RESEARCH—BEST PRACTICES Putting Research …€”BEST PRACTICES Putting Research into Practice ... can compare fractions such as 2_ 5 ... Examples from Unit 7 MP.2 Reason Quantitatively

$Page 1: RESEARCH—BEST PRACTICES Putting Research …€”BEST PRACTICES Putting Research into Practice ... can compare fractions such as 2_ 5 ... Examples from Unit 7 MP.2 Reason Quantitatively$
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RESEARCH—BEST PRACTICES

Putting Research into Practice

Dr. Karen C. Fuson, Math Expressions Author

From Our Curriculum Research Project: Understanding Decimal Numbers

Relating decimals to fractions helps with reading decimal fractions because they are read as if they were written as regular fractions: 0.43 is 43 ___ 100 read as a whole number with a fraction label “forty-three hundredths.”

Students use a place value chart to discuss the symmetry of the base-ten system about the ones place (not about the decimal point) and to discuss the general “multiply by ten” relationships as one moves one place to the left and the “divide by ten” relationships as one moves one place to the right. Students also use a length model and the Number Path to visualize decimal fractions.

Students use their understanding of decimal fractions to understand the big idea about adding, subtracting, and comparing decimal fractions: you can only add, subtract, and compare like decimal fractions.

Throughout the unit, word problems allow students to use their ideas about decimal fractions in problem-solving situations. As always, students generate as well as solve such problems and situations.

From Current Research: Representing Decimal Numbers

Decimals with one digit to the right of the decimal point partition each unit interval on the number line into subintervals of length 1 __ 10 , and decimals with two digits to the right of the decimal point refine this to intervals of length 1 ___ 100 , with 10 of these fitting into each interval of length 1 __ 10 … As you can see, space between these numbers is already rather small. It would be very difficult to draw a picture of the next division, defined by decimals with three digits to the right of the decimal point. Nonetheless, you can imagine this subdivision process continuing on and on, giving finer and finer partitions of the line.

UNIT 7 | Overview | 597Q

Research & Math BackgroundContents Planning

Fuson, Karen C., SanGiovanni, John, Adams, Thomasina Lotte. Focus in Grade 5: Teaching with Curriculum Focal Points. NCTM Reston, VA 2009

Van de Walle, John A., Karp, Karen., Bay-Williams, Jennifer. M. Elementary and Middle School Mathematics: Teaching Developmentally (Seventh ed.). 2010. Boston: Allyn and Bacon.

National Research Council. “Developing Proficiency with Whole Numbers.” Adding It Up: Helping Students Learn Mathematics. Washington, D.C.: National Academy Press, 2001. pp. 88–89, 236.

Modeling Fractions

During grades 3–5, students should build their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths. By using an area model in which part of a region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as 1 _ 2 and 1. For example, fifth-graders can compare fractions such as 2 _ 5 and 5 _ 8 by comparing each with 1 _ 2 — one is a little less than 1 _ 2 , and the other is a little more. By using parallel number lines, each showing a unit fraction and its multiples (see fig. 5.1), students can see fractions as numbers, note their relationship to 1, and see relationships among fractions, including equivalence.

They should also begin to understand that between any two fractions, there is always another fraction.

Figure 5.1:

0 1–2 1

0 2–4

1–4

3–4 1

0 4–8

2–8

3–8

5–8

6–8

7–8

1–8 1

National Council of Teachers of Mathematics Principles and Standards for School Mathematics (Number and Operations Standard for Grades 3–5) Reston: NCTM, 2000. pp. 148, 149

Other Useful References: Decimal Numbers

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Getting Ready To Teach Unit 7Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages students to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.

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

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

TEACHER EDITION: Examples from Unit 7

MP.1 Make Sense of Problems Justify Reasoning Discuss why 3 tenths = 30 hundredths. Students should mention the following ideas:

→ 3 dimes represent the same amount as 30 pennies.

→ Dividing each of the 3 tenths of the bar into 10 equal parts makes 30 smaller, equal hundredths parts.

→ Multiplying the numerator and the denominator of 3 __ 10 by 10 makes the equivalent fraction 30 ___ 100 because the multiplication divides each tenth into ten equal parts.

Lesson 9

MP.1. Make Sense of Problems Analyze the Problem Have students solve Problems 25–30 on page 260. They need to analyze the problem situations to recognize the following.

Problems 25 and 26: 1 week is the same as 7 days.

Problems 27 and 28: 1 dollar is the same as 100 cents.

Problems 29 and 30: Students may convert the decimals to fractions to add or multiply. Also, 1.0 mile is the decimal equivalent of 1 mile.

Lesson 10

Mathematical Practice 1 is integrated into Unit 7 in the following ways:

Make Sense of ProblemsAnalyze RelationshipsAnalyze the ProblemUse Appropriate Methods

Look for a PatternDraw a DiagramMake a GraphMake a Table

Check AnswersUse a Different MethodJustify ReasoningReasonable Answers

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Mathematical Practice 2Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves comparing fractions with like and unlike denominators and comparing decimals.


MP.2 Reason Quantitatively On the board or overhead, draw the number line shown below and write the fraction 1 _ 8 nearby. Ask students to think about how 1 _ 8 is related to the numbers on the number line.

0 1 2

1

Lesson 2

MP.2 Reason Abstractly and Quantitatively Connect Symbols and Models To complete decimal comparisons, students might choose to use their Decimal Secret Code Cards or the MathBoard squares. Those students who use MathBoards should write the decimal numbers in the middle of the board, then write a > or < sign. Before writing the sign, some students may find it helpful to first underline the larger number.

Have students begin the comparisons by identifying any equivalent pairs that may be present. Comparison methods include: using the .00 card, the MathBoard squares, thinking about the numbers as money amounts, or writing an equivalent number of hundredths for each number in tenths.

Lesson 10


Reason QuantitativelyConnect Symbols and WordsConnect Symbols and Models

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

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

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

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


MP.3 Construct a Viable Argument Compare Methods Students should work in Small Groups to repeat Exercises 10–14, but finding a common numerator. Have students discuss whether they prefer one method over the other and why they might choose a certain method. Student discussion may include that sometimes the computation to find a common numerator involves lesser numbers.

Lesson 6

What’s the Error? W H O L E C L A S S

MP.3, MP.6 Construct Viable Arguments/Critique Reasoning of Others Puzzled Penguin Give students an opportunity to read and respond to Puzzled Penguin on page 240. Students should notice that the numerator and denominator of 5 _ 6 are not multiples of the numerator and denominator of 2 _ 3 by the same number.

Lesson 4


Construct a Viable ArgumentPuzzled Penguin

Justify ConclusionsCompare Methods

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Mathematical Practice 4Model with mathematics.

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


MP.1, MP.4 Make Sense of Problems/Model with Mathematics Make a Graph Have students work in Student Pairs to solve Exercises 7–11 on page 250. Students should discuss how the two line plots help them find repeated values or clusters of values in the data.

Lesson 7

MP.4 Model with Mathematics Draw a Diagram Send students to the board and have students at their seats make an informal sketch showing 8 _ 4 = 2. (Each of two congruent rectangles should be divided into four equal shaded parts.) Students should discuss how their diagrams relate to the number of pennies in $2. Ask them to write out equations using 1 _ 4 and 0.25 to support their answers.

Lesson 8


Model with MathematicsDraw a Diagram

Decimal Secret Code CardsMake a Graph

Make a Table

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Mathematical Practice 5Use appropriate tools strategically.

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

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


MP.5 Use Appropriate Tools Draw a Diagram Students may want to draw a number line or use the diagrams from Exercises 6–8 to help them complete Exercises 9–11. Students should circle lengths on the number lines to compare the fractions.

Lesson 2

MP.5 Use Appropriate Tools Decimal Secret Code Cards Working together as a class, complete exercises. Encourage students to use their Decimal Secret Code Cards from Lessons 9 and 11 either to complete exercises or to check their work. The goal of the exercises is to remind students that the general method they used to compare whole numbers can also be used to compare decimals.

Lesson 12


Use Appropriate ToolsMathBoardClass Multiplication Table

Paper ModelDecimal Secret Code Cards

Draw a DiagramModel the Math

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Mathematical Practice 6Attend to precision.

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


MP.6 Attend to Precision Explain a Method Have students solve Exercises 6–11 and discuss the answers. Ask them to explain why the two fractions in each exercise are equivalent. They should recognize that they are multiplying the numerator and denominator of the original fraction by the same number to produce more but lesser unit fractions.

Lesson 4

MP.6 Attend to Precision Explain a Representation Invite a volunteer to count the number of students in your class, and then plot and label that number on the number line. Discuss the position of the number by asking students to decide if the number is closer to 10, closer to 25, or closer to 50, and give reasons to support their answer.

Point out that comparing the number of students in class to 10, 25, and 50 is an example of using “benchmarks.”

Lesson 2

MATH TALK Ask students to study the Place Value Chart on Student Book page 261 and look for symmetries and relationships. Then discuss the symmetries and relationships students see.

Lesson 11

MATH TALKin ACTION

Students discuss how to read and write 0.28 and how different representations are related.

Damon: The number represents 28 ___ 100 and a fraction with a denominator of 100 is a fraction in hundredths.

Jorge: Think of the number as an amount of money, but say hundredths instead of cents—$0.28 is twenty-eight cents.

Emily: So 0.28 is twenty-eight hundredths because a penny is one hundredth of a dollar.

Lesson 9


Attend to PrecisionDescribe a Method

Explain a MethodExplain a Representation

Puzzled Penguin

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Mathematical Practice 7Look for structure.

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


MP.7 Look for Structure Ask students to suggest reasons why there is no “oneths” place. The decimal point misleads some students into thinking that the symmetry is around the decimal point, but instead it is around the ones place—the base for all of the other places. The discussion should point out that as we move to the right on the place value chart, we divide each place by 10. When we move from the ones place to the tenths place, we find that the tenths place represents 1 divided by 10, which produces 10 equal parts of 1, or tenths.

Lesson 11

MP.7 Look for Structure Identify Relationships Ask students to look carefully at the denominators of the fractions in each pair, and discuss how they are related. Be sure that the discussion includes the following. For 3 _ 5 and 5 __ 10 ,

• One denominator is a factor of the other (5 is a factor of 10).

• The greater denominator, 10, is the least denominator that will work as the common denominator.

• Using 10 as the common denominator requires us to rewrite only one of the fractions.

For 5 _ 8 and 4 _ 5 ,

• 1 is the only number that is a factor of both denominators.

• The product of 5 and 8 is the least denominator that will work as the common denominator.

For 5 _ 8 and 7 __ 12 ,

• 4 is a factor of both denominators.

• 24, which is less than the product of the denominators, will work as a common denominator.

Lesson 6


Look for Structure Use Structure Identify Relationships

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► Math and Autumn LeavesThe weather in different parts of the United States has a noticeable effect on plants and trees. In warm parts of the country, trees can keep their leaves all year long. In the northern states, fall weather causes leaves to change color. People from around the country plan trips to see and photograph the red, yellow, orange, and brown leaves. A fall leaf-viewing trip could involve driving through a national forest, biking along a rail trail, or hiking into the mountains.

Solve.

1. One popular park to photograph leaves in autumn is Macedonia Brook State Park in Kent, Connecticut. The Yellow Trail is the shortest hiking trail and is 51 ____ 100

mile long. What is this fraction written as a decimal?

2. The Rogers family is visiting Massachusetts to see the leaves change color. The Old Eastern Marsh Trail is 1 2 __

5 miles long. The Bradford Rail Trail is 1 3 ___

10 miles long.

The Rogers family wants to take the longer trail. Which trail should they take?

Show your work.

Name Date

0.51 mile

The Old Eastern Marsh Trail

UNIT 7 LESSON 13 Focus on Mathematical Practices 265

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► Clarkston ParkJoshua and Lily are going north to participate in a walking tour in Clarkston Park to photograph the leaves. Here is the trail map of the different walking trails.

3. Write the length of the West Trail as afraction.

4. Which trail is shorter: the West Trail or the North Trail? Write the comparison using >, <, or =.

5. Write the length of the Lower Trail as a fraction.

6. Which trail is longer: the Lower Trail or the South Trail? Write the comparison using >, <, or =.

7. Write a fraction that is equivalent to the length of the North Trail.

8. Use the number line below and the benchmark fractions to name which trail is represented by each point.

A:

B:

C:

D:

E:

F:

G:

Name Date

Mid Trail Upper Trail

East Trail North Trail

Lower Trail West Trail

South Trail

2 9 ___ 10

miles

The North Trail; 2 3 __ 5 < 2 9 ___

10

1 4 ___ 10

miles

The South Trail; 1 4 __ 8 > 1 4 ___

10

Possible answer: 2 6 ___ 10

266 UNIT 7 LESSON 13 Focus on Mathematical Practices


STUDENT EDITION: LESSON 13 PAGES 265–266

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

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


MP.8 Use Repeated Reasoning Ask students to work together to describe how to write the expanded form of a decimal number. Their results should include the following.

• Moving from left to right, find the total value of each digit in the number by multiplying the place value by the digit.

Lesson 11

MP.8 Use Repeated Reasoning Generalize Again, encourage students to generalize what they have learned about comparing two fractions with the same denominator and different numerators. Students should be able to verbalize that for fractions that have the same denominator, the fraction with the greater numerator is the greater fraction because it is made of more unit fractions.

Lesson 1


Use Repeated Reasoning GeneralizeDraw Conclusions

FOCUS on Mathematical Practices Unit 7 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson students use what they know about fractions and decimals to solve problems about autumn leaves and walking trails.

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Getting Ready to Teach Unit 7Learning Path in the Common Core StandardsIn this unit, students build upon the concepts of fractions presented in Unit 6. The number line is presented as a powerful model to represent fractions. Students use their conceptual knowledge of fractions to develop procedures to compare fractions and to find equivalent fractions using common denominators.

Another goal of Unit 7 is to develop an understanding of decimal numbers by relating decimals to fractions and whole-number place values. Students are introduced to a variety of models for representing fractions and decimals as parts of a set (dimes and pennies), parts of a whole (decimal bar), and as a model of distance (number line). Students use the relationship between decimals and fractions to build decimal concepts, including decimal place value and comparing decimals less than and greater than 1.

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

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

→ Lesson 1: Incorrectly using denominators to compare fractions

→ Lesson 2: Forgetting to consider whether an overestimate or an underestimate is more appropriate

→ Lesson 4: Forgetting to multiply both the numerator and the denominator by the same number to write equivalent fractions

→ Lesson 6: Forgetting to multiply the numerator and denominator in a fraction by the same number

→ Lesson 10: When comparing decimals, not recognizing that the decimals being compared must refer to the same whole

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

UNIT 7 | Overview | 597AA


Fraction Concepts

Lessons

1 2 3 4 5 6

Fractions on the Number Line This unit introduces the number line as another way to model fractions. Students compare the number line model with the fraction bar models they used in Unit 6. Students can see that the number of divisions between 0 and 1 determines the unit fraction. In the number line below the interval between 0 and 1 is divided into four equal parts, so each mark indicates 1 _ 4 . The labels above the number line after 1 show numbers greater than 1. The labels below the line show the numbers as mixed numbers. Fractions greater than 1 and mixed numbers are different ways to represent the same distance on the number line.

0 1 2 3 52–4

3–4

1–4

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3–41 2–

42

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1–42 2–

43 3–43 2–

44 3–441–

43c db

The number line can be used to help students find relationships between mixed numbers and fractions greater than 1. For example, by looking at the number line above, it is clear that 13 __ 4 = 3 1 _ 4 .

Students also use the number line to explore fraction benchmarks by estimating whether a fraction is closer to 0 or 1. It is beneficial for students to develop the ability to compare fractions to 0, 1 _ 2 , and 1 because it will help them use mental math to compare fractions. For example, if students are comparing 3 _ 8 and 7 __ 12 , and they are able to reason that if 3 _ 8 is less than 1 _ 2 and 7 __ 12 is greater than 1 _ 2 , then 3 _ 8 is less than 7 __ 12 , they will be able to compare the fractions more easily.

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

Benchmarks Students also reason

using benchmarks such as 1 _ 2 and 1.

For example, they see that 7 _ 8 < 13 __ 12

because 7 _ 8 is less than 1 (and is

therefore to the left of 1) but 13 __ 12 is

greater than 1 (and is therefore to

the right of 1).

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Equivalent Fractions As students explore the number line, they observe that there are multiple ways to label the marks on a number line. This leads to a discussion that the different labels for the same point are equivalent fractions. For example, on these number lines, students can see that 1 _ 2 and 2 _ 4 are the same distance from 0, so 1 __

2 = 2 _ 4 .

0 1 2 3 51–2

1–2

2–2

3–2

4–2

5–2

6–2

7–2

8–2

10–2

0–2

41–21 1–

22 1–23 a

0 1 2 3 52–4

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3–41 2–

42

9–4

1–42 2–

43 3–43 2–

44 3–441–

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Fraction bar models and the multiplication table are also used to help students conceptualize equivalent fractions. In the fraction bar model below, students can see that the gray sections are the same length, so 1 _ 3 = 2 _ 6 = 3 _ 9 = 4 __ 12 = 5 __ 15 = 6 __ 18 .

1–9

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Students begin to see that, for example, the 1 _ 3 is partitioned into 2 smaller parts to get 2 _ 6 , into 3 smaller parts to get 3 _ 9 , into 4 smaller parts to get 4 __ 12 , and so on. This partitioning idea helps them conceptualize why you can multiply and divide the numerator and denominator by forms of 1 ( 2 _ 2 , 3 _ 3 , 4 _ 4 , and so on) to find equivalent fractions. They connect their understanding of fraction bar models to the multiplication table to further solidify this understanding.

× 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 101

2

3

4

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10

2

3

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6 9 12 15 18 21 24 27 30

× 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 101

3 3 6 9 12 15 18 21 24 27 30

8 12 16 20 24 28 32 36 40

10 15 20 25 30 35 40 45 50

12 18 24 30 36 42 48 54 60

14 21 28 35 42 49 56 63 70

16 24 32 40 48 56 64 72 80

18 27 36 45 54 63 72 81 90

20 30 40 50 60 70 80 90 100

× 6

Two rows from the multiplication table make series of equivalent fractions so that students see that these processes are general and can work for any numbers.


Equivalent Fractions Grade 4

students learn a fundamental

property of equivalent fractions:

multiplying the numerator

and denominator of a fraction

by the same non-zero whole

number results in a fraction that

represents the same number as

the original fraction. This property

forms the basis for much of their

other work in Grade 4, including

the comparison, addition, and

subtraction of fractions and the

introduction of finite decimals.


Partitioning Students can use area

models and number line diagrams

to reason about equivalence. They

see that the numerical process of

multiplying the numerator and

denominator of a fraction by

the same number, n, corresponds

physically to partitioning each

unit fraction piece into n smaller

equal pieces. The whole is then

partitioned into n times as many

pieces, and there are n times as

many smaller unit fraction pieces as

in the original fraction.

1 × 6 ____ 3 × 6

= 6 __ 18

6 ÷ 6 _____ 18 ÷ 6

= 1 _ 3

UNIT 7 | Overview | 597CC


Comparing Fractions In Lesson 1, students revisit the concept of unit fractions. Students understand that the greater the number of parts (denominator), the smaller the fraction. They leverage this understanding to compare unit fractions and fractions with the same denominator or same numerator. For example, to compare 2 _ 3 and 2 _ 5 , they reason that since fifths are smaller than thirds, 2 fifths is smaller than 2 thirds and 2 _ 5 < 2 _ 3 , or 2 _ 3 > 2 _ 5 . Fraction bar models are presented to help students visualize this concept.

1–5

1–5

1–5

1–5

1–5

1–3

1–3

1–3

Fraction bars are also used to help students compare fractions with different numerators and the same denominator. This model shows that 2 _ 5 < 3 _ 5 .

1–5

1–5

1–5

1–5

1–5

Students also explore comparing fractions of different-sized wholes. Models help them visualize that, for example, 1 _ 8 of a bigger whole is greater than 1 _ 8 of a smaller whole.

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Comparing Fractions Grade 4

students use their understanding

of equivalent fractions to compare

fractions with different numerators

and different denominators. For

example, to compare 5 _ 8 and 7 __ 12

they rewrite both fractions as

60 ___ 96

(= 12 × 5 ______ 12 × 8

) and 56 __ 96 (= 7 × 8 ______ 12 × 8

)

Because 60 __ 96 and 56 __ 96 have the same

denominator, students can compare

them using Grade 3 methods and

see that 56 __ 96 is smaller, so 7 ___ 12

< 5 __ 8 .

Students use their understanding of the number line to identify that lesser fractions and mixed numbers are located to the left and greater fractions and mixed numbers are located to the right. So, on the number lines below, since 3 _ 5 is to the right of 5 __ 10 , 3 _ 5 > 5 __ 10 .

0 810

910

610

710

410

510

210

110

310

1

0 15

25

35

45

1

Once students have developed a conceptual understanding of comparing fractions, they learn how to compare by finding equivalent fractions with a common denominator. Different strategies are introduced depending on the relationship between the denominators.

Case 1: One denominator is a factor of the

other.

Possible Strategy: Use the greater

denominator as the common denominator.

Example Compare 3 __ 5 and 5 ___

10 .

Use 10 as the common denominator.

3 × 2 _____ 5 × 2

= 6 ___ 10

6 ___ 10

> 5 ___ 10

, so 3 __ 5 > 5 ___

10 .

Case 2: The only number that is a factor of

both denominators is 1.

Possible Strategy: Use the product of the

denominators as the common denominator.

Example Compare 5 __ 8 and 4 __

5 .

Use 5 × 8, or 40, as the common

denominator.

5 × 5 _____ 8 × 5

= 25 ___ 40

4 × 8 _____ 5 × 8

= 32 ___ 40

25 ___ 40

< 32 ___ 40

, so 5 __ 8 < 4 __

5 .

Case 3: There is a number besides 1 that is a

factor of both denominators.

Possible Strategy: Use a common

denominator that is less than the product of

the denominators.

Example Compare 5 __ 8 and 7 ___

12 .

24 is a common multiple of 8 and 12. Use 24

as the common denominator.

5 × 3 _____ 8 × 3

= 15 ___ 24

7 × 2 ______ 12 × 2

= 14 ___ 24

15 ___ 24

> 14 ___ 24

, so 5 __ 8 > 7 ___

12 .

UNIT 7 | Overview | 597EE


Decimal Concepts

Lessons

8 9 10 11 12

Models for Fractions and Decimals In Lesson 8, students use what they know about dimes, pennies, and dollars and fractional parts of a set to relate fractions and decimals. They learn that fraction and decimal numbers are different ways of writing the same value. The goal is for students to think of hundredths, whether written as a fraction or as a decimal, as pennies in a dollar and think of tenths as dimes in a dollar, when a dollar is used to represent 1 whole.

Students extend the models to represent halves, fourths, and numbers greater than 1. The following array models show the relationships between money, fractions, and decimals.

110

0.100.1

1 of 10 equal parts

10100

10 of 100 equal parts

1100

1 penny = = 0.01

1whole

4 of 4 equalparts

==++

14

14

+44

14

14

=100100

++25

100+

25100

25100

25100

0.25 + 0.25 + 0.25 + 0.25 = 1.00

0.25 + 0.25 + 0.25 + 0.25 + 0.25 + 0.25 + 0.25 = 1.75

25100

25100

25100

+25

100+

25100

+25

100+ +

25100

+175100

=100100

= = =75

100+

75100

1 +751001

7 of 4 equal parts

14

14

14

74

+ +14

+

44

34

+

14

14

+ +14

+ =314

=

597FF | UNIT 7 | Overview

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Decimals Fractions with

denominator 10 and 100, called

decimal fractions, arise naturally

when students convert between

dollars and cents, and have a

more fundamental importance,

developed in Grade 5, in the base

10 system. For example, because

there are 10 dimes in a dollar, 3

dimes is 3 __ 10 of a dollar; and it is

also 30 ___ 100 of a dollar because it is 30

cents, and there are 100 cents in a

dollar. Such reasoning provides a

concrete context for the fraction

equivalence

3 ___ 10

= 3 × 10 _______ 10 × 10

= 30 ____ 100

Students also explore decimal numbers and their fractional equivalents by using bars divided into tenths and hundredths. This bar model connects back to the fraction strips and number lines that students used to represent fractions. This model emphasizes thinking of fractions and decimals as parts of a whole. Tenths written as fractions and decimals are on a bar model divided into 10 equal parts. Hundredths written as fractions and decimals are on a bar divided into 100 equal parts.

1—10

0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.100 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

+ 0.1 1—10

1—10

++ 0.1 1—10

++ 0.1++ 0.1 1—10

++ 0.1

10100

20100

30100

40100

50100

60100

70100

80100

90100

100100

D D D D D D D D D D

From the models above, students can visualize the relationship between tenths and hundredths and the equivalence between these numbers written as fractions and decimals. The D on the top bar model emphasizes that each section of the bar represents a dime, so the bar model also helps students connect the concept back to money.

Decimal Secret Code Cards Students explore decimal place value by assembling Secret Code Cards to form decimal numbers. The cards are used to show tenths, hundredths, and decimals both less than and greater than one. These cards help students remember the underlying concept of place value as they build decimal numbers. To make the decimal number 0.37, for example, students select the cards representing 3 tenths and 7 hundredths.

0 0 7.0 07.

0 3.0 3.

The 0.3 card is placed over the 0.07 card to create the number 0.37.

0 0 7.0 07.

0 3.0 3.

UNIT 7 | Overview | 597GG


Secret Code Card Backs The backs of the cards show the corresponding number of dimes or pennies so students can make the connection to the models they used in Lesson 8.

3 dimes are on the back of the 0.3 card.

7 pennies are on the back of the 0.07 card.

Using the cards is beneficial for students because they are able to connect their understanding of whole number place value and the whole number Secret Code cards to the new concept of decimal place value.

Students also use the Secret Code Cards and the other models used in this unit, including their MathBoard, to help them compare decimal numbers. By looking at the back of the cards, students are able to visualize the comparison using the money representation. Lesson 12 shows students how to write zeros in decimal numbers so that the value of the number does not change, but so that it is easier to compare.


Comparing Decimals Students

compare decimals using the

meaning of a decimal as a fraction,

making sure to compare fractions

with the same denominator. For

example, to compare 0.2 and 0.09,

students think of them as 0.20

and 0.09 and see that 0.20 > 0.09

because

20 ____ 100

> 9 ____ 100

The argument using the meaning

of a decimal as a fraction

generalizes to work with decimals

in Grade 5 that have more than

two digits, whereas the argument

using a visual fraction model,

shown in the margin, does not. So

it is useful for Grade 4 students to

see such reasoning.

With the places aligned and the extra zeros added, we can see which is greatest.

Problem:

Which of these numbers is the greatest: 2.35, 2.3, or 2.4

Solution:

2.352.302.40

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Decimal Place Value Lesson 11 presents a place value chart to help students understand how to show and read decimals. The Math Expressions place value chart is designed to emphasize that the “ones place” not the decimal point is the center of the chart. Students observe that tens and tenths and hundreds and hundredths are symmetric about the ones place.

× 10 (Greater) ÷ 10 (Lesser)

1.10. 0.1100. 0.01

1001

101

11

110

1100

$1.00$10. 0$00 .10$100. 0$00 .01

1 101 101 101 10

Students further model place value by placing the Decimal Secret Code Cards on the Place Value frame.

ONES

Hu

nd

reds

Tens

Tenth

s

Hu

nd

redth

s

andtenths

hundredths

Place value

Make numbers

Read numbers

Using the frame is beneficial to students because it helps them make and read numbers. The cards also help students to write decimal numbers in expanded form. Notice that the value of the number on the card still appears when the numbers are assembled, so students can more easily see the number as a sum.

20 + 3 + 0.5

0 5.0 5.

2 020

33

2 020

33

0 5.0 5.

UNIT 7 | Overview | 597II


Problem Solving

Lessons

3 6 7

Problem Solving Plan In Math Expressions a research-based problem solving approach that focuses on problem types is used.

• Interpret the problem• Represent the situation• Solve the problem• Check that the answer makes sense.

Real World Applications of Fractions and Decimals Throughout the unit, real world scenarios are used to develop meanings for fraction and decimal concepts. Students learn to solve problems involving finding factions of different-size wholes. They solve problems in which it is necessary to compare fractions or determine if fractions are equivalent. Students apply their understanding of decimal numbers to determine, for example, what part of a mile will be run in 8 days if 0.1 of a mile is run one day.

Line Plots In Unit 7, as well as in Unit 6, students apply their understanding of fractions to solve problems using line plots. They make line plots given fractional data and use the line plots to analyze the data and solve problems.

Hand Width (in inches)

278

234

258

212

238

214

0

Width (in inches)

Number of Students

2 1 __ 4 1

2 3 __ 8 2

2 1 __ 2 2

2 5 __ 8 4

2 3 __ 4 2

2 7 __ 8 1

Focus on Mathematical Practices

Lesson

13

The standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, students use what they know about fractions and decimals to solve problems involving autumn leaves and walking trails.

597JJ | UNIT 7 | Overview

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RESEARCH—BEST PRACTICES Putting Research …€”BEST PRACTICES Putting Research into Practice ... can compare fractions such as 2_ 5 ... Examples from Unit 7 MP.2 Reason Quantitatively