Mathematics Task Arcs

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Mathematics Task Arcs

Overview of Mathematics Task Arcs:

A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number of standards within a domain of the Common Core State

Standards for Mathematics. In some cases, a small number of related standards from more than one domain may be addressed.

A unique aspect of the task arc is the identification of essential understandings of mathematics. An essential understanding is the underlying mathematical truth in the lesson. The essential understandings are critical later in the lesson guides, because of the solution paths and the discussion questions

outlined in the share, discuss, and analyze phase of the lesson are driven by the essential understandings.

The Lesson Progression Chart found in each task arc outlines the growing focus of content to be studied and the strategies and representations students may use. The lessons are sequenced in deliberate and intentional ways and are designed to be implemented in their entirety. It is possible for students to develop a deep understanding of concepts because a small number of standards are targeted. Lesson concepts remain the same as the lessons progress; however the context or representations change.

Bias and sensitivity:

Social, ethnic, racial, religious, and gender bias is best determined at the local level where educators have in-depth knowledge of the culture and values of the community in which students live. The TDOE asks local districts to review these curricular units for social, ethnic, racial, religious, and gender bias before use in local schools.

Copyright:

These task arcs have been purchased and licensed indefinitely for the exclusive use of Tennessee educators.

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U N I V E R S I T Y O F P I T T S B U R G H

Multiplication:

Exploration of Multiplicative Comparisons and the Link to Division

A SET OF RELATED LESSONS

mathematics

Grade 4

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Table of Contents 3

Introduction

Overview ... 7

Identified CCSSM and Essential Understandings ... 9

Tasks’ CCSSM Alignment ... 11

Lesson Progression Chart ... 12

Tasks and Lesson Guides TASK 1: Pattern Blocks ... 17

Lesson Guide ... 18

TASK 2: Playing Baseball ... 23

Lesson Guide ... 24

TASK 3: Jack and the Beanstalk ... 28

Lesson Guide ... 29

TASK 4: House of Cards ... 32

Lesson Guide ... 33

TASK 5: Fair Tickets ... 36

Lesson Guide ... 37

TASK 6: Smoky Mountains ... 41

Lesson Guide ... 42

TASK 7: Generations ... 46

Lesson Guide ... 47

TASK 8: Earning Money ... 51

Lesson Guide ... 52

mathematics

Grade 4

Introduction

Multiplication:

Exploration of Multiplicative Comparisons and the Link to Division

A SET OF RELATED LESSONS

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Introduction 7

Overview

This set of related lessons provides a study of multiplicative comparisons, problems in which students have to solve for “how many times greater/less.” They will use visual representations and reasoning about numbers, as well as repeated addition/subtraction to link to the concept of multiplication/division. The standards addressed by this set of related lessons are 4.OA.A.1-2.

Note: Due to the grade level and the placement of this type of multiplication situation type in the trajectory of student learning, prior knowledge of multiplication as “number of groups” by “number in the group”

(3.OA.A.1) is necessary.

Task 1 introduces students to multiplicative comparison situations by asking them to compare areas of pattern blocks, a very hands-on approach with familiar counters.

Tasks 2 and 3 are situations in which the students are asked to solve for how many times greater something is when they are provided both of the factors. Students can solve the task by using their prior knowledge of repeated addition or multiplication and visual representations. Task 4 provides an opportunity for students to solidify their knowledge by solving scaling-up problems, as well as classifying numbers that are and are not multiples of the scale factor.

Tasks 5 and 6 introduce students to the concept of a missing factor in comparative multiplication problems.

Students are provided a diagram of the total (product) and the constant, but not the scale factor. When solving these tasks, students can use multiplicative thinking or division/repeated subtraction in order to find the missing factor. Task 6 first gives students a set of numbers from which they have to determine the multiplicative relationship. Then the task gives students an opportunity to disprove a claim about the multiplicative relationship between a factor and product.

Students are provided another opportunity to determine the multiplicative relationship in Task 7, but students are also asked to deepen understanding by recognizing when a non-multiplicative example is given. Task 8 is a solidifying understanding task. Students are asked to synthesize what they have learned in the set of tasks to scale both up and down.

Through engaging in the lessons in this set of related lessons, students will:

• explore the meaning of the two factors in comparison multiplication problems;

• recognize the inverse relationship between multiplication and division, and determine that division can be used to solve comparison multiplication problems when either the group size or the scaling factor is provided; and

• use a variety of representations to respond to a problem situation, and weigh the advantages and disadvantages of such representations.

By the end of these lessons, students will be able to answer the following overarching questions:

• What is the meaning of each of the two factors in comparison multiplication problems?

• How is the inverse relationship between multiplication and division helpful in solving multiplication problems where only one factor is given?

The questions provided in the guide will make it possible for students to work in ways consistent with the Standards for Mathematical Practice. It is not the Institute for Learning’s expectation that students will name the Standards for Mathematical Practice. Instead, the teacher can mark agreement and disagreement of mathematical reasoning or identify characteristics of a good explanation (MP3). The teacher can note

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8 Introduction

and mark times when students independently provide an equation and then re-contextualize the equation in the context of the situational problem (MP2). The teacher might also ask students to reflect on the benefit of using repeated reasoning, as this may help them understand the value of this mathematical practice in helping them see patterns and relationships (MP8). In study groups, topics such as these should be discussed regularly because the lesson guides have been designed with these ideas in mind. You and your colleagues may consider labeling the questions in the guide with the Standards for Mathematical Practice.

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Introduction 9

Identified CCSSM and Essential Understandings

CCSS for Mathematical Content:

Operations and Algebraic Thinking

Essential Understandings

Represent and solve problems involving addition and subtraction.

3.OA.A.1 (Prior Knowledge)

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

Use the four operations with whole numbers to solve problems.

4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

In the multiplicative expression A x B, A can be defined as a scaling factor. (NCTM, Essential Understanding, 2011).

A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

In a comparative relationship, there is an amount that is constant and other amounts are compared to it.

A repeated relationship exists within an amount that is x times more than or x times less than a given amount and this relationship remains constant in a set that is scaled.

In a scalar relationship, the amount that remains constant can be determined by using division when one of the factors and the total amount are known.

A multiplicative comparison involves a constant increase that is x times more or x times less;

whereas an additive comparison only involves determining how many more than or how many less than another set.

Multiples are what you get after you multiply a number by an integer and a given factor has infinite multiples. When referring to “how many times more/less,” multiples are being used.

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

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10 Introduction

The CCSS for Mathematical Practice

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

2. Reason abstractly and quantitatively.

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

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

*Common Core State Standards, 2010, NGA Center/CCSSO

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Introduction 11

Tasks’ CCSSM Alignment

Task

4.OA.A.1 4.OA.A.2 MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8

Task 1

Pattern Blocks

Developing

Understanding

     

Task 2 Playing Basketball

Developing Understanding

      

Task 3 Jack and the Beanstalk

       

Task 4

House of Cards

Solidifying

Understanding

      

Task 5 Fair Tickets

Developing

Understanding

       

Task 6 Smoky Mountains

       

Task 7 Generations

Developing

Understanding

      

Task 8

Earning Money

Solidifying

Understanding

      

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12 Introduction

Lesson Progression Chart

Overarching Questions

• What is the meaning of each of the two factors in comparison multiplication problems?

• How is the inverse relationship between multiplication and division helpful in solving multiplication problems where only one factor is given?

TASK 1 Pattern Blocks

Developing Understanding

TASK 2 Playing Basketball

TASK 3 Jack and the

Beanstalk Developing Understanding

TASK 4 House of Cards

Solidifying Understanding

Cont ent

Explores the idea of something being

“x times greater/

as large.” Discuss multiplicative versus additive situations.

Deepens understanding of

“times how many”

with an unknown product, builds clarity of meaning of the factors: one is the scale factor and one is the constant.

Continue to build understanding of

“how many times greater” with unknown product, the meaning of factors, and the relationship with multiples.

Solidifies understanding of solving multiplicative comparisons when the products are unknown by applying their knowledge of the meaning of factors in multiplicative comparisons to numerous situations.

Str at eg y

Students use manipulatives to represent how many times greater the area of one shape is than another.

Students use a diagram to represent the constant and create a scaled diagram.

Students use multiplicative thinking and what they know about multiples.

Students use multiplicative thinking, known multiples, and the meaning of each factor/product.

Repr esentations

Pattern blocks and equations.

Presented in a context, makes use of diagrams, a table, and equations.

Presented in a context, a comparison of diagrams, and equations.

Pictures and equations.

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Introduction 13

TASK 5 Fair Tickets

TASK 6 Smoky Mountains

TASK 7 Generations

TASK 8 Earning Money

Solidifying Understanding

Cont ent

Introduces the concept of a missing factor in comparison problems; explores the idea of scaling down to find the constant.

Links division to multiplication to solve a problem of

“how many times fewer” by providing the total and one factor only.

Deepens understanding of comparison multiplication when the multiplier (scale factor) is unknown by providing examples and non- examples.

Solidifies the structure of comparison multiplication through both scaling up and scaling down.

Str at eg y

Students use division, visual partitioning, or multiplicative thinking to solve for a missing factor.

Students divide via partitioning a number line to find the missing scale factor.

Students divide to find the missing scale factor in a multiplication comparison.

Students use both multiplication and division when working from a missing product and missing factor, respectively.

Repr esentations

Provided with a diagram that must be manipulated, and equations.

Number line and equations.

Diagrams and equations.

Table and equations.

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14 Introduction

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mathematics

Tasks and Lesson Guides

Multiplication:

Exploration of Multiplicative Comparisons and the Link to Division

A SET OF RELATED LESSONS

Grade 4

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Tasks and Lesson Guides 17

TASK

1

Name__________________________________________________________________________

Pattern Blocks

Jackson compared the areas of different pattern blocks.

1. We have a yellow hexagon. How many times greater is the area of the yellow hexagon block than the area of the green triangle? How do you know?

2. Jackson has a yellow hexagon. How many times greater is the area of the yellow hexagon block than the red trapezoid? Explain the comparison.

3. Jackson has a blue rhombus. How many times greater is the yellow hexagon than the blue rhombus?

Explain the comparison.

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18 Tasks and Lesson Guides

GUIDE

1 Pattern Blocks

Rationale for Lesson: Introduce students to the concept of comparing areas to explore comparison multiplication.

Task: Pattern Blocks

Jackson compared the areas of different pattern blocks.

1. We have a yellow hexagon. How many times greater is the area of the yellow hexagon block than the area of the green triangle? How do you know?

2. Jackson has a yellow hexagon. How many times greater is the area of the yellow hexagon block than the red trapezoid? Explain the comparison.

3. Jackson has a blue rhombus. How many times greater is the yellow hexagon than the blue rhombus? Explain the comparison.

Common Core Content Standards

Standards for Mathematical Practice

MP1 Make sense of problems and persevere in solving them.

MP6 Attend to precision.

MP7 Look for and make use of structure.

MP8 Look for and express regularity in repeated reasoning.

Essential

Understandings • A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

• A repeated relationship exists within an amount that is x times more than or x times less than a given amount and this relationship remains constant in a set that is scaled.

• A multiplicative comparison involves a constant increase that is x times more or x times less; whereas an additive comparison only involves determining how many more than or how many less than another set.

Materials

Needed • Reproducible for students.

• Pattern blocks. (A reproducible sheet of pattern blocks have been provided at the end of the lesson guide if the blocks are not available at the school. One copy is needed per every two students.)

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LESSON GUIDE

SET-UP PHASE

1

Read the task to yourself. What is meant by “compare the areas?” Now turn and talk with your partner about what the task means and what you are to do.

EXPLORE PHASE Possible Student

Pathways

Assessing Questions Advancing Questions Group can’t get started. What pattern blocks are you

comparing?

How can you use the green triangles to find out how much larger the yellow hexagon is?

Group writes about comparisons, but does not express the comparison with an equation.

How many times greater is the area of the hexagon than the area of the green triangle?

If you used an “H” for the hexagon and a “T” to stand for the triangle, how can you write an equation to show that a hexagon is 6 times greater than a triangle?

Group records the comparison of the triangle to other shapes as a multiplicative relationship.

T x 6 = H T x 3 = TR T x 2 = R

Why is the T the same in all your equations but the other factor changes?

What do the factors, the 6, the 3, and the 2 represent in this multiplication model?

Write a sentence explaining what they mean.

Group doesn’t use the scale factor between the shapes to solve part 3.

When you compared the shapes, you wrote T x 6 = H.

How many times greater is the area of the hexagon than the triangle?

Compare the area of the trapezoid to the area of the hexagon. How many times greater is it?

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GUIDE

1

SHARE, DISCUSS, AND ANALYZE PHASE

EU: A repeated relationship exists within an amount that is x times more than or x times less than a given amount and this relationship remains constant in a set that is scaled.

• Tell us what you did when comparing the area of the green triangle and the hexagon pattern block.

• It took 6 triangles to make a hexagon. (Revoicing)

• What does the 6 represent? (The triangles that fill up the hexagon.)

• What did we have to keep repeating in order to fill up the area of the hexagon?

(The triangles.)

• If I use the trapezoid, how many do I need to fill up the same area as the hexagon? (Two.)

• I got one trapezoid, but I had to get one more. I had to repeat it and get a second one.

EU: A situation that can be represented by multiplication has an element that

represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

• In each of these situations, what remained constant or the same? (The area of the hexagon.)

• This is one of the factors. (Marking) 1 x 6 = 6

1 x 2 = 2 1 x 3 = 3

• The other factor told us how many times we had to increase the figure so that it would have the same area as the hexagon. How many times did we increase the triangle, or how many did we need to be the same area as the hexagon? (6)

• We scaled up 6 triangles to be the same area as the hexagon. (Marking)

• How many times did we scale up or how many more of the trapezoids did we need? (2)

• When we get two trapezoids, what do we know? (We know that we have the same area as the triangles.)

• Tell me about the rhombi.

• So when we work with comparison multiplication, one factor remains constant and the other factor tells us how to scale up the other factor.

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EU: A multiplicative comparison involves a constant increase that is x times more or x

1

times less; whereas an additive comparison involves asking how many more than or how many less than another set.

• Look at the picture of the triangle used to scale up to the hexagon. It took 6. Look at the picture of the trapezoids needed to scale up to the hexagon. The hexagon is two times larger than the trapezoid. The hexagon is three times larger than the rhombi.

• Listen to this situation. How is it different from the hexagon situations that we just worked with?

• Mary has 8 hexagons and Tom has 6 triangles. How many more blocks does Mary have than Tom?

• How does this situation differ from the kinds of comparison problems that we solved when doing the hexagon = 6 triangles, the hexagon = 2 trapezoids, and the hexagon = 3 rhombi?

(We would do 8 – 6 to figure out that Mary has 2 more blocks than Tom. The other problems said how many TIMES greater is the area.)

Application (Display the decomposition of the green triangle.) If the green equilateral triangle is cut down the middle to form two right triangles, how many times greater will the hexagon be than the cut rhombus?

Summary We discovered that multiplication was important when making comparison between the different pattern blocks. We called the factor that represented the size of the group the constant factor.

Quick Write Write an equation to compare the area of the rhombus to the area of the hexagon. Label the parts of your equation to show what they represent.

Support for students who are English Learners (EL):

1. Use of counters in the set-up of the lesson.

2. Private think time so that students have time to organize their thoughts and struggle with the material individually. Cooperative learning so that students have an opportunity to work through ideas in a small group before sharing out to the class.

3. Students are pressed to explain the meaning of area using the patterns blocks.

4. Students are pressed to physically point to relationships on the overhead projector/board.

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GUIDE

1

PATTERN BLOCKS

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TASK

2

Name__________________________________________________________________________

Playing Basketball

The Chamberlains are having a family reunion and every year they play basketball.

• Gabe made 4 baskets in a row in the basketball shooting contest.

• Uncle David made 2 times as many baskets as Gabe.

• Aunt Doris made 4 times as many baskets as Gabe.

• Grandfather Morris made 3 times as many baskets as Gabe.

1. How many baskets did each person make?

2. How is the total number of baskets each person made related?

3. How would the problem be the same if Gabe made 5 baskets? How would it be different?

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GUIDE

2 Playing Basketball

Rationale for Lesson: Introduce students to the concept of multiplicative comparison when the product is unknown.

Task: Playing Basketball

The Chamberlains are having a family reunion and every year they play basketball.

• Gabe made 4 baskets in a row in the basketball shooting contest.

• Uncle David made 2 times as many baskets as Gabe.

• Aunt Doris made 4 times as many baskets as Gabe.

• Grandfather Morris made 3 times as many baskets as Gabe.

1. How many baskets did each person make?

2. How is the total number of baskets each person made related?

3. How would the problem be the same if Gabe made 5 baskets? How would it be different?

MP4 Model with mathematics.

MP5 Use appropriate tools strategically.

Essential

Understandings • In the multiplicative expression A x B, A can be defined as a scaling factor. (NCTM, Essential Understanding, 2011).

• A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

• In a comparative relationship there is an amount that is constant and other amounts are compared to it.

Materials

• More than 24 counters if students choose to model with counters.

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SET-UP PHASE

Read the task to yourself. If you drew a picture of this problem, who would be in the picture and what would they be doing? Now turn and talk with your partner about what the task means.

Pathways

Assessing Questions Advancing Questions Group can’t get started. How can you show how

many baskets Gabe made?

Did Uncle David make more baskets or fewer baskets than Gabe?

How can you show the baskets Uncle David made?

Group uses counters. Tell me what your picture shows.

How many baskets would someone make who made 5 times as many basket as Gabe? How do you know?

Group uses equations.

4 x 1 = 4 4 x 2 = 8 4 x 3 = 12 4 x 4 = 16

What does the “4” represent in the problem?

Why do you keep multiplying by 4?

Group does not see a relationship between the products.

How can you arrange the number of baskets each relative made?

What is the difference between the number of baskets Gabe and Uncle David made?

2

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GUIDE

2

EU: A repeated relationship exists within an amount that is x times more than or x times less than a given amount and this relationship remains constant in a set that is scaled.

• Tell me about the baskets that Gabe made in the context. (4 baskets.)

• Tell us about the diagram that you made for Uncle David’s shots. (4 baskets for Gabe and 4 and 4 baskets for Uncle David.)

• What made you write 4 + 4 or 2 groups of 4? (Revoicing) (Uncle David made two times the number of shots that Gabe made.)

• Look at Uncle David’s number of shots. Where do you see in the diagram that Uncle David made two times the number of Gabe’s shots?

• How can we write this as a multiplication equation? (4 x 2)

• Why can we write 4 x 2? (We can write this because he did 4 and then 4 more.)

• We can write this because 4 is repeated twice. We can see two 4s repeated in Uncle David’s shots. (Revoicing)

EU: In a comparative relationship there is an amount that is constant and other amounts are compared to it.

• Someone wrote a set of equations showing each person’s shots. Let’s look at these. What do you notice about the equations? (They all have a 4.)

4 x 2 = 8 4 x 3 = 12 4 x 4 = 16

• Why do they all have a 4? Why does the 4 remain constant? Why is it always there? (Every person’s shots are being compared with Gabe.) Say more. (And Gabe has 4 shots.)

• I like how you completed your reasoning by saying, “Every person’s shots are being compared with Gabe’s shots and Gabe has 4 shots.” (Revoicing)

• We call 4 the constant factor because it is like the anchor or it remains there and others’

shots are compared and scaled up based on this amount. (Recapping)

EU: In the multiplicative expression A x B, A can be defined as a scaling factor. (NCTM, Essential Understanding, 2011).

• If the factor 4 is the constant, then what is the other factor in the problems? (It is the other person’s shots. STUDENT ERROR.)

4 x 2 = 8 4 x 3 = 12 4 x 4 = 16

• Who agrees? Who disagrees? How many shots did Uncle David make? How many shots did Aunt Doris make? (Uncle David made 8 and Aunt Doris made 12.)

• Let me label these in our equation. Labels really help us keep this straight in our heads.

• So what does the other factor in the equation mean? (Frequently, students cannot answer this question.)

• It is “how many times more baskets.” We can call it the scalar factor; it tells how many times to scale up from Gabe’s amount.

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• Let’s pull all of the ideas together. Let’s refer to these equations: 4 x 2 = 8 and 4 x 3 = 12.

One factor is the constant, one factor says how many times more and is the scalar, and the product is the number of shots that the person made more than Gabe. Does this make sense?

• Tell me about Grandfather’s shots, then. He made 3 times more. Who can give me an equation and write about all of the factors? Stop and jot – private work time.

Application 1. Donel spit a watermelon seed 5 feet. His cousin, Bradley, spit a seed 3 times farther. How many feet did Bradley spit his seed?

2. Suppose Emily spit a seed 4 times father than Donel. How far did she spit her seed?

3. Suppose John spit a seed 6 times farther than Donel. How far did he spit his seed?

Summary See recapping above.

Quick Write Write a multiplication story problem that requires a comparison. Provide a solution to your own problem.

1. Press students to identify the patterns shown in the visual representation as they describe their thinking to others.

2. Encourage students to continually reference the context when thinking through the problem and explaining their thinking.

2

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Name__________________________________________________________________________

Jack and the Beanstalk

Shaunte is reading Jack and the Beanstalk to her little brother and starts thinking about the math involved!

“Jack traded the family cow for a magic bean. He planted the bean and went to bed. The next day, the beanstalk was 72 feet tall.”

1. Jack is 4 feet tall. How many times taller than Jack is the beanstalk if it is 72 feet tall? Make a diagram and label the diagram to show the relationship of Jack’s height to the height of the beanstalk.

2. Explain why multiplication or division can be used to solve this problem.

3. If the giant in Jack and the Beanstalk is 9 feet tall, how many times taller is the beanstalk than the giant?

4. There is a cat that is 12 inches tall. If the beanstalk is 360 inches tall, then how many times taller is it than the cat? Write a multiplication and division problem that could be used to solve this problem. Label the factor that is the constant and the one that is the scalar factor.

TASK

3

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Jack and the Beanstalk 3

Rationale for Lesson: Introduce students to the concept of comparison multiplication when the multiplier (scaling factor) is unknown.

Task: Jack and the Beanstalk

Shaunte is reading Jack and the Beanstalk to her little brother and starts thinking about the math involved!

“Jack traded the family cow for a magic bean. He planted the bean and went to bed. The next day, the beanstalk was 72 feet tall.”

1. Jack is 4 feet tall. How many times taller than Jack is the beanstalk if it is 72 feet tall? Make a diagram and label the diagram to show the relationship of Jack’s height to the height of the beanstalk.

2. Explain why multiplication or division can be used to solve this problem.

3. If the giant in Jack and the Beanstalk is 9 feet tall, how many times taller is the beanstalk than the giant?

4. There is a cat that is 12 inches tall. If the beanstalk is 360 inches tall, then how many times taller is it than the cat? Write a multiplication and division problem that could be used to solve this problem. Label the factor that is the constant and the one that is the scalar factor.

MP5 Use appropriate tools strategically.

Essential

Understandings • A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

• In a scalar relationship the amount that remains constant can be determined by using division when one of the factors and the total amount are known.

Materials

• A copy of Jack and the Beanstalk.

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SET-UP PHASE

Raise your hand if you are familiar with the fairy tale Jack and the Beanstalk. Let’s make sure everyone is familiar with the story. Who can explain the story without all of the details? (Select a student or students to share the basic plot to ensure everyone has the necessary background knowledge to comprehend the context of the task.) Read the task to yourself. Now turn and talk with your partner about what the task means.

Pathways

Assessing Questions Advancing Questions Group does not use a

diagram to explain the comparisons.

Tell me about Jack’s height.

Tell me about the height of the beanstalk.

Make a sketch of Jack’s height and the height of the beanstalk.

Group makes a diagram but does not express the comparison with an equation.

Tell me about your diagram and how it tells about Jack’s height and the height of the beanstalk.

Can you write an equation that describes your diagram?

Make sure I know the meaning of the equation so write a sentence that explains what it means.

Group shows a

multiplication equation but does not express the comparison with a division equation.

How did you find the scale factor using multiplication?

If I told you I could use division to figure out one of the relationships, then can you think about my thinking?

How can I use division?

SHARE, DISCUSS, AND ANALYZE PHASE MP5: Use appropriate tools strategically.

• Someone used a bar model and someone used a number line model. Let’s listen to each group’s explanation of how their model represents the Jack and the Beanstalk context.

• What makes each of these models good for this context?

• Someone in another class used this model, 4 circles, as Jack’s height. Why would representing Jack as a set of 4 circles be a model that is less applicable and maybe even harder to understand? (Display this model.) (Challenging)

EU: A repeated relationship exists within an amount that is x times more than or x times less than a given amount and this relationship remains constant in a set that is scaled.

• Someone wrote 4 x __ = 72.

• What does the 4 represent in the problem?

• It is the constant. (Marking)

• How does this help you think about the 72 feet?

3

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EU: In a scalar relationship the amount that remains constant can be determined by

3

using division when one of the factors and the total amount are known.

• What are we trying to figure out in this situation that is different from other situations that we have examined previously? (We don’t know how many times taller the beanstalk is than Jack. We know how tall the beanstalk is. This is like knowing the total number of Uncle David’s shots.)

• Someone used division. Someone wrote 72 ÷ 4 = 18. Tell me about the beanstalk and how it relates to the division problem.

• What does the quotient represent? (The scalar factor.)

• Another name for the scalar is multiplier.

• Why can either word be used? Do you hear a relationship between the two words?

• Earlier, someone used multiplication with a missing factor and someone used division. Why can we use either operation?

• The problem expected you to write a multiplication and a division equation to show how the problem can be solved. Why did the problem ask you for both? Which operation do you think is more efficient and why? (Challenging) Which operation would be more efficient if the beanstalk was 24 feet tall? Why?

• In the past two lessons we talked about a constant and a factor that is a scalar. Does this problem have these, too? (The scaling factor is 18, 4 is the constant, and the 72 is how tall the beanstalk is compared to Jack’s height.)

Application • If the giant in Jack and the Beanstalk is 9 feet tall, how many times taller is the beanstalk than the giant?

• What if I told you that there is a cat and the cat is 12 inches tall? If the beanstalk is 360 inches tall, then how many feet taller is it than the cat?

• Write a multiplication and division problem that could be used to solve this problem. Label the factor that is the constant and the one that is the scalar factor.

Summary We discovered that because of the inverse relationship between multiplication and division, both can be used to solve comparison problems. Division can be used when the multiplier, also known as the scaling factor, is unknown.

Quick Write If Jack was 3 feet tall, how many times taller would the beanstalk be than Jack?

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4

Name__________________________________________________________________________

House of Cards

Delia, Tran, and Helene are making houses using playing cards. They use 6 cards to build each level. Delia built a house with only one level. Tran built a house 2 times taller than Delia’s house. Helene built a house 2 times taller than Tran’s house.

1. Use a diagram and equations to show how many total cards each of the friends used to make their house.

2. Which of the following numbers could be used to make a house of cards with 6-card levels?

a. 36 b. 14 c. 62 d. 48 e. 52 f. 66

3. If Delia, Tran, and Helene made the same size houses but used 8 cards for each level, how many total cards did each use?

4. Which of the following could be used to make a house of cards with 8-card levels?

a. 36 b. 16 c. 68 d. 48 e. 52 f. 78

5. If Helene has 24 playing cards, how many levels of each kind of house can she build? How do you know?

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House of Cards 4

Rationale for Lesson: Solidify students’ understanding of multiplicative comparisons when the product or one of the factors is unknown. Students will also consider how knowing multiples can help you determine an unknown factor.

Task: House of Cards

Delia, Tran, and Helene are making houses using playing cards. They use 6 cards to build each level.

Delia built a house with only one level. Tran built a house 2 times taller than Delia’s house. Helene built a house 2 times taller than Tran’s house.

1. Use a diagram and equations to show how many total cards each of the friends used to make their house.

2. Which of the following numbers could be used to make a house of cards with 6-card levels?

a. 36 d. 48 b. 14 e. 52 c. 62 f. 66

3. If Delia, Tran, and Helene made the same size houses but used 8 cards for each level, how many total cards did each use?

See student paper for complete task.

MP2 Reason abstractly and quantitatively.

Essential

Understandings • In the multiplicative expression A x B, A can be defined as a scaling factor. (NCTM, Essential Understanding, 2011).

• A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies. (NCTM, Essential Understanding, 2011)

• Multiples are what you get after you multiply a number by an integer and a given factor has infinite multiples. When referring to “how many times more/less,” multiples are being used.

Materials

• Playing cards to model the context for students (optional).

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GUIDE

4

SET-UP PHASE

Read the task to yourself. Think about this while you read: If you took a photograph of the people in the problem, who would be in the picture and what would they be doing? Has anyone made a house of playing cards before? Can you explain how you did it? Why is making a house of cards so tricky?

Now turn and talk with your partner about what the task is asking you to do.

Pathways

Assessing Questions Advancing Questions Group can’t get started. How many cards are used to

build the first level?

How many cards are used to build the first and second floors?

Group uses diagrams. What can you tell me about your diagram?

How many cards would someone who made a house 3 times as tall as Delia’s house use? How do you know?

Group uses equations.

6 x 1 = 6 6 x 2 = 12 6 x 4 = 24

What does the “6” represent in the problem?

Why do you keep multiplying by 6?

Group does not see a relationship between the products of several multiplicative comparisons.

How can the number of cards used to build each level be arranged?

What is the difference between the number of cards used to build the first and second level of the house of cards?

6 6

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4

EU: In the multiplicative expression A x B, A can be defined as a scaling factor. (NCTM, Essential Understanding, 2011) AND

• Let’s look a little more at the equations. What pattern do you see in the equations?

• Are there any patterns that you notice?

• What is the difference between the products 6, 12, 18, and 24?

• Where else do we see a 6 in the problem? How can we describe the relationship between the factor 6 and all of the products?

• The products, or multiples, of 6 result when the 6 is multiplied by another factor.

(Revoicing)

• Let me make sure I understand everything that has been said.

• The numbers that are multiplied to find a product are called the factors. Where are the factors in these equations?

• What is the constant factor? What is the difference between these products?

• I wonder why the constant factor and the difference between the products are both 6?

• Does anyone have an idea why this occurred? We really need to understand this relationship. (Marking)

• Where are the products? What is the difference between one product to the next?

• The difference between the products is 6 because the product changes according to how many times the constant factor is repeated by the scale factor.

EU: Multiples are what you get after you multiply a number by an integer and a given factor has infinite multiples. When referring to “how many times more/less,” multiples are being used.

• The products are the multiples of the constant factor. Look at the list of numbers. Which are multiples of the constant factors of 6?

• What makes these numbers multiples of 6? What does this have to do with comparison multiplication?

Application If 3 cubes are used to make a tower, how many cubes are needed to make a tower 4 times as tall?

Summary Let’s look at what we did yesterday and connect it to today’s learning. How was finding the number of baskets each person shot made relative to finding the number of cards used for each house of cards?

Quick Write No quick write for students.

1. Use counters in the set-up of the lesson.

2. Private think time allows the students to make sense of the problem in different ways before collaborating.

3. Collaboration allows students the opportunity to work through ideas and practice discussions first before moving to a whole group discussion.

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5

Name__________________________________________________________________________

Fair Tickets

Lily and Saul went to the fair. They each bought 140 tickets. Lily said 140 tickets are 5 times more tickets than she bought last year. Saul said 140 tickets are 4 times more tickets than he bought last year.

1. Use the diagram below to compare the tickets Lily bought last year, the tickets Saul bought last year, and the tickets they bought this year.

2. Saul bought more tickets than Lily last year. Explain why this is true. Be sure to use the diagram.

3. How many fair tickets did Lily buy last year?

4. How many fair tickets did Saul buy last year?

Saul’s tickets last year:

Lily’s tickets last year:

140 tickets

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Fair Tickets

Rationale for Lesson: Introduce students to comparison multiplication when the set size (constant factor) is unknown; students must find a missing factor.

Task: Fair Tickets

Lily and Saul went to the fair. They each bought 140 tickets. Lily said 140 tickets are 5 times more tickets than she bought last year. Saul said 140 tickets are 4 times more tickets than he bought last year.

1. Use the diagram below to compare the tickets Lily bought last year, the tickets Saul bought last year, and the tickets they bought this year.

2. Saul bought more tickets than Lily last year. Explain why this is true. Be sure to use the diagram.

3. How many fair tickets did Lily buy last year?

4. How many fair tickets did Saul buy last year?

5

Saul’s tickets last year:

Lily’s tickets last year:

140 tickets

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GUIDE

5

^EssentialUnderstandings

Materials Needed

• Reproducible for students.

• Tickets or picture of tickets to clarify the context for students (optional).

SET-UP PHASE

Read the task to yourself. Think about this while you read: If you took a photograph of the people in the problem, who would be in the picture and what would they be doing? (Select students to share the characters’ names and the setting to ensure everyone has a basic understanding of the context of the problem.) Now turn and talk with your partner about what the task means.

Pathways

Assessing Questions Advancing Questions Group can’t get started. How much does the diagram

(rectangle) represent?

What do we know about how many tickets Lily bought last year? How can the diagram help us find out how many she bought last year?

Group does not express the comparison with an equation.

What operation does the phrase “140 is 4 times more” suggest?

How would you write 140 equals 4 times something?

Group does not express the comparison with a division equation.

What strategy did you use to find the other factors in these multiplication equations?

How can you use the same values and write a division equation to represent the problem?

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5

EU: A multiplicative comparison involves a constant increase that is x times more or x times less; whereas an additive comparison only involves determining how many more than or how many less than another set.

• Tell us about the diagram. What did you do to the diagram to make it represent the context?

(The diagram shows 140 but we had to find 4 times and 5 times less than 140, so we spit up the 140 into sections.)

• Who understands what this team did? Can you say it back?

• What does 140 represent on each of the diagrams? (The total number of tickets; the product.) If 140 is the product, what do the 4 and 5 represent? (The 4 and 5 represent the number of times bigger or the number of groups in our diagram.)

• We have the product and the number of groups; the number of groups in comparison multiplication is called the multiplier or the scale factor. (Revoicing)

• What are we missing? (The size of the group; how many tickets each person had last year.) What is another term for the size of the group?

EU: In a scalar relationship the amount that remains constant can be determined by using division when one of the factors and the total amount are known.

• The problem uses the phrases “4 times more” and “5 times more,” but your representation of 140 has been divided into sections. Why did you divide the 140 into sections?

• Wait a minute. When I hear “times as many,” I think multiplication. Why didn’t you multiply?

(Challenging) (We knew we needed to multiply, but because we are missing a factor, we couldn’t multiply.)

• Can we write a multiplication equation with a missing factor? What would that look like for this problem?

• What is the relationship between multiplication and division? (Division is like undoing multiplication, division is the opposite of multiplication.) So, division is the undoing, or inverse, of multiplication. (Revoicing)

• Let me make sure we are all together in our understanding. Because division is the inverse of multiplication, when one of the factors is missing in a problem, either the constant or the scale factor, we can use division to solve the problem. (Recapping)

• How did you know from your diagrams that Saul bought more tickets than Lily last year?

(When you divide the 140 into 4 sections for Saul and into 5 sections for Lily, you can see Saul’s sections are larger.) Why is that? (Fourths are larger than fifths when the whole is the same.) When you did the actual division, did the answers support your thinking?

Application At the fair this year, Lily rode 12 rides. That is 4 times as many as she rode last year. How many did she ride last year?

Summary Because division is the undoing of multiplication, when either the constant or the scale factor is missing in a multiplication comparison problem, we can use division to find the unknown factor.

Quick Write Billy claims 140 tickets is 7 times more than the tickets he bought last year.

How many fair tickets did Billy buy last year? The answer is 20. Why is division necessary for solving this problem?

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GUIDE

5

1. Students are pressed to identify the patterns shown in each representation as they describe them.

2. The teacher marks verbally and in writing key ideas from the lesson as they are revealed.

3. Students are asked to write out and reference the relationships within the context of the problem.

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TASK

6

Name__________________________________________________________________________

Smoky Mountains

Four families drove to the Great Smoky Mountains National Park to vacation.

• The Nyguen family drove 504 miles to the park.

• The Nyguen family drove 3 times more miles than the Peterson family.

• The Dorsey family drove 2 times fewer miles than the Nyguen family.

• The Dorsey family drove 4 times more miles than the Lang family.

1. Draw a number line. Place the names of the families on the number line from the least to the greatest number of miles driven. Explain how you know where to place each family’s name.

2. Write an equation and explain how far each family drove to vacation.

Extension

Another family, the Morriseys, claims they drove more miles than any of the other families. How many times more miles than the Langs did the Morriseys have to drive to be correct?

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GUIDE

6 Lesson Guide 6: Smoky Mountains

Rationale for Lesson: Continue to develop students’ understanding of comparison multiplication when the set size (constant factor) is unknown; concept of “times fewer” is introduced.

Task: Smoky Mountains

Four families drove to the Great Smoky Mountains National Park to vacation.

• The Nyguen family drove 504 miles to the park.

• The Nyguen family drove 3 times more miles than the Peterson family.

• The Dorsey family drove 2 times fewer miles than the Nyguen family.

• The Dorsey family drove 4 times more miles than the Lang family.

1. Draw a number line. Place the names of the families on the number line from the least to the greatest number of miles driven. Explain how you know where to place each family’s name.

2. Write an equation and explain how far each family drove to vacation.

Extension

Another family, the Morriseys, claims they drove more miles than any of the other families. How many times more miles than the Langs did the Morriseys have to drive to be correct?

Essential Understandings

Materials Needed

• Reproducible for students.

• Map and pictures of the Smoky Mountains (optional).