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Whole Number Operations and

Mastering Basic Facts

•NCTM Number and Operations Standard

NCTM Number and Operations Standard

http://standards.nctm.org/document/chapter3/numb.htm

Instructional programs from prekindergartenthrough grade 12 should enable all students to--understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; compute fluently and make reasonable estimates.

Four PrerequisitesFacility with Counting

An integral aspect of children’s beginning work with operationsCounting forward, backward, by 2s, 3s, 4s

Experiences with a Variety of Concrete Situations

Manipulative materials provide a link to connect each operation to real- world problem- solving situations.

Four Prerequisites

Familiarity with Many Problem- solving Contexts

Using Word Problems

Experiences in Using Language to Communicate Mathematical Ideas

Children need to read, talk, listen, and write about mathematics

Word Problems

Young children develop an understanding of operations by solving a variety of word problems.Real world experience provide students with personal relevance.Real world problems can arise from classroom activities, events or children’s literature.

My generation will be running the world soon. If we say 4 + 4 = 9 then that’s the way it’s going to be!

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Introducing Addition

The process of joining things togetherCombining disjointed sets to form a total group. New terminology: ComposingThere are 100 facts that involve two one-digit addends and their sum.Progresses from the concrete to the pictorial to the symbolic

Addition Properties

Commutative property (order property)a + b = b + a

Associative property (grouping property) (a + b) + c = a + (b + c)

Addition property of zero -Identify element for addition

a + 0 = a

Addition StoriesFish Eyes by Lois EhlertHow Many Snails by Paul GigantiRed Riding Hood’s Math Adventure by Lalie HarcourtSplash by Ann JonasMission: Addition by Loreen LeedyDealing with Addition by Lynette LongDomino Addition by Lynette LongReady, Set, Hop! by Stuart Murphy Annie’s One to Ten by Ann OwenHershey's Kisses Addition Book by Jerry Pallotta

Subtraction

The inverse relationship of additionShould be approached as a part- part-whole connection to additionNew terminology: DecomposingThere are 100 factsThree interpretations

Take awayComparisonMissing addend

Subtraction Properties

The commutative, associative and identity property result in inequalities when working with subtraction.

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Subtraction Stories

Ten, Nine, Eight by Molly BangSeven Little Rabbits by John Becker Five Little Monkeys Jumping on the Bed by Eileen ChristenlowBlast Off! A Space Counting Book byNorma Cole,Ten Little Mice by Joyce DunbarRoll Over! A Counting Song by Edith FowlerHow Many Feet in the Bed? By Diane Johnston Hamm Ten Little Bunnies by Nurit KarlinElevator Magic by Stuart Murphy Hershey's Kisses Subtraction Book by Jerry Pallotta

Types of Addition and Subtraction Word Problems

There are 4 addition and subtraction problem types

JoinSeparatePart-part-wholeCompare

Examples of problem types

JoinElements are being added or joined togetherThere are three quantities in a join problem

Result Unknown 4 + 7 = ?Change Unknown 4 + ? = 11Start Unknown ? + 7 = 11

Examples of problem types

SeparateElements are being removed from a setThere are three quantities in a separate problem

Result Unknown 11 - 7 = ?Change Unknown 11 - ? = 4Start Unknown ? - 7 = 4

Examples of problem types

Part-Part wholeThere is no actionThere is no change over timeProblems focus on the relationship between a set and its two subsets.The three quantities include the two parts and the whole.

Part-part whole with the whole unknownPart-part whole with the part unknown

Examples of problem types

CompareThere is no actionInvolves comparisons between two different sets.The three quantities include the two wholes and the difference.

Compare – Difference unknownCompare- Larger whole is unknownCompare – Smaller whole is unknown

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Fact Families: Addition and Subtraction

Demonstrates relationship between addition and subtractionEach family consists of four related factsWhen children know the addition facts, they can more readily recall the related subtraction fact.3 + 5 = 8 8 – 5 = 35 + 3 = 8 8 – 3 = 5

Introducing Multiplication

Multiplication PropertiesCommutative property (order property)

a x b = b x aAssociative property (grouping property)

(a x b) x c = a x (b x c)Distributive property

a x (b + c) = (a x b) + (a x c)Identity element

(a x 1 = a)Zero property

(a x 0 = 0)

Multiplication Table

1441321201089684726048362412012

132121110998877665544332211011

120110100908070605040302010010

10899908172635445362718909

9688807264564840322416808

8477706356494235282114707

7266605448423630241812606

6055504540353025201510505

484440363228242016128404

36333027242118151296303

2422201816141210864202

12111098765432101

00000000000000

1211109876543210x

Multiplication PatternsMultiplication Patterns

1 x 1 = 11 x 2 = 21 x 3 = 31 x 4 = 41 x 5 = 51 x 6 = 61 x 7 = 71 x 8 = 81 x 9 = 91 x 10 = 10

5 x 1 = 55 x 2 = 105 x 3 = 15 5 x 4 = 20 5 x 5 = 255 x 6 = 305 x 7 = 355 x 8 = 405 x 9 = 455 x 10= 50

10 x 1 = 1010 x 2 = 2010 x 3 = 3010 x 4 = 4010 x 5 = 5010 x 6 = 6010 x 7 = 7010 x 8 = 8010 x 9 = 9010 x 10 = 100

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Multiplication Patterns2 x 1 = 22 x 2 = 42 x 3 = 6 2 x 4 = 82 x 5 = 102 x 6 = 122 x 7 = 142 x 8 = 162 x 9 = 182 x 10 = 20

4 x 1 = 44 x 2 = 84 x 3 = 124 x 4 = 16 4 x 5 = 204 x 6 = 244 x 7 = 284 x 8 = 324 x 9 = 364 x 10 = 40

8 x 1 = 88 x 2 = 168 x 3 = 248 x 4 = 328 x 5 = 40 8 x 6 = 48 8 x 7 = 568 x 8 = 648 x 9 = 728 x 10 = 80

Multiplication Patterns

3 x 1 = 33 x 2 = 63 x 3 = 93 x 4 = 12 3 x 5 = 153 x 6 = 183 x 7 = 213 x 8 = 243 x 9 = 273 x 10 = 30

6 x 1 = 66 x 2 = 126 x 3 = 186 x 4 = 246 x 5 = 30 6 x 6 = 36 6 x 7 = 42 6 x 8 = 48 6 x 9 = 54 6 x 10 = 60

Multiplication Patterns

9 x 1 = 99 x 2 = 189 x 3 = 27 9 x 4 = 36 9 x 5 = 459 x 6 = 549 x 7 = 639 x 8 = 729 x 9 = 819 x 10 = 90

Multiplication Patterns•7 x 1 = 7•7 x 2 = 14•7 x 3 = 21•7 x 4 = 28•7 x 5 = 35 •7 x 6 = 42 •7 x 7 = 49•7 x 8 = 56 •7 x 9 = 63 •7 x 10 = 70

Lattice MethodIntroduced in Italy in 1494

Lattice Method

8 3

3

5

9

51

24

0

04

2

90 5

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John Napier

Scottish Mathematician (1550-1617)Developed Tables of Logarithms of trigonometric functionsDeveloped method of multiplication using rods with numbers marked off on them.Introduced decimal notions for fractions.

Napier’s Rods

Napier’s Bones

http://www.cee.hw.ac.uk/~greg/calculators/napier/simulation.html

Circle Multiplication

Multiplication StoriesAnno's Mysterious Multiplying Jar by Mitsumasa AnnoSpaghetti and Meatballs for All! By Marilyn BurnsEach Orange Had 8 Slices by Paul GigantiBunches and Bunches of Bunnies by Louise MatthewsThe Sunday Scoop By Stuart MurphyToo Many Kangaroo Things to Do! By Stuart MurphyAmanda Bean’s Amazing Dream by Cindy NeuschwanderHershey's Milk Chocolate Bar Multiplication Book by Jerry PallottaOne Hundred Hungry Ants by Elinor J. PinczesGrain of Rice by Helena Clare Pittman The Best of Times by Greg Tang

Introducing Division

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Division

The inverse relationship of multiplicationThere are only 90 facts. Why?Two interpretations

Measurement divisionPartitive division

Division Properties

Identity property (6 ÷ 1 = 6)

Commutative and Associative property do not apply

Zero property is a special case

Division Stories

The Doorbell Rang by Pat HutchinsDivide and Ride by Stuart Murphy A Reminder of One by Elinor J. Pinczes

Understanding Multiplication and Division

Types of Multiplication Word ProblemsThere are 4 problem types

Equal groupArea and arrayMultiplicative comparisonsCombination problems

Examples of problem types

Equal groupsInvolve making a certain number of equal sized groups.The three numbers in the problem

Number of groupsSize of the groupsTotal number of objects

For example: 2 x 3 means 2 groups of 3

Examples of problem typesArea and Array problems

Involve finding the area of a rectangular region or finding the total number of objects in a rectangular array or arrangement.Arrays are rectangular arrangements of discrete, countable objects such as desks arranged in rows in a classroom.

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Examples of problem types

Multiplicative comparisonsInvolve comparing two quantities multiplicatively.These problems describe how many times as much one quantity is as compared to another quantity

Examples of problem types

Comparison problemsAlso known as Cartesian productsThey involve different combination that can be made from sets of objects.Example: number of outfits that can be made from two shirts and three pairs of pants.

Understanding Multiplication and Division

Types of Division Word ProblemsThere are 5 problem types

Equal groups: Partitive DivisionEqual groups: Measurement DivisionArea and arrayMultiplicative comparisonsCombination problems

Examples of problem typesEqual groups

Involve splitting a larger group into several smaller groups.The three numbers represent the

Number of groupsThe size of the groupsThe total number of objects

There are 2 different types of equal groups division problems.

Partitivemeasurement

Examples of problem typesPartitive Division

The process of dividing items into groups oneone-- atat-- aa-- time.time.The total number of objects is partitioned into a specified number of groupsReferred to as Fair Sharing

When you know the number of groups When you know the number of groups or parts to make, the problem is known or parts to make, the problem is known as a as a PartitivePartitive division problem.division problem.

Examples of problem types

Measurement DivisionThe total number of objects is measured out into groups of a certain size.Referred to as repeated subtraction.

When you know the size of the groups to be measured out, the problem is known as a measurement division problem.

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Fact Families: Multiplication and Division

Demonstrates relationship between multiplication and divisionEach family consists of four related facts

Except when zero is a member of the familyDivision by zero is undefined

3 x 7 = 21 21 ÷ 7 = 37 x 3 = 21 21 ÷ 3 = 7

Basic Facts InstructionAllow students time to explore their own methods and solutions.Have manipulatives available for modeling and to support children’s thinkingHave children develop fact strategies along with fact knowledgePresent problems in context of meaningful situationsEncourage children to share their strategies, discuss and compare them and to learn from each other.

SummaryStudents must be provided with ample opportunities to develop a sound understanding of whole number operations.An understanding of whole number operations begins with concrete experiences.Solving a variety of types of word problems is necessary for developing “operation sense.”