research—best practices putting research into practice · 2013-01-22 · problem is n + 6 1_ 2 =...

Research & Math BackgroundContents Planning

Dr. Karen C. Fuson, Math Expressions Author

ReseaRch—Best PRactices

Putting Research into Practice

From Our Curriculum Research Project

Problem Solving Process Math Expressions uses a research-based algebraic problem-solving approach: Students understand the problem situation, represent the situation with a math drawing and/or a situation equation, solve that representation (possibly by writing a solution equation), and check that the answer makes sense.

Problem Solving Types Students solve all of the problem types for addition, subtraction, multiplication, and division, and they solve all subtypes in which each of the three quantities in each problem type is the unknown. Some of these subtypes are more difficult to represent and solve than others. The more difficult subtypes may require a first equation or drawing to represent the situation (a situation equation), and then a second equation or computation to represent the solution (a solution equation).

Math Drawings Students make their own math drawings for problem situations and they learn to use the Math Expressions research-based numerical drawings for the problem types. These drawings provide coherence across the grades because students can use them for any quantities (small and large whole numbers, decimals, or fractions). Relating math drawings to equations helps students understand where in the drawing the total and the product are represented for each operation, which helps students solve equations with difficult unknowns.

Across the Grades Students in Grades 2–5 benefit from using math drawings and equations to represent problem situations with multidigit numbers, fractions, and decimals when they first begin to solve problems with these quantities. With experience, students become more fluent at understanding, representing, and solving all unknowns for all problem types, and the need for math drawings, and even for equations, drops. As they progress, students may develop the ability to mentally solve problems. For example, in Grade 3, students may be able to mentally solve multiplication and division problems with single-digit factors.

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Fuson, Karen C. “Developing Power in Whole Number Operations.” A Research Companion to Principles and Standards for School Mathematics. Reston, VA: NCTM, 2003. 68–70.

Carpenter, Thomas P., Fennema, Elizabeth, Franke, Megan Loef, Levi, Linda, & Empson, Susan B. Children's Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann, 1999.

Barbu, Otilia C. & Beal, Carole R. “Effects of Linguistic Complexity and Math Difficulty on Word Problem Solving by English Learners.” International Journal of Education, 2.2 (2010): E6.

Nortvedt, Guri A. “Coping Strategies Applied to Comprehend Multistep Arithmetic Word Problems by Students with Above-Average Numeracy Skills and Below-Average Reading Skills.” Journal of Mathematical Behavior, 30.3 (Sept. 2011): pp. 255–269.

Two-Step or Multistep Problems There are no algorithmic one-way approaches for all problems. Encouraging individual approaches may lead to new solutions to new problems not yet even imagined. For a given problem, some students may use a forward approach, almost immediately finding an answer for part of the problem and then deciding if that is a step toward the overall solution, while others may work backwards. Students may write two or more equations or make one or more math drawings. Math drawings are especially helpful for problems with comparisons; for two-step comparison problems, even college students will reverse the equations and improve performance considerably by making a drawing.

Language of Equations The algebraic language of equations is a useful problem-solving tool that students use beginning in Kindergarten. At Grades 2–5, students may be able to write a single equation for problems involving simple subtypes and simple numbers. It can be difficult to write one equation for a two-step or a multistep problem, especially initially. It is important to emphasize individual meaning-making approaches to problem solving rather than requiring that everyone write one equation for the whole two-step or multistep problem. Students need to see and understand such single equations for a problem, and they need to build facility with writing, interpreting, and solving multistep equations, including using the Order of Operations; however, most students will do better by solving in steps: writing more than one equation and/or making one or more math drawings. Students can then work together to discuss and solve a single equation a student has written or write a single equation from the separate steps they have used in solving.

Other Useful Resources

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Getting Ready to Teach Unit 6Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages students to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.

Mathematical Practice 1Make sense of problems and persevere in solving them.

Students analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning.

TeaCher ediTion: examples from Unit 6

MP.1 Make Sense of Problems Reasonable Answers Remind students of the importance of checking their work, then ask them to explain how addition can be used to decide if an answer of 1 _ 8 lb is reasonable. Possible answer: If each of the smaller pieces weighs 1 _ 8 lb, then the weight of 4 smaller pieces should be 1 _ 8 + 1 _ 8 + 1 _ 8 + 1 _ 8 , or 4 _ 8 lb. Since 4 _ 8 = 1 _ 2 , the answer is reasonable.

Lesson 2

MP.1 Make Sense of Problems Analyze the Problem Read aloud Problem 14 on Student Book page 202. Discuss and complete the solution using the following prompts.

• What steps are needed to solve the problem?

• What operations are needed to solve the problem?

• Can the operations be performed in any order, or must the operations be performed in a specific order?

• What equation will you write to solve the problem?

Lesson 8

Mathematical Practice 1 is integrated into Unit 6 in the following ways:

Analyze the ProblemAnalyze Relationships

Connect Symbols and WordsDraw a Diagram

Reasonable AnswersWrite an Equation

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Mathematical Practice 2Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves reasoning about problem situations and misleading language to connect contexts and understanding to symbols and situations.

TeACHeR eDITION: examples from Unit 6

MP.2 Reason Abstractly and Quantitatively Connect Diagrams and Equations Give Student Pairs an opportunity to discuss and complete Problem 1 on Student Book page 187. Be sure they recognize that Step c (finding an unknown addend) involves subtraction.

312 + n = 961 961 961 − 312

__

649

312 n

Lesson 1

MP.2 Reason Abstractly and Quantitatively Connect Symbols and Words Arrange students in Small Groups to complete Student Book page 191.

Some groups may find it helpful to identify real world contexts in which the denominators of the fractions might occur. In Exercise 1, for example, point out that a denominator of 4 might represent measurement units such as fractions of an inch, fractions of a cup, or fractions of an hour. Or you might ask questions about the denominators, such as:

• A denominator of 4 represents fourths, and another name for fourths is quarters. What contexts can you think of that involve fourths or quarters?

Lesson 3


Connect Diagrams and EquationsConnect Symbols and WordsReasonable Answers

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Mathematical Practice 3Construct viable arguments and critique the reasoning of others.

Students use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Students are also able to distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Students can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Math Talk is a conversation tool by which students formulate ideas and analyze responses and engage in discourse. See also Mathematical Practice 6: Attend to Precision.

TeaCher ediTion: examples from Unit 6

MP.3 Construct Viable arguments Compare Strategies Arrange students in Small Groups to discuss and solve Problems 2–4 on Student Book page 205. Upon completion of the work, ask a volunteer from each group to share the methods they used to solve each problem.

Lesson 9

What’s the Error? W H O L E C L A S S

MP.3, MP.6 Construct Viable arguments/Critique reasoning of others Puzzled Penguin Write the following problem on the board.

What number, when increased by 6 1 _ 2 , is 10? Puzzled Penguin wrote the solution equation n = 10 + 6 1 _ 2 .

Encourage students to discuss the problem, identify the error, and then write an equation to solve the problem. Be sure students recognize that the problem consists of two addends and a total. One addend, n, is unknown. The other addend is 6 1 _ 2 . A situation equation to represent the problem is n + 6 1 _ 2 = 10, which leads to the solution equation n = 10 - 6 1 _ 2 . So, n = 3 1 _ 2 .

Lesson 1


Compare Methods Compare Strategies Puzzled Penguin

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Mathematical Practice 4Model with mathematics.

Students can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Students might draw diagrams to lead them to a solution for a problem. Students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent their relationships using such tools as diagrams, tables, graphs, and formulas.


MP.1, MP.4 Make Sense of Problems/Model with Mathematics Draw a Diagram Drawing comparison bars will help all students solve Problem 6 with some students working at the board so that everyone can compare and discuss.

Carlos

Isaac

195

987

c

Emphasize to your class that the two most important questions to ask in a comparison problem are:

• Who has more (or less)?

• How much more (or less)?

If they focus on these questions, instead of just on the words more or less, they will be more likely to solve the problem correctly.

Tell students to draw comparison bars to help them solve the rest of the problems with misleading language.

Lesson 5

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Write an Equation Read aloud Problem 1 on Student Book page 207. Give students an opportunity to restate the problem using their own words, which should demonstrate an understanding that the number of miles Xavier, Yuri, and Zack drove subtracted from 1,050 is the number of miles Walter drove. On the board, write the solution equation below to represent this fact.Walter = 1,050 - (Zack + Yuri + Xavier)

Discuss Steps A–E of the solution, and be sure students understand the steps.

Point out to students that they do not need to write only one equation to solve a multistep problem. They can use any method they choose, and any number of equations.

Lesson 10


Draw a DiagramMathBoardWrite an Equation

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Mathematical Practice 5Use appropriate tools strategically.

Students consider the available tools and models when solving mathematical problems. Students make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. They recognize both the insight to be gained from using the tool and the tool’s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations.

Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Students learn and develop models to solve numerical problems and to model problem situations. Students continually use both kinds of modeling throughout the program.

Teacher ediTion: examples from Unit 6

MP.4, MP.5 Model with Mathematics/Use appropriate Tools MathBoard Introduce the activity by discussing this statement.

4 ÷ 1 _ 2 means “How many halves in 4?”

• Think about the division 4 ÷ 1 _ 2 . There are many stories you could invent that include that fact. For example, a group of friends having a sleepover divide each of 4 pizzas in half. Altogether they create 8 halves.

• What other stories can you invent to represent 4 ÷ 1 _ 2 ?

• On your MathBoard, draw a model to represent 4 ÷ 1 _ 2 . Accept any model that represents 4 ÷ 1 _ 2 . Sample model:

Invite volunteers to sketch different models for 4 ÷ 1 _ 2 on the board. Discuss the models.

Lesson 3

MP.4, MP.5 Model with Mathematics/Use appropriate Tools On Student Book page 210, students make a bar graph to display a given set of data. Encourage them to use a straightedge or ruler when constructing the graph. If the activity is to be completed outside your classroom, some students may find it helpful to first discuss possible intervals for the vertical axis of the graph.

Lesson 11


MathBoard

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Mathematical Practice 6Attend to precision.

Students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other.


MP.6 Attend to Precision Explain Solutions Since the explanations for Problems 1–4 are likely to vary, invite a volunteer from each group to share their explanations with the class. Lead students to infer from those explanations that there are different ways to estimate reasonableness, and any method that provides a general sense of what the answer should be is a good method.

Lesson 4

MP.6 Attend to Precision Explain an Example Give students an opportunity to write examples on the board to represent these three scaling generalizations. Examples will vary; possible examples shown.

• Multiplying a number n by a factor of 1 gives a product equal to n. 10 ⋅ 1 = 10

• Multiplying a number n by a factor < 1 gives a product < n. 10 ⋅ 1 > 10 ⋅ 0.5 ; 10 > 5

• Multiplying a number n by a factor > 1 gives a product > n. 10 ⋅ 1 < 10 ⋅ 1.5; 10 < 15

Lesson 6

MATH TALK Use Solve and Discuss. For each problem, ask students to decide if the language is leading because it suggests the correct operation, or misleading because it suggests an incorrect operation. Encourage students to say the comparing sentence or question both ways to decide if the language is leading.

Lesson 5

MATH TALKin ACTION

We’ve talked about additive comparisons and multiplicative comparisons. What kind of comparison is this problem, and how do you know?

Hailey: Additive comparisons are problems that are solved by adding or subtracting.

Jose: In this problem, the population of Cincinnati is given, and it is bigger than the population of Dayton by about 155,000 people. So we subtract about 155,000 people from the population of Cincinnati to find the population of Dayton.

How is a problem that is a multiplicative comparison the same as Problem 5, and how is it different?

Alexa: It’s the same because both kinds of problems compare two quantities.

Dakota: It’s different because in a multiplicative comparison, multiplication or division is used to find the answer.

Lesson 7


Explain an Example Explain Solutions Puzzled Penguin

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Mathematical Practice 7Look for and make use of structure.

Students analyze problems to discern a pattern or structure. They draw conclusions about the structure of the relationships they have identified.

Teacher ediTion: examples from Unit 6

MP.7 Look for Structure Identify Relationships Complete Problems 1–3 on Student Book page 197 as a class activity. The models that are given and sketched provide a way for students to see how the two quantities compare, and help them recognize if a greater quantity is being compared to a lesser quantity, or vice versa.

Lesson 6

MP.7 Look for Structure Identify Relationships If students understand that the numerator and denominator of a fraction share a relationship by division (symbolized by the fraction bar), they can divide to change a fraction to a decimal. An alternative is writing equivalent fractions in tenths or hundredths, which involves multiples of the denominators and finding the least common multiple.

Lesson 11


Identify Relationships

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Class Activity6-11

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► Math and GymnasticsIn a gymnastics competition, gymnasts compete in events such as the balance beam, parallel bars, vault, and floor exercise.

Leigh earned the following scores from the judges for her balance beam routine.

9.20 9.30 9.20 9.30 9.20 9.00

Follow these steps to find Leigh’s final score.

1. Order the scores from least to greatest.

2. Cross off the lowest score and the highest score.

3. Find the average of the remaining scores by adding the scores and dividing the sum by 4.

4. Calculate Leigh’s final score by adding 7.0 (the difficulty rating of her routine) to the average you found in Exercise 3.

The judges’ scores for Olivia’s balance beam routine are shown below.

9.40 9.40 9.50 9.50 9.40 9.40

5. Calculate Olivia’s average score by following the steps described in Exercises 1–3 above.

6. Calculate Olivia’s final score by adding 6.6 (the difficulty of her routine) to the average you found in Exercise 5.

Name Date

9.00; 9.20; 9.20; 9.20; 9.30; 9.30

9.00; 9.20; 9.20; 9.20; 9.30; 9.30

(9.20 + 9.20 + 9.20 + 9.30) ÷ 4 = 9.225

9.225 + 7.0 = 16.225

9.40; 9.40; 9.40; 9.40; 9.50; 9.50

(9.40 + 9.40 + 9.40 + 9.50) ÷ 4 = 9.425

9.425 + 6.6 = 16.025

UNIT 6 LESSON 11 Focus on Mathematical Practices 209

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► Math and DivingIn diving competitions, divers compete in springboard and platform events.

Follow these steps to find the total score for a dive.

→ Order the judges’ scores from least to greatest.

→ Cross off the lowest score and the highest score.

→ Find the sum of the remaining scores.

→ Multiply the sum by the difficulty of the dive.

7. Suppose a diver earned the following scores from judges on his first of five platform dives.

9.5 10.0 9.0 10.0 10.0

a. In the space at the right, sketch a bar graph to display the scores.

b. The difficulty of the dive was 3.8. Follow the steps above to calculate the total score for the dive.

8. The table below shows the scores the diver received from the judges for his four remaining dives.

Dive Scores Diffi culty

2 9.4 8.9 9.0 9.5 9.4 3.2

3 8.8 8.6 8.0 8.0 8.5 3.5

4 9.6 9.5 9.5 9.6 9.4 2.8

5 7.0 7.5 6.5 7.5 7.0 3.0

For each dive, follow the steps above to calculate the dive’s total score.

a. Dive 2 total score:

c. Dive 4 total score:

b. Dive 3 total score:

d. Dive 5 total score:

9. How many points altogether were scored on the five dives?

Name Date

433.49 points

Graphs may vary. Possible answer:

29.5 ⋅ 3.8 = 112.1

88.96 87.85

80.08 64.5

1 2 3 4 5

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Dive 1 Scores

Judge

8.5

9.0

9.5

10.0

210 UNIT 6 LESSON 11 Focus on Mathematical Practices

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STUDeNT eDITION: LeSSON 11, pAGeS 209–210

Mathematical practice 8Look for and express regularity in repeated reasoning.

Students use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.


Mp.8 Use Repeated Reasoning Generalize Invite a volunteer from each group to share their explanations for Problems 5–8 on Student Book page 194 with the class, and then challenge your students to demonstrate other ways those same strategies can be used to decide reasonableness.

Lesson 4

Mp.8 Use Repeated Reasoning Conclude Problem 10 involves the concept of fraction equivalence. Be sure students conclude that multiplying both the numerator and the denominator of a fraction by the same number

a_ b = (n ⋅ a)

_____ (n ⋅ b)

is the same as multiplying a_ b by 1 because n_ n

(any number divided by itself) is 1.

Lesson 6


ConcludeGeneralize

Focus on Mathematical practices Unit 6 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson, students use what they know about operating with decimals to answer questions about gymnastics and diving scores.


Math Expressions VOCABULARY

As you teach this unit, emphasize

understanding of these terms:

• situation equation• solution equation

See the Teacher Glossary

Getting Ready to Teach Unit 6 Learning Path in the Common Core StandardsIn previous units, the lessons focused on math concepts and skills as students applied these skills in word problems. In this unit, the focus is on solving problems as students use whole numbers, fractions, and decimals to find answers. Their work involves writing real world contexts and writing situation and solution equations for one-step, two-step, and multistep problems representing addition, subtraction, multiplication, and division.

Visual models and real world situations are used throughout the unit to represent problems and illustrate solutions.

Help Students Avoid Common ErrorsMath Expressions gives students opportunities to analyze and correct errors, explaining why the reasoning was flawed.

In this unit, we use Puzzled Penguin to show typical errors that students make. Students enjoy teaching Puzzled Penguin the correct way, and explaining why this way is correct and why the error is wrong. The following common errors are presented to students as letters from Puzzled Penguin and as problems in the Teacher Edition that were solved incorrectly by Puzzled Penguin:

→ Lesson 1: Writing incorrect situation and solution equations

→ Lesson 4: Identifying an estimate that is not reasonable

→ Lesson 6: Not choosing the correct operation

→ Lesson 10: Using division to find a fractional part of a whole

In addition to Puzzled Penguin, there are other suggestions listed in the Teacher Edition to help you watch for situations that may lead to common errors. As part of the Unit Test Teacher Edition pages, you will find a common error and prescription listed for each test item.

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The Problem Solving Process

Using the Mathematical Practices Throughout the program, Math Expressions integrates a research-based algebraic problem solving approach that focuses on problem types. Problem solving is a complex process that involves all eight of the CCSS Mathematical Processes. It is also an individual process that can vary considerably across students. Students may conceptualize, represent, and explain a given problem in different ways.

Mathematical Process Student Actions

Understand the Problem Situation

MP.1 Make sense of the problem.

MP.2 Reason abstractly and quantitatively.

Make Sense of the Language

Students use the problem language to conceptualize the real world situation.

Represent the Problem Situation

MP.4 Model with mathematics.

MP.7 Look for and make use of structure.

Mathematize the Situation

Students focus on the mathematical aspects of the situation and make a math drawing and/or write a situation equation to represent the relationship of the numbers in the problem.

Solve the Problem

MP.5 Use appropriate tools.

MP.8 Use repeated reasoning.

Find the Answer

Students use the math drawing and/or the situation/solution equation to find the unknown.

Check That the Answer Makes Sense

MP.3 Critique the reasoning of others.

MP.6 Attend to precision.

Check the Answer in the Context of the Problem

Students write the answer to the problem, including a label. They explain and compare solutions with classmates.

Students are taught to make their own math drawings. Relating math drawings to equations helps them understand where in the drawing the total and the product are represented for each operation, and helps them solve equations with difficult unknowns.

UNIT 6 | Overview | 473AA


Math Talk Learning CommunityResearch In the NSF research project that led to the development of Math Expressions, much work was done with helping teachers and students build learning communities within their classrooms. An important aspect of doing this is Math Talk. The researchers found three levels of Math Talk that go beyond the usual classroom routine of students simply solving problems and giving answers, and the teacher asking questions and offering explanations. It is expected that at Grade 5, students will engage in Math Talk at all levels.

Math Talk Level 1 A student briefly explains his or her thinking to others. The teacher helps students listen to and help others, models fuller explaining and questioning by others, and briefly probes and extends students’ ideas.

Example Word Problem

Anja has worked at her job for 6 1 _ 2 years. Each year she works 48 weeks, and each week she works 37 1 _ 2 hours. How many hours does Anja work each year?

What information is not needed to solve this problem?

Trevor: 6 1 _ 2 years—the length of time Anja has worked at her job.

What information is needed to solve the problem?

Trevor: She works 48 weeks each year and 37 1 _ 2 hours each week.

What would you do to find the number of hours?

Gabrielle: Multiplying 37 1 _ 2 by 48 will tell us the number of hours she works each year. 48 times 37 1 _ 2 is 1,800.

48 × 37 1 __ 2 = 48 ___ 1 × 75 ___

2

= 3,600 _____

2

= 1,800

How would you estimate to check the answer?

Anthony: I would use rounding; 37 1 _ 2 rounds to 40 and 48 rounds to 50. The exact answer should be close to 40 × 50, which is 2,000.

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Math Talk Level 2 A student gives a fuller explanation and answers questions from other students. The teacher helps students listen to and ask good questions, models full explaining and questioning (especially for new topics), and probes more deeply to help students compare and contrast methods.


A shopping mall has 215 rows of parking spaces with 35 spaces in each row. A special permit is required to park in the first 6 spaces of 75 rows. How many parking spaces (s) do not require a special permit?

How can you restate this problem in different ways?

Jordan: A parking lot has 215 parking spaces. Some of the spaces require a permit. Some of the spaces don't.

Maria: The total number of parking spaces is made up of two parts, and one part—the number of spaces that do not require a special permit—is unknown.

How can we find the total number of parking spaces?

Destiny: I would multiply the number of rows by the number in each row. So I would multiply 215 by 35.

Miguel: 35 times 215 is 7,525. But we also need to find how many special permit spaces there are.

215 × 35

_

1075 + 6450

__

7,525

Jasmine: We could find that number if we multiply 6 times 75. That number is 450.

75 × 6

_

450

So how do we find the number of parking spaces that do not require a special permit?

Jada: We have to subtract 450 from 7,525 which is 7,075.

7,525 - 450

__

7,075

Austin: So 7,075 parking spaces do not require a special permit.

UNIT 6 | Overview | 473CC


Math Talk Level 3 The explaining student manages the questioning and justifying. Students assist each other in understanding and correcting errors and in explaining more fully. The teacher monitors and assists and extends only as needed.


The charge for an automobile repair was $328.50 for parts and $64 per hour for labor. The repair took 3 3 _ 4 hours. What was the total cost (c) of the repair?

Kai, explain how to solve the problem.

Kai: We want to find the total cost of the repair. So I would multiply $64 times 15 __ 4 and add the product to $328.50.

64 ___ 1 × 15 ___ 4 = 960 ____ 4

= 240

$328.50 + 240.00

__

$568.50

Larissa: Why did you multiply $64 by 15 __ 4 ?

Kai: Because 3 3 _ 4 is the same as 15 __ 4 . I'm finding the cost of labor.

Muriel: Couldn't we have multiplied $64 by 3 and $64 by 3 _ 4 and then added the products together?

Can we use both ways to solve the problem?

Hector: 3 times $64 is $192 and 3 _ 4 times $64 is $48, and the total of those two amounts is $240.

64 × 3

_

192

3 __ 4 × 64 ___ 1 = 192 ____ 4 or 48

Alexa: But we could have also multiplied $64 times 3.75 to find that same amount.

3.75 × $64

__

1500 + 22500

__

$240.00

Kai: Yes, we could have done that. But I multiplied $64 times 15 __ 4 and also found that the cost of labor was $240. Then I added $240 to $328.50 for a total of $568.50.

Did everyone get the same answer to this problem?

Class: Yes.

Summary Math Talk is important not only for discussing solutions to word problems, but also for any kind of mathematical thinking students do, such as simplifying an expression or solving an equation.

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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON OPERATIONS AND ALGEBRAIC THINKING

An Algebraic Perspective Students

begin developing an algebraic

perspective many years before

they will use formal algebraic

symbols and methods. They

read to understand the problem

situation, represent the situation

and its quantitative relationships

with expressions and equations,

and then manipulate that

representation if necessary, using

properties of operations and/or

relationships between operations.

Linking equations to concrete

materials, drawings, and other

representations of problem

situations affords deep and flexible

understandings of these building

blocks of algebra.

Writing Equations to Solve Problems

Lessons

1 2

Situation and Solution Equations for Addition and Subtraction Throughout the lessons in Unit 6, students write one or two equations to solve word problems. Writing two equations involves writing a situation equation and a solution equation.

A situation equation shows the structure or relationship of the information in a problem. Students who write a situation equation to represent a problem use their understanding of inverse operations to then rewrite the equation as a solution equation.

A solution equation shows the operation that is used to solve a problem. If only one equation is written by students and used to solve a problem, the equation is a solution equation.

Model the Math Students work within addition and subtraction contexts and represent these problems with a break apart drawing. In a break apart drawing for addition and subtraction, the total is placed at the top. For example, the break apart drawing below models four related addition and subtraction equations.

10.1

2.4 7.7

2.4 + 7.7 = 10.1 10.1 - 7.7 = 2.4

7.7 + 2.4 = 10.1 10.1 - 2.4 = 7.7

Students model decimal and fraction problems using break apart drawings and write situation and solution equations to solve the problems.

Jalen had 4 2 __ 3 pounds of modeling clay and used

3 1 __ 2 pounds for a craft project. How many pounds

of clay were not used?

3 1 __ 2 c

4 2 __ 3

UNIT 6 | Overview | 473EE



Problem Types In Grades 3, 4,

and 5, students extend their

understandings of addition and

subtraction problem types [...] to

situations that involve fractions

and decimals. Importantly, the

situational meanings for addition

and subtraction remain the same

for fractions and decimals as for

whole numbers.

Problem Types for Addition and Subtraction At this grade level, students solve addition and subtraction problem types involving whole numbers, fractions, and decimals.

Put Together, Result Unknown

A 5 _ 8 -inch thick paperback book is on top of a 15 __ 16 -inch thick paperback book. What is the total thickness of books?

Put Together, Change Unknown

Enrique has two packages to mail. The weight of one package is 12 1 _ 4 pounds. What is the weight of the second package if the total weight of the packages is 15 1 _ 8 pounds?

Put Together, Start Unknown

A shopper spent $53.50 for a sweater and a T-shirt. What was the cost of the sweater if the cost of the T-shirt was $16.50?

Take Apart, Result Unknown

At a track and field meet, Cody’s time in a sprint event was 17.6 seconds. What was Shaina’s time if she completed the event in 1.08 fewer seconds?

Take Apart, Change Unknown

Altogether, 91,292 people live in Waterloo and Muscatine, two cities in Iowa. The population of Waterloo is 68,406 people. What is the population of Muscatine?

Take Apart, Start Unknown

Altogether, the fourth graders jumped 345,127 times. If the fifth graders had done 2,905 fewer jumps, there would have been a tie. How many jumps did the fifth graders do?

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Add To, Result Unknown

Walt is running to get in shape. He ran 1.5 miles the first day. On the second day, he ran 2.75 miles. How far did he run during the two days?

Add To, Change Unknown

Matt is competing in the long jump event. His first jump was 3.56 m. So far, the longest jump in the event is 4.02 m. How much farther must he jump to be in first place?

Add To, Start Unknown

Walt is running for exercise. He ran around Lake Blue and then ran 2.75 miles home. He ran for a total of 4.25 miles. How far did he run around Lake Blue?

Take From, Result Unknown

A bag contained 6 2 _ 3 cups of flour. Scott used 2 1 _ 3 cups to make some bread. How much flour was left in the bag after he made the bread?

Take From, Change Unknown

Maryl measured her younger brother on January 1 of this year and found that he was 40 3 __ 16 inches tall. On January 1 of last year, he was 37 15 __ 16 inches tall. How much did he grow in a year’s time?

Take From, Start Unknown

Sarita has some ribbon. After she used 23.8 cm of it, she had 50 cm left. How much ribbon did Sarita start with?

UNIT 6 | Overview | 473GG


Situation and Solution Equations for Multiplication and Division Multiplication and division contexts are presented in Lesson 2. To help recognize and generate the situation and solution equations that represent the various contexts, students model the contexts using Rectangle Models. In a Rectangle Model, the product (or dividend) is placed inside the rectangle.

Model the Math The Rectangle Model below represents four related equations.

36

2 72

2 × 36 = 72 72 ÷ 36 = 2

36 × 2 = 72 72 ÷ 2 = 36

A Rectangle Model helps students see the information that is unknown and recognize the operation that is used to find that information. For example, when a completed rectangle shows a product in the middle and a factor as the length or width, students recognize division as the operation that is used to find a missing factor.

Inverse Operations Students build upon their understanding of inverse operations by using Rectangle Models as springboards for writing the situation and solution equations that are used to solve multiplication and division problems involving whole numbers, fractions, and decimals.

The musicians in a marching band are arranged in equal rows, with 8 musicians in each row. Altogether, the band has 104 musicians. In how many rows are the musicians marching?

Elena has chosen carpet that costs $4.55 per square foot for a rectangular floor that measures 12 1 _ 2 feet by 14 1 _ 2 feet. How many square feet of carpet is needed to cover the floor?

8

n

104

12 1 __ 2 n

14 1 __ 2

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Connections Students extend their

whole number work with adding

and subtracting and multiplying

and dividing situations to decimal

numbers and fractions. Each of

these extensions can begin with

problems that include all of the

subtypes of [problem] situations.

The operations of addition,

subtraction, multiplication, and

division continue to be used in

the same way in these problem

situations when they are extended

to fractions and decimals (although

making these extensions is not

automatic or easy for all students).

Problem Types for Multiplication and Division Problems The following multiplication and division problem types appear throughout Grade 5.

Equal Groups, Product Unknown

On the first day of soccer practice, 2 _ 5 of the players were wearing new shoes. The team has 20 players. How many players were wearing new shoes?

Equal Groups, Group Size Unknown

To get ready for her first semester of school, Jayna spent a total of $7.92 for eight identical notebooks. What was the cost of each notebook?

Equal Groups, Number of Groups Unknown

How many individual pieces of cheese, each weighing 1 _ 4 lb, can be cut from a block of cheese weighing 5 pounds?

Arrays, Product Unknown

A theater has 39 rows of seats. Each row has 54 seats. How many seats are in the theater?

Arrays, Group Size Unknown

In a school gymnasium, 588 students were seated for an assembly in 21 equal rows. What number of students were seated in each row?

Arrays, Number of Groups Unknown

The musicians in a marching band are arranged in equal rows, with 8 musicians in each row. Altogether, the band has 104 musicians. In how many rows are the musicians marching?

UNIT 6 | Overview | 473II


Multiplication and Division Problem Types (continued)

Area, Product Unknown

Elena has chosen carpet that costs $4.55 per square foot for a rectangular floor that measures 12 1 _ 2 feet by 14 1 _ 2 feet. How many square feet of carpet is needed to cover the floor?

Area, Side Unknown

A sidewalk covers 3,372 square feet. If the sidewalk is 4 feet wide, what is its length?

Compare, Product Unknown (Measurement Example)

The length of an unstretched spring is 120 cm. How long (l) will the spring be if it is stretched to 3 times that length?

Compare, Group Size Unknown (Measurement Example)

The length of a collapsed fishing pole (c) is 1 _ 8 times as long as its extended length. The extended length of the pole is 16 feet. What is its collapsed length?

Compare, Number of Groups Unknown (Measurement Example)

A maple tree in the backyard of a home has a height of 0.75 meters. How many times as tall (t) is a nearby hickory tree that is 9 meters tall?

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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS—FRACTIONS

Writing Problems In Grade 5,

[students] connect fractions with

division, understanding that

5 ÷ 3 = 5 _ 3 ,

or, more generally, a _ b  = a ÷ b

for whole numbers a and b, with

b not equal to zero. They can

explain this by working with their

understanding of division as equal

sharing. They also create story

contexts to represent problems

involving division of whole numbers.

Write Multiplication and Division Word Problems

Lesson

3

Fractions and Decimals A variety of multiplication and division equations involving fractions and decimals are presented in Lesson 3. For each equation, students draw a model to represent the product or quotient and then, using words, write a problem for which the equation is sensible.

Model the Math Students may infer that the division equation 3 ÷ 1 _ 2 = 6 represents the idea of cutting 3 whole sheets of paper into halves, creating 6 halves. A possible model that may be drawn for this context and equation is shown below.

2 halves

1 whole 1 whole 1 whole

2 halves 2 halves

Other equations that are presented in Lesson 3, and some possible models for those equations, are shown below.

3 __ 4 ⋅ 2 = 6 __

4

0 1 2

5 __ 6 ⋅ 1 __

3 = 5 ___

18

$6 ⋅ 3.5 = $21

1 __ 4 ÷ 2 = 1 __

8

÷21–4

1–4

$16.50 ÷ 3 = $5.50

$16.50

$5.50

$5.50

$5.50

50¢$5$5 $5 50¢ 50¢

UNIT 6 | Overview | 473KK


Determine Reasonableness

Lessons

1 2 4

Addition and Subtraction An important part of performing computations and solving equations is checking exact answers for reasonableness. Lesson 1 provides opportunities for students to develop number sense and reasoning skills.

Suppose you were asked to add the decimals at the right, and you wrote 2.07 as your answer. Without using pencil and paper to actually add the decimals, give a reason why an answer of 2.07 is not reasonable.

Suppose you were asked to subtract the fractions at the right, and you wrote 5 __

6 as your answer. Without using pencil and

paper to actually subtract the fractions, give a reason why an answer of 5 __

6 is not reasonable.

Multiplication and Division Lesson 2 also provides opportunities for students to develop number sense and reasoning skills.

Suppose you were asked to multiply the numbers at the right, and you wrote 15,000 as your answer. Without using pencil and paper to actually multiply the numbers, give a reason why an answer of 15,000 is not reasonable.

Suppose you were asked to divide the numbers at the right, and you wrote 30 as your answer. Without using pencil and paper to actually divide the numbers, give a reason why an answer of 30 is not reasonable.

2.65 + 0.42

__

1 __ 2 - 1 __

3

2,500 × 0.6

90 ÷ 1 __ 3

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Using Benchmarks Students also

reason using benchmarks such as 1 _ 2

and 1. For example, they see that 7 _ 8 is less than 13 __ 12 because 7 _ 8 is less

than 1 (and is therefore to the left

of 1 [on a number line]) but 13 __ 12 is

greater than 1 (and is therefore to

the right of 1).

Using Number Sense Although students can check exact answers by performing computations a second time, a more efficient method involves using number sense to gain a general expectation of what exact answers should be. In Lesson 4, students are asked to use number sense and a variety of strategies to help decide the reasonableness of their exact answers.

Using Rounding One such strategy involves rounding. Students determine reasonableness by comparing an exact difference to a rounded difference.

Altogether, 91,292 people live in Waterloo and Muscatine, two cities in Iowa. The population of Waterloo is 68,406 people. What is the population of Muscatine?

Using Estimation and Mental Math Estimation and mental math are also components of number sense, and used by students to determine the reasonableness of a computed quotient.

In a school gymnasium, 588 students were seated for an assembly in 21 equal rows. What number of students were seated in each row?

Using Basic Facts Another number sense strategy involves using a pattern of basic facts to predict two consecutive whole numbers the exact answer should be between.

A $45 award will be shared equally by 6 friends. In dollars, what is each friend’s share of the award?

Using Benchmarks It is especially important for students to check computations involving fractions because generally speaking, many students are less comfortable working with fractions than with decimal or whole numbers, and less comfort often translates to more errors.

A way for students to check their computations involving fractions is to use benchmark values. Whole numbers and halves are examples of benchmark values.

A 5 _ 8 -inch thick paperback book is on top of a 15 __ 16 -inch thick paperback book. What is the total thickness of books?

Summary All of the strategies in Lesson 4 involve number sense, and the goal of each strategy is for students to gain a sense of what to expect for an exact answer without having to check for reasonableness by performing computations a second time.


Reasonable Answers Students

make sense of fractional quantities

when solving word problems,

estimating answers mentally to see

if they make sense.

UNIT 6 | Overview | 473MM



Comparison Problems In

an additive comparison, the

underlying question is what

amount would be added to

one quantity in order to result

in the other. In a multiplicative

comparison, the underlying

question is what factor would

multiply one quantity in order to

result in the other.

Comparison Word Problems

Lessons

5 6 7

Language of Comparison Problems Students work with the language of comparison word problems in Lesson 5, and are presented with language that is leading and language that is misleading.

Leading Language The word more in the problem below suggests addition, and addition is used to solve the problem.

Altogether, the Blue Team jumped 11,485 times. The Red Team did 827 more jumps than the Blue Team. How many jumps did the Red Team do?

The phrase times as many in the situation below suggests multiplication, and multiplication is used to solve the problem.

Maria scored 6 points in the basketball game. Suzanne scored 4 times as many points as Maria. How many points did Suzanne score?

Misleading Language Although the word more in the situation below suggests addition, subtraction is the operation used to solve the problem.

Ted jumped 1,300 times. He did 100 more jumps than Mario. How many jumps did Mario do?

In contrast to the problem about Maria above, division is the operation used to solve other times as many problems.

Mr. Wagner has 32 horses on his farm. He has 4 times as many horses as Mr. Cruz. How many horses does Mr. Cruz have?

Summary The goal for students completing Lesson 5 is to understand that the language of a problem must be considered in the whole context of the problem because the meaning of the language may be different in context than by itself.

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Multiplicative Comparison

Problems In Grade 5, unit fractions

language such as “one third as

much” may be used. Multiplying

and unit fraction language change

the subject of the comparing

sentence, e.g., “A red hat costs

A times as much as the blue hat”

results in the same comparison as

“A blue hat costs 1 __ A times as much

as the red hat,” but has a different

subject.

Multiplicative Comparison Problems Relating math drawings to equations helps students understand where the total and the product are for each operation. This is especially true for multiplicative comparison problems that involve misleading language. Drawing comparison bars helps students show which quantity is larger, and helps them identify a solution equation and computation that can be used to solve the problem.

Writing Comparison Equations Students learn in Lesson 6 that both multiplication and division can be used to compare two quantities.

The comparison bars below represent the time a student worked on spelling (s) and math (m) homework.

s

m

1. Write a comparison sentence that includes the words “as long as” and compares

a. m to s. m is seven times as long as s

b. s to m. sis one-seventh as long as m

2. Write a comparison equation that compares

a. m to s. m= 7 ⋅ s or m = 7s

b. s to m. s = 1 _ 7 ⋅ m or s = 1 _ 7 m

3. Write a division equation that compares sto m. s = m ÷ 7 or s = m__ 7

Multiplicative Comparison Problem Types

Compare, Product Unknown (Measurement Example)

The length of an unstretched spring is 120 cm. How long (l) will the spring be if it is stretched to 3 times that length?

Compare, Group Size Unknown (Measurement Example)

The length of a collapsed fishing pole (c) is 1 _ 8 times as long as its extended length. The extended length of the pole is 16 feet. What is its collapsed length?

Compare, Number of Groups Unknown (Measurement Example)

A maple tree in the backyard of a home has a height of 0.75 meters. How many times as tall (t) is a nearby hickory tree that is 9 meters tall?

l

120

16

c

9 m

0.75 m

UNIT 6 | Overview | 473OO



Scaling As students' notions of

quantity evolve and generalize

from discrete to continuous

during Grades 3–5, their notions

of multiplication evolves and

generalizes. This evolution deserves

special attention because it begins

in [Operations and Algebraic

Thinking] but ends in [Numbers

and Operations—Fractions]. Thus,

the concept of multiplication

begins in Grade 3 with an entirely

discrete notion of “equal groups.”

By Grade 4, students can also

interpret a multiplication equation

as a statement of comparison

involving the notion "times as

much." This notion has more

affinity to continuous quantities,

e.g., 3 = 4 × 3 _ 4 might describe how

3 cups of flour are 4 times as

much as 3 _ 4 cup of flour. By Grade 5,

when students multiply fractions

in general, products can be larger

or smaller than either factor, and

multiplication can be seen as

an operation that "stretches or

shrinks" by a scale factor. This view

of multiplication as scaling is the

appropriate notion for reasoning

multiplicatively with continuous

quantities.

Multiplicative Comparison Problem Types (continued)

Compare, Product Unknown (Equal Groups Example)

At Southtown High School, the number of students in band is 1 3 _ 4 times the number in orchestra. If 56 students are in orchestra, how many are in band?

Compare, Group Size Unknown (Equal Groups Example)

Dana has 9 CDs. She has 1 _ 5 as many as Sonya. How many CDs does Sonya have?

Compare, Number of Groups Unknown (Equal Groups Example)

To prepare for a test, Esmerelda studied 0.8 times as long as Mallory. Mallory studied for 50 minutes. How long (l) did Esmerelda study?

Multiplication and Scaling Students are also introduced to the idea of scaling in Lesson 6. Scaling is a number sense concept, and involves predicting the effect of resizing one factor of a multiplication.

Gina and Mario each receive a weekly allowance. So far this year, Gina has saved $20 and Mario has saved 0.4 times that amount. Who has saved the greatest amount of money?

Last week Camila worked 40 hours. Sergio worked 4 __ 5 that length of time. Which person worked more

hours last week?

On a math quiz, Juan was asked to find these two products:

3 × 10.6 2.7 × 10.6

a. Without using pencil and paper to actually find the products, how will the product of 3 × 10.6 compare to the product of 2.7 × 10.6?

b. How will the product of 2.7 × 10.6 compare to the product of 3 × 10.6?

Along with rounding, estimation, mental math, and benchmarks, scaling is another strategy students can use to help decide the reasonableness of an exact answer.

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Additive Comparison Problems

Compare situations can be

represented with tape diagrams

showing the compared quantities

(one smaller and one larger) and

the difference. Other diagrams

showing two numbers and the

unknown can also be used. Such

diagrams are a major step forward

because the same diagrams

can represent the adding and

subtracting situations for all of

the kinds of numbers students

encounter in later grades (multi-

digit whole numbers, fractions,

decimals, variables).

Identify Comparison Problem Types Students may conceptualize, represent, and explain a given problem in different ways. Drawings help students show which quantity is larger, and identify a solution equation and computation that can be used to solve the problem. Rather than use the subtraction equation, students often prefer to represent an addition equation as Smaller + Difference = Larger as the only representation for comparison situations.

Writing Equations Students identify and solve real world one-step additive and multiplicative comparison problems in Lesson 7.

Newborn baby Lila is 44.5 centimeters tall. Her older brother Tremaine is 4 times as tall. How tall (t) is Tremaine?

Type of comparison: multiplicative

Equation and answer: 44.5 • 4 = t; t = 178 cm tall

Brandon has 1 _ 4 cup of flour, and would like to make a recipe that requires 1 5 _ 8 cups of flour. How many more cups (c) of flour are needed for the recipe?

Type of comparison: additive

Equation and answer: 1 _ 4 + c = 1 5 _ 8 ; c = 1 3 _ 8 more cups

Additive Comparison Problem Types

Compare, Difference Unknown

Julia jumped rope 1,200 times. Samantha jumped 1,100 times. How many more jumps did Julia do?

Compare, Bigger Unknown

Altogether, the Blue Team jumped 11,485 times. The Red Team did 827 more jumps than the Blue Team. How many jumps did the Red Team do?

Compare, Smaller Unknown

Ted jumped 1,300 times. He did 100 more jumps than Mario. How many jumps did Mario do?

1 5 _ 8 cups

1 _ 4 cup c

UNIT 6 | Overview | 473QQ



Two-Step and Multistep Problems

Students use the new kinds of

numbers, fractions and decimals,

in geometric measurement and

data problems and extend to some

two-step and multi-step problems

involving all four operations. In

order to keep the difficulty level

from becoming extreme, there

should be a tradeoff between the

algebraic or situational complexity

of any given problem and its

computational difficulty taking

into account the kinds of numbers

involved.

Problems With More Than One Step

Lessons

8 9 10

Different Approaches for Solving Problems There are multiple entry points to solving two-step or multistep problems. In other words, there are no algorithmic one-way approaches for solving those problems. Encouraging individual approaches to solve problems may lead to new solutions to new problems not yet imagined. For a given problem, some students may use a forward approach, immediately finding an answer for a first part of a problem and then deciding if that answer is a needed step toward the overall solution. Others may work backwards, using two or more drawings, and/or writing two or more equations.

All of the problems in Lessons 1–7 involved one step. The problems in Lessons 8–10 involve more than one step. Problems involving more than one step typically involve more than one operation, and often the order in which those operations are performed is important.

Two-Step Problems In Lesson 8, students learn that parentheses in an expression or equation represent grouping symbols that indicate which operation should be performed first. Students begin their work by writing equations to represent problems that are solved using two steps, and include parentheses to indicate which operation should be performed first. The two-step problems include whole numbers, decimals, and fractions.

A recipe that makes 6 servings requires 1 1 __ 4 cups of flour.

How much flour (f) would be needed to make the recipe for one-half the number of servings?

An apple orchard in Minnesota has 8 rows of 26 honeycrisp trees and 14 rows of 23 red delicious trees. How many honeycrisp and red delicious trees (t) are in the orchard?

An investor purchased 250 shares of stock. Calculate the investor’s total cost (c) if the price per share was $18.40 and a fee of $65.75 was charged for the transaction.

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Multistep Problems Although the problems in Lesson 9 again represent whole numbers, decimals, and fractions in real world contexts, the problems differ in that they are multistep. In multistep problems, more than two steps must be completed to generate the solutions.

An investor purchased 150 shares of stock at $13.60 per share and sold the shares later for $11.92 per share. Calculate the profit or loss of the transaction.

a. What equation can be used to find the amount of money needed to buy (b) the shares?

b. What equation can be used to find the amount of money received for selling (s) the shares?

c. Does the transaction represent a profit or loss? Why?

d. What equation can be used to calculate the loss (l)? Solve your equation to calculate the loss.

This week an employee is scheduled to work 7 1 __ 2 hours each

day, Monday through Friday, and 2 hours on Saturday morning. If the employee’s goal is to work 40 hours, how many additional hours (h) must be worked?

Other Types of Problems Included in this unit are problems that involve too much information, too little information, and interpreting remainders. Problems involving hidden information are taught in Unit 8.

Too Much Information

A wallpaper border is being pasted on the walls of a rectangular room that measures 12 feet by 14 1 _ 2 feet. The cost of the border is $6.50 per foot. How many feet of border is needed for the room?

Too Little Information

Ms. Bleyleven has 11 windows in her house. The heights in centimeters of 4 windows are shown below.

160.2 cm 163 cm 155.9 cm 158.5 cm

How many windows in her house have a height that is a whole number of centimeters?

UNIT 6 | Overview | 473SS


Interpret Remainders Lesson 10 is a practice lesson that includes all of the skills and strategies students learned in this unit. In addition, students interpret division remainders.

At closing time, 55 adults and 89 students are waiting in line to ride an amusement park roller coaster. The capacity of the coaster is 38 riders. How many trips (t) must the coaster make to give all of the people in line a ride? How many people will be on the last trip of the day?

Six teachers, seventy-eight students, and ten parents are boarding buses for a school field trip. Each bus can carry 32 passengers. If the passengers board each bus until it is full, how many passengers (p) will be on the bus that is not full?

Focus on Mathematical Practices

Lesson

11

The Standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, students use what they know about multiplying whole numbers and decimals to complete computations related to gymnastics and diving scores.

473TT | UNIT 6 | Overview

research—best practices putting research into practice · 2013-01-22 · problem is n + 6 1_ 2 =...

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