Computation Of Fractions - Intensive Intervention

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National Center onINTENSIVE INTERVENTIONat American Institutes for ResearchComputation of Fractions11000 Thomas Jefferson Street, NWWashington, DC 20007E-mail: NCII@air.org

While permission to reprint this publication is not necessary, the citation should be:National Center on Intensive Intervention. (2014).Computation of Fractions. Washington, DC:U.S. Department of Education, Office of Special Education Programs, National Center onIntensive Intervention.This document was produced under the U.S. Department of Education, Office of Special EducationPrograms, Award No. H326Q110005. Celia Rosenquist serves as the project officer. The views expressedherein do not necessarily represent the positions or polices of the U.S. Department of Education. Noofficial endorsement by the U.S. Department of Education of any product, commodity, service orenterprise mentioned in this website is intended or should be inferred.

Contents1.Intensive Intervention: Computation of Fractions . . . . . . . . . . . . . 52.Fraction Addition and Subtraction Concepts . . . . . . . . . . . . . . . .20Sample Fraction Addition and Subtraction Concepts Activities 1–3. . . . . . .21a. Activity One: Using Fraction Tiles and Fraction Circles . . . . . . . . . . . 21b. Activity Two: Using Fraction Tiles and Fraction Circles . . . . . . . . . . .24c. Activity Three: Using Fraction Tiles and Fraction Circles . . . . . . . . . . 27Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30a. Fraction Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 30b. Fraction Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . 323.Addition and Subtraction With Unlike Denominators. . . . . . . . 36Sample Adding and Subtracting With Unlike Denominators Activities 1–3 . . . 37a. Activity One: Writing Equivalent Fractions(Finding a Common Denominator) . . . . . . . . . . . . . . . . . . . 37b. Activity Two: Writing Equivalent Fractions to Solve Addition Problems(Finding a Common Denominator) . . . . . . . . . . . . . . . . . . . 41c. Activity Three: Writing Equivalent Fractions to Solve Subtraction Problems(Finding a Common Denominator) . . . . . . . . . . . . . . . . . . . 43Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45a. Adding and Subtracting Fractions With Unlike Denominators . . . . . . . 454.Using Addition and Subtraction to Convert Mixed Numbersand Improper Fractions . . . . . . . . . . . . . . . . . . . . 48Sample Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49a. Using Addition and Subtraction to Convert Mixed Numbersand Improper Fractions . . . . . . . . . . . . . . . . . . . . . . . . 491000 Thomas Jefferson Street, NWWashington, DC 20007E-mail: NCII@air.org

Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56a. Using Addition and Subtraction to Convert Mixed Numbersand Improper Fractions . . . . . . . . . . . . . . . . . . . . . . . . 56b. Using Addition and Subtraction to Convert Mixed Numbersand Improper Fractions (Scaffolded) . . . . . . . . . . . . . . . . . . 585.Supplemental Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 64a. Fraction Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65b. Fraction Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 66c. Fraction Addition Flash Cards . . . . . . . . . . . . . . . . . . . . . 68d. Fraction Subtraction Flash Cards . . . . . . . . . . . . . . . . . . . 74e. Fraction Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80f. Multiplication Chart . . . . . . . . . . . . . . . . . . . . . . . . . 86g. Improper Fraction Cards . . . . . . . . . . . . . . . . . . . . . . . 87h. Mixed Number Cards . . . . . . . . . . . . . . . . . . . . . . . . . 931208a 07/15i. Prompt Cards for Converting Improper Fractions and Mixed Numbers . . . 984Mathematics Training Materials: Computation of Fractions Contents

Computation of Fractions:Considerations for InstructionPurpose and Overview of GuideThe purpose of this guide is to provide strategies and materials for developing andimplementing lessons for students who need intensive instruction in the area of fractions.Special education teachers, mathematics interventionists, and others working withstudents struggling in the area of fractions may find this guide helpful.Within college- and career-ready standards, fractions are taught in Grades 3–5. This guidemay be used as these concepts are introduced or with students at higher grade levelswho continue to struggle with the concepts. Sample activities, worksheets, andsupplemental materials also accompany this guide and are available for download athttp://www.intensiveintervention.org.The guide is divided into four sections:1. The sequence of skills as defined y college- and career-ready standards2. A list of important vocabulary and symbols3. A brief explanation of the difficulties that students m y have with fractions4. Suggested strategies for teaching fraction computation conceptsSequence of Skills—College- and Career-Ready StandardsBuild fractions from unit fractions—applying and extending operations of whole numbers.¡ Add and subtract fractions.¡ Decompose fractions in more than one way.¡ Add and subtract mixed numbers with like denominators.¡ Solve word problems involving addition or subtraction.¡ Multiply fractions.¡ Solve word problems involving multiplication.Intensive Intervention: Computation of Fractions1000 Thomas Jefferson Street, NWWashington, DC 20007E-mail: NCII@air.org

¡ Use equivalent fractions.¡ Add and subtract fractions with unlike denominators.¡ Solve word problems involving addition or subtracting of unlike denominators.¡ Divide fractions.¡ Solve word problems involving division of fractions.¡ Continue multiplication.Vocabulary and SymbolsThe following terms are important for students to understand when working with fractions:Fraction: A part of a whole, withall parts equivalent.Numerator: How many parts ofthe whole.Denominator: How many partsmake up the whole.1 1 2 1 4, , , ,4 2 3 8 556Common Denominator: One ormore fractions have the samedenominator. Necessary foradding and subtracting fractions.12 88Least Common Multiple (LCM):The smallest common multipleof two or more denominators.Used to determine commondenominator.Multiples of 3: 3, 6, 9, 12, 15Multiples of 5: 5, 10, 15, 20LCM is 15.Improper Fraction: A fractionthat is greater than one.6, 9, 14 8,4 6 5 26123444Equivalent Fractions:Fractions with equal value.Simplify/Reduce: Puttingfractions in lowest terms.42 63123 4 4 153 5 521 84Greatest Common Factor (GCF): Proper Fraction: A fraction that isLargest common factor for theless than one.numerator and the denominator.1, 1, 2, 1, 4Used to simply/reduce fractions.4 2 3 8 5Factors of 12: 1, 2, 3, 4, 6, 12Factors of 15: 1, 3, 5, 15GCF is 3.Mixed Number: A number thathas a whole number and afraction.4141023Unit Fraction: A fraction with1 in the numerator.1, 1, 1, 1, 112 8 5 3 2Intensive Intervention: Computation of Fractions

Common Areas of DifficultyPrerequisite Skills Not Mastered¡ Basic fact retrieval (for computation and comparison of fractions with unlikedenominators)Specific raction Skills¡ Reading fractions¡ Writing fractions¡ Understanding that the larger the denominator, the smaller the value¡ Poor understanding of multiples¡ Understanding the four models of fractions and when to use them: Area Sets Measurement DivisionConceptual UnderstandingFraction tiles and fraction circles can be used to help students visualize and conceptuallyunderstand many fraction concepts. These manipulatives represent 1 whole, 1/2, 1/3, 1/4,1/5, 1/6, 1/8, 1/10, and 1/12.Adding Fractions4/8 3/4 118181818141414Start with the whole.Place 4/8 under the whole.Place 3/4 under the 4/8.Intensive Intervention: Computation of Fractions7

14118181818141414Combine 4/8 3/4 under the whole.Students should realize that 1/4 more is needed to complete the problem.4/8 3/4 1 and 1/4.The same can be done using fraction circles.1/4 1/3 Start with 1/4 and 1/3 pieces.Place the pieces on the whole circle.Determine that 1/12 pieces need to be used.Show that 5/12 of the whole remains, which means 1/4 1/3 equals 7/12.8Intensive Intervention: Computation of Fractions

Subtracting Fractions1/2 – 2/5 121515Start with 1/2 and show subtracting 2/5.1/2 – 2/5 is the difference between a pink tile and two green tiles.121515110Place a 1/10 next to 2/5.The difference is one purple tile, or 1/10.3/8 – 1/4 – Start with 3/8 and show subtracting 1/4.This shows that 1/4 is equal to 2/8, so the difference is 1/8.Intensive Intervention: Computation of Fractions9

Multiplying Fractions2 2/3Show two sets of 2/3 tiles.13131313113131313Show 1 whole.Show two sets of 2/3 tiles.This shows 2 2/3 equals 1 and 1/3.3 3/4Show three sets of 3/4 tiles.Combine the sets, showing that they are equal to 2 and 1/4.Note to Teachers: Fraction tiles and circles will not work with all problems. You shouldensure that the manipulatives will work for the problems you plan to demonstrate. Forexample, you cannot use these manipulatives to show 1/5 5/6 26/30 because youdo not have pieces that show 30ths.10Intensive Intervention: Computation of Fractions

Representing Multiplication of Fractions With Grids2/3 1/4Using the denominators, build a grid—the fi st denominator represents the number ofrows, and the second denominator represents the number of columns.The numerator in each fraction tells how many rows and columns to shade in.For 2/3, shade in two rows.For 1/4, shade in one column.The total number of boxes will be your new denominator: 12.The total number of boxes shaded for both fractions will be your new numerator:2/3 1/4 2/12.Using Fraction Bars to Determine Least Common DenominatorEach bar has a number, 1–9, at the beginning, and each of its multiples is listed (up to 9) on the bar.2/3 3/4 3434689121216152018242128243227361. Select the fraction bars for 3 and 4. Line them up under each other.2. By looking at the two fraction bars, determine the LCM, which is 12.3. Convert each fraction to a new fraction with the denominator of 12.a. 2/3 ?/12 è 12 3 4 è 4 2 8 è 2/3 8/12b. 3/4 ?/12 è 12 4 3 è 3 3 9 è 3/4 9/124. 2/3 3/4 now becomes 8/12 9/12.5. 8/12 9/12 17/126. 17/12 can be reduced to 1 and 5/12.Intensive Intervention: Computation of Fractions11

Adding Fractions With a Number Line3/4 3/401/42/43/411/42/43/424/5 4/501/52/53/54/511/52/53/54/5Multiplying Fractions With a Number Line2/3 80121/32/33/34/35/36/37/38/3 9/310/3 11/3 12/3 13/3 14/3 15/3 16/3 17/3 18/3 19/3 20/3 21/3 22/3Intensive Intervention: Computation of Fractions

Procedural UnderstandingFlowcharts can assist students in understanding the algorithm of a concept.Flowchart for Dividing FractionsStartInvert the divisor(2nd fraction).noAre there anymixed numbers?yesConvert mixednumbers toimproperfractions.Simplify ifpossible.Simplify s the solutionan improperfraction?noSimplify ifneeded.yesReduce theimproper fraction.Simplify ifneeded.Intensive Intervention: Computation of Fractions13

AlgorithmsKnowing algorithms is an important component for solving fraction problems. Studentsmust understand the steps needed to solve each type of problem. Teachers may usethese step-by-step procedures to determine where a student may be having difficulty in thealgorithm. After a teacher has determined where the issue is, then teaching can begin atthat point.Adding Fractions With Like Denominators(a) Make sure the denominators are the same.(b) Add the numerators.(c) Put the answer over the same denominator.(a)3 1553 1(b)54(c)5Subtracting Fractions With Like Denominators(a) Make sure the denominators are the same.(b) Subtract the second numerator from the fi st.(c) Put the answer over the same denominator.75–12127–5(b)122(c)12(a)Adding Fractions With Unlike Denominators(a) Find the least common denominator (LCD) of the fractions: 4 3 12(b) Rename the fi st fraction so it has the LCD.(c) Rename the second fraction so it has the LCD.(d) Rewrite the problem with the new fractions.14Intensive Intervention: Computation of Fractions

(e) Add the numerators.(f) Put the answer over the LCD.12 43(a) 4 3 12 LCD(b)1 33 4 3122 43 43 (d)123 8(e)1211(f)12(c) 812812 Subtracting Fractions With Unlike Denominators(a) Find the LCD of the fractions.(b) Rename the fi st fraction so it has the LCD.(c) Rename the second fraction so it has the LCD.(d) Rewrite the problem with the new fractions.(e) Subtract the second numerator from the fi st.(f) Put the answer over the LCD.31– 52(a) 5 2 10 LCD(b)3 26 5 2101 52 56–(d)106–5(e)101(f)10(c) 510510 Intensive Intervention: Computation of Fractions15

Multiplying Fractions(a) Multiply the numerators of the fractions.(b) Multiply the denominators of the fractions.(c) Put the product of the numerators on top of the fraction.(d) Put the product of the denominators on the bottom of the fraction.33 84(a) 3 3 9(b) 8 4 32(c, d)932Dividing Fractions(a) Invert the second fraction (the divisor).(b) Change the division sign to a multiplication sign.(c) Multiply the numerators of the fractions.(d) Multiply the denominators of the fractions.(e) Put the product of the numerators on top of the fraction.(f) Put the product of the denominators on the bottom of the fraction.2 3 5 42 4(b) 5 3(a)2 4 5 38(e, f)15(c, d)16Intensive Intervention: Computation of Fractions

Adding Mixed Numbers(a) Convert the fi st mixed number to an improper fraction.(b) Multiply the denominator by the whole number.(c) Put the product over the denominator.(d) Add the new fraction’s numerator to the original fraction’s numerator.(e) Use the same denominator for the new numerator.(f) Convert the second mixed number to an improper fraction.(g) Multiply the denominator by the whole number.(h) Put the product over the denominator.(i) Add the new fraction’s numerator to the original fraction’s numerator.(j) Use the same denominator for the new numerator.(k) Rewrite the problem with improper fractions.(l) Find the LCD of the fractions.(m) Rename the fi st fraction so it has the LCD.(n) Rename the second fraction so it has the LCD.(o) Rewrite the problem with the LCD.(p) Add the numerators.(q) Put the answer over the LCD.(r) Simplify. Reduce improper fraction to a mixed number.33 6483(a) 242(b) 4 2 8(c)848 3(d) 411(e)43(f) 68(g) 8 648848 3(i) 851(j)81151(k) 48(h) (l) 4 2 8, 8 1 8 LCDIntensive Intervention: Computation of Fractions(m)(n)(o)(p)(q)(r)11 222 4 2851 151 8 182251 8822 51 873 819817

Multiplying Mixed Numbers(a) Convert the fi st mixed number to an improper fraction.(b) Multiply the denominator by the whole number.(c) Put the product over the denominator.(d) Add the new fraction’s numerator to the original fraction’s numerator.(e) Use the same denominator for the new numerator.(f) Convert the second mixed number to an improper fraction.(g) Multiply the denominator by the whole number.(h) Put the product over the denominator.(i) Add the new fraction’s numerator to the original fraction’s numerator.(j) Use the same denominator for the new numerator.(k) Rewrite the problem with the improper fractions.(l) Multiply the numerators of the improper fractions.(m) Multiply the denominators of the improper fractions.(n) Put the product of the numerators on top of the fraction.(o) Put the product of the denominators on the bottom of the fraction.(p) Simplify. Reduce improper fraction to a mixed number.Example:323 4 94(a) 3(f) 434(b) 9 3 27(g) 4 4 1627927 2(d) 929(e)916416 3(i) 419(j)4(c)1829(h)(k)2919 94(l) 29 19 551(m) 9 4 36551 3611(p) 1536(n, o)Intensive Intervention: Computation of Fractions

Dividing Mixed Numbers(a) Convert the fi st mixed number to an improper fraction.(b) Multiply the denominator by the whole number.(c) Put the product over the denominator.(d) Add the new fraction’s numerator to the original fraction’s numerator.(e) Use the same denominator for the new numerator.(f) Convert the second mixed number to an improper fraction.(g) Multiply the denominator by the whole number.(h) Put the product over the denominator.(i) Add the new fraction’s numerator to the original fraction’s numerator.(j) Use the same denominator for the new numerator.(k) Rewrite the problem with the improper fractions.(l) Invert the second fraction (the divisor).(m) Change the division sign to a multiplication sign.(n) Multiply the numerators of the fractions.(o) Multiply the denominators of the fractions.(p) Put the product of the numerators on the top of the fraction.(q) Put the product of the denominators on the bottom of the fraction.(r) Simplify. Reduce improper fraction to a mixed number.Example:12 3 231(a) 62612212 1(d) 213(e)2(c)23(g) 3 3 9939 2(i) 311(j)313 11(k) 23(h)Intensive Intervention: Computation of Fractions(l, m)133 211(n) 13 3 39(o) 2 11 22(p, q)39 22(r) 117221208b 07/15(b) 2 6 12(f) 319

2. Fraction Addition and Subtraction ConceptsSample Activitiesa. Activity One: Using Fraction Tiles and Fraction Circlesb. Activity Two: Using Fraction Tiles and Fraction Circlesc. Activity Three: Using Fraction Tiles and Fraction CirclesWorksheetsa. Fraction Additionb. Fraction Subtraction

Sample Fraction Addition and SubtractionConcepts Activities 1–3College- and Career-Ready Standards:Build fractions from unit fractions by applying and extending previous understandings ofoperations on whole numbers.4.NF.3. Understand a fraction a/b with a 1 as a sum of fractions 1/b.¡ Understand addition and subtraction of fractions as joining andseparating parts referring to the same whole.Activity One:Using Fraction Tiles and Fraction CirclesPurpose:¡ To show addition concepts (joining) with fraction tiles (or circles).¡ Give the student a visual representation of adding fractions alongwith an equation that matches the visual.Principles of Intensive Intervention Illustrated:¡ Provide concrete learning opportunities (including use ofmanipulatives).¡ Provide explicit error correction and have the student repeat thecorrect process.¡ Use precise, simple language to teach key concepts or procedures.¡ Use explicit instruction and modeling with repetition to teach aconcept or demonstrate steps in a process.¡ Provide repeated opportunities to practice each step correctly.Materials (available for download from NCII):¡ Fraction tiles or fraction circles (see Supplemental MaterialsSection)¡ Fraction addition flash cards (see Supplemental Materials Section)¡ Worksheet: Fraction Addition (for extra practice)Sample Fraction Addition and Subtraction Concepts Activities 1–31000 Thomas Jefferson Street, NWWashington, DC 20007E-mail: NCII@air.org1

Modeling:1/81/81/81/81. Place one tile in front of the student and review numerator anddenominator vocabulary.2. Place four of the 1/8 tiles in front of the student (not pushedtogether).3. Explain that one tile is 1/8.4. Sample language: “When you put two tiles together, you have2/8. By joining the two 1/8 pieces, you are adding them to get2/8.” (Show equation saying 1/8 1/8 2/8.)5. Sample language: “When you add a third 1/8 tile, you get 3/8.By joining the three 1/8 pieces, you are adding them to get 3/8.”(Show equation saying 1/8 1/8 1/8 3/8.)6. Repeat for steps to show 4/8.7. Explain: To get any fraction with a number that is greater than 1 inthe numerator, you join or add together fractions with 1 in thenumerator.Guided Practice:1/61/61/61/61/61/61. Place all six of the 1/6 pieces in front of the student. Also placea piece of paper in front of the student so that he or she canrecord equations.2. Ask the student to show 1/6.3. Have the student record 1/6 1/6.4. Ask the student to show 2/6.5. Have the student record 1/6 1/6 2/6.6. Repeat until all combinations with 6 in the denominator havebeen

Intensive Intervention: Computation of Fractions Intensive Intervention: Computation of Fractions 9 Subtracting Fractions 1/2 – 2/5 1 2 1 5 1 5 . Start with . 1/2 and show subtracting 2/5. 1/2 – 2/5 is the difference between a pink tile and two green tiles. 1 2 1 5 1 5 1 10 . Place a . 1/10 . next to . 2/5. The difference is one purple .

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