand for the type of wood the furniture was constructed of. What was the original cost of the table to the nearest penny?

The original cost of the piece of furniture was approximately $478.26. 13. Write your own application percent of change problem. Answers will vary.


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SJ Page 95 Practice Exercises 1. In a certain algebra class, 4 out of 5 students are planning on a career in engineering. Write this ratio in

three forms. Also, write a ratio of students who will be engineers compared to those who are not planning to be engineers.

4:5, 4 to 5, 4/5 The ratio of students who will be engineers to those who will not is 4 to 1, 4:1, or 4/1. 2. For the object drawn on the left, draw a second similar object according to the ratio of 2 to 1. 3. If the current exchange rate has one U.S. dollar worth 0.68042 Euros, how many U.S. dollars can you get

for exchanging 125 Euros? Round your answer to the nearest cent. 125 Euros exchange for 183.71 US dollars. 4. For the following similar triangles, find the length of the unknown side to the nearest tenth of a

centimeter. The length of x is 5.2 cm. 5. For each of the following, calculate the percent of change from the original amount and state whether

the percent of change is an increase or a decrease. Round all answers to the nearest tenth of a percent.

a. Socks: Original cost per pair is $6.00 b. Suit: Original cost per suit is $175.95 New cost per pair is $4.80 New cost per suit is $96.77 Decrease of 20% Decrease of 45%

b. Clock radio: Original cost per radio is $35.00 New cost per pair is $39.00 Increase of 11.4%

9 cm

8 cm

5.85 cm

x

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SJ Page 96 6. Write a percent of change problem with an increase of 55%. Answers will vary. 7. How are ratios used to model real-world phenomenon? Answers will vary. 8. Solve the following proportions for the unknown variable. If needed, round to the nearest tenth.

a. = 812 15x x = 6.4 b. =5.1

12.2 6.8z z 2.8≈

c. =16 59y

y = 28.8 d. =12 57 x

x 2.9≈

9. Solve the following proportions for the unknown variable. If needed, round to the nearest tenth.

a. − =5 0.2080

x x = 21 b. −=2 23 10

x x 8.7≈

c. −=8 3015 24

x x = 17.2 d. − =20 0.25xx

x = 16

e.

− =20 0.2520

x x = 15

f.

−=5 68 28

x x = 23.5


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SJ Page 97 Outcome Sentences I know that a ratio is I know that a proportion The difference between a ratio and a fraction is The difference between a ratio and a proportion is For me, the best way to solve a proportion is to I am still having trouble with

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Teacher Reference Lesson 7 Quiz Answers 1. Answers will vary 2. The x – 1 was not distributed properly. The correct solution is

5 x 16 15

75 6x 681 6x81 x6

27 x2

−=

= −=

=

=

3. The original price was $100 4. Answers will vary


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Lesson 7 Quiz Name: 1. Write a ratio application problem for the ratio 5 to 3. 2. Find and correct the mistake below.

−=

= −=

=

=

5 16 15

75 6 176 6766383

x

xx

x

x

3. The percent of change increase for a product was 35%. If the new price is $135, what was the original price of the product?

4. What are the similarities and differences between ratios, proportions, and percent of change?

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SJ Page 98 Vocabulary Organizer Template

Word:

Definition:

Clue (Picture or Words):

Examples:

Word:

Definition:


Examples:

Word:

Definition:


Examples:

Word:

Definition:


Examples:


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Tile Pad SJ Page 99

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SJ Page 101 Equal Tile Pad

=


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Inequality Tile Pad SJ Page 103

= <

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Assessment: Solving One-Variable Equations

I. Lesson 1 The following diagram represents a series of operations with numbers. Each operation is completed before going to the next operation. 1. Complete the following table with the correct input and output that matches the diagram of

operations and numbers above.

Input Output –4 28

2. If dividing a number by 8 and then multiplying by 12 results in the number 144, what was the original

number? II. Lessons 2 and 3 3. Write an equation to solve the problem below. Make sure to label all variables. Solve the problem and explain each step used in solving. Justify your equation and solution.

A dining table is discounted 20% from its original price. If the discounted price of the table is $60.40, what is the original cost of the dining table?

4. For the following equation, solve for the variable. − + = −2( 3) 5 3( 1)w w

×3 INPUT OUTPUT +6 ÷3 −9 ×7


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III. Lessons 4 – 6 5. Solve the following system using the graphical method. Verify the solution using your graphing calculator.

− + = −

+ =2 8

1x yx y

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6. The following problem was solved using the addition method. Find the mistake in the solution of the following system and correct it:

2 3 123 3

x yx y

+ =+ = −

system

3 9x = equations added 3 93 33 333

x

x

x

=

=

=

simplify and solve for x

2 3 122(3) 3 126 3 12

6 6 3 12 63 63 63 33 23

2

x yyyyyy

y

y

+ =+ =+ =

− + = −=

=

=

=

solve for y

Solution is (3, 2)

7. The music teacher needs to raise $42,000, for new band instruments, from the sale of 1800 tickets for the annual spring concert. Normally the spring concert is free. But since the music teacher is trying to raise money, she has decided to charge $20 for some tickets and $30 for others. a. If there are x of the $20 tickets and y of the $30 tickets sold, write an equation that states

that the sum of the tickets sold is 1800.

b. How much money is received from the sale of x tickets for $20 each?

c. How much money is received from the sale of y tickets for $30 each?

d. Write an equation that states that the total amount received from the sale is $42,000

e. Use the substitution method to find how many tickets of each type must be sold to raise

the $42,000.


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IV. Lessons 7 and 8 8. Draw the following inequality, − ≤ >10 OR 15x x , on the number line provided. Make sure to scale the

number line accordingly:

9. The total amount of money spent in the US for wireless communication services, C (in billions of dollars), can be modeled by the equation 6.205 11.23C t= + , where t is the number of years past 1995 (ie. t = 0 for the year 1995; t = 1 for the year 1996 and so on) (source: Cellular Telecommunications and Internet Association)

a. What value of t represents the year 2005?

b. What values of t give a cost of more than $150,000,000,000? Round your answer to the nearest whole year.

c. In what year does the equation project that the money spent on wireless communication services will exceed $150,000,000,000?

10. Write a single compound inequality for the following number graph. Then make up a story for your inequality.

0

0 6 12 –6 –12

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Assessment Answer Key

1. Input Output

–4 –77 11 28

2. The original number was 96.

3.

Let x be the original cost of the dining table. Since the discounted price was 20% off the original cost, then we have

=

=

. .. .. .

.

0 80x 60 400 80x 60 400 80 80

0 80.

x0 80

=

=

.

.

75 50

x 75 50

4.

− + = −− + = −

− = −− − + = − − +

=

=

( ) ( )2 w 3 5 3 w 12w 6 5 3w 3

2w 1 3w 32w 2w 1 3 3w 2w 3 3

2 wor

w 2

5. The solution is (3,–2).


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6. The mistake is that when the equations are added the y’s do not cancel; should either subtracted the equations or multiple one equation by -1 (both sides). The correct solution should be:

+ =+ = −

−+ =

− − =

=

+ = −

− + = − −= −

−=

2x 3y 12x 3y 3

Multiply the second equation by 12x 3y 12

x 3y 3Add equations

x 15Substitute value for x

15 3y 3Solve for x

15 15 3y 3 153y 183y 183 3

3y3

= −

= −−

6

y 6Solution is 15 6( , )

+ =

+ − =

− ==

?

?

( ) ( )

Check using first equation 2x 3y 12

2 15 3 6 12

30 18 12 12 12

7a. + =x y 1800 7b. 20x 7c. 30y 7d. + =20x 30y 42000 7e. Solve the first equation for y.

+ =− + = −

= −

x y 1800x x y 1800 x

y 1800 x

Now substitute this value of y into the second equation.

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+ =+ − =+ − =− + =

− + − = −− = −− −=− −

−

( )20x 30y 42000

20x 30 1800 x 4200020x 54000 30x 42000

10x 54000 4200010x 54000 54000 42000 54000

10x 1200010x 1200010 10

10−

x10

=

=

1200

x 1200

Now solve for y. = −= −=

y 1800 xy 1800 1200y 600

The solution is (1200 tickets, 600 tickets) or 1200 $20 tickets must be sold along with 600 $30 tickets.

8.

9a. For the year 2005, t = 2005 – 1995 = 10 9b. The values of t which give a cost of more than “150” is t > 12.8 or 13 years. 9c. The year that wireless communication services will exceed $150,000,000,000 is 2008. 10. − < ≤8 7x AND x ; Answers will vary. A sample response might be: “It was a very cold

January day. So cold, that the temperatures ranged from a high of 7°F to a low of almost –8°F”.

-10 0 10


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References & Resources The authors and contributors of Algebra II Foundations gratefully acknowledges the following resources: Donovan, Suzanne M.; Bransford, John D. How Students Learn Mathematics in the Classroom. Washington, DC: The

National Academies Press. 2005. Driscoll, Mark. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999. Eves, Howard. An Introduction to the History of Mathematics (5th Edition) Philadelphia, PA: Saunders College Publishing,

1983. Harmin, Merrill. Inspiring Active Learning: A Handbook for Teachers. Alexandria, VA: Association for Supervision and

Curriculum Development, 1994. Harshbarger , Ronald J. and Reynolds, James J., Mathematical Applications for the Management, Life, and Social

Sciences Eighth Edition, Houghton Mifflin Boston, MA 2007. Hoffman, Mark S, ed. The World Almanac and Book of Facts 1992. New York, NY: World Almanac. 1992. Kagan, Spencer. Cooperative Learning. San Clemente, CA: Resources for Teachers. 1994. Karush, William. Webster’s New World Dictionary of Mathematics. New York: Simon & Schuster. 1989. McIntosh, Alistair, Barbara Reys, and Robert Reys. Number Sense: Simple Effective Number Sense Experiences. Parsippany,

New Jersey: Dale Seymour Publications. 1997. McTighe, Jay; Wiggins, Grant. Understand by Design. Alexandria, VA: Association for Supervision and Curriculum

Development. 2004. Marzano, Robert J. Building Background Knowledge for Academic Achievement. Alexandria, VA: Association for

Supervision and Curriculum Development. 2004. Marzano, Robert J.; Pickering, Debra J.; Jane E. Pollock. Classroom Instruction that Works. Alexandria, VA:

Association for Supervision and Curriculum Development. 2001. National Research Council. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

2001. Ogle, D.M. (1986, February). “K-W-L: A Teaching Model That Develops Active Reading of Expository Text.” The Reading

Teacher, 39(6), 564–570. The National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: The

National Council of Teachers of Mathematics. 2000. Van de Walle, Jon A. Elementary and Middle School Mathematics: Teaching Developmentally (4th Edition). New York: Addison

Wesley Longman, Inc. 2001. The authors and contributors Algebra II Foundations gratefully acknowledges the following internet resources: http://www.metalprices.com/# http://nationalzoo.si.edu/Animals/AsianElephants/factasianelephant.cfm http://hypertextbook.com/facts/1998/JuanCancel.shtml http://www.conservationinstitute.org/ocean_change/predation/salmonsharks.htm http://blogs.payscale.com/ask_dr_salary/2007/03/starting_salari.html http://www.mpaa.org/FlmRat_Ratings.asp (December 2008) http://federaljobs.net/fbijobs.htm (December 2008) http://www-pao.ksc.nasa.gov/kscpao/release/2000/103-00.htm (December 2008) www.seaworld.org http://www.dailyherald.com/story/?id=92571 http://www.infoplease.com/ipa/A0004598.html http://www.washingtonpost.com/wp-dyn/content/article/2006/10/05/AR2006100501782.html http://www.dxing.com/frequenc.htm http://www.wjhuradio.com/

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