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Quotations and illustrations from Principles and Standards for School Mathematics regarding the teaching of fractions.

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and

 

 

 

* * * * * * * * * * * * * * * *

Excerpts from Fraction Bars Program showing activities that satisfy the Standards for teaching fractions.

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Standards for School Mathematics, page 33

Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. …

As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers.

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Fraction Bars Step-By-Step Teacher's Guide Equality Step 1, page 13

Activities Overview of Equality

1. Have students place their orange bars in a row from smallest to largest, as shown below.

 

 

 

2. Have students sort their red, blue, yellow, and green bars onto these 13 piles, so bars with the same shaded amount are in each pile. (see figure) Have students use these piles of bars to complete the table below.

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Standards for Grades 3-5, page 144

Equivalence should be another central idea in grades 3-5. Students' ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, ¾ can be thought of as a half and a fourth, as 6/8, or as 0.75.

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Fraction Bars Step-By-Step Teacher's Guide Equality Step 3, page 15

Activities Obtaining Equivalent Fractions

Place a clear transparency over the Fraction Bars transparencies before drawing the dotted lines. Water-based pens will wash off the plastic Fraction Bars. Keep a list of the equations generated in Activity 1 for use in Activity 2.

1. Show students and have them find, a blue bar with 1 part shaded. (1/4) Split each part of the bar into 2 equal parts, by drawing dotted lines as shown in the figure.

Ask students the following: How many total parts does the split bar now have? (8) How many shaded parts? (2) What fraction does the bar now represent? (2/8) What does this show about the fractions 1/4 and 2/8? (1/4 = 2/8)

2. Repeat the process of splitting each part of a bar into 2 equal parts to obtain equal fractions with several types of bars. (See figures for examples.)

Guide students to see that splitting each part of a bar into 2 equal parts is the same as multiplying by 2. It doubles the number of parts and doubles the number of shaded parts, but the total shaded amount remains the same.

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Standards for Grades 3-5, page 149

Numbers and Operations

Through the study of various means and models of fractions – how fractions are related to each other and to the unit whole and how they are represented – students can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1.

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Fraction Bars Step-By-Step Teacher's Guide Inequality Step 2, page 29

Activities Determining Inequalities by Comparing Fractions to 1/2

1. Ask students to determine if 5/12 is greater than 1/2 or less than 1/2 and to explain their reasoning. (less than 1/2 because a bar with 5 parts out of 12 is less than half shaded; or 1/2 = 6/12) Some students may wish to compare the shaded amounts of the 5/12 and 1/2 bars.

2. Repeat this activity by asking students to compare the following fractions to 1/2 and write the inequalities. Ask them to describe the bar for each fraction and explain their reason for each inequality. Some students may find it helpful to find or sketch the bar for the fractions.

(6/10, 8/14, 3/5, and 11/20 are greater than 1/2 and 4/9 and 6/15 are less than 1/2)

3. Summarize this activity by asking students to write a statement which tells how to determine when a fraction is less than 1/2 or greater than 1/2. (A fraction is less than 1/2 if its bar is less than half shaded and greater than 1/2 if its bar is more than half shaded.)

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Standards for Grades 6-8, page 216

Number and Operations

Visual images of fractions as fraction strips should help many students think flexibly in comparing fractions. As shown in figure 6.2, a student might conclude that 7/8 is greater than 2/3 because each fraction is exactly "one piece" smaller than 1 and the missing 1/8 piece is smaller than the missing 1/3 piece.

Fig. 6.2

 

 

 

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Fraction Bars Step-By-Step Teacher's Guide Inequality Step 3, pages 31-32

Activities Inequality of Fractions

1. Have students place markers on their mats for 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, and 1/12. Encourage them to share observations about these inequalities. (Possibilities: 1/2 is the greatest fraction and 1/12 is the smallest; the fractions get smaller as you move down the mat; as the denominator gets larger the fraction gets smaller.) Have them write these fractions in order from smallest to largest and explain their reasoning. (1/12 < 1/11 < 1/10 and so on. A bar divided into 12 equal parts has smaller parts than a bar divided into 11 equal parts.)

2. Repeat this process on the TOWER OF BARS mat for 2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9, 2/10, 2/11 and 2/12.

3. Have students place markers on their mats for 1/2, 2/3, 3/4, 4/5, and 5/6. Encourage them to share observations about these inequalities. (Possibilities: 1/2 is the smallest fraction and 5/6 is the greatest; the fractions get larger as you move down the right-hand side of the mat; the fractions get larger as the bars get closer to a whole bar.)

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Standards for School Mathematics, page 35

The development of rational-number concepts is a major goal for grades 3-5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4) …

When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981).

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Fraction Bars Step-By-Step Teacher's Guide

Addition Step 2, page 37

Activities Adding Fractions

1. Show students, and have them find, the blue 1/4 and 2/4 bars. Have them place the shaded amounts of the bars end to end and write the addition equation. (1/4 + 2/4 = 3/4)

2. Show students, and have them find, the orange 7/12 and 8/12 bars. Have them place the shaded amounts end to end, and write the resulting equation. (7/12 + 8/12 = 15/12 = 1 3/12) This is a good time to discuss improper fractions and how to write improper fractions as whole or mixed numbers.

3. Have students select several pairs of bars of the same color, place their shaded amounts end to end, and write the equations. Ask them to demonstrate the results with their bars. Discuss which pairs total less than 1 whole bar, which equal 1 whole bar, and which total more than 1 whole bar. For totals of more than 1 whole bar (improper fractions) have students write their answers as whole or mixed numbers.

4. Have students select pairs of the same colored bars and place the shaded parts end to end on appropriate lines of the NUMBER LINES mat to determine sums of the fractions. Division lines on the NUMBER LINES mat match division lines on the bars.

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Standards for School Mathematics, page 33

In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2.

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Fraction Bars Step-By-Step Teacher's Guide

Addition Step 3, page 39

Activities Adding Fractions Using Common Denominators

1. Have students select the green bar for 1/2 and the yellow bar for 1/3, and place them end to end to determine if the total shaded amount is less than or greater than 1 whole bar. (less)

Have them find the red bar with the same shaded amount as the green 1/2 bar (red 3/6 bar) and the red bar with the same shaded amount as the yellow 1/3 bar. (red 2/6 bar) Discuss that they have found the common denominator for 1/2 and 1/3. Have them write the addition equation for this process. (1/2 + 1/3 = 3/6 + 2/6 = 5/6)

Use this example to show how adding numerator to numerator and denominator to denominator results in an unreasonable answer. (1/2 + 1/3 is not 2/5. This sum is smaller than one of the addends. Demonstrate by comparing the 1/2 and 2/5 bars.)

2. Have students select the red bar for 1/6 and the blue bar for 3/4, and place them end to end to determine if the total shaded amount is less than or greater than 1 whole bar. (less)

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Standards for School Mathematics, page 218

Teachers can also help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation … . Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation.

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Fraction Bars Step-By-Step Teacher's Guide

Subtraction, Step 3, page 47

Activities Subtracting Mixed Numbers

The figure shows a NUMBER LINES mat, with markers above the points for 1 5/6 and 3/6.

Ask students how they could use this mat to find the difference between 1 5/6 and 3/6. (They could count the spaces between the two points. The distance is 8/6 or 1 2/6. They could also "count up" from 3/6, first adding 3/6 to get to 1, then adding 5/6 to get from 1 to 5/6. The total distance again would be 8/6 or 1 2/6.)

Have students use the vertical form of subtraction to compute the answer to the same problem, and compare the answer with the difference from the NUMBER LINES mat.

 

Activities Subtracting Mixed Numbers: Regrouping

Have students turn over their mats, place markers above the points for 1 3/8 and 5/8, and find the difference between the two points. (The distance is 6/8. They could also "count up" from 5/8, first adding 3/8 to get 1, then adding 3/8 to get from 1 to 1 3/8. The total distance again would be 6/8.)

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Standards for School Mathematics, page 34

In grades 6-8, operations with rational numbers should be emphasized. Students' intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g., 8 ´ 1/2) produces a result less than the whole number. This is counter to students' prior experience (with whole numbers) that multiplication always results in a greater number.

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Fraction Bars Step-By-Step Teacher's Guide

Subtraction, Step 3, page 57-58

Activities Unit Fraction Times Whole Numbers

You will need to keep a list of equations generated in this activity for use in Activity 4.

1. Have students list their observations about the DOTS mat. (Some possibilities: dots are contained inside hexagons; there are 48 sets of dots; the mat is a tessellation of squares and hexagons; the number of dots in each blue region is divisible by 5, in each white region by 4, in each green region by 3, in each red region by 2.)

2. Ask students to find the blue region with 15 dots. Have them demonstrate and explain how to determine 2/5 of 15. (Split the dots into 5 equal groups, so each group has 3 dots. Take 2 of the groups for a total of 6 dots.) Have students write the resulting multiplication equation. (2/5 x 15 = 6)

3. Repeat this process to find 3/4 of 20. (Split the blue region of 20 dots into 4 equal groups of 5. Take 3 of the groups for a total of 15.) The equation is 3/4 x 20 = 15.

4. List some of the equations for Activities 1 and 2. Have students examine the equations and state a generalization for multiplying a fraction times a whole number. (Multiply the numerator times the whole number and keep the denominator.) Some students may notice that in each of the equations, the whole number can be divided evenly by the denominator. This leads to a discussion of canceling presented in Part 2 of this step.

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Standards for Grades 3-5, page 152

Number and Operations

Examining the effect of multiplying or dividing numbers can also lead to a deeper understanding of these operations. For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient. With models or calculators, students can explore dividing by numbers between 0 and 1, such as 1/2, and find that the quotient is larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."

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Fraction Bars Step-By-Step Teacher's Guide

Division, Step 1, pages 65-66

Activities Division of Fraction Parts

Place a clear transparency over the Fraction Bars transparencies before drawing dotted lines. Water-based pen will wash off plastic Fraction Bars.

1. Show students, and have them find, a yellow bar with 1 part shaded and a red bar with 1 part shaded. By drawing dotted lines, demonstrate that the shaded amount of the yellow bar is twice as big as the shaded amount of the red bar. Further, demonstrate that the shaded amount of the red bar "fits into" the shaded amount of the yellow bar 2 times. This can be stated as 1 part out of 3 divided by 1 part out of 6 is equal to 2.

2. Ask students to visualize or draw a bar with 7 equal parts and 1 part shaded. Have them complete the division statement, 1 whole bar divided by 1 part out of 7 is equal to -------. (7)

3. Continue this line of reasoning by dividing a whole bar by smaller and smaller shaded amounts such as 1 part out of 10 and 1 part out of 12.

4. Have students write a statement about what happens when a whole bar is divided by smaller and smaller shaded amounts. (The result is larger and larger numbers.)

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Standards for School Mathematics, page 33

Teachers should help students develop an understanding of fractions as division of numbers. And in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers.

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Fraction Bars Step-By-Step Teacher's Guide

Division, Step 3, pages 69-70

Activities Dividing Whole Numbers by Whole Numbers

This activity requires strips of paper the width of a Fraction Bar and lengths of 6 inches, 12 inches, and 18 inches. Each student or group will need two strips of each length. Save the equations generated in this activity for use in Activity 3.

1. Ask students to use their Fraction Bars to measure how many whole Fraction Bars each strip of paper represents and to label their bars as shown in the figure (6-inch strip = 1 bar; 12-inch strip = 2 bars; 18-inch strip = 3 bars)

2. Ask students to determine what part of a whole bar each person would receive if the paper 2-bar were divided equally among 4 people. (1/2 of a whole bar) Have students explain their answers. (Half of a 2-bar is a 1-bar and half of a 1-bar is 1/2 of a whole bar.) Have students fold a 2-bar to obtain 4 equal parts and compare 1 of these parts to the green 1/2 bar. (They are the same size.) Have students write a division equation for 2 divided by 4. (2 ¸ 4 = 1/2)

3. Repeat this activity, dividing a 3-bar into 4 equal parts. Have students compare 1 of the 4 equal parts and the 3/4 blue bar. (They are the same size.) Have students write a division equation for 3 ¸ 4. (3 ¸ 4 = 3/4)
    1. Ask students what part of a whole bar each person would receive if the 2-bar were divided equally among 3 people, and to explain their reasoning. (2/3 of a whole bar because 1 of these parts has the same size as the shaded part of a yellow 2/3 bar) Have them write a division equation for 2 divided by 3. (2 ¸ 3 = 2/3)

Standards for School Mathematics, page 33

Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. … As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers.

Standards for Grades 3-5, page 144

Equivalence should be another central idea in grades 3-5. Students' ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, ¾ can be thought of as a half and a fourth, as 6/8, or as 0.75.

 

Standards for Grades 3-5, page 149

Numbers and Operations

Through the study of various means and models of fractions – how fractions are related to each other and to the unit whole and how they are represented – students can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1.

 

 

Standards for Grades 6-8, page 216

Number and Operations

Visual images of fractions as fraction strips should help many students think flexibly in comparing fractions. As shown in figure 6.2, a student might conclude that 7/8 is greater than 2/3 because each fraction is exactly "one piece" smaller than 1 and the missing 1/8 piece is smaller than the missing 1/3 piece.

Fig. 6.2

 

 

 

 

 

Standards for School Mathematics, page 35

The development of rational-number concepts is a major goal for grades 3-5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4) …

When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981).

Standards for School Mathematics, page 33

In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2.

 

Standards for School Mathematics, page 218

Teachers can also help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation … . Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation.

 

Standards for School Mathematics, page 34

In grades 6-8, operations with rational numbers should be emphasized. Students' intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g., 8 ´ 1/2) produces a result less than the whole number. This is counter to students' prior experience (with whole numbers) that multiplication always results in a greater number.

 

Standards for Grades 3-5, page 152

Number and Operations

Examining the effect of multiplying or dividing numbers can also lead to a deeper understanding of these operations. For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient. With models or calculators, students can explore dividing by numbers between 0 and 1, such as 1/2, and find that the quotient is larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."

 

Standards for School Mathematics, page 33

Teachers should help students develop an understanding of fractions as division of numbers. And in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers.

 

 

 

 

 

 

 

 

 

 

 

 

 W H A T D O T H E S T A N D A R D S Home

 

 

W H A T D O T H E S T A N D A R D S

S A Y A B O U T T E A C H I N G

F R A C T I O N S

 

 

 

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼

Quotations and illustrations from Principles and Standards for School Mathematics regarding the teaching of fractions.

▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

 

and

 

 

 

* * * * * * * * * * * * * * * *

Excerpts from Fraction Bars Program showing activities that satisfy the Standards for teaching fractions.

* * * * * * * * * * * * * * * *

?▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼

Standards for School Mathematics, page 33

Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. …

As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers.

?▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

 

 

Fraction Bars Step-By-Step Teacher's Guide Equality Step 1, page 13

Activities Overview of Equality

1. Have students place their orange bars in a row from smallest to largest, as shown below.

 

 

 

2. Have students sort their red, blue, yellow, and green bars onto these 13 piles, so bars with the same shaded amount are in each pile. (see figure) Have students use these piles of bars to complete the table below.

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Standards for Grades 3-5, page 144

Equivalence should be another central idea in grades 3-5. Students' ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, ¾ can be thought of as a half and a fourth, as 6/8, or as 0.75.

?▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

 

 

Fraction Bars Step-By-Step Teacher's Guide Equality Step 3, page 15

Activities Obtaining Equivalent Fractions

Place a clear transparency over the Fraction Bars transparencies before drawing the dotted lines. Water-based pens will wash off the plastic Fraction Bars. Keep a list of the equations generated in Activity 1 for use in Activity 2.

1. Show students and have them find, a blue bar with 1 part shaded. (1/4) Split each part of the bar into 2 equal parts, by drawing dotted lines as shown in the figure.

Ask students the following: How many total parts does the split bar now have? (8) How many shaded parts? (2) What fraction does the bar now represent? (2/8) What does this show about the fractions 1/4 and 2/8? (1/4 = 2/8)

2. Repeat the process of splitting each part of a bar into 2 equal parts to obtain equal fractions with several types of bars. (See figures for examples.)

Guide students to see that splitting each part of a bar into 2 equal parts is the same as multiplying by 2. It doubles the number of parts and doubles the number of shaded parts, but the total shaded amount remains the same.

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼

Standards for Grades 3-5, page 149

Numbers and Operations

Through the study of various means and models of fractions – how fractions are related to each other and to the unit whole and how they are represented – students can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1.

?▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

 

 

Fraction Bars Step-By-Step Teacher's Guide Inequality Step 2, page 29

Activities Determining Inequalities by Comparing Fractions to 1/2

1. Ask students to determine if 5/12 is greater than 1/2 or less than 1/2 and to explain their reasoning. (less than 1/2 because a bar with 5 parts out of 12 is less than half shaded; or 1/2 = 6/12) Some students may wish to compare the shaded amounts of the 5/12 and 1/2 bars.

2. Repeat this activity by asking students to compare the following fractions to 1/2 and write the inequalities. Ask them to describe the bar for each fraction and explain their reason for each inequality. Some students may find it helpful to find or sketch the bar for the fractions.

(6/10, 8/14, 3/5, and 11/20 are greater than 1/2 and 4/9 and 6/15 are less than 1/2)

3. Summarize this activity by asking students to write a statement which tells how to determine when a fraction is less than 1/2 or greater than 1/2. (A fraction is less than 1/2 if its bar is less than half shaded and greater than 1/2 if its bar is more than half shaded.)

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Standards for Grades 6-8, page 216

Number and Operations

Visual images of fractions as fraction strips should help many students think flexibly in comparing fractions. As shown in figure 6.2, a student might conclude that 7/8 is greater than 2/3 because each fraction is exactly "one piece" smaller than 1 and the missing 1/8 piece is smaller than the missing 1/3 piece.

Fig. 6.2

 

 

 

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Fraction Bars Step-By-Step Teacher's Guide Inequality Step 3, pages 31-32

Activities Inequality of Fractions

1. Have students place markers on their mats for 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, and 1/12. Encourage them to share observations about these inequalities. (Possibilities: 1/2 is the greatest fraction and 1/12 is the smallest; the fractions get smaller as you move down the mat; as the denominator gets larger the fraction gets smaller.) Have them write these fractions in order from smallest to largest and explain their reasoning. (1/12 < 1/11 < 1/10 and so on. A bar divided into 12 equal parts has smaller parts than a bar divided into 11 equal parts.)

2. Repeat this process on the TOWER OF BARS mat for 2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9, 2/10, 2/11 and 2/12.

3. Have students place markers on their mats for 1/2, 2/3, 3/4, 4/5, and 5/6. Encourage them to share observations about these inequalities. (Possibilities: 1/2 is the smallest fraction and 5/6 is the greatest; the fractions get larger as you move down the right-hand side of the mat; the fractions get larger as the bars get closer to a whole bar.)

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Standards for School Mathematics, page 35

The development of rational-number concepts is a major goal for grades 3-5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4) …

When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981).

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Fraction Bars Step-By-Step Teacher's Guide

Addition Step 2, page 37

Activities Adding Fractions

1. Show students, and have them find, the blue 1/4 and 2/4 bars. Have them place the shaded amounts of the bars end to end and write the addition equation. (1/4 + 2/4 = 3/4)

2. Show students, and have them find, the orange 7/12 and 8/12 bars. Have them place the shaded amounts end to end, and write the resulting equation. (7/12 + 8/12 = 15/12 = 1 3/12) This is a good time to discuss improper fractions and how to write improper fractions as whole or mixed numbers.

3. Have students select several pairs of bars of the same color, place their shaded amounts end to end, and write the equations. Ask them to demonstrate the results with their bars. Discuss which pairs total less than 1 whole bar, which equal 1 whole bar, and which total more than 1 whole bar. For totals of more than 1 whole bar (improper fractions) have students write their answers as whole or mixed numbers.

4. Have students select pairs of the same colored bars and place the shaded parts end to end on appropriate lines of the NUMBER LINES mat to determine sums of the fractions. Division lines on the NUMBER LINES mat match division lines on the bars.

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Standards for School Mathematics, page 33

In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2.

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Fraction Bars Step-By-Step Teacher's Guide

Addition Step 3, page 39

Activities Adding Fractions Using Common Denominators

1. Have students select the green bar for 1/2 and the yellow bar for 1/3, and place them end to end to determine if the total shaded amount is less than or greater than 1 whole bar. (less)

Have them find the red bar with the same shaded amount as the green 1/2 bar (red 3/6 bar) and the red bar with the same shaded amount as the yellow 1/3 bar. (red 2/6 bar) Discuss that they have found the common denominator for 1/2 and 1/3. Have them write the addition equation for this process. (1/2 + 1/3 = 3/6 + 2/6 = 5/6)

Use this example to show how adding numerator to numerator and denominator to denominator results in an unreasonable answer. (1/2 + 1/3 is not 2/5. This sum is smaller than one of the addends. Demonstrate by comparing the 1/2 and 2/5 bars.)

2. Have students select the red bar for 1/6 and the blue bar for 3/4, and place them end to end to determine if the total shaded amount is less than or greater than 1 whole bar. (less)

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Standards for School Mathematics, page 218

Teachers can also help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation … . Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation.

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Fraction Bars Step-By-Step Teacher's Guide

Subtraction, Step 3, page 47

Activities Subtracting Mixed Numbers

The figure shows a NUMBER LINES mat, with markers above the points for 1 5/6 and 3/6.

Ask students how they could use this mat to find the difference between 1 5/6 and 3/6. (They could count the spaces between the two points. The distance is 8/6 or 1 2/6. They could also "count up" from 3/6, first adding 3/6 to get to 1, then adding 5/6 to get from 1 to 5/6. The total distance again would be 8/6 or 1 2/6.)

Have students use the vertical form of subtraction to compute the answer to the same problem, and compare the answer with the difference from the NUMBER LINES mat.

 

Activities Subtracting Mixed Numbers: Regrouping

Have students turn over their mats, place markers above the points for 1 3/8 and 5/8, and find the difference between the two points. (The distance is 6/8. They could also "count up" from 5/8, first adding 3/8 to get 1, then adding 3/8 to get from 1 to 1 3/8. The total distance again would be 6/8.)

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Standards for School Mathematics, page 34

In grades 6-8, operations with rational numbers should be emphasized. Students' intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g., 8 ´ 1/2) produces a result less than the whole number. This is counter to students' prior experience (with whole numbers) that multiplication always results in a greater number.

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Fraction Bars Step-By-Step Teacher's Guide

Subtraction, Step 3, page 57-58

Activities Unit Fraction Times Whole Numbers

You will need to keep a list of equations generated in this activity for use in Activity 4.

1. Have students list their observations about the DOTS mat. (Some possibilities: dots are contained inside hexagons; there are 48 sets of dots; the mat is a tessellation of squares and hexagons; the number of dots in each blue region is divisible by 5, in each white region by 4, in each green region by 3, in each red region by 2.)

2. Ask students to find the blue region with 15 dots. Have them demonstrate and explain how to determine 2/5 of 15. (Split the dots into 5 equal groups, so each group has 3 dots. Take 2 of the groups for a total of 6 dots.) Have students write the resulting multiplication equation. (2/5 x 15 = 6)

3. Repeat this process to find 3/4 of 20. (Split the blue region of 20 dots into 4 equal groups of 5. Take 3 of the groups for a total of 15.) The equation is 3/4 x 20 = 15.

4. List some of the equations for Activities 1 and 2. Have students examine the equations and state a generalization for multiplying a fraction times a whole number. (Multiply the numerator times the whole number and keep the denominator.) Some students may notice that in each of the equations, the whole number can be divided evenly by the denominator. This leads to a discussion of canceling presented in Part 2 of this step.

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Standards for Grades 3-5, page 152

Number and Operations

Examining the effect of multiplying or dividing numbers can also lead to a deeper understanding of these operations. For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient. With models or calculators, students can explore dividing by numbers between 0 and 1, such as 1/2, and find that the quotient is larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."

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Fraction Bars Step-By-Step Teacher's Guide

Division, Step 1, pages 65-66

Activities Division of Fraction Parts

Place a clear transparency over the Fraction Bars transparencies before drawing dotted lines. Water-based pen will wash off plastic Fraction Bars.

1. Show students, and have them find, a yellow bar with 1 part shaded and a red bar with 1 part shaded. By drawing dotted lines, demonstrate that the shaded amount of the yellow bar is twice as big as the shaded amount of the red bar. Further, demonstrate that the shaded amount of the red bar "fits into" the shaded amount of the yellow bar 2 times. This can be stated as 1 part out of 3 divided by 1 part out of 6 is equal to 2.

2. Ask students to visualize or draw a bar with 7 equal parts and 1 part shaded. Have them complete the division statement, 1 whole bar divided by 1 part out of 7 is equal to -------. (7)

3. Continue this line of reasoning by dividing a whole bar by smaller and smaller shaded amounts such as 1 part out of 10 and 1 part out of 12.

4. Have students write a statement about what happens when a whole bar is divided by smaller and smaller shaded amounts. (The result is larger and larger numbers.)

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Standards for School Mathematics, page 33

Teachers should help students develop an understanding of fractions as division of numbers. And in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers.

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Fraction Bars Step-By-Step Teacher's Guide

Division, Step 3, pages 69-70

Activities Dividing Whole Numbers by Whole Numbers

This activity requires strips of paper the width of a Fraction Bar and lengths of 6 inches, 12 inches, and 18 inches. Each student or group will need two strips of each length. Save the equations generated in this activity for use in Activity 3.

1. Ask students to use their Fraction Bars to measure how many whole Fraction Bars each strip of paper represents and to label their bars as shown in the figure (6-inch strip = 1 bar; 12-inch strip = 2 bars; 18-inch strip = 3 bars)

2. Ask students to determine what part of a whole bar each person would receive if the paper 2-bar were divided equally among 4 people. (1/2 of a whole bar) Have students explain their answers. (Half of a 2-bar is a 1-bar and half of a 1-bar is 1/2 of a whole bar.) Have students fold a 2-bar to obtain 4 equal parts and compare 1 of these parts to the green 1/2 bar. (They are the same size.) Have students write a division equation for 2 divided by 4. (2 ¸ 4 = 1/2)

3. Repeat this activity, dividing a 3-bar into 4 equal parts. Have students compare 1 of the 4 equal parts and the 3/4 blue bar. (They are the same size.) Have students write a division equation for 3 ¸ 4. (3 ¸ 4 = 3/4)
    1. Ask students what part of a whole bar each person would receive if the 2-bar were divided equally among 3 people, and to explain their reasoning. (2/3 of a whole bar because 1 of these parts has the same size as the shaded part of a yellow 2/3 bar) Have them write a division equation for 2 divided by 3. (2 ¸ 3 = 2/3)

Standards for School Mathematics, page 33

Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. … As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers.

Standards for Grades 3-5, page 144

Equivalence should be another central idea in grades 3-5. Students' ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, ¾ can be thought of as a half and a fourth, as 6/8, or as 0.75.

 

Standards for Grades 3-5, page 149

Numbers and Operations

Through the study of various means and models of fractions – how fractions are related to each other and to the unit whole and how they are represented – students can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1.

 

 

Standards for Grades 6-8, page 216

Number and Operations

Visual images of fractions as fraction strips should help many students think flexibly in comparing fractions. As shown in figure 6.2, a student might conclude that 7/8 is greater than 2/3 because each fraction is exactly "one piece" smaller than 1 and the missing 1/8 piece is smaller than the missing 1/3 piece.

Fig. 6.2

 

 

 

 

 

Standards for School Mathematics, page 35

The development of rational-number concepts is a major goal for grades 3-5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4) …

When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981).

Standards for School Mathematics, page 33

In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2.

 

Standards for School Mathematics, page 218

Teachers can also help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation … . Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation.

 

Standards for School Mathematics, page 34

In grades 6-8, operations with rational numbers should be emphasized. Students' intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g., 8 ´ 1/2) produces a result less than the whole number. This is counter to students' prior experience (with whole numbers) that multiplication always results in a greater number.

 

Standards for Grades 3-5, page 152

Number and Operations

Examining the effect of multiplying or dividing numbers can also lead to a deeper understanding of these operations. For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient. With models or calculators, students can explore dividing by numbers between 0 and 1, such as 1/2, and find that the quotient is larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."

 

Standards for School Mathematics, page 33

Teachers should help students develop an understanding of fractions as division of numbers. And in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers.