Task Arc - TN Core

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 :

These math task arcs are peer-reviewed and have been vetted for content by experts. However, it is the responsibility of local school districts to review these units for social, ethnic, racial, 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.

mathematics

Grade

3

Understanding

Fractions as Numbers

A SET OF RELATED LESSONS

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

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© 2013 University of Pittsburgh – Third Grade Set of Related Lessons: Understanding Fractions as Numbers

3

Table of Contents 5

Table of Contents

Introduction

Overview ................................................................................................................................................ 9

Identified CCSSM and Essential Understandings ............................................................ 11

Tasks’ CCSSM Alignment ............................................................................................................. 14

Lesson Progression Chart ............................................................................................................ 16

Tasks and Lesson Guides

TASK 1: Shares of Pizza ................................................................................................................. 21

Lesson Guide ............................................................................................................................................. 23

TASK 2: Shares of Clay ................................................................................................................... 29

Lesson Guide ............................................................................................................................................. 31

TASK 3: Shares of Fudge ............................................................................................................... 36

Lesson Guide ............................................................................................................................................. 37

TASK 4: Fraction of a Number Line ......................................................................................... 43

Lesson Guide ............................................................................................................................................. 44

TASK 5: Rolls of Bubble Gum ..................................................................................................... 50

Lesson Guide ............................................................................................................................................. 52

TASK 6: Shaded and Unshaded ................................................................................................. 57

Lesson Guide ............................................................................................................................................. 59

TASK 7: A Fraction of a Whole .................................................................................................... 63

Lesson Guide ............................................................................................................................................. 64

TASK 8: Pulling Ideas Together .................................................................................................. 68

Lesson Guide ............................................................................................................................................. 71


mathematics

Grade

3

Introduction

Understanding Fractions as Numbers


Introduction 9

Overview

This set of related tasks provides a study of fractions as numbers. Students will learn the meaning of fractions and compare amounts determining those that are equivalent or greater or less than each other.

They will work with a fraction of a whole, using both area models and the number line models. Students will develop an understanding of the 3.NF.1, 3.NF.2a, 3.NF.2b, 3.NF3a–d Common Core State Content Standards and Standards for Mathematical Practice. Note: These lessons do not require students to work with a fraction of a set; additional lessons that require students to work with a fraction of a set will be needed.

Tasks 1–3 are presented within a situational context as a means of scaffolding student learning. Since students are used to solving different types of situational problems, this builds on their prior learning. In these lessons, students work with a fraction of a whole model and work with familiar related fractions of halves and fourths, as well as thirds and sixths. A great deal of emphasis is placed on early in the set of lessons.

In several lessons, addition and multiplication of fractions is modeled by the teacher, even though third grade students are not expected to use these operations in their independent problem-solving work. Modeling addition and multiplication of fractions for students makes sense in these lessons because students are familiar with these operations and understand that 1/ b is comprised of parts that are 1/ b in size and iterations of 1/ b add up to a / b , or a / b can be decomposed into a x 1/ b . Although writing addition and multiplication equations with fractions is not a student expectation in third grade, some students may begin to write equations as the result of the teacher modeling the process during the discussions of the tasks.

In Tasks 4–5, students solve tasks that do not have a context. Students are challenged to represent points on a number line model. Both tasks show more than one whole on the number line so students come to understand that a fraction, such as , represents the whole and that is more than a whole. Students also begin to think about multiple names for points and realize that equivalent fractions can be written for a point.

Tasks 6–7 do not have a contextual situation, but instead a visual model is provided. Students do not have to figure out how to construct a model, thus freeing them to engage in making connections and sharing their mathematical reasoning related to the concept. In these lessons, students work with an area model and grapple with equivalent fractions that can be written to describe the same area of the figure.

The prerequisite knowledge necessary to enter these lessons is an understanding of addition, making equal shares of a whole, and sharing fairly.

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

• solve situational story problems involving a variety of contexts;

• work with a fraction of a whole and a fraction of a number line labeled 0–1;

• determine if there should be iterations of 1/ b equal to a / b ; and

• recognize that iterations of 1/ b can be added and the size of the piece remains constant but the number of pieces changes.

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

• What are different situations in which we talk about a fraction of the whole or a fraction of a number line?

• How do models of fractions differ from each other? (A fraction of a whole and a fraction of a number line?)

• How do we compare fractions to determine which is more?


10 Introduction

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 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.


Introduction 11

Identified CCSSM and Essential Understandings

1

CCSS for Mathematical Content:

Number and Operations – Fractions

Develop understanding of fractions as numbers.

Essential Understandings

3.NF.A.1

3.NF.A.2

3.NF.A.2a

3.NF.A.2b

Understand a fraction 1/ b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a / b as the quantity formed by a parts of size 1/ b .

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Represent a fraction 1/ b on a number line diagram by defining the interval from 0 to

1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/ b and that the endpoint of the part based at 0 locates the number

1/ b on the number line.

Represent a fraction a / b on a number line diagram by marking off a lengths 1/ b from

0. Recognize that the resulting interval has size a / b and that its endpoint locates the number a / b on the number line.

A fraction describes the division of a whole or unit (area/region, set, linear/measurement) into equal parts. A fraction is relative to the size of the whole or unit.

Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of ( = + + or 3 x , =

5 x ).

The denominator tells how many equal parts into which the whole or unit is divided. The numerator tells how many equal parts of that subdivided whole are indicated.

When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator.

Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of ( = + + ) or revoicing of the iterations as 3 x (exposure level).

The denominator tells how many equal parts into which the whole or unit is divided. The numerator tells how many equal parts of that subdivided whole are indicated.

1 Council of Chief State School Officers (CCSSO) & National Governors Association Center for Best Practices (NGA Center). (2010).

Mathematics. Common core state standards for mathematics . Retrieved from http://www.corestandards.org/Math


12 Introduction

CCSS for Mathematical Content:

Number and Operations – Fractions

3.NF.A.3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

3.NF.A.3a

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Essential Understandings

Rational numbers (fractions) have an infinite number of equivalent forms, and the forms are equivalent if the same portion of the set or area of the figure is represented or they represent the same point on the number line.

If the numerator is half the quantity in the denominator, then the fraction is equal to a half.

3.NF.A.3b

3.NF.A.3c

Recognize and generate simple equivalent fractions, e.g., 1/2

= 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples:

Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate

4/4 and 1 at the same point of a number line diagram.

When creating equivalent fractions all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way.

What is created is an equivalent fraction.

A fraction in which the total number of pieces in the whole is the same as the total number of the pieces is called one whole because all of the pieces of the whole are accounted for (e.g., , , ).

A fraction with a denominator of one says that the size of the piece is one whole; therefore, a number larger than the denominator lets us know the number of whole pieces.

3.NF.A.3d

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Fractions with the same size pieces, or common denominators, within the same size whole can be compared with each other because the size of the pieces is the same.

Comparison to known benchmark quantities can help determine the relative size of a fractional piece because the benchmark quantity can be seen as greater, less than, or the same as the piece.


The CCSS for Mathematical Practice

2

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.

Introduction 13

2 Council of Chief State School Officers (CCSSO) & National Governors Association Center for Best Practices (NGA Center). (2010).

Mathematics. Common core state standards for mathematics . Retrieved from http://www.corestandards.org/Math


14 Introduction

Tasks’ CCSSM Alignment

Task

Task 1

Shares of Pizza

Developing Understanding

Task 2

Shares of Clay


Task 3

Shares of Fudge


Task 4

Fraction of a Number Line


Task 5

Rolls of Bubble Gum


Task 6

Shaded and Unshaded


Task 7

A Fraction of a Whole


Task 8

Pulling Ideas Together

Solidifying Understanding

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





 



     











 

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

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


Introduction 15

Task

Task 1

Shares of Pizza


Task 2

Shares of Clay


Task 3

Shares of Fudge


Task 4

Fraction of a Number Line


Task 5

Rolls of Bubble Gum


Task 6

Shaded and Unshaded


Task 7

A Fraction of a Whole


Task 8

Pulling Ideas Together

Solidifying Understanding

     

      

      

      

      

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   

      


16 Introduction

Lesson Progression Chart

Overarching Questions

• What are different situations in which we talk about a fraction of the whole, a fraction of a set, or a fraction of a number line?

• How do models of fractions differ from each other? (A fraction of a whole, a fraction of a set, and a fraction of a number line?)

• How do we compare fractions to determine which is more?

Task 1

Shares of Pizza

Developing

Understanding

Task 2

Shares of Clay

Developing

Understanding

Shares of pizza are shown and compared by students to determine who has the most pizza. Fractions with like and unlike denominators are compared. Equivalent fractions are discussed.

Determine fractions of a whole.

Addition with like denominators to bring about an awareness of the meaning of the numerator and denominator.

A comparison of fractions with different denominators in relationship to the whole as well as a discussion of equivalent fractions.

Partitioning a whole into equal pieces.

Counting iterations of equal pieces.

Comparing amounts.



Comparing amounts.

Task 3

Shares of Fudge

Developing

Understanding

Task 4

Fraction of a

Number Line

Developing

Understanding

Representing portions of fudge and comparing amounts based on the use of 1 and as benchmark quantities. Equivalent fractions are discussed again.

Partitioning fractions, determining if fractions are equivalent fractions, or improper fractions on a number line bring about the awareness that there is more than one whole on the number line.



Comparing amounts by using a benchmark of 1 and .

Using benchmark quantities of 1 and to make comparisons.

A story situation that requires students to partition figures into equal shares and name the amounts.

A story situation that requires students to partition figures into cover figures with square tiles that represent square feet and to write multiplication equations.

A story situation that requires students to share a fraction of a whole set of individual items.

No contextualize situation is provided; numbers are placed on the number line.


Introduction 17

Task 5

Rolls of Bubble

Gum

Developing

Understanding

Locate fractions, including fractions, improper fractions, and equivalent fractions at points on a number line.

Students use benchmark quantities of and

1 to determine which amount is greatest or least.

A story situation and a number line model are provided.

Students are asked to locate fractions on a number line.

Task 6

Shaded and

Unshaded

Developing

Understanding

Determine the portion of the figure that is shaded and unshaded.

Recognize that a square is a unit or a row of the figure is a unit and write equivalent fractions.

Count shaded and unshaded portions and name the fraction. Divide the whole number of squares into equal groups and name the shaded and unshaded portions with a fraction.

No story situation is provided. Two area models showing shaded areas that can be named in more than one way are provided.

No story situation is provided. Two area models are shaded. One irregular shading requiring students to name the size of the pieces as 32nds or move the to form .

Task 7

A Fraction of a

Whole

Developing

Understanding

Shade named fractions on a figure partitioned into more pieces than those identified in the denominator.

Write equivalent fractions and then compare the amounts on two figures.

Count the unit as individual squares or as a row in one figure or a column in another figure as a means of naming equivalent fractions.

Task 8

Pulling Ideas

Together

Solidifying

Understanding

Write many names for as well as using and 1 whole as a benchmark quantity to compare fractions. Provide more than one name for a shaded figure.

Using and 1 whole to compare fractions.

A context, the area model, and the number line model.


18 Introduction


mathematics

Grade

3

Tasks and Lesson Guides

Understanding Fractions as Numbers


Tasks and Lesson Guides 21

Name_________________________________________________________

Shares of Pizza

1. Sally has of a cheese pizza and of a pepperoni pizza. Which does she have more of, the cheese pizza or the pepperoni pizza?

TASK

1

2. Mary and Dominic have the same size pizza but their pizzas are cut differently.

Mary has of her pizza. Dominic has of his pizza. Show their pizzas. Who has the most pizza and how do you know?


22 Tasks and Lesson Guides

TASK

1

3. Is it easier to compare Sally’s pizza or Mary’s and Dominic’s pizzas? How are the two comparisons different from each other?

Extension: John has of a pizza. Jean has of a pizza. John claims that he can use what he knows about halves to think about his amount of pizza and Jean’s amount of pizza. Who has the most pizza and how do you know? Use the > or < sign to compare their amount of pizza.



Shares of Pizza

Rationale for Lesson: In this lesson, students will solve a situational problem involving shares of pizza and the comparison of two pizzas. Equivalent amounts, the comparison of amounts to a half, and both fractions with like denominators and familiar fractions (halves, fourths, and thirds and sixths) with unlike denominators are compared with each other.

Task 1: Shares of Pizza

1. Sally has of a cheese pizza and of a pepperoni pizza. Which does she have more of, the cheese pizza or the pepperoni pizza?

2. Mary and Dominic have the same size pizza but their pizzas are cut differently. Mary has of her pizza.

Dominic has of his pizza. Show their pizzas. Who has the most pizza and how do you know?

3. Is it easier to compare Sally’s pizza or Mary’s and Dominic’s pizzas? How are the two comparisons different from each other?

See student paper for complete task.

Common

Core Content

Standards

3.NF.A.1

3.NF.A.3a

Understand a fraction 1/ when a whole is partitioned into fraction a / b b as the quantity formed by 1 part b equal parts; understand a

as the quantity formed by a parts of size 1/ b .


Standards for

Mathematical

Practice

3.NF.A.3d


MP1 Make sense of problems and persevere in solving them.

MP2 Reason abstractly and quantitatively.

MP3 Construct viable arguments and critique the reasoning of others.

MP4 Model with mathematics.

MP6 Attend to precision.

MP7 Look for and make use of structure.

LESSON

GUIDE

1



LESSON

GUIDE

1

Essential

Understandings

Materials

Needed

• A fraction describes the division of a whole or unit (area/region, set, linear/measurement) into equal parts. A fraction is relative to the size of the whole or unit.

• The denominator tells how many equal parts into which the whole or unit is divided. The numerator tells how many equal parts of that subdivided whole are indicated.

• Rational numbers (fractions) have an infinite number of equivalent forms, and the forms are equivalent if the same portion of the set or area of the figure is represented or they represent the same point on the number line.

• Comparison to known benchmark quantities can help determine the relative size of a fractional piece because the benchmark quantity can be seen as greater, less than, or the same as the piece.

• Student reproducible task sheet

• Six pizzas/circles per group, scissors



SET-UP PHASE

Listen as I read the task to you. You can draw a picture of the pizza or you can use the pieces of art paper to represent the pizza.

LESSON

GUIDE

1

EXPLORE PHASE

Possible Student

Pathways

Group can’t get started.

Shows two different sizes of pizza when showing the thirds and the sixths.

Assessing Questions

Tell me what you are trying to figure out about Sally’s pizza.

Tell me about the shares of the two pizzas.

Shows two pizzas, one with shaded and one with shaded.

Tell me about the pizzas you are showing. Which shaded area is more and how do you know?

Advancing Questions

Here are two pizzas. Can you show the share of fourths that Sally has? Then let me know about her eighths.

The pizzas are the same size.

Why does this matter? Leave this drawing so we can compare the new drawing using the two pizzas that are the same size.

Which amount is closest to a whole pizza? Write a sentence explaining how you know that is closest to the whole.



LESSON

GUIDE

1

SHARE, DISCUSS, AND ANALYZE PHASE

Sally’s Pizza:

EU: A fraction describes the division of a whole or unit (area/region, set, linear/ measurement) into equal parts. A fraction is relative to the size of the whole or unit.

EU: The denominator tells how many equal parts into which the whole or unit is divided. The numerator tells how many equal parts of that subdivided whole are indicated.

• Tell us about and show us Sally’s pizzas. (Each of her pizzas is cut into four pieces.)

• How did you know to cut each of the pizzas into four equal pieces? (The bottom number tells us the way to cut the pizza.) (Revoicing)

• You’re correct; the denominator tells the number of the size of the pieces in the whole.

(Revoicing and Marking)

• What does the numerator tell us? (The number of each kind of pizza that Sally has.)

EU: Rational numbers (fractions) have an infinite number of equivalent forms, and the forms are equivalent if the same portion of the set or area of the figure is represented or they represent the same point on the number line.

• Which pizza does Sally have more of, cheese or pepperoni? (Challenging) ( is more than .)

• How do you know is more than ?

• Say more. She claims that > . Who agrees?

(I agree because 3 is more than 2 pieces of pizza.)

• So you looked at the numerators. Why didn’t you need to compare the denominators? (They are the same denominators so we only need to compare the numerators.) (Marking)

EU: Comparison to known benchmark quantities can help determine the relative size of a fractional piece because the benchmark quantity can be seen as greater, less than, or the same as the piece.

• I heard someone claim that they thought about Sally’s pizza as of cheese pizza and of pepperoni pizza and when they think of it this way, they claim they know which is more right away. What is this person thinking? Does anyone know? (Challenging) ( is more. I can have of a dollar and it is more. is the same as a half so 2 is less than 3.)

• Does anyone else want to add on? (4 pieces is more than 2 and you have almost all of them.

You have 3 of them. 2 of the fourths would be the same as half.)

• So and are equivalent. (Record = .) (Revoicing and Marking)

• Stop and jot. Use the greater than and less than sign to tell me which is more or which is less, or . ( > or < .)



Mary’s and Dominic’s Pizza:


• Mary and Dominic have the same size pizza but their pizzas are cut differently. Mary has of her pizza. Dominic has of his pizza. Show their pizzas. Who has the most pizza and how do you know?

• Who has more pizza, Mary or Dominic?

(They have the same. Dominic has the most [error].)

• Can someone show us Mary’s and Dominic’s pizza? (If students are struggling when drawing thirds, then show students an easy way to make thirds by drawing a Y in a circle or folding paper so the right and the left side of the paper overlap with the center of the paper.)

• Let’s look at Dominic’s pizza. What is this piece called? ( )

• How many sixths does Dominic have? (

• Stop and jot. What does + = ___? (

• So is made up of how many pieces? (2)

) (Record + = ___) (Marking)

)

• Who has more pizza, Mary or Dominic? Stop and jot. Use the <, >, or = sign to tell me who has the most pizza. ( = )

• Can anyone just look at their fractions, the and the , and immediately know who had the most? (I knew was the most because there are 6 pieces. [Error])

• What do you think of her thinking? (I disagree. Thirds is a really big piece but sixths are small pieces and it takes two of them to make the third so they are equal.)

• What makes = ? It looks like is so much more. (Challenging) (It takes two small pieces to fit into the third.)

• Who understands what she is saying and can put it into their own words?

• Can someone show us with the fractions?

• So you said you took the three pieces and cut each one into two pieces. When you did this it automatically cut the one piece.

(Record x = .) (Note: Students are not expected to do this independently.)

LESSON

GUIDE

1



LESSON

GUIDE

1

Application Can you use the same kind of reasoning as above and compare these fractions? Put the > or < sign between each to tell me about the two fractions.

(Since the pieces are the same size pieces it is easy to compare them, because all we have to do is look at the number of pieces and compare the two fractions. In the case of and we know that is one whole. is close to the whole so it is more than because = .)

____

____

____

Summary

____

We can easily compare fractions with like denominators because the pieces in both fractions are the same size so then we just have to look at the number of pieces we have of each pizza. When the pieces or the denominators are not the same, then we have to look more closely at the size of the pieces and how many of those pieces that we have.

If we know how close the amount is to or to the whole then we can use this information to tell us which piece is the largest.

Quick Write

Which is greater and how do you know? ____

Support for students who are English learners (EL):

1. Slow down discussions for students who are English learners by asking other students to repeat ideas and to put ideas into their own words.

2. Have students continually point to the model and the fractions during the discussion so that it is perfectly clear what is being discussed.

TASK

2


TASK

2


Name_________________________________________________________

Shares of Clay

Students are building clay figures in art class. The teacher gives students a chart that identifies the amount of clay needed for each portion of the figure and the total amount of clay needed for each figure.

Figure Fraction of a Stick

Person

for the head

for the body

for the legs and feet

Duck

Dinosaur

for the body

for the head and legs

for the head

for the body

for the legs and feet

TASK

2

1. How much clay does each figure require?



TASK

2

2. Which figure requires the most clay and how do you know?



Shares of Clay

Rationale for Lesson: In this lesson, students deal with three sets of information in a table. For the first time, students discuss one whole as a fraction and amounts that are more than one whole, making it easy for students to make comparisons of the amounts for the first time as well.

Task 2: Shares of Clay

Students are building clay figures in art class. The teacher gives students a chart that identifies the amount of clay needed for each portion of the figure and the total amount of clay needed for each figure.

1. How much clay does each figure require?

2. Which figure requires the most clay and how do you know?


Common

Core Content

Standards

3.NF.A.1

3.NF.A.3c



Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

3.NF.A.3d


Standards for

Mathematical

Practice







MP8 Look for and express regularity in repeated reasoning.

LESSON

GUIDE

2



LESSON

GUIDE

2

Essential

Understandings

Materials

Needed


• Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of.

• A fraction in which the total number of pieces in the whole is the same as the total number of the pieces is called one whole because all of the pieces of the whole are accounted for.



• Two colors of paper cut into four pieces. Students need eight pieces of each color. (Each piece will serve as one whole piece of clay.)



SET-UP PHASE

Listen as I read the task to you. You can use the strips of paper to represent the clay.

EXPLORE PHASE


Pathways


Claims that s/he needs a whole stick of clay for the dinosaur, but provides no evidence of how s/he arrived at the total.

Claims to need a whole stick of clay for the person.

Writes , , and is one whole stick.


Which figure do you want to make? How much clay do you need?

Tell me how you know you need one whole stick.

Tell me how you arrived at the total.


How can you figure out how much clay you need? Here’s the clay. Mark off each of the pieces that you need.

Write what you said as a number sentence so that others can see how you arrived at the total.

What equation can you write to describe how you know what the total amount of clay is?

LESSON

GUIDE

2




LESSON

GUIDE

2

Clay needed for the dinosaur: for the head, for the body, for the legs.

(Begin discussion with the clay needed for the dinosaur because students are more comfortable working with one half, or , and .)


EU: Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of.

• How much clay is needed for the dinosaur?

( )

• How did you know this? ( + + = = 1)

• Who agrees? Can someone put this into their own words and show us with the strips of paper? (I cut the stick into four parts and just said 1, 2, and 1.)

• Where did you get the 1, 2, and 1 from? (The top number.)

• So you only looked at the numerator or the top number. (Revoicing)

• Why didn’t you have to think about the denominator?

(All of the pieces are the same size so you can just add them together.)

EU: A fraction in which the total number of pieces in the whole is the same as the total number of the pieces is called one whole because all of the pieces of the whole are accounted for.

• How do you know that is the same as a whole stick of clay?

( is all of the fourths.)

• Who understands what she is saying and can put it into their own words? (There are 4 pieces and you need the whole stick because you need all of them.)

• Someone wrote . Why isn’t this correct? (Challenging) (This isn’t right because this means 4 wholes.)

• You’re right, means 4 wholes and we know this because the denominator tells us that the size of the piece is one whole. (Marking)

Clay needed for the duck, for the body, for the head and legs.

Repeat previous EUs and discussion cycle.



Clay needed for the person, for the head, for the body, for the legs and feet.



• How much clay is needed for the person?

( + + + = , more than one stick.)

• Who agrees and can add on to her thinking? (I knew right away that it was more than one.)

• Say more. (I counted 5 parts.)

• Say more, I’m not sure what you mean by 5 parts.

( + + = so this is one whole stick of clay. Then there is one more needed.)

• When we have the same number of pieces in the numerator as the size of the pieces in the denominator, then this represents the whole.

(Marking)

• So you said it is more than one whole, how would we write this as a fraction? How many fifths do we need? ( , 1 ) (Marking and Revoicing)

• You’re correct, = 1 . How do you know by looking at the fraction that it will be more than a whole? (Challenging)


• Which figure requires the most clay and how do you know? (The person needs the most clay.

The is more because and are each one whole strip of clay.)

• Say more, your claim tells us why and are incorrect, but not why is the correct response.

( is more than .)

• You’re correct, – = .

(Marking)

Application Write an addition equation for the amount of clay needed for the duck.

Summary Adding fractions with like denominators, or pieces of the same size, means that we can just count the total number of like pieces to arrive at the total.

A fraction where the numerator and the denominator are the same is another name for one whole (e.g., , , ).

A fraction with a numerator that is greater than the denominator will be greater than the whole.

Quick Write Circle the fractions that show more than one. Explain how you know the fraction shows more than one whole. Put a box around the fraction that shows exactly one whole and explain how you know it is one whole.


1. Ask students to talk about what is the same and what is different about the responses.

Noting patterns and relationships will help students build an understanding of the concept

(e.g., , , versus , , ).


LESSON

GUIDE

2


TASK

3

Name_________________________________________________________

Shares of Fudge

Four friends each have a fraction of a bar of fudge. All of the bars of fudge are the same size. Who has the most fudge and how do you know? You may choose your own tools and models for solving this problem.

Jill: of a bar of peanut butter fudge

James:

Fred:

Juan:

of a bar of mint fudge

of a bar of chocolate fudge

of a bar of butterscotch fudge

TASK

1

Their friend Russell claims that he has the most fudge because he bought two different kinds of fudge. He bought of a bar of cherry fudge and of a bar of vanilla fudge. Do you agree or disagree with Russell? Why or why not?



Shares of Fudge

Rationale for Lesson: In this lesson, students work with information in a table again but this time, a new context is used and the size of the numbers change. One familiar amount is used to link to prior knowledge of a half since the emphasis in this lesson is on making comparisons of amounts by using benchmarks of 1 or .

Task 3: Shares of Fudge

Four friends each have a fraction of a bar of fudge. All of the bars of fudge are the same size. Who has the most fudge and how do you know? You may choose your own tools for solving this problem.

Jill: of a bar of peanut butter fudge

James: of a bar of mint fudge

Fred: of a bar of chocolate fudge

Juan: of a bar of butterscotch fudge

Their friend Russell claims that he has the most fudge because he bought two different kinds of fudge.

He bought of a bar of cherry fudge and of a bar of vanilla fudge. Do you agree or disagree with

Russell? Why or why not?

Common

Core Content

Standards

3.NF.A.1

3.NF.A.3a




Standards for

Mathematical

Practice

3.NF.A.3b

3.NF.A.3d

Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.









LESSON

GUIDE

3



LESSON

GUIDE

3

Essential

Understandings

Materials

Needed



• Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of.


• If the numerator is half the quantity in the denominator, then the fraction is equal to a half.



• Strips of art paper to represent fudge, rulers to represent fudge, or for drawing



SET-UP PHASE

Today we are going to talk about bars of fudge. All of the bars of fudge are the same size. Take a look at each student’s share of the fudge. Make a prediction right now. Which student do you think got the most fudge? Turn and talk with your partner. Put a circle around a student’s name. Who do you think and why?

(Some are close to half but some are close to the whole bar and this is how I knew which amount was the most.) Now work together to make a drawing and write an explanation saying how you know.

Note: Do not engage in the conversation at length so as to get into the details of the task. Call on students who seem to be using benchmark amounts to determine which is the largest.

LESSON

GUIDE

3

EXPLORE PHASE


Pathways

Makes a diagram and represents the students’ fudge.

Identifies as the most fudge, but provides no reasoning.

Claims that Russell has the most fudge.


Tell us about your diagram and what it shows about the fudge.

Tell me why you think is the most fudge.

How do you know that

Russell has the most fudge?


Which student has the most fudge? How can you figure this out?

I hear what you’re saying about being one away from the whole. Look at

. How many away from the whole is it?

So both students’ amounts are one part away from the whole. Which is the most or the greatest amount of fudge?

Can you write an explanation that captures what you just said? You compared his amount to the other students’ amounts.

How can you show this on your paper so that everyone knows your thinking?



LESSON

GUIDE

3


Which is the most and how do you know?

Juan: of a bar of butterscotch fudge.


EU: Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of.

• Who has the most fudge and can you show us how you know that person has the most fudge?

(Juan.)

• Who made a sketch of the amounts of fudge? Come up and show us each of their amounts.

• Say more to support your claim that Juan has the most.

(Juan almost has the whole thing.)

• Say more. How do you know Juan almost has a whole bar of fudge? ( + is the whole bar.)

• (Record + = = 1) I notice that you said = 1 whole. (Marking) and this is

• Can someone say back his reasoning and point to the diagram so we can all understand what he said?

• James’s fudge, the of a bar, is also one piece away from the whole. Why isn’t it the closest to the whole? (Challenging) (The size of is bigger than so is closer to one.)

• Can someone add on to his reasoning and point to the diagram?

( has fewer pieces than the one with tenths so since there are fewer pieces this means that the pieces are larger.)

• So we determined that and were each one piece away from one whole, but was more because is a smaller piece than , because ten pieces means that the pieces will be smaller than when there are six pieces. This means that the pieces will be larger because there are fewer pieces. is closer to one whole than . (Recapping)


Jill: of a bar of peanut butter fudge.


EU: If the numerator is half the quantity in the denominator, then the fraction is equal to a half.

• How much fudge does Jill have? ( )

• Do a stop and jot. Write down three different names that we can give to Jill’s amount of .

(Challenging) ( , , , , , .)

• What is true about each of these fractions that are called halves? (Challenging) (The top number is half of the bottom number. Take half of the bottom number and put it in the top number.)

• So you see a relationship between the numerator and the denominator. The numerator is half of the number in the denominator.

(Revoicing)

• Sounds like you are saying that the numerator, the number on top, is half the size of the denominator, the number on the bottom, which would mean that 10 is half of 20 and that 2 is half of 4. Is that what you are saying? (Marking)




Fred: of a bar of chocolate fudge.


• Which other amount of fudge is close to half a bar of fudge and how do you know? Turn and talk with a partner.

• Which is close to ? ( ) How do you know this is close to half?

( is half so has one over.)

• Be more specific. ( and is .)

• So can I write + ? Why? (Challenging) (Yes, because is so you can add + .)


Russell: + =


• Someone claimed that Russell has the most. Who agrees that Russell has the most? Why do you agree? (Russell has one whole bar of fudge because he has of one kind and of another kind. If you add this together, you get or 1 whole bar of fudge.)

• Why can we combine Russell’s two kinds of fudge? (They are both the same size whole.

They are both cut into thirds. We can put both in the same pan.)

• Who understands the reason why we can put the vanilla and the butterscotch fudge together and can say back what they think they heard? (The kinds of fudge don’t make a difference.)

• Since both types of fudge are cut into thirds, the type of fudge, vanilla and butterscotch, doesn’t really matter in the problem. (Marking)

• What if the pan for the vanilla fudge looked like this (draw a pan) and the pan for the butterscotch fudge looked like this (draw a smaller pan) ? Why would we NOT be permitted to put these thirds together? (Challenging) (The thirds are not equal to each other. Even though they are thirds they aren’t the same.)

• So the size of the whole must be the same.

(Revoicing)

• If the whole is the same and the size of the pieces in each is also the same, then all we need to do is add the numerators. (Point to the model of 2/3 and 1/3.) (Marking)

LESSON

GUIDE

3



LESSON

GUIDE

3

Application

Circle the fractions that are close to . Choose one and explain how you know the fraction is close to .

Summary

We can compare fractions to to determine if the fraction is close to , either less than , or more than so we can determine the size of the fraction.

We can also compare fractions that represent one whole. , , , and all represent a whole because the numerator and denominator are the same which means that all of the pieces of the named size of the whole are accounted for. All of the halves in two halves or all of the thirds in are accounted for in the situation.

When comparing fractions that are one piece from either the half or the whole, then compare the size of the one piece to determine which is the largest piece.

Quick Write Write two different fractions for the circled amount.



Name_________________________________________________________

Fraction of a Number Line

Place each fraction at a point on the number line. If more than one fraction is placed at the same point, then explain how two fractions can represent the same point on the number line.

TASK

4

0 1 2 3



LESSON

GUIDE

4

Fraction of a Number Line

Rationale for Lesson: In this lesson, students continue to work with halves, fourths, and fractions greater than a whole. This time, however, a different model, the number line model, is used. With the use of the number line model, students are challenged to think about wholes that are connected to each other.

Task 4: Fraction of a Number Line

Place each fraction at a point on the number line. If more than one fraction is placed at the same point, then explain how two fractions can represent the same point on the number line.


Common

Core Content

Standards

3.NF.A.1

3.NF.A.2a



Represent a fraction 1/ b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/ b and that the endpoint of the part based at 0 locates the number 1/ b on the number line.

3.NF.A.2b

3.NF.A.3a

3.NF.A.3b

3.NF.A.3d

Represent a fraction a / b on a number line diagram by marking off a lengths 1/ b from 0. Recognize that the resulting interval has size a / b and that its endpoint locates the number a / b on the number line.


Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.



Standards for

Mathematical

Practice

Essential

Understandings

Materials

Needed










• If the numerator is half the quantity in the denominator, then the fraction is equal to a half.

• When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator.





LESSON

GUIDE

4



LESSON

GUIDE

4

SET-UP PHASE

Look at the directions. I’ll read them to you. This is a simple task so do this independently and write about why you placed the numbers where you placed them on the number line.

EXPLORE PHASE


Pathways


Places all of the numbers on the number line correctly except for .


Can you tell me what this number is? ( )

Tell me why you put each of the numbers where you put them.


Where does this number go on the number line?

This , where does it go?

You know where and are, so where would be?

Places at the same place as .

You have and at the same point on the number line.

Can you tell me about these two points?

is placed incorrectly. It isn’t the same as . It is more. Can you figure out what four halves is?




Placing on the number line.


• Tell us where you placed on the number line. (In between 0 and 1.)

• What made you place it here? (There are two equal parts, here and here.)

• Someone placed at the 2. What do you think about this? (Challenging) (It is not 2.)

• What is the difference between and 2? ( is a small piece and two is big.)

• Say more. Can you talk about candy bars to help us understand? (Two candy bars is more than if you get a half of one candy bar.)

• You’re right, two whole candy bars (record 2) is way more than if you have one candy bar that you get half of (record on the board). (Recapping and Revoicing)


EU: When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator.

• Where did you place on the number line? (At the same place as .)

• Who agrees with her? She claims that and share the same point on the number line.

How can this be? (Challenging)

• So + is what? ( )

• How many fourths can make up the whole? ( )

• Can you show us on the number line? ( + + + )

• + + + = or one whole. (Revoicing and Marking)

Comparing and .

EU: If the numerator is half the quantity in the denominator, then the fraction is equal to a half.

• Explain how the two fractions can share the same point on the number line. (2 is half of the

4 parts.)

• Who understands what she just said? Can someone explain how = ? (There are four parts and when you take half of them it is the same as .)

• You’re correct. There are 4 equal pieces and 2 is half of 4. (Revoicing and Marking)

• Does this always work? Can you give me other names for this point? For example, which of these two fractions says one half and how do you know? or ? ( because 4 is half of 8.)

• 4 is half of 8 so we know that = . (Revoicing and Marking)

• Give me two more fractions that equal . Do a stop and jot.

LESSON

GUIDE

4



LESSON

GUIDE

4

Placing on the number line and considering the meaning of .

EU: The denominator tells how many equal parts into which the whole or unit is divided.

The numerator tells how many equal parts of that subdivided whole are indicated.


• We heard that = 2. What does everyone think about that? (Four pieces is what this says.)

• Can you show us on the number line where you see four pieces that are each a half in size?

• Can I write + + + ? What is the total or the sum? Do a stop and jot. How do you know this is ?

(You just count the pieces and you get four.)

• I don’t understand how makes 2 wholes. Can someone explain this? (Challenging)

( + = or one whole.)

• So when the numerator is larger than the denominator, what do we know? (We know it is more than a whole because you only need two halves to make a whole.) (Marking and

Revoicing)

• If you know that = 2, then what does represent? Where is it located on the number line?

(Challenging) (1 )

• Who can tell us why 1 is another name for ? ( is 1 and then there is left over.)

• Let’s use some of our reasoning that we have shared already. What if I have ? Where is this located? Everyone show it on the number line.

(Challenging)

• What if I give you , then what number is this?

(3) How do you know?

( + + is 3 because

is one whole.)

• So how do you know when it is going to be a whole number like 2 or 3, or a mixed number like 1 ?

• Anytime that the numerator is bigger than the denominator you will be able to make a whole because as soon as you have a fraction with a numerator and the denominator the same, for example or , this makes a whole and any left over is more than the whole. (Marking and

Revoicing)

• Do a stop and jot. Write a fraction using fourths that is more than a whole.

Comparing and .


• Where did you put and ? Who can come up and show us where to put these and explain why you put them where you put them?

• Some people put these two fractions at the same place on the number line. Who agrees?

Who disagrees? (Challenging) (I disagree, is the same as , so I know this one goes here.

is the same as 2.)

• Who understood what she said and can put her ideas into their own words? ( is equal to so they go on the same point on the number line.)

• This is correct, = and they share this point on the number line.

(Marking)

• If this is a candy bar (draw a candy bar) can someone shade or of the candy bar?

• Give me some other names for this point (the ) . ( , , , )

• Just like yesterday when we had many names for , we can come up with many names for the point of on the number line. (Marking)



Application Look at the six numbers below. Put a box around the numbers that are less than one. Circle the numbers that are greater than one.

Summary Today we worked with a number line and we found that points can have more than one name. We also found that when the numerator is smaller than the denominator, then the number is less than one. When the numerator is greater than the denominator, then the amount is greater than one. Who understands what I’m saying and can say it back?

Quick Write Circle the fractions that are less than one whole.


1. Slow down discussions for students who are English learners by asking other students to repeat ideas, to put ideas in their own words, and continually point to the model and the equation simultaneously.

2. Ask students to repeat and put others’ ideas into their own words so that they have many opportunities to talk and share their reasoning.

LESSON

GUIDE

4



TASK

5

Name_________________________________________________________

Rolls of Bubble Gum

•

•

Four friends each bought a roll of bubble gum tape.

Carlos chewed of his gum.

Helen chewed of her gum.

• Jamal chewed of his gum.

• Lisbeth chewed the most gum.

1. Use the number line below to illustrate:

• Which friend chewed the greatest amount of gum? _______________

• Which friend chewed the least amount of gum? _______________

• Which two friends chewed the same amount of gum? _______________



2. Explain how you can use to help you determine the value of and when each fraction refers to the same whole.

TASK

5



LESSON

GUIDE

5

Rolls of Bubble Gum

Rationale for Lesson: In this lesson, students have another opportunity to work on a number line model. Students are challenged to use 1 whole and as a benchmark quantity again. For the first time, students look at and as equivalent fractions and they are challenged to explain how the two amounts can be located on the same point on the number line. Prior to this point in time, students have identified fractions that are equivalent to .

Task 5: Rolls of Bubble Gum

Four friends each bought a roll of bubble gum tape.

• Carlos chewed of his gum.

• Helen chewed of her gum.

• Jamal chewed of his gum.

• Lisbeth chewed the most gum.

1. Use the number line below to illustrate:

• Which friend chewed the greatest amount of gum?

• Which friend chewed the least amount of gum?

• Which two friends chewed the same amount of gum?


Common

Core Content

Standards

3.NF.A.2b

3.NF.A.3a

Represent a fraction off a lengths 1/ has size a / b

_______________

_______________

_______________

and that its endpoint locates the number the number line.

b a / b on a number line diagram by marking

from 0. Recognize that the resulting interval a / b on


Standards for

Mathematical

Practice

3.NF.A.3d










Essential

Understandings

Materials

Needed






LESSON

GUIDE

5



LESSON

GUIDE

5

SET-UP PHASE

Look at the directions. I’ll read them to you. Take five minutes to work on the task and then you can work in small groups to see if you agree on how you marked the bubble gum tape. I wonder who has the most bubble gum? See if you can figure this out.

EXPLORE PHASE


Pathways


Struggling to place all of the markings on the one number line.

Student has represented each amount except for .


Where is the whole amount of bubble gum? If this whole amount is cut into fourths, then how many pieces is it cut into?

Tell me about and . How did you know where to place these?


I’ll be back. See if you can mark the gum into fourths and then you get of the gum. Can you show me of the gum?

I’m going to draw four different lines for each student’s bubble gum.

Although the task only shows one number line you need to make several number lines, one for each student; you can do so on your paper. See if you can show each student’s amount on a different line.

Tell me about each of the students’ bubble gum. How did you know where to mark each of their amounts?

When you look at what does this tell you about how the bubble gum is partitioned? How many pieces belong to Lisbeth?

Can you show her amount?







• Tell us where you placed and how you knew where to put it. (I cut four pieces and then took three of them.)

• How did you know to do this? (The bottom number says how many pieces and the top number says the number of pieces that we get.)

• The bottom number or the denominator says how many equal pieces are in the whole, and the top number or the numerator says how many pieces you get. (Revoicing and Marking)

Repeat for each student’s amount of bubble gum.

Comparing and .


• Some of you placed and at the same point. I don’t understand. How can they be at the same point? (Challenging) (8ths are just smaller pieces but it is the same.)

• Can someone add on to her thinking?

(There are 4 here and each one of these gets cut into two pieces so this makes it into 8. When you get three pieces here [ ] this means you are getting 6 pieces here [ ].)

• How many students understand what they said? Who can put the ideas into their own words?

• It sounds like you said that each fourth is cut into 2 equal pieces. When each fourth is cut it means that the 3 pieces in the are cut. That makes 6 on top and 8 on the bottom.

(Teacher records x = .) So now we have .


Placing on the number line and comparing with one whole.


• Tell me about . Where is it placed?

• Will it be closer to one whole or to a half?

(It will be closer to one whole because 5 is almost 6.)

• I don’t understand what you mean by it is 5 and close to 6. Can someone put the ideas into their own words?

( is the whole and this is one piece away from the whole.)

• So must be the most? What do you think?

• is one piece away from the whole and so is the . Which is closest to the whole?

(Challenging) (A sixth is a small piece but a fourth is a bigger piece.)

• How does what she said help us determine which is closer to the whole, or ?

( is closer because it is one smaller piece away from the whole.)

• Who agrees? Who disagrees? Can someone come up and point to the portions on the number line?

• is one small piece away from the whole. If we have and add , we will have the whole.

It really helped us to think about where each amount was in comparison to the whole.

(Marking)

LESSON

GUIDE

5


LESSON

GUIDE

5

Placing on the number line and comparing the fraction to .


• Let’s look at . Is more than or less than ?

(Challenging) (It is more than half because 3 is more than half of 5.)

• Say more. How do you know that 3 is more than half of five? (2 is half of 5.)

• Who understands and agrees? Can you say more? (2 and 2 make 5.)

• That was good math thinking because you showed how both halves, the 2 and 2 , add up to the whole. (Revoicing and Marking)

Application Which of these pieces is closest to a whole? Explain your reasoning.

or ____

or ____

Summary Today we used benchmark amounts of a whole and a half to make comparisons. Once we realized that pieces were each one piece away from the whole, then we compared the size of the pieces in the denominators to determine which was the smallest piece. This told us which was closest to the whole. We used similar thinking to think about how far away from a half pieces were.

Quick Write Which is largest and how do you know?

or ?


1. Ask students to point to the point on the number line for each fraction and then ask them how far from one the fraction is. This will help students make connections between the visual diagram and the fractions. Press students to tell you how they know that one amount is closer to one than the other amount.



Name_________________________________________________________

Shaded and Unshaded

1. What fraction of the figure is shaded gray? Name the gray portion in two different ways. How did you determine how to name the fraction of the figure that is shaded gray?

TASK

6



TASK

6

2. What fraction of the figure is shaded gray? How did you determine how to name the fraction of the figure that is shaded gray?



Shaded and Unshaded

Rationale for Lesson: In this lesson, students work with an area model for the first time. Rather than shading a portion of the figure, students are asked to name the shaded and unshaded areas. Multiple responses are possible which opens the door to a discussion of equivalent fractions. Previously, students scaled up a fraction but this time they will discuss scaling up or scaling down a fraction.

Task 6: Shaded and Unshaded

1. What fraction of the figure is shaded gray? Name the gray portion in two different ways. How did you determine how to name the fraction of the figure that is shaded gray?

LESSON

GUIDE

6


Common

Core Content

Standards

Standards for

Mathematical

Practice

Essential

Understandings

Materials

Needed

3.NF.A.1

3.NF.A.3a












LESSON

GUIDE

6

SET-UP PHASE

There are three different names for the first fractions. Can you see if you can come up with three different names? Then look at the second figure. Look, it has parts of some squares shaded. What are you going to have to do to figure out the size of those pieces and name them?

EXPLORE PHASE (SMALL GROUP TIME, APPROXIMATELY 10 MINUTES)


Pathways

Assessing Questions Advancing Questions

Identifies for the first figure.

Tell me how many squares are shaded on the first figure.

I want to challenge you to give me two different names for the first figure.

Writes , , and for the first figure.

Writes and for the second figure.

Tell me how you know that the first figure can be called by all of these names, the

, , and .

Tell me how many squares are shaded altogether in the second figure.

Can you come up with two names for the second figure?

Suppose I want to name these smaller pieces. What size or what fraction of the whole do they show?




Part 1, figure with shaded.


• Tell me about the first figure.

(I see . I see . I see .)

• How can we have all of these names for the shaded portion of the figure? Which is correct and how do you know? Who can show us where they see these amounts? (Challenging)

(There are 16 equal pieces and 8 of them are shaded.)

• Who understands what she said and can point to the figure and say back what you heard?

• Someone else called this and someone else called it . Are we permitted to call the shaded portion or ? If so, explain your thinking. (8 is half of 16 so we can call it , too. If we call it we can call it , too, because 2 is half of 4.)

• If we call it then can we see fourths in the figure? Who can point to and or the ?

• They all name the same shaded amount so they are equivalent names for the amount.

(Marking)

• How is each person thinking about the amount differently? (Challenging) (The sizes of their pieces are different.)

• We can name the shaded portion as , , and .

(Marking)

• If I wanted to name it another way, could I? Turn and talk with your partner.

• How else could we name this figure? Do a sketch and see if you can come up with a fourth name for this figure. (Challenging) ( )

• How can we change to ? (You can multiply the 8 by 2 and the 16 by 2 and this gives you .)

• So x = . (Marking and Revoicing)

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LESSON

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Part 2, figure with shaded.


• Tell us about the shaded portion in the second figure. How do you name the shaded portion?

(I see 12 whole squares shaded and then I moved the halves so I made two wholes.)

• Who can show us what this would look like? There are 12 and then halves are put together. I think I need to see what happens. Come up and show us on your diagram.

• Who understood what she just said and can put the ideas into their own words and talk about fractions?

(Challenging) (She sees + .)

• (Teacher records + .) (Marking) I like how you said + . How many 16ths is this? Stop and jot down a fraction that tells about the total number of squares shaded. ( )

• There is another way of naming the shaded portion of the figure. Can anyone think of a way to do this? (Make 32 pieces.)

• If we change the size of each piece, making the denominator 32, what fraction of the figure is shaded? ( )

• What would have to happen to the in order to have 32 pieces? Do a turn and talk and see if you and your partner can figure this out. (Cut each of the 16 pieces into two pieces.)

• What happens to the 14 pieces?

(The 14 pieces get cut, too.)

• What do you mean, they get cut too?

(14 is part of the 16, so when you cut the 16 pieces the

14 got cut already, so you have 14 that turn into 28 pieces.)

• x = because when you cut the number of pieces in the whole, the 16, you also cut the number of pieces in the numerator.

(Marking and Revoicing)

See Quick Write below.

Application What part of the figure is shaded? Write two fractions that show the shaded portion of the figure.

Summary

Change the size of the pieces and write another fraction to represent the shaded area.

There are multiple names for a fraction and each of the fractions are equivalent because they both have the same amount of area.

Quick Write

What do you know about and ? How are these two figures related?


1. Have students point to the diagram. When referring to the pieces in the figures, remember that there are 16 squares; therefore, it may be difficult for students to see that as four rows/ pieces. To help students understand this idea, you can cut colored acetate into four pieces and overlay the pieces on the figure with 16ths.



Name_________________________________________________________

A Fraction of a Whole

In each figure below, the area of the whole rectangle is 1. Shade an area equal to the fraction underneath each rectangle. Compare the two fractions. Which is more and how do you know?

Figure 1 Figure 2

TASK

7



LESSON

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A Fraction of the Whole

Rationale for Lesson: In this lesson, students work with an area model again. Comparing amounts is at the center of the lesson, as well as equivalence. Students are challenged to consider one square unit as a part of the whole, or a row or column as a part of the whole. Students are not constructing a figure for each fraction but instead, they are asked to work with an existing figure that is partitioned.

Task 7: A Fraction of the Whole

In each figure below, the area of the whole rectangle is 1. Shade an area equal to the fraction underneath each rectangle. Compare the two fractions. Which is more and how do you know?

Figure 1 Figure 2

Common

Core Content

Standards

Standards for

Mathematical

Practice

Essential

Understandings

Materials

Needed

3.NF.A.1

3.NF.A.3a










• Fractions with the same size pieces, or common denominators, within the same size whole can be compared with each other because the size of the pieces is the same.




SET-UP PHASE

Look at the directions. I will read the task to you. The rectangles are not cut into eight pieces, so how are you going to think about these figures so you can get 8ths?

LESSON

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EXPLORE PHASE


Pathways

Shades of the second figure.

Shades 6 squares of the

24 for . (Error)

Shades and correctly.


Tell me how you know that you have of the figure shaded.

Tell me about your thinking.

What part of this whole did you shade?

Tell me what you shaded.


You said that two of the three columns are shaded.

If this is true, can you write another name for this fraction?

You shaded 6 of these 24 squares. This doesn’t show

. When you look at , how many groups do you know you need? (8) How can you work with all of the squares to find out how many are needed in each of the 8 groups?

Which is more, or , and how do you know?



LESSON

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Figure 2, shaded.


• Tell us about how you knew how to shade of Figure 2. (I said that a column is one group and the second column is two groups and the third column is three groups. So I shaded two groups of the three for .)

• Who understands what she is saying and can put her ideas into their own words?

• Can someone go point so we know where is and the other is on the figure? (Points to the first column of the figure.)

• You are seeing as this whole column. (Marking)

• What is + ? Stop and jot. Write it down on your paper.

• What did you write down? ( )


• What is another name for ? ( )

• How did you know that was another name for ? (16 squares make up the same amount of space as the 2 columns.)

• Someone come up and show us .

• We can write = because both fractions describe the same amount of shaded area.


Figure 1, shaded.

• Let’s look at . Tell me what you know about this fraction.

(It is cut into 8 parts.)

• How can you do this when there are so many squares? (We can make 8 groups and pass out the squares. There are 8 rows and each row is a group.)

• Let’s make 8 groups and pass out the 24 squares. Work in groups to do this. Tell me how many squares will go in each group. (3)

• There are 3 squares in each group; how many groups are there? This row counts as what? ( )

• How many eighths do we need? ( )

• Shade of the figure and turn to your partner and explain how you know you shaded of the figure.

• Who can explain their thinking to the class? Why do we call this ?

(8 rows and 6 of the rows are shaded.)

• So the denominator tells us that there are 8 rows or groups. The numerator tells us how many rows or groups are shaded. (Marking and Revoicing)

• What is another name for ? Turn and talk and see if you can come up with another name for .

• What is another name for ? ( )

• How do you know = ?

(They take up the same amount of space. I did 6 x 3 = 18.)

• You can say 8 rows and each one has three squares and this is 24. 6 of the rows are shaded and each of these rows has 3 squares and this is 18.

(Record x = .) (Note: Students are not expected to master this concept.)



Comparing and .

EU: Fractions with the same size pieces, or common denominators, within the same size whole can be compared with each other because the size of the pieces is the same.

• Which is more, or ? Turn and talk with your partner. Put the greater than or less than sign in to show which is more. Be prepared to explain how you know which is more.

• Can someone write the sentence on the board?

• Can someone explain their thinking? ( >

( > )

)

• What made you change each fractions to 24ths and then compare them with each other?

(I compared the 24ths because they had the same size pieces so it was easier to compare them.)

• It is easier to compare fractions that have the same common denominator or the same size pieces. (Marking)

• Did anyone compare the pieces by looking at the unshaded portion of the figure? Tell us about your thinking.

• So you can compare figures by looking at the shaded or the unshaded portion of the figure.

I noticed that even when you compared the unshaded portion, you talked about 24ths.

(Marking) Why do you think this happened?

Application No application.

Summary It is easier to compare fractions when you have a common denominator. You can compare amounts by analyzing the portion of the figure not shaded or the portion that is shaded.

Quick Write

Give another name for .



2. As you repeat or revoice ideas, also write them on the board so students can hear and see the fractions.

LESSON

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TASK

8

Name_________________________________________________________

Pulling Ideas Together

1. What fraction of the figure is shaded gray? Name the gray portion in three different ways. Explain the meaning of the denominator and the numerator for each fraction.

__________ _________ _________

Are the three names that you gave the shaded portion of the figure equivalent fractions? Why or why not?


2. Shade of the figure below. Give another name for .


TASK

8

3. What fraction of the figure below is shaded? Write a fraction to describe the shaded portion of the figure. Explain the meaning of the denominator and the numerator.



TASK

8

4. Place each of the fractions on the number line below. Circle the smallest fraction.

5. Give four different fractions for .

________ __________ __________ _________

6. Tim claims that he can use what he knows about benchmark fractions to compare the two fractions in problem a–c. Insert the greater than or less than sign to compare the two fractions. Explain how benchmarks of or 1 can help you compare the fractions below. a. _________ b. _________ c. _________



Pulling Ideas Together

Rationale for Lesson: In this lesson, students have an opportunity to apply all of the lessons learned in this set of related lessons. Students work with an area model and a number line model. Students are working with halves and many names for one half as well as identifying two names for the same point on a number line. In addition to exploring equivalent fractions, students are also challenged to compare fractions by making use of benchmark fractions of and 1.

Task 8: Pulling Ideas Together

1. What fraction of the figure is shaded gray? Name the gray portion in three different ways. Explain the meaning of the denominator and the numerator for each fraction.

Are the three names that you gave the shaded portion of the figure equivalent fractions? Why or why not?

2. Shade of the figure below. Give another name for .


Common

Core Content

Standards

3.NF.A.2b

3.NF.A.3a

Represent a fraction off a lengths 1/ has size a / b and that its endpoint locates the number the number line.

b a / b on a number line diagram by marking

from 0. Recognize that the resulting interval a / b on


Standards for

Mathematical

Practice

3.NF.A.3b

3.NF.A.3d

Recognize and generate simple equivalent fractions, e.g., 1/2

= 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.




MP3 Create viable arguments and critique the reasoning of others.




MP8 Look for and make use of repeated reasoning.

LESSON

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LESSON

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Essential

Understandings

Materials

Needed







SET-UP PHASE

Read the directions and complete the worksheet privately first. You will be able to talk with your group after everyone has had time to complete the worksheet.

LESSON

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8

EXPLORE PHASE


Pathways

Struggling to put all of the fractions on the number line.

Writes for the shaded portion of the figures in question 1, but has not written two other equivalent fractions.

Student indicates that is greater than because there are 8 pieces.


Which of the fractions do you recognize? Can you put this fraction on the number line first?

Tell me about . Where do you see this in the picture?


It is hard to put all of these different fractions on the same number line. You always have the right to make other number lines underneath this one. See if you can put the eighths on this number line.

What do you notice about the shaded and unshaded portion of the figure? What is another name for 12 pieces of the 24 pieces?

Tell me what you know about

. If I compare to or 1, which fraction is it closest to and how do you know?

Can you use what you know about in comparison to

1 whole to compare this amount with now that you know it is equal to one half?

Show the greater than or less than sign and write an explanation.



LESSON

GUIDE

8


The shaded portion of question 1 and 3.


• Tell us how you named the shaded portion of Figure 1. ( , , or .)

• The denominator describes the size of the piece. How many students agree with this statement? If this is true, then point to of the figure. Point to of the figure and tell us about of the figure.

• What do you know about , , and ? What is the same? What is different? (They are equivalent.)

• How do you know this?

(2 is half of 4, 12 is half of 24.)

• Yes, one way to determine if they are equivalent is to ask, “What is the relationship between the numerator and the denominator?” If the fraction is half, then the numerator will be half of the denominator. (Marking)

• Who agrees? Who disagrees?

Discussion of Question 3:

• What fractions did you write for question 3?

• How can this figure show half when all of the shaded squares are not in a row?

(Challenging) What do you think about this?

Questions 3 and 4.


• Were there any fractions that shared the same point on the number line? If so, which ones?

What does this mean when they share the same point?

( = 1 and = and = .)

• Can you explain each of these? What do you know about = 1? How can = 1?

(Challenging) (Since all five of the fifths are shown in the fraction , this shows 1 whole.)

• Stop and jot. Is this true for other fractions? Write three fractions that equal one whole.

(Challenging)

• Can anyone point to on the number line?

• Anytime the numerator is the same as the denominator we have one whole shown as a fraction. (Marking)

• Does this fraction, , show one whole? (No.) Why not?

(It is one fifth more.)

• Who understands what she said and can put this in their own words?

(It is one piece more than the whole.)

• Yes, it shows 1 whole and . (Marking) Stop and jot and tell me about this fraction, .

• Tell me about and . (These fractions show halves.) How did you know these were names for ?

• Give me all of the names for that you can name. Stop and jot. I will be grading this.



Comparisons using benchmarks, questions 3 and 5.


• Did anyone use the benchmarks of and 1 to place any of the fractions on the number line?

If so, which fractions?

• How can you use benchmark fractions to compare the fractions in number 5?

(I know that is away from the whole. I know that is a little over half so it is less than .)

• Slow down. How many students understood what she said and can explain this to their partner? Turn and talk. How did she use two different benchmarks to compare the fractions?

Application No application.

Summary What part of our work today was the easiest for you? Turn to your partner and describe what you understand. What part of our work today was difficult for you?

Quick Write No Quick Write for students.



2. Ask students to listen as you read the names for . Ask students to tell you what is the same about each of the fractions.

(They are all names for . The numerator is half of the number in the denominator.)

LESSON

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LESSON

GUIDE

1

Institute for Learning

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