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BERNARDS TOWNSHIP PUBLIC SCHOOLSBASKING RIDGE, NEW JERSEYFRAMEWORK FOR COMPUTATIONAL FLUENCYGRADE 4Summer 2008Supervisor:Marian PalumboCommittee:Cindy CicchinoPaul DavisPat GambinoDiana KoeckertMegan MacMahonLinda MullenMaureen O’NeilAmy PersilyDavid PersilyKirstin PetersNoreen Quinn-FoyDeborah ReynoldsKathy SimonKathy Van NattaMegan Van PeltTerry Vena

In order to develop students’ math skills, the mathematics curriculum shouldinclude a balance and connection between conceptual understanding and computationalfluency. “Fluency refers to having efficient, accurate and generalizable methods(algorithms) for computing that are based on well-understood properties and numberrelationships” (Principles and Standards for School Mathematics, p.144). Developing aconceptual understanding of mathematical reasoning is essential. Students need toacquire computational fluency in order to be successful problem solvers.Not all students develop automatic recall of basic facts at the same time.However, teachers should work with students so that each student acquires anunderstanding of several computational strategies and implements them appropriatelywith the goal of gaining automaticity with basic facts and computational algorithms. Forexample, a focus in the primary grades is to master computational fluency with additionand subtraction facts through twenty. Students should develop multiplication and divisionfact power between third and fourth grade.Algorithms are important tools that help students become fluent and flexible incomputing. In addition to the algorithm instruction provided in Everyday Mathematics,students should learn the appropriate “traditional” algorithm. In order to facilitate asmooth articulation of the teaching of the “traditional” algorithms, Grade 2 teachers areresponsible for teaching the multi-digit addition algorithm with regrouping, Grade 3teachers are responsible for teaching the multi-digit subtraction algorithm withregrouping, Grade 4 teachers are responsible for teaching the multi-digit multiplicationalgorithm, and Grade 5 teachers are responsible for teaching the long division algorithm.Sometimes students bring the “traditional” algorithms from home and introduce them

into the instructional setting at various other times during the course of the school year.Teachers should allow the students to utilize the “traditional” algorithm (even if thetiming is not congruent with that listed above) as long as the student demonstrates anunderstanding of and competency with the algorithm itself. As always, teachers shouldencourage the students to practice a variety of appropriate computational algorithms asthe use of various algorithms will increase the students’ computational fluency. On anindividual student basis, teachers can also make suggestions for use of a particularalgorithm for those students who appear to lack fluency with computational algorithms.The Framework for Computational Fluency (FCF) provides a variety of materials to usein addition to the materials already provided in Everyday Mathematics. Teachers shoulduse the FCF book for developing and practicing computational fluency and basic factsprior to accessing other math resources. Teachers can utilize the FCF book in a varietyof ways. The pages in the booklet are organized by grade level, however teachers are freeto use pages from other units or grade levels to differentiate instruction in order to bettermeet the needs of the learners. The activities in the booklet can be used in place of oralong with a Math Message or the Mental Math and Reflexes. They can be used aspractice or as assessment, timed or not timed. Teachers are encouraged to present FCFworksheets via the Smartboard with students using slates and/or notebooks to record theirwork. For ease of implementation some of the pages are aligned with the lessons inEveryday Mathematics. Each grade level within the FCF has a sheet that aligns the FCFpages with the Everyday Mathematics lessons.

ReferencesBell, J., et al. (2007). Everyday mathematics the University of Chicago School ofMathematics project: Teacher’s lesson guide. Chicago, IL: McGraw Hill WrightGroup.National Council of Teachers of Mathematics (NCTM) (2006). Curriculum focal pointsfor prekindergarten through grade 8 mathematics. Retrieved July 8, 2008, ifier id&itemid 270National Council of Teachers of Mathematics (NCTM) (2000). Principles and standardsfor school mathematics. Reston, VA: The National Council of Teachers ofMathematics, Inc.Primary mathematics textbook 1A/B. (2007). Singapore: Marshall Cavendish Education.Primary mathematics textbook 2A/B. (2007). Singapore: Marshall Cavendish Education.Primary mathematics textbook 3A/B. (2007). Singapore: Marshall Cavendish Education.Primary mathematics textbook 4A/B. (2007). Singapore: Marshall Cavendish Education.Primary mathematics textbook 5A/B. (2007). Singapore: Marshall Cavendish Education.

Multiplication with Regrouping (use during unit 5)Objectives: To guide students as they develop regrouping strategies for multiplying 2and 3-digit numbers and to encourage using estimation to check if answers arereasonable.Key ActivitiesStudents solve 2-digit multiplicationproblems, record their work with paperand pencil, and share regroupingstrategies. Students use ballparkestimates to check whether their answersare reasonable. Students practice usingregrouping methods to multiply 2-, 3-,and 4-digit numbers.Key Concepts and Skills Share solution strategies forfinding the product of 2-digitnumbers using the traditionalregrouping method Estimate products by changingthe factors to “close but easier”numbersKey Vocabularyregrouping MaterialsClass Data Pad Activity sheet

Mental Math and ReflexesPose pairs of problems similar to the following:30 * 40 ? 60 * 3020 * 400 ? 50 300100 40 ?Math MessageSolve. Be prepared to tell how you found your answer.58* 24I. Teaching the Lesson¾ Math Message Follow-UpHave students share and explain their answers. Explain to the class that they will use a new strategyto solve double-digit multiplication problems with regrouping. To support English language learners,discuss the meaning of regrouping.¾ Discussing the Use of the Regrouping Strategy to Solve Multi-Digit MultiplicationProblemsReview with the class the place value of each digit in a double-digit number. Discuss how 30 * 40 isthe same as 3 * 4, just with the zeros put in to show that the numbers are in the tens place.¾ Solving Multiplication Problems; Keeping a Paper-and-Pencil RecordRewrite the Math Message on the board and model the Paper-and-Pencil record for Regrouping withMultiplication. Highlight the importance of lining up the tens and ones columns when using thisstrategy. Demonstrate multiplying the ones column of the bottom factor first and “carrying” a tenover to the tens column when necessary. Show the “carrying” of the ten by writing a small 3 directlyover the tens column. Remind students that the small 3 is representative of 3 tens and should beadded to the tens column product when finding the answer.Write problems like the following on the board, some in a horizontal format and some in a verticalformat. Explain to students that horizontal problems should be rewritten in the vertical format.29 * 7 76 * 4 53 * 28 163 * 58 26 * 85 219 * 352

Have students work on the problems on their slates. Remind them to check whether each answer isreasonable by making a ballpark estimate.¾ Finding the Product of Two Multi-Digit NumbersHave partners work together to solve the multiplication problemsII. Ongoing Learning and PracticeStudents should continue to practice these concepts using the worksheet below and thecorresponding pages in the Framework for Computational Fluency.III. Differentiation OptionsReadiness: For students who need more practice, pull them aside in small groups. Start withproblems with a 2-digit factor multiplied by a 1-digit factor.Enrichment: For students who grasp the concept easily, challenge them to make a crosswordpuzzle where the clues are the problems and the answers in the puzzle are the products.

Name:Date:Multiplication with Regrouping27 * 34 325 * 9 532 * 8 3204 * 43 Mr. Jarwoomie has 9 houses. Each house has 4 rooms. Each room has 4 electrical outlets and eachoutlet has 2 plugs. How many plugs are in all of Mr. Jarwoomie’s houses? PlugsMrs. Coldhands has 1986 pages in her stamp collection book. On each page there are 9 stamps.How many stamps does she have? Stamps

Suggested Implementation Guide for Framework for Computational FluencyTeachers should feel free to implement pages at their own professional discretion.Unit 1: Naming and Constructing Geometric tion to Student Reference BookPoints, Line Segments, Lines, and RaysAngles, Triangles, and QuadranglesParallelogramsPolygonsDrawing a Circle with a CompassCircle ConstructionsHexagon and triangle ConstructionsSupplemental MaterialsUnit 2: Using Numbers and Organizing DataLesson2.12.22.32.42.52.62.72.82.9TitleA Visit to Washington D.C.Many Names for Many NumbersPlace Value in Whole NumbersPlace Values with a CalculatorOrganizing and Displaying DataThe MedianAddition of Multi-Digit NumbersDisplaying Data with a Bar GraphSubtraction of Multi-Digit NumbersSupplemental Materials4-1 through 4-54-6 through 4-94-10 through 4-13Unit 3: Multiplication and Division; Number Sentences and actions“What’s My Rule?”Multiplication FactsMultiplication Facts PracticeMore Multiplication Facts PracticeMultiplication and DivisionWorld Tour: Flying to AfricaFinding Air DistancesA Guide for Solving Number StoriesTrue or False Number SentencesParentheses in Number SentencesOpen SentencesSupplemental Materials4-14 through 4-15OMITOMIT

Unit 4: Decimals and Their mal Place ValueReview of Basic Decimal ConceptsComparing and Ordering DecimalsEstimating with DecimalsDecimal Addition and SubtractionDecimals and MoneyThousandthsMetric Units of LengthPersonal References for Metric LengthMeasuring in MillimetersSupplemental Materials4-16 through 4-184-194-20 through 4-30OMITUnit 5: Big Numbers, Estimation, and 11TitleExtended Multiplication FactsMultiplication WrestlingEstimating SumsEstimating ProductsPartial Products Multiplication (part 1)Partial Products Multiplication (part 2)Lattice MultiplicationBig NumbersPowers of TenRounding and Reporting LargeNumbersComparing DataSupplemental Materials4-314-32 through 4-34, 4-394-35 through 4-38OMIT

Unit 6: Division; Map Reference Frames; Measures of AnglesLessonTitle6.1Multiplication and Division NumberStories6.2Strategies for Division6.3The Partial-Quotients DivisionAlgorithm (part 1)6.4Expressing and InterpretingRemainders6.5Rotations and Angles6.6Using a Full Circle Protractor6.7The Half Circle Protractor6.8Rectangular Coordinate Grids for Maps6.9Global Coordinate Grid SystemSupplemental MaterialsOMIT6.10The Partial-Quotients DivisionAlgorithm (part 2)Unit 7: Fractions and Their Uses; Chance and ProbabilityLessonTitle7.1Review of Basic Fraction Concepts7.2Fractions of Sets7.3Probabilities When Outcomes areEqually Likely7.4Pattern Block Fractions7.5Fraction Addition and Subtraction7.6Many Names for Fractions7.7Equivalent Fractions7.8Fractions and Decimals7.9Comparing Fractions7.10The ONE for Fractions7.11Probability, Fractions, and Spinners7.12A Cube-Drop ExperimentSupplemental Materials4-40 through 4-42, 4-524-43 through 4-47OMIT

Unit 8: Perimeter and AreaLesson8.18.28.38.4TitleKitchen Layouts and PerimeterScale DrawingsAreaWhat is the Area of My SkinSupplemental MaterialsOMIT8.58.68.7Formula for the Area of a RectangleFormula for the Area of aParallelogramFormula for the Area of a Triangle8.8Geographical Area MeasurementsOMITUnit 9: Fractions, Decimals, and PercentsLessonTitleSupplemental Materials9.1Fractions, Decimals, and Percents9.2Converting “Easy” Fractions toDecimals and Percents9.3Using a Calculator to Convert Fractionsto Decimals9.4Using a Calculator to RenameFractions as Percents9.5Conversions among Fractions,4-46 through 4-51Decimals, and Percents9.6Comparing the Results of a Survey9.7Comparing Population DataOMIT9.8Multiplication of Decimals4-53 through 4-569.9Division of Decimals4-57 through 4-61Unit 10: Decimals and Place ValueLessonTitle10.1Explorations with a Transparent MirrorSupplemental MaterialsOPTIONAL10.210.310.410.510.6Finding Lines of ReflectionProperties of ReflectionsLine SymmetryFrieze PatternsPositive and Negative NumbersOPTIONALOMIT

Unit 11: 3-D Shapes, Weight, Volume, and plemental MaterialsWeightGeometric SolidsConstructing Geometric SolidsA Volume ExplorationA formula for the Volume ofRectangular PrismsSubtraction of Positive and NegativeNumbersCapacity and WeightOMITUnit 12: g RatesSolving Rate ProblemsConverting Between RatesComparison Shopping: Part 1Comparison Shopping: Part 2World Tour and 50-Facts Test WrapUpsSupplemental MaterialsOMIT

Computational FluencyName:Date:Time:4–11. Write the numbers in 000101010001000101011110a) The number is 01001001001000TensOnes11111b) The number is .2. Mr. Barn sold his car for this amount of money. 10,000 10,000 10,000 1,000 1,000 100 100 100 100a) Write 100 a)a) Write the amount of money in standard notation:b) Write the amount of money in words:

Computational FluencyName:Date:Time:4 – 1a3. Write the following in standard notation.a) Eight thousand, four hundred two dollarsb) Twelve thousand, seven hundred ninety-three dollarsc) Ninety thousand, five hundred eleven dollarsd) Eighty-eight thousand, eight dollarse) Ninety-nine thousand, nine hundred ninety-nine dollars4. Write the following in words.a) 2,070b) 9,217c) 47,030d) 98,104e) 40,600f) 78,999

Computational FluencyName:Date:Time:4–21. Complete the number patterns.a) 6,000 ; 7,000 ; ; 9,000 ;b) 2,400 ; 4,400 ; ; ; 10,400c) 4,065 ; 14,065 ; 24,065 ; ;d) 9,843 ; 9,943 ; ; 10,143 ;2. Write the values of the digits in each of the following numbers.a) 23,529b) 40,6183. Fill in the blanks.a) 5623 5,000 600 20 b) 16,048 10,000 40 8c) 40,180 100 80d) 72,005 70,000 5e) 63,100 63,000

Computational FluencyName:Date:4 – 2a4. Fill in the blanks.a) 4,000 300 7 b) 50,000 6,000 400 c) 30,000 700 60 8 d) 90,000 90 5. Fill in the blanks.a) is 1000 more than 42,628.b) 26,324 is 1000 more than .c) is 100 less than 90,000.d) 86,000 is 100 less than .e) 45,600 is more than 45,500.f) 38,400 is less than 39,400.g) 29,409 30,409h) 24,830 – 24,820i) 37,526 is more than 37,000.j) 37,526 is more than 7,526.Time:

Computational FluencyName:Date:Time:4–3Write the answers on the line.1. Which one of the following numbers has the digit 4 in the hundreds place?92,40524,92749,25050,9422. In 25,364, the digit 5 is in the place.3. Write the next number in the following number pattern.26,49531,49536,49541, 4954. Write the missing number in each of the following.a) 56,180 50,000 100 80b) 40,000 2,000 90 6 c) is 1000 more than 89,800.d) is 1000 less than 28,481.5. Which one of the following is the greatest.70,58278,50275,80278,2056. Which one of the following is the smallest30.30.03307. There were about 24,500 people at a football game. Which one of thefollowing could be the actual number of people?24,56124,39124,51924,083

Computational FluencyName:Date:Time:4–4Write the answers on the line.1. What is the greatest 5-digit number that can be formed using all of thedigits 0, 2, 9, 5 and 7?2. What does each of the digits in 86,373 stand for?86,3733X7X3X6X8X3. The value of the digit 6 in 68.64 is .4. In 19.49, which digit is in the hundredths place?5. Write the missing number in each of the following number patternsa) 50,230 ; ; 46,230 ; 44,230b) 53.54 ; 53.04 ; 52.04

Computational FluencyName:Date:Time:4–5Write the answers on the line.1. What number does each letter represent?ll79,000IIIAA :II79,500IIIIBIII80,000B :IIC 80,500C :2. Write the missing number in each of the following.a)36,795 30,000 700 95b)is 100 more than 29,912.c)is 10,000 less than 83,045.3. When 57,329 is written 57,300, it is rounded off to the nearest.4. Round off 15,247 to the nearest 10.5. Mrs. Cohen bought a shirt, which cost about 33. Which one of thefollowing could be the actual cost of the shirt? 33.10 33.95 33.50 32.406. In 4.73, the value of the digit 3 is .7. In 84.92, which digit is in the hundredths place?

Computational FluencyName:Date:Time:4–6Add.1.5 0 1 4.9 71 6 5 7.5 43 8 4 5.2 46 0 1 2.1 28 1 0 8.4 37 3 9 6.4 67 4 6 3.5 74 2 2 9.7 39 4 1 3 0

Computational FluencyName:Date:Time:4–7Add.1.6 4 72. 2 0 14.5 4 04 3 63. 2 4 25. 2 5 96 3 6 1 4 27 0 0 2 6 86.2 3 3 1 5 37. David has 410 blue marbles. He has 59 red marbles. How many marblesdoes he have altogether?8. Lucy has 125 stickers. Her brother has 64 stickers more than her. Howmany stickers does her brother have?9. After selling 242 baseball cards, Joe had 304 cards left. How many cardsdid he have at first?

Computational FluencyName:Date:Time:4–8Add.A.7 8 3 D.G.5 74 7 6 B. E.7 72 6 7 4 3 52 8 7C.3 92 7 8 F. 1 9 6H.5 9 5 2 6 67 0 29 96 6 1 2 7 9I.3 6 7 5 5 9

Computational FluencyName:Date:Time:4–9Add.1.6 5 42. 3 4 74.4 7 55 5 6 2 8 43. 3 8 45. 2 6 97.5 8 66 0 8 1 6 96. 1 9 28.2 8 9 5 7 66 9 56 3 7 2 7 79.4 9 7 3 1 4

Computational FluencyName:Date:Time:4 – 10Subtract.A.6 8 9–D.6 81 7 5–G.5 37 8 6–E.3 58 9 7–B.8 53 7 9–H.1 49 7 9–F.4 72 5 8–C.6 19 8 9–I.5 07 6 2–3 1

Computational FluencyName:Date:Time:4 – 11Subtract.1.4 4 72.– 1 3 14.9 4 8– 2 2 68 9 73.– 2 4 05.5 9 6– 1 4 27 6 3– 2 6 16.2 9 3– 1 5 37. A tailor bought 78 buttons. He used 43 of them. How many buttons didhe have left?8. Morgan saved 276. She saved 54 more than Mary. How much moneydid Mary save?9. Kristin went stopping with 245. She bought a watch and had 102 left.How much did she pay for the watch?

Computational FluencyName:Date:Time:4 – 12Subtract.1.5 22.– 3 74.9 65 0–43.– 3 65.– 5 77.7 46 2– 4 66.– 5 88.8 7– 5 98 34 5– 3 99.9 0– 6 4

Computational FluencyName:Date:Time:4 – 12aSubtract.A.9 7 3B.6 0 6– 2 3 8L.– 2 6 34 3 5–R.M.7 24 4 0U.7 5 0– 7 2 4N.– 1 0 77 8 4–E.6 9 2– 5 7 66 1 53 9–7 5What goes up when the rain comes down?Write the letters in the boxes below to find out.73511654033334374526363363735

Computational FluencyName:Date:Time:4 – 13Subtract.1.3 1 02.– 2 8 94.6 3 27 4 6– 6 6 93.– 3 2 85.– 4 7 37.5 2 53 3 4– 4 5 96.– 1 3 98.9 3 7– 8 7 96 1 84 5 3– 1 5 59.7 5 2– 2 7 8

Computational FluencyName:Date:Time:4 – 141. Find the missing factors.a)b)4 x 8x 5 15c) 7 x 56d) 4 X 32e) 5 x 45f) 6 X 42g) x 6 54h) x 9 27i) x 7 70j) x 8 642. Fill in the blanks.a)8 1 x8 2 xThe factors of 8 are , , , and .b)15 1 x15 3 xThe factors of 15 are , , , and .

Computational FluencyName:Date:Time:4 – 151.Is 2 a factor of 35?2.Is 3 a factor of 45?3. Write yes or no.Number303648607584Is 3 a factor ofthe number?Is 4 a factor ofthe number?Is 5 a factor ofthe number?

Computation

for school mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc. Primary mathematics textbook 1A/B. (2007). Singapore: Marshall Cavendish Education. Primary mathematics textbook 2A/B. (2007). Singapore: Marshall Cavendish Education. Primary mathematics textbook 3A/B. (2007).

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