ReseaRch—BesT PRacTIces Putting Research into Practice · For example, fifth-graders can compare fractions such as 2_ 5 and 5_ 8 by comparing each with 1_ 2 — ... Elementary and

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ReseaRch—BesT PRacTIces

Putting Research into Practice

Dr. Karen C. Fuson, Math Expressions Author

From Our Curriculum Research Project:

TMEP (Toward a Mathematics Equity Pedagogy) sets high-level goals for learning with understanding, for high-level oral language competencies, and for the sophisticated use of mathematical modeling and mathematical symbolizing. These high-level goals are achieved by enabling all children to enter the mathematical activity at their own level. Teachers accomplish this by using rich and varied language about a given problem so that all children come to understand the problem situation, by mathematizing (focusing on the mathematical features of) a situation to which all children can relate (and that may be generated by a child), and by having children draw models of the problem situation.

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 figure 5.1 on p. 513P), 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.

UNIT 6 | Overview | 513O

Research & Math BackgroundContents Planning

Mack, Nancy K. Connecting to Develop Computational Fluency with Fractions. Teaching Children Mathematics, November, 2004, Vol. 11, Issue 4, p. 226.

Van de Walle, John A. Computation with Fractions. Elementary and Middle School Mathematics: Teaching Developmentally (Third Edition). New York: Longman, 2001. pp. 260–273.

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:

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

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

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

TeACHeR eDITION: examples from Unit 6

MP.1 Make Sense of Problems When adding mixed numbers, the wholes and the fractions are added separately. After the addition, the sum of the fractions might be a fraction greater than 1. This fraction then needs to be converted to a new mixed number. The additional whole is added to the other wholes.

Students can use Fraction Strips, draw fraction bars, or solve each problem numerically. If necessary, show students that they can add the wholes and fractions separately.

Lesson 5

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Draw a Diagram Problem 35 asks students to make a diagram to show that one of their solutions is correct. Students can make fraction bars or use another type of drawing that makes sense to them. Give students a couple of minutes to make their drawings and then choose volunteers to present and explain their work.

Lesson 6

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

Make Sense of ProblemsLook for a Pattern

Draw a DiagramMake a Graph

Analyze the ProblemMake a Model

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ACTIVITY 3


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 understanding the meaning of fractions and mixed numbers in real world situations.

TeacheR ediTion: examples from Unit 6

MP.2 Reason abstractly and Quantitatively Connect Symbols and Words Most fraction denominator words are made by adding “th” or “ths” to the name of the number (for example, fourths, fifths, sixths, and so on).

Write several whole numbers greater than 3 on the board and then write fractions using the numbers as denominators. Have the students say the names of the fractions as you point to them.

Lesson 1

MP.2 Reason abstractly and Quantitatively Connect Symbols and Models Discuss with students how to make each mixed number with their Fraction Strips. Ask:

• How do you know how many 1 whole strips are needed to make the mixed number? The whole number in the mixed number tells me how many 1 whole strips to use.

• How do you know how many 1 _ 5 parts of a Fraction Strip are needed to make the mixed number? The numerator in the mixed number tells me how many fifths are needed.

Student pairs should build each mixed number with their Fraction Strips.

Lesson 4


Reason Abstractly and QuantitativelyReason QuantitativelyConnect Diagrams and Equations

Connect 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 Some students may have devised numeric methods for changing the mixed numbers to fractions. Invite these students to share their methods with the class. Be sure the following methods are discussed.

• Decompose the Mixed Number Write the whole number part as a sum of 1s. Then write each 1 as d _

d  (in this case, 5 _ 5 ).

Add all of these wholes to the fraction part to get a fraction.

2 2 _ 5 = 1 + 1 + 2 _ 5 = 5 _ 5 + 5 _ 5 + 2 _ 5 = 12 __ 5 or 2 2 _ 5

• Multiply and then Add Multiply the whole number part by 5 to find the number of fifths in the whole part, then add the number of fifths in the fraction part. If needed, give students additional mixed numbers to build with their fifth strips and convert to mixed numbers.

Lesson 4

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 Read the first Puzzled Penguin letter on Student Book page 205 aloud and discuss possible responses as a class. Puzzled Penguin added both the numerators and the denominators. Emphasize to students that a unit fraction is a unit, just like cups or miles, or even fish. When we add like fractions, the unit does not change. Write the following on the board to help make this point.3 miles + 2 miles = 5 miles

3 fish + 2 fish = 5 fish

3 fifths + 2 fifths = 5 fifths

3 __ 5 + 2 __ 5 = 5 __ 5

Lesson 3


Construct a Viable ArgumentCritique the Reasoning of Others

Compare MethodsPuzzled Penguin

Compare Strategies

UNIT 6 | Overview | 513S

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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.

Teacher ediTion: examples from Unit 6

MP.4 Model with Mathematics Write an Equation Students are asked to write equations for Problems 32–35. In some cases, the situation equation will not be the solution equation. Students can use the relationship between addition and subtraction to write a solution equation. For example, in Problem 32, students may write 4 _ 5 - k = 2 _ 5 . This can be rewritten as k = 4 _ 5 - 2 _ 5 .

Lesson 3

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Make a Model Problems 20 and 21 ask students to draw a model. Problem 21 is a comparison problem, so some students may make comparison bars like those shown on the student page. Others will draw models similar to the others in this lesson, showing circles divided in halves or fraction bars with 7 halves shaded. Either type of drawing is fine.

Lesson 7


Model with MathematicsWrite an EquationMathBoard

Develop a FormulaDraw a DiagramMake a Graph

Make a ModelPaper ModelUse a Model

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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 Use a Concrete Model Write 2 _ 7 + 4 _ 7 on the board.

Instruct students to use a dry-erase marker to circle the addends on the sevenths bar. Work with students to write the addition equation under the bar as shown below. Be sure to include the circled step. It can help students remember not to add the denominators. In this case, they are adding 2 sevenths plus 4 sevenths. The result is 6 sevenths; the denominators are not added but stay the same.

1–7

1–7

2–7

4–7

6–7

1–7

1–7

1–7

1–7

1–7

2 + 4———7

+ = =

Lesson 3

MP.5 Use Appropriate Tools Class Fraction Cards Several students with the same Class Fraction Cards come to the front and stand in two groups. The Student Leader writes the addition problem and the class reads it.

1 8

1 8

1 8

1 8

1 8

1 8

1 8

78323.U05L07.02M Student Leader writes: 3 __ 8 + 4 __

8

Class “Three eighths plus four eighths equals seven eighths.”

The Student Leader writes the rest of the equation:

3 __ 8 + 4 __

8 = 7 __

8

Lesson 4


Use Appropriate ToolsClass Fraction Cards

Use a Concrete ModelPaper Model

MathBoardFraction Strips

UNIT 6 | Overview | 513U

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ACTIVITY 3


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 and express numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other.

TeAcher ediTion: examples from Unit 6

MP.6 Attend to Precision Explain Solutions Select some of the exercises and choose volunteers to present the solutions. Continue to emphasize that the unit fraction does not change when we add or subtract fractions, so the denominator stays the same.

Lesson 3

MP.6 Attend to Precision Describe Methods While students are doing their folding and marking, encourage them to share their strategies for folding. Also have them discuss what they notice about fractional parts as they fold. For example, they might observe that:

• Fourths are easy to make by folding a strip in half and then in half again.

• Thirds are harder to fold. One strategy is to form an S-shape and then press the shape flat, being careful to keep the three sections equal.

• Folding thirds in half or folding a half into thirds are both easy ways to make sixths.

Lesson 1

MATH TALK Exercises 5–22 on Student Book page 208 provide practice with converting between mixed numbers and fractions. You might use Solve and Discuss for one or two problems of each type. Be sure more than one method is presented for each type of conversion. Then let pairs work on the remaining problems.

Lesson 4

MATH TALKin ACTION

Have students share how they remember which symbol to use to compare fractions. A sample conversation follows.

Garret: The bigger, open end of the symbols goes near the greater number.

Marcia: The pointed, small end of the symbol points toward the smaller number.

Paolo: I use the “hungry fish” when I compare numbers. Because the fish is very hungry, it wants to eat the biggest thing it can find.

8 > 5

Write the opposite comparison on the board and have students read it aloud.

5 < 8

Lesson 2


Attend to PrecisionDescribe Methods

Explain SolutionsPuzzled 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 Write 2 _ 6 = 1 _ 6 + 1 _ 6 on the board and point out that it is the sum of two copies of 1 _ 6 . Ask students how they could show this sum as a multiplication. Write = 2 ⋅ 1 _ 6 after the sum. Repeat this for 3 _ 6 , 4 _ 6 , 5 _ 6 , and 6 _ 6 .

Lesson 1

MP.7 Look for Structure Identify Relationships Ask students how adding and subtracting fractions greater than 1 with like denominators is like adding and subtracting fractions less than 1 with like denominators. Only the numerators are added or subtracted; the denominators stay the same.

Lesson 5


Look for Structure Identify Relationships

UNIT 6 | Overview | 513W

ACTIVITY 1

ACTIVITY 1

Class Activity

4_2_MNLESE824543_37A.aiSam Valentino2.6.122nd pass

vegetables wheat

vegetablesItalianherbs

vegetables

dairycows

fruit

fruit

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► Math and Vegetarian Pizza FarmsA pizza farm is a circular region of land divided into eight pie-shaped wedges or slices, such as those you would see in a pizza. There are hundreds of such farms across the United States. At a vegetarian pizza farm, each wedge or slice grows a different vegetarian ingredient used to make a pizza. Some things you might find on a vegetarian pizza farm include wheat, fruit, vegetables, Italian herbs, and dairy cows.

Write an equation to solve.

A farmer created a vegetarian pizza farm with these wedgesor slices: 3 __

8 for vegetables, 1 __

8 for wheat, 2 __

8 for fruit,

1 __ 8 for dairy cows, and 1 __

8 for Italian herbs.

1. What fraction of the farm is made up of fruitor vegetables?

2. What fraction of the farm is not made up of wheat?

3. Which wedge of the farm is bigger, the wedge forfruit or the wedge for Italian herbs? Explain.

Show your work.

Name Date

Equations may vary.

v = 3__8+ 2__

8; v = 5__

8; 5__

8 of the farm

n = 8 __ 8 - 1 __

8 ; n = 7 __

8 ; 7 __

8 of the farm

the wedge for fruit; 2__8> 1__

8

UNIT 6 LESSON 10 Focus on Mathematical Practices 223

4_MNLESE824543_U06L10.indd 223 10/04/12 8:41 AM

Class Activity

Time Spent ResearchingDuring Study Time (in hours)

51

52

53

54

55

6-10

© H

oughton Mifflin H

arcourt Publishing C

ompany • Im

age Credits: ©

PhotoD

isc/Getty Im

ages

Write an equation to solve.

4. On Monday, two of the workers at the pizza farm each filled a basket with ripe tomatoes. Miles picked 15 1 __

6 pounds of tomatoes, and Anna picked

13 5 __ 6 pounds of tomatoes. How many more pounds

of tomatoes did Miles pick than Anna?

For Problems 5–6, use the line plot to solve.

After a field trip to a vegetarian pizza farm, Mrs. Cannon asked each of her students to use some of their study time to research different vegetarian ingredients for pizzas. The line plot below shows the amount of time each student spent researching during study time.

Show your work.

5. How many students spent at least 3 __ 5 hour

researching? Explain how you know.

6. How many hours in all did the students who researched for 2 __

5 hour spend researching?

Write a multiplication equation to solve.

Name Date

p = 15 1 __ 6 - 13 5 __

6 ; p = 1 2 __

6 more pounds of tomatoes

Equations may vary.

8 students; I counted the number of dots

4 × 2 __ 5 = h; h = 8 __

5 ; 8 __

5 or 1 3 __

5 hours

above 3 __ 5 , 4 __

5 , and 5 __

5 . 5 + 2 + 1 = 8

224 UNIT 6 LESSON 10 Focus on Mathematical Practices

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Student edItIOn: LeSSOn 10 pageS 223–224

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.

teacher edItIOn: examples from unit 6

Mp.8 use repeated reasoning Generalize Work with the class to write a general formula for multiplying a whole number, w, times a unit fraction, 1 _

d.

w ⋅ 1 __ d = w ⋅ 1 _____

d = w__

d

Lesson 7

Mp.8 use repeated reasoning Draw Conclusions Review the answers and discuss any problems that caused difficulty for students. Some students may have trouble with Exercise 7 because there is no fraction part to subtract from. Ask these students if they can make a fraction part by ungrouping the 7. Students should conclude that 7 can be rewritten as 6 4 _ 4 .

Lesson 9


Draw ConclusionsGeneralize

Conclude

Focus on Mathematical practices Unit 6 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson students use what they know about fractions to solve problems about pizza farms.

513X | UNIT 6 | Overview

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Getting Ready to Teach Unit 6Learning Path in the Common Core StandardsIn this unit, students build upon their previous understanding of fractions. The activities in this unit help students gain a conceptual and practical understanding of the parts of fractions, relationships among fractions, mixed numbers and fractions greater than 1, and operations with fractions and mixed numbers. Students are expected to apply their understanding of fractions to numerical calculations and real world problem solving situations.

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 understand and perform operations with fractions. 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 2: Incorrectly using denominators to compare fractions

→ Lesson 3: Adding or subtracting both numerators and denominators when adding or subtracting fractions

→ Lesson 5: When subtracting mixed numbers, subtracting the lesser fraction from the greater fraction without considering the order of the mixed numbers or ungrouping

→ Lesson 6: When ungrouping to subtract mixed numbers, forgetting to subtract 1 from the whole number part

→ Lesson 7: When multiplying a fraction by a whole number, multiplying both the numerator and the denominator by the whole number

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.

Math Expressions VOCABULARY

As you teach the unit, emphasize

understanding of these terms.

• numerator• denominator

See the Teacher Glossary.

UNIT 6 | Overview | 513Y

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Fraction Concepts

Lessons

1 2 4

Unit Fractions This unit is structured to build the understanding of unit fractions that students developed previously. Students begin by folding fraction strips to see each unit fraction, 1 _

d, as one of d equal

parts of the whole.

1 whole

1–3

1–3

1–3

1–4

1–4

1–4

1–4

1–6

1–6

1–6

1–6

1–6

1–6

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1—12

1–8

1–8

1–8

1–8

1–8

1–8

1–8

1–8

Folding the fraction strips allows students to visualize how to think of fractions as a sum of unit fractions. For example, folding the sixths strip so that 2 one sixths are showing demonstrates that 1 _ 6 + 1 _ 6 = 2 _ 6 .

1–6

1–6

Students apply their understanding of multiplication as repeated addition to write the sum of unit fractions as multiplication, as shown below.

9. 3 __ 4 = =

14

14

1414

14

14

1 __ 4 + 1 __

4 + 1 __

4 3 × 1 __

4

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Fraction bars are a powerful model that are used throughout the unit. Fraction bars help students represent the inverse relationship between the number and the size of unit fractions: a larger d in 1 _

d means more, but smaller, equal parts. Fraction bars also show

equivalent fractions ( 2 _ 4 = 3 _ 6 = 4 _ 8 ).

Conceptualizing unit fractions is beneficial because it helps students overcome typical errors in adding and subtracting fractions.

Building One Whole Students’ ability to visualize one whole as d_ d

( 4 _ 4 , 7 __ 7 , and so on) is crucial in understanding mixed numbers and ungrouping when subtracting mixed numbers. In Lesson 2, students write wholes as sums of unit fractions. For example, to find pairs of numbers that add to one whole, they think of putting together different combinations of one-fifth.

1–5

1–5

1–5

1–5

1–5 5 __ 5 = 3 __ 5 + 2 __ 5

1–5

1–5

1–5

1–5

1–5 5 __ 5 = 2 __ 5 + 3 __ 5

1–5

1–5

1–5

1–5

1–5

5 __ 5 = 1 __ 5 + 4 __ 5

Compare and Order Fractions Students use the fraction bar model to help them compare and order fractions. The models allow students to compare unit fractions as they begin to conceptualize why a unit fraction gets smaller as the denominator gets larger. This is a foundational understanding that students will continue to build upon as they move into more and more complex fraction concepts. Once students have developed an understanding of the magnitude of unit fractions, they can apply that understanding to compare, as well as order, unit fractions.

Mixed Numbers and Fractions Greater than 1 Fraction bar models also help students understand and visualize the relationship between mixed numbers and fractions greater than 1.

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1–5

1 whole

1 whole

1–5

1–5

1–5

The models above help students visualize that 2 wholes and 3 fifths is the same amount as 5 fifths, 5 fifths, and 3 fifths, or 13 fifths, therefore, 2 3 _ 5 = 13 __ 5 .

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

Mixed Numbers Converting a

mixed number to a fraction should

not be viewed as a separate

technique to be learned by rote,

but simply as a case of fraction

addition. Similarly, converting a

fraction greater than 1 to a mixed

number is a matter of decomposing

the fraction into a sum of a whole

number and a number less than 1.

Students can draw on their

knowledge from Grade 3 of whole

numbers as fractions. For example,

knowing that 1 = 3 _ 3 , they see

5 __ 3 = 3 __

3 + 2 __

3 = 1 + 2 __

3 = 1 2 _ 3 .

UNIT 6 | Overview | 513AA



Adding Fractions This simple

understanding of addition as

putting together allows students to

see in a new light the way fractions

are built up from unit fractions.

The same representation that

students used in Grade 3 to see a

fraction as a point on the number

line now allows them to see a

fraction as a sum of unit fractions:

just as 5 = 1 + 1 + 1 + 1 + 1, so

5 __ 3 = 1 __

3 + 1 __

3 + 1 __

3 + 1 __

3 + 1 __

3

because 5 _ 3 is the total length of

5 copies of 1 _ 3 . Armed with this

insight, students decompose and

compose fractions with the same

denominator. They add fractions

with the same denominator.

Add and Subtract Fractions

Lessons

3 5 6 9

Add and Subtract Fractions with Like Denominators This unit presents fraction strips and fraction bars along with unit fractions as methods of modeling and understanding fraction addition and subtraction.

1–7

1–7

2–7

4–7

6–7

1–7

1–7

1–7

1–7

1–7

2 + 4———7

+ = =

In the example above, students see that 2 sevenths plus another 4 sevenths gives 6 sevenths in all.

1–7

1–7

5–7

3–7

2–7

1–7

1–7

1–7

1–7

1–7

5 - 3———7

- = =

In the example above, students see that if they subtract 3 sevenths from 5 sevenths there are 2 sevenths left.

Note that the addition and subtraction of the numerators is written as a sum or difference above the denominator. This placement of the sum or difference emphasizes that only the numerators are added or subtracted. Additionally, the critical step of adding or subtracting only the numerators is circled as a transition for understanding. That step can be omitted later.

As students continually discuss how the numerical computations relate to the visual models, the computations become meaningful.

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Subtracting Fractions Using

the understanding gained from

work with whole numbers of the

relationship between addition

and subtraction, they also

subtract fractions with the same

denominator. For example, to

subtract 5 _ 6 from 17 __ 6 , they decompose

17 ___ 6 = 12 ___

6 + 5 __

6 , so 17 ___

6 - 5 __

6 = 17-5 _____

6 = 12 ___

6

= 2.

Add and Subtract Fractions Greater Than 1 In Lesson 5, students connect their understanding of fractions greater than 1 and adding and subtracting fractions to add and subtract fractions greater than 1. The fraction bar model is used to help students internalize the concept. This model shows that 4 _ 5 + 3 _ 5 = 7 _ 5 .

+

+4–5

3–5

This model shows that 8 _ 5 – 4 _ 5 = 4 _ 5 . Notice that to subtract 4 _ 5 , students cover up 4 fifths. They can see that there are 4 fifths left.

Add Mixed Numbers with Like Denominators As students move on to adding mixed numbers with like denominators, they begin to see that sometimes the sum of the fraction part of the mixed numbers is greater than 1. They learn that if the sum includes a fraction greater than 1, they need to convert it to a new mixed number. Using fraction strips helps students to conceptualize why this happens and how to make the conversion.

The lesson also presents the fraction greater than 1 method for adding mixed numbers. Students see that if they change the mixed numbers into fractions greater than 1 first, they will get the same answer as if they add the mixed numbers.

1 2 __ 5 + 3 4 __ 5 = 7 __ 5 + 19 ___ 5 = 26 ___ 5 = 5 1 __ 5

1 2 __ 5

+ 3 4 __ 5

_

4 6 __ 5

= 4 + 1 + 1 __ 5 = 5 1 __ 5

UNIT 6 | Overview | 513CC


Subtract Mixed Numbers with Like Denominators As students explore subtracting mixed numbers, they learn that sometimes the greater mixed number has a fraction part that is less than the lesser mixed number’s fraction part. In this instance, they need to ungroup a whole to make enough fractional parts to add. As with addition of mixed numbers, working with fraction bars helps students gain a conceptual understanding of this ungrouping. Students are shown two ways to record the ungrouping and subtraction.

6 2 __ 8 = 5 + 8 __

8 + 2 __

8 = 5 10 ___

8

-4 5 __ 8 = 4 5 __

8

_____

1 5 __ 8

As with addition, the method of first converting the mixed numbers to fractions greater than 1 is also discussed.

4 1 __ 5 - 1 3 __ 5 = 21 ___ 5 - 8 __ 5 = 13 ___ 5 = 2 3 __ 5

6 2 __ 8 = 5 10 ___

8

-4 5 __ 8 = 4 5 __

8

__

1 5 __ 8

5

8 + 2

513DD | UNIT 6 | Overview

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Multiplication of a fraction by a

whole number Previously in

Grade 3, students learned that 3 ⋅ 7

can be represented as the number

of objects in 3 groups of 7 objects,

and write this as 7 + 7 + 7. Grade 4

students apply this understanding

to fractions, seeing

1 __ 3 + 1 __

3 + 1 __

3 + 1 __

3 + 1 __

3 as 5 × 1 __

3 .

In general, they see a fraction

as the numerator times the

unit fraction with the same

denominator, e.g.,

7 __ 5 = 7 × 1 __ 5 , 11 ___ 3 = 11 × 1 __

3 .

The same thinking, based on the

analogy between fractions and

whole numbers, allows students to

give meaning to the product of a

whole number and a fraction, e.g.,

they see

3 × 2 __ 5 as 2 __ 5 + 2 __ 5 + 2 __ 5 = 3 × 2 _____ 5 = 6 __ 5 .

Multiply Fractions by Whole Numbers

Lessons

7 8 9

A Whole Number Times a Unit Fraction The understanding of how to use unit fractions as sums to create fractions and whole numbers thatstudents develop in the first part of this unit allows them to more readily understand the product of a whole number and a unit fraction.

In this example, students apply their understanding of adding 1 _ 8 to itself 3 times to help them multiply 3 by 1 _ 8 .

1 __ 8 + 1 __

8 + 1 __

8 = 1 + 1 + 1 ________

8 = 3 __

8

3 ⋅ 1 __ 8 = 3 ⋅ 1 ____

8 = 3 __

8

Students then generalize this procedure to write a formula for multiplying a whole number, w, times a unit fraction, 1 _

d.

w ⋅ 1 __ d = w ⋅ 1 _____

d = w__

d

Models continue to be used so that students can connect the concrete representation with the symbolic one.

It is the action of shading in 3 one-eighths sections of the fraction circle model above that helps students represent 1 _ 8 + 1 _ 8 + 1 _ 8 .

UNIT 6 | Overview | 513EE

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B


A Whole Number Times a Non-Unit Fraction Multiplying a whole number times a non-unit fraction is also presented in Lesson 7. Students use models to represent this concept before representing it symbolically.

Week 1 Week 2 Week 3

Week 4 Week 5

Week 1 Week 2 Week 3

3⋅2__6=3⋅2____6 =6__6 5⋅2__6=5⋅2____6 =10___6

As with multiplying a whole number by a unit fraction, students write a general formula for multiplying a whole number, w, times a fraction, n _

d  . The product is w groups of n  1 _

d  .

w ⋅ n __ d   = w ⋅ n ⋅ 1 __

d   or w ⋅ n __ 

d   = w ⋅ n _____

d   

513FF | UNIT 6 | Overview

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Problem Solving

Lessons

6 7 8 9

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 Mixed Numbers Throughout the unit, real world scenarios are used to develop meanings and computational methods for operations with fractions and mixed numbers. Students’ previous work with equations to solve problems involving whole numbers is applied as they write equations to solve problems involving fractions and mixed numbers. Addition, subtraction, and multiplication problem types, including multiplicative comparison situations, are revisited.

Line Plots Students apply their understanding of fractions to solve problems using line plots. They create line plots with fractional data and use the line plot to analyze the data. Students find patterns in line plots, including where there are data clusters, where there are gaps in the data, and the range of the data.

Distance (inches)

0 1 412

34

114

112

314

2 124

122

324

3 134

132

334

14

Focus on Mathematical Practices

Lesson

10

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 to solve problems involving vegetarian pizza farms.


Problem Solving with

Fractions When solving word

problems students learn to attend

carefully to the underlying unit

quantities. In order to formulate an

equation of the form A + B = C or

A – B = C for a word problem, the

numbers A, B, and C must all refer

to the same (or equivalent) wholes

or unit amounts.

UNIT 6 | Overview | 513GG


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Building a Math Talk CommunityMATH TALK Frequent opportunities for students to explain

their mathematical thinking strengthen the learning community of your classroom. As students actively question, listen, and express ideas, they increase their mathematical knowledge and take on more responsibility for learning. Use the following types of questions as you build a Math Talk community in your classroom.

Elicit student thinking

• So, what is this problem about?

• Tell us what you see.

• Tell us your thinking.

Support student thinking

• What did you mean when you said _______?

• What were you thinking when you decided to do _______?

• Show us on your drawing what you mean.

• Using wait time: Take your time… We’ll wait…

Extend student thinking

• Revoicing: So you’re saying that…

• Now that you have solved the problem in that way, can you think of another way to work on this problem?

• How is your way of solving like _______’s way?

• How is your way of solving different from _________’s way?

Increase participation of other students in the conversation

• Prompt students for further participation: Would someone like to add on?

• Ask students to restate someone else’s reasoning: Can you repeat what s/he just said in your own words?

• Ask students to apply their own reasoning to someone else’s reasoning:

• Do you agree or disagree and why?

• Did anyone think of this problem in a different way?

• Does anyone have the same answer, but got it in a different way?

• Does anyone have a different answer? Will you explain your solution to us?

What are specific math topic probes:

• What would happen if…?

• How can we check to be sure that this is a correct answer?

• Is that true for all cases?

• What pattern do you see here?

513HH | UNIT 6 | Overview

Documents

ReseaRch—BesT PRacTIces Putting Research into Practice · For example, fifth-graders can compare fractions such as 2_ 5 and 5_ 8 by comparing each with 1_ 2 — ... Elementary and