a story of ratios

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 7 – Module 3

Session 2


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios

Session Objectives

• Examine the development of mathematical understanding across the module using a focus on concept development within the lessons.

• Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.



AgendaIntroduction to the ModuleConcept DevelopmentModule Review

Introduction to the Module

Concept Development

Module Review

Introduction to the Module

Concept Development

Module Review



Progressions for the Common Core State Standards in MathematicsWhat to expect in Grade 7:• More complex expressions; containing signed numbers • Various forms of equivalent expressions to model contextual

relationships• Relating numeric solution process to algebraic steps• Using mental math and estimation to check answers to complex

problems• Using variables and equation models to represent real-world problems• Extending the process of solving equations to inequalities, while realizing

that multiplying or dividing both sides of an inequality by a negative number reverses the comparison order.

Expressions and Equations Grades 6-8



Curriculum Overview of A Story of Ratios



Grade 7 Module 3 Overview

• Lessons and topics• Narrative• Focus Standards• Builds upon Foundations• Mathematical Practices• Terminology (old and new)• Tools and Representations



Topic AUse Properties of Operations to Generate Equivalent Expressions

• Take a moment to read through:

• Module Overview paragraphs 2-3

• Topic A Opener

• What do you expect to see conceptually as Topic A unfolds?



Important TerminologyEquivalent Expressions

An Expression in Expanded Form

An Expression in Standard Form

An Expression in Factored Form



Equivalent ExpressionsExample 1 (a): Rewrite and by combining like terms.

Write the original expressions and expand each term using addition. What are the new expressions equivalent to?

Lesson 1 / Example 1(a)



Equivalent ExpressionsExample 1 (a): (cont’d)

Because both terms have the common factor of , we can use the distributive property to create an equivalent expression.

Expressions and Equations Progression Document (p. 6)

Lesson 1 / Example 1 (a)



Example 1 (b): Find the sum of and .

Associative property of addition Commutative property of addition Associative property of addition Combined like terms (distributive property)

Expressions and Equations Progression Document (p. 5)

Equivalent Expressions

Lesson 1 / Example 1 (b)

Example 1 (b): Find the sum of and .

Any order, any grouping property

Combined like terms (distributive property)



Equivalent ExpressionsCan the any order, any grouping property be used with other operations?Subtraction? Yes!

Multiplication? Yes! Multiplication is both associative and commutative.

Division? Yes again!



Equivalent ExpressionsExample 4 (f): Alexander says that is equivalent to because of any order,

any grouping. Is he correct? Why or why not?

Sample response:If I let and let , then:

Lesson 1 / Example 4 (f)



Writing Products as SumsExample 1:a. Draw a tape diagram representing the

sum using square units.

b. Redraw the diagram without squares, as two rectangles that are 1 unit high.

c. Draw a rectangular array representing .

Lesson 3 / Example 1

𝟑 𝟐

𝟑

𝟑

𝟐

𝟑 𝟐𝟏

𝟗𝟔3(3 + 2) = 9 + 6



Writing Products as Sums(continued)d. Draw a tape diagram representing the

sum using square units.

e. Redraw the diagram without squares, as two rectangles that are 1 unit high.

f. Draw a rectangular array representing .


𝒙

𝟑

𝟐

𝒙 𝟐𝟏

𝒙 𝟐…

𝟑 𝒙 𝟔3(x + 2) = 3x + 6



Writing Products as SumsExample 6

A square fountain area with side length is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of three different ways.




Writing Products as Sums

Lesson 3 / Example 64(s + 1) = 4s + 4 = 2s + 2(s + 2)



Writing Sums as ProductsExample 2: and are positive integers. , , and stand for the number of

unit squares in a rectangular array.

What does the large rectangle that contains the three smaller rectangles represent?

2x + 12y + 8 square units

How many rows are there in this rectangular array?

2 rows

What are the missing values and how do you know?


2x + 12y + 8 = 2(x + 6y + 4)



Using the Identity and Inverse to Write Equivalent ExpressionsLesson Highlights:

• Students come to recognize the identity properties of 0 and 1.

• Students come to recognize the existence of inverses (opposites and reciprocals) that together make 0’s and 1’s in expressions.

• Why do we emphasize student awareness of these properties?



Collecting Rational Number Like Terms

Example 4:Model how to write the expression in standard form using rules of rational numbers.





• Module Overview paragraph 4

• Topic B Opener

• What do you expect to see conceptually as Topic B unfolds?

Topic BSolve Problems Using Expressions, Equations, and Inequalities



Understanding EquationsWhere does an equation come from?

Expressions and Equations Progression p. 4

How will we approach equations?Lots and lots of wonderful word problems!



Understanding EquationsExample 1:

The ages of three sisters are consecutive integers. The sum of their ages is 45. Find their ages.a. Use a tape diagram to find their ages.

, and The sisters are 14, 15, and 16.


45

Youngest Sister

Middle Sister

Oldest Sister

1

1 1



Understanding Equations(continued)b. If the youngest sister is years old, describe the ages of the other two 𝒙

sisters in terms of , write an expression for the sum of their ages in 𝒙terms of , and use that expression to write an equation that can be used 𝒙to find their ages

Youngest sister’s age in years:Middle sister’s age in years: Oldest sister’s age in years: Sum of their ages: Sum of their ages is 45:




Understanding Equations(continued)c. Determine if your answer from part (a) is a solution to the equation you

wrote in part (b).

TRUE!




Using If-Then Moves in Solving EquationsWhat is an If-Then move?

Any order, any grouping If , then

Why if-then moves?• If-then moves promote reasoning about the equality of expressions.



Using If-Then Moves in Solving EquationsExample 1: Julia, Keller, and Israel are volunteer firefighters. On Saturday the

volunteer fire department held its annual coin drop fundraiser at a streetlight. After one hour, Keller had collected $42.50 more than Julia and Israel had collected $15 less than Keller. Altogether, the three firefighters collected $125.95. How much did each person collect? Find the solution using a tape diagram.




Using If-Then Moves in Solving EquationsThe amount that Keller collected in dollars:

The amount Isreal collected in dollars: or

Write an equation in terms of j that can be used to find the amount each person collected.

The sum of the amounts collected by all three people equals .




Using If-Then Moves in Solving Equations any order, any grouping If a=b, then to make 0 any order, any grouping and additive inverse additive identity property of 0

If a=b, then ; to make 1

any order, any grouping multiplicative inverse




Using If-Then Moves in Solving Equations


In terms of Julia Collected Isreal Collected Keller Collected

Julia’s Amount

Isreal’s Amount

Keller’s Amount i - 27.50

i + 15k - 27.50 k - 15



Using If-Then Moves in Solving EquationsExample 2:

Shelby is seven times as old as Bonnie. If in 5 years, the sum of Bonnie and Shelby’s ages is 98, find Bonnie’s present age.


Present Age (years) Future Age (years)

Bonnie

Shelby

x + 5

7x + 5

Bonnie’s present age is 11 years old.



Lessons 1 through 9

Lessons 1- 9/ Concepts and Representations




Session 3



Angle Problems and Solving EquationsExample 4:

Two line intersect in the following figure. The ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair is 2:1.


2xᵒ xᵒ

Acute angle: ,

Obtuse angle:



Angle Problems and Solving EquationsExercise 2

In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for x and y. Confirm your answers by measuring with a protractor.

Lesson 11 / Exercise 2

Angles and are complimentary and sum to ; angles and are complimentary and sum to .



Fluency SprintOpening Exercise:

Round 1: You have 1 minute to complete as many as possible…do not skip around, go in order! You only need to provide the answer.• If you have the answer correct, clap once!• Finish the remaining problems.

Lesson 12 / Opening Exercise



Properties of InequalitiesExample 1:

Using the number cubes at your tables, complete the station exercises by recording each trial of integers in the first and third columns. Station 1: If c. Station 2: If then . Station 3: If and , then .Station 4: If and , then .




InequalitiesOpening Exercise:

Tarik is trying to save to buy a new tablet. Right now he has and can save a week from his allowance. Write and evaluate an expression to represent the amount of money saved after:2 weeks:3 weeks:4 weeks:5 weeks:6 weeks:7 weeks:8 weeks:




Solving InequalitiesExample 1

A youth summer camp has budgeted $ for the campers to attend a carnival. 𝟐𝟎𝟎𝟎The cost for each camper is $ . which includes general admission to the 𝟏𝟕 𝟗𝟓carnival and meals. The youth summer camp must also pay $ for the 𝟐 𝟐𝟓𝟎chaperones to attend and $ for transportation to and from. What is the 𝟑𝟓𝟎greatest number of campers that can attend the carnival if the camp must stay within their budgeted amount?


The camp may take up

to 77 campers.



Graphing Solutions to InequalitiesExample 1

A local car dealership is trying to sell all of the cars that are on the lot. Currently it has cars on the lot and the general manager estimates that they will 𝟓𝟐𝟓consistently sell cars per week. Estimate how many weeks it will take for the 𝟓𝟎number of cars on the lot to be less than ? 𝟕𝟓




Mid-Module AssessmentTopics A and B

Mid Module Assessment / Problem #6



Mid-Module AssessmentTopics A and B

Mid-Module Assessment / Problem #6




• Module Overview paragraphs 5-7

• Topic C Opener

• What do you expect to see conceptually as Topic C unfolds?

Topic CUse Equations and Inequalities to Solve Geometry Problems



The Most Famous Ratio of AllOpening Exercise

Draw a circle using a compass.

Write your own definition for the term circle.

Circle: Given a point in the plane and a number , the circle with center and radius is the set of all points in the plane that are a distance from the point .




The Most Famous Ratio of AllExample 1:

1. Mark a point on the wheel with a piece of masking tape or chalk.2. Mark a starting point on the floor, align it with the mark on the wheel,

and carefully roll the wheel so that is rolls one complete revolution.3. Mark the endpoint on the floor with a piece of masking tape or chalk.




The Most Famous Ratio of AllFrom the Educator Guide to the 2013 Grade 7 Common Core Mathematics Test:

This curriculum utilizes the exact value of pi as well as three approximations of the value of pi,

including:

• Calculator approximation



The Area of a CircleDiscussion:

Cut a circle into 16 equal size sectors.Arrange the “triangles” by alternating their directions and sliding them together to form a parallelogram.

Lesson 17 / Discussion



The Area of a CircleThe circumference is , where is the radius. Therefore, half of the circumference is .

What is the area of the rectangle using the side lengths shown?

Lesson 17 / Discussion



The Area of a CircleExample 1

Michael is laying tile in the floor of a square walk-in shower. He wants a circle pattern as shown. He needs to determine the area of the circle so that he can order the special tile for the shower project. Michael drew the tile pattern on graph paper so that he could have a visual model to help guide his thinking.




The Area of a CircleProblem Set #10

If the diameter of a circle is double the diameter of a second circle, what is the ratio of area of the first circle to the area of the second?

Lesson 17 / Problem Set #10



More Problems on Area and CircumferenceOpening Exercise:

Draw a circle of diameter 12 cm and a square of side length 12 cm on grid paper. Determine the area of the square and the circle.




More Problems on Area and CircumferenceExample 3

The circumference of a circle is . What is the exact area of the circle? Draw a diagram to assist you in solving the problem. If , then .Which yields .

The exact area of the circle is




Unknown Area Problems in the Coordinate Plane

Example 1:The coordinate plane shown contains figure , parallelogram .c. Area of the rectangle: d. Area of each triangle: e. Area of parallelogram :

The coordinate plane shown contains figure , a rectangle with the same base as the parallelogram above.f. Draw triangles and next to so that you have a rectangle that is the

same size as the one you created on the first coordinate plane .





Find the unknown value labeled as on the figure.





(Continued)




Composite Area ProblemsExample 3:

Find the area of the shaded region labeling known values.

The area of the shaded region is .




Composite Area ProblemsProblem Set

5. Find the area of the region which is formed by three semicircles.




Lessons 10 through 20

Lessons 10 - 20/ Concepts and Representations




Session 4



Topic CUse Equations and Inequalities to Solve Geometry Problems



Surface Area

Outcomes:• Students find the surface area of three-dimensional

objects whose surface area is composed of triangles and quadrilaterals. They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s).

Lessons 21-22



Surface Area

Lesson 21/Example 1



Surface Area – Day 2

Lesson 22



Surface Area

Lesson 22/Exit Ticket



The Volume of a Right Prism

Lesson 23

Outcomes:• Students use the known formula for the volume of a right

rectangular prism (length x width x height).

• Students understand the volume of a right prism to be the area of the base times the height.

• Students compute volumes of right prisms involving fractional values for length.



The Volume of a Right PrismQuestion: How could we create a right triangular prism with five times the volume of the right triangular prism pictured, without changing its base? What would its volume be?

Lesson 23 / Discussion and Concept




Lesson 23/Exit Ticket



The Volume of a Right Prism – Day 2

Lesson 24

Outcomes:• Students will use the known formula for the volume of a right

rectangular prism (length x width x height) and find the sum (repeated use of the distributive property) to give an approximation of the volume. (Length width) height or (Area of the base) height. Students will understand the volume of a right prism to be the area of the base times the height.

• Students compute volumes of right prisms involving fractional values for length.




Lesson 24/Example 3



Volume and Surface Area

Lesson 25

Outcomes:• Students solve real-world and mathematical problems

involving volume and surface areas of three-dimensional objects composed of cubes and right prisms.



Volume and Surface Area

Lesson 25/Exercise 1



Volume and Surface Area – Day 2

Lesson 26

Outcomes:• Students solve real-world and mathematical problems

involving volume and surface areas of three-dimensional objects composed of cubes and right prisms.



Volume and Surface Area – Day 2

Lesson 26/Exercise 1



End-of-Module AssessmentTopics A, B, and C



End-of-Module Assessment

Questions 3



Biggest TakeawayTurn and Talk:• What questions were answered for you?

• What new questions have surfaced?



Key Points• Properties of operations are used to generate equivalent expressions; and different forms of expressions serve different purposes.

• “Any order, any grouping” and the distributive property are important for generating equivalent expressions (Grade 6); in Grade 7 an understanding of operations

involving negative numbers is also necessary.

• The steps in solving an equation mirror the numeric process; if-then statements justify the making of zeros and ones. Visual models relate algebraic steps to

numeric steps.

• As equations get more complex, estimation skills and mental math verify that an answer is reasonable.

• The solution process for inequalities is similar to that of equations, with the exception of multiplying or dividing by negative numbers.

• Conceptual understanding of pi and the use of equations, allow us to solve real-life problems related to circles.

• Facts about angle relationships can be represented with equations and used to find unknown angles.

• Nets serve as visual connectors between the area of 2-dimensional objects and the surface area of 3-dimensional prisms.

a story of ratios

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