195

C H A P T E R 9

Operations: Meaningsand Basic Facts

She let the children choose the kind of candythey wanted, but she realized there could be aproblem—everyone might want the largest. So shemade a rule: each child could have

• 4 yellow, or• 3 orange, or• 2 red

Put each grouping on the board.

Snapshot of a Lesson

Objective

Children develop multiplication ideas as re-peated addition and division ideas as repeated sub-traction using groups of candies.

Needed Materials

Flannelboard or magnetic board; flannel ormagnetic “candies”: 20 small yellow, 15 mediumorange, and 12 large red; disks for each child: 25yellow, 20 orange, and 15 red.

Procedures

Read this story to children, stopping to askquestions and model each situation on the board:

Once upon a time there was a little old ladywho lived in a little old house in the middle of alittle old town. She was such a nice little old ladythat all the children who lived in the town liked tocome and visit her. For a special treat, she wouldoften give them candy. She had three kinds, eachwrapped in a different color:

• small yellow ones• medium orange ones• large red ones

Present these situations and questions for con-sideration, using the board:

1. One day 3 children came to visit. They all tookred candies. How many red candies could eachchild take? Who can show how many piecesthe children took altogether?

Expected response: 2 � 2 � 2 � 6; possibly 3groups of 2, 3 twos are 6, or 3 � 2 � 6. Havechild show and count.

2. Another day, 2 children took red.(a) How many pieces altogether?

Put a sample of each on the board.

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196 Chapter 9 • Operations: Meanings and Basic Facts

(b) How do you know?(c) Is there any other way to know?

Ask questions (a), (b), and (c) for the nextexamples. After several examples with the wholegroup watching the board, have each child usedisks.

3. 3 children took orange4. 2 children took yellow5. 4 children took orange6. 3 children took yellow7. If needed:

2 children took orange4 children took yellow

8. 6 children—3 took yellow, 3 took red9. 4 children—2 took red, 2 took orange

10. 5 children—2 took orange, 3 took yellow11. If needed:

5 children—all took yellow6 children—all took red

12. Suppose the old lady had 8 pieces of yellowcandy. How many children could choose yel-low? How do you know?

After several examples using the board, haveeach child solve the following problems with disks.

1. 15 orange candies (how many children?)2. 10 red candies3. 12 red candies

of addition, subtraction, multiplication, and divi-sion—and knowledge of the basic number facts foreach of these operations—provides a foundationfor all later work with computation. To be effectivein this later work, children must develop broadconcepts for these operations. This development ismore likely to happen if you present each opera-tion with multiple representations using variousphysical models. Such experiences help childrenrecognize that an operation can be used in severaldifferent types of situations. Children also must un-derstand the properties that apply to each opera-tion and the relationships between operations.

Learning the basic number facts is one of thefirst steps children take as they refine their ideasabout each operation. By using these facts, plus anunderstanding of place value and mathematical

4. 12 yellow candies5. 12 orange candies6. 9 orange candies7. 8 red candies

Practice

Worksheet paralleling lesson (use to evaluate):

Introduction

The Snapshot of a Lesson incorporates severalessential components of a well-planned classroomactivity involving computation. First and foremost,it involves the student actively in manipulatingobjects to answer questions. It provides problem-solving experiences that promote reasoning and dis-cussion. “How do you know?” is an importantquestion because it encourages students to thinkabout “why” and not just “what.” Also, computa-tional ideas are posed in a potentially real situation.

These components are important in elementaryschool mathematics lessons, particularly as chil-dren develop understanding of the relevance andmeaning of computational ideas. An understanding

Extension

1. If the old lady gave away 2 candies, how manychildren came? What color candies did shegive? What if she gave 9 candies? 15?

2. If she gave away 6 candies, how many childrenmight have come, and what color candies didthey get? Can you find more than one answerto this question? Find some other numbers ofcandies that have more than one answer.

3. Pretend you see boots all lined up outside aclassroom. How many children are in theroom if there are 18 boots and each child leftboots?

4. Ask children to make up similar problems.

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Introduction 197

properties, a child can perform any addition, sub-traction, multiplication, or division with wholenumbers. Understanding the operations and havingimmediate recall of number facts are essential indoing estimation, mental computation, and pencil-and-paper algorithms; but these skills are just as es-sential when using calculators and computers.Without such devices, the basic facts form thebuilding blocks for performing more difficult, mul-tidigit calculations. When calculators are readilyavailable, the basic facts (along with operationsense and understanding of place value) provide ameans for quickly checking the reasonableness ofanswers. Moreover, knowing the basic facts lets

children perform calculations or estimate answersin many everyday situations where it would beslower to use a calculator. So no matter what typeof computation a child is using—mental computa-tion, estimation, paper-and-pencil, or a calculator—quick recall of the basic number facts for each oper-ation is essential.

In addition to remembering basic number facts,students need to make gains in the ability to an-swer questions requiring computational accuracyand in situations where efficiency is useful. Figure9-1 provides a partial listing of the expectationsidentified in Principles and Standards for SchoolMathematics (NCTM, 2000) description of the

Instructional programs from prekindergarten through grade Pre-K–2 Expectations Grades 3–5 Expectations 12 should enable all students to: All students should: All students should:

• Understand meanings of • Understand various meanings • Understand various meanings of operations and how they relate of addition and subtraction of multiplication and division to one another whole numbers and the • Understand the effects of

relationship between the two multiplying and dividing whole operations numbers

• Understand the effects • Identify and use relationshipsof adding and subtracting whole between operations, such asnumbers division as the inverse of

• Understand situations multiplication, to solve problemsthat entail multiplication and • Understand and use propertiesdivision, such as equal groupings of operations, such as theof objects and sharing equally distributivity of multiplication

over division • Describe classes of numbers

according to characteristicssuch as the nature of theirfactors

• Compute fluently and make • Develop and use strategies • Develop fluency with basicreasonable estimates for whole-number computations, number combinations for

with a focus on addition multiplication and division andand subtraction use these combinations to

• Develop fluency and basic mentally compute relatednumber combinations for problems, such as 30 � 50addition and subtraction • Develop fluency in adding,

• Use a variety of methods and subtracting, multiplying, andtools to compute, including dividing whole numbers objectives, mental computation, • Develop and use strategies to estimation, paper and pencil, estimate the results of and calculators dividing whole numbers

FIGURE 9-1 Number and Operation Standard with expectations for children ingrades prekindergarten–2 and 3–5.

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198 Chapter 9 • Operations: Meanings and Basic Facts

Number and Operations Standard that we will fo-cus on throughout this chapter.

FOCUS QUESTIONS

1. What different models can be used to representeach of the operations: addition, subtraction,multiplication, and division?

2. What word problems and other situations helpdevelop meaning for each of the operations?

3. What mathematical properties pertain to eachoperation?

4. What is the three-phase process for helpingchildren learn basic facts?

5. How should thinking strategies for the basicfacts be taught?

6. How does number theory provide opportunitiesto discover mathematics in the elementarymathematics curriculum?

number concepts they have already constructed(Kouba and Franklin, 1993). Most children enter-ing school are ready in some ways and not ready inothers for formal work on the operations. Four pre-requisites for such work seem particularly impor-tant: (1) facility with counting, (2) experience witha variety of concrete situations, (3) familiarity withmany problem-solving contexts, and (4) experienceusing language to communicate mathematicalideas.

Facility with Counting

Children use counting to solve problems in-volving addition, subtraction, multiplication, anddivision long before they come to school, as re-search has indicated (Baroody and Standifer,1993). Any problem with whole numbers can besolved by counting, provided there is sufficienttime. Because there is not always the time to solveproblems by counting, children need to be able touse more efficient operations and procedures thathelp them cope with more difficult computation.Figure 9-2 illustrates this idea by comparing thecounting method with the multiplication operation.

Counting nevertheless remains an integral as-pect of children’s beginning work with the opera-tions. They need to know how to count forward,backward, and by twos, threes, and other groups

How many bottles?

We can count —1, 2, 3, 4, 5, 6

How many bottles?

We could count, but it’s more efficient to use acombination of counting (6 bottles in a carton,5 cartons) and multiplication — 5 × 6 = 30.

Math Links 9.1The full-text electronic version of NCTM’s Number

and Operation Standard, with electronic examples(interactive activities), is available at the NCTM’s website, which you can access from this book’s web site.Click on pre-K–2, 3–5, or 6–8 to read more about theNumber and Operation Standard.

Math Links 9.2A variety of Web resources for basic facts that may be

used by teachers and children may be found at the NCTM’sIlluminations web site (accessible from this book’s website) under the Number and Operation Standard.

FIGURE 9-2 An example showing the efficiency ofusing operations.

Helping Children DevelopNumber Sense andComputational Fluency

Ultimately, the instructional goal is that chil-dren not only know how to add, subtract, multiply,and divide, but, more important, know when to ap-ply each operation in a problem-solving situation.Children also should be able to recall the basicfacts quickly when needed.

How can teachers help children attain theseskills and understandings? Begin by finding outwhat each child knows. Then capitalize on theirknowledge while continuing to build on the

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Helping Children Develop Number Sense and Computational Fluency 199

Experience with a Variety of Concrete Situations

Children need to have many experiences inreal-life situations and in working with physicalobjects to develop understanding about mathemati-cal operations. Research has indicated that workwith actual physical objects promotes achievementfor most children (Sowell, 1989). Understandingimproves if they can relate mathematical symbolsto some experience they have had or can visualize.A basic fact cannot be learned with meaning un-less it has been experienced in a situation thatgives it meaning.

Manipulative materials serve as a referent forlater work with the operations, as well as for con-structing the basic facts. They also provide a linkto connect each operation to real-world problem-solving situations. Whenever a child wants to besure that an answer is correct, materials can beused for confirmation.

Familiarity with Many Problem-Solving Contexts: Using WordProblems

Word problems are an essential part of learningto use mathematics effectively. Word problems areused in mathematics classes for developing concep-tual understanding, teaching higher-level thinkingand problem-solving skills, and applying a varietyof mathematical ideas. Word problems requirereading, comprehension, representation, and calcu-lation. Children generally have little difficulty withsingle-step word problems but have more difficultywith multistep, complex problems.

As with other mathematical content, a varietyof problem-solving contexts or situations should beused to familiarize students with the four basic op-erations, continuing all along the way until compu-tational mastery is achieved. Children need to

think of mathematics as problem solving—as ameans by which they can resolve problemsthrough applying what they know, constructingpossible routes to reach solutions, and then verify-ing that the solutions make sense. Students mustrealize that mathematics is a tool that has real-lifeapplications. Most children already know that com-putation is used in everyday life. Mathematics les-sons need to be connected to those experiences,but students also need to realize that 6 � 8 � ❐ or9 � 2 � ❐ also may be problems—ones that theycan solve. They need to have the attitude, “I don’tknow the answer, but I can work it out.”

Experience in Using Language toCommunicate Mathematical Ideas

Children need to talk and write about mathe-matics; experiences to develop meanings need to beput into words. Manipulative materials and prob-lems can be vehicles for communicating aboutmathematics. Such discussion of mathematics is acritical part of meaningful learning. All early phasesof instruction on the operations and basic factsshould reflect the important role language plays intheir acquisition. The Principles and Standards forSchool Mathematics (NCTM, 2000) discusses theroles of language in great depth in presenting therecommendations on communication, and thus pro-vides a valuable source of additional information.

The move to symbols is often made too quicklyand the use of materials dropped too soon. Instead,the use of materials should precede and then paral-lel the use of symbols. Children should be recordingsymbols as they manipulate materials. As illustratedin the lesson that opened this chapter, languageshould be used to describe what is happening in agiven situation. Then and only then will childrensee the relation of symbols to manipulation of mate-rials and problem setting (Carey, 1992).

The language that children learn as they com-municate about what they are doing, and whatthey see happening as they use materials, helpsthem understand the symbolism related to opera-tions. Thus the referent for each symbol isstrengthened. Children should begin their workwith operations after talking among themselvesand with their teacher about a variety of experi-ences. They need to be encouraged to continuetalking about the mathematical ideas they meet asthey work with the operations. As soon as feasible,they need to put their ideas on paper—by draw-ings alone at first. As soon as they are able towrite, children should also be encouraged to writenumber sentences and narrative explanations oftheir thinking.

Math Links 9.3You can use electronic manipulatives, such as Num-

ber Line Arithmetic and Rectangle Multiplication (acces-sible from this book’s website), to illustrate basic facts.

(see Chapter 7). They need to count as they com-pare and analyze sets and arrays as they affirm theirinitial computational results, but they need morethan counting to become proficient in computing.

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200 Chapter 9 • Operations: Meanings and Basic Facts

Developing Meanings for the Operations

As we explained earlier, children encounterthe four basic operations in natural ways whenthey work with many diverse problem situations.By representing these problem situations (e.g., act-ing them out, using physical models, or drawingpictures), they develop meanings for addition, sub-traction, multiplication, and division. Mastery ofbasic facts and later computational work with mul-tidigit examples must be based on a clear under-standing of the operations.

Thus both computational proficiency andunderstanding of operations are desired outcomesof mathematics instruction. The following generalsequence of activities is appropriate for helpingchildren develop meaning for the four basicoperations:

1. Concrete—modeling with materials: Use a vari-ety of verbal problem settings and manipulativematerials to act out and model the operation.

2. Semiconcrete—representing with pictures:Provide representations of objects in pictures, di-agrams, and drawings to move a step away fromthe concrete toward symbolic representation.

3. Abstract—representing with symbols: Usesymbols (especially numeric expressions andnumber sentences) to illustrate the operation.

In this way, children move through experiencesfrom the concrete to the semiconcrete to the ab-stract, linking each to the others.

The four operations are clearly different, butthere are important relationships among them thatchildren will come to understand through model-ing, pictorial, and symbolic experiences:

• Addition and subtraction are inverse opera-tions; that is, one undoes the other:

5 � 8 � 13 ;9: 13 � 5 � 8

• Multiplication and division are inverse opera-tions:

4 � 6 � 24 ;9: 24 � 4 � 6

• Multiplication can be viewed as repeatedaddition:

4 � 6 ;9: 6 � 6 � 6 � 6

• Division can be viewed as repeated subtraction:

24 � 4 ;9: 24 � 6 � 6 � 6 � 6

These relationships can be developed throughcareful instruction with a variety of different expe-riences.

Addition and Subtraction

Figure 9-3 illustrates a variety of models (in-cluding counters, linking cubes, balance scale, andnumber line) that can be used to represent addi-tion. Each model depicts the idea that additionmeans “finding how many in all.”

The models for addition can also be used forsubtraction. Each model can be applied in the threefollowing different situations that lead to subtrac-tion.

1. Separation, or take away, involves having onequantity, removing a specified quantity from it, andnoting what is left. Research indicates that thissubtraction situation is the easiest for children tolearn; however, persistent use of the words takeaway results in many children assuming that thisis the only subtraction situation and leads to mis-understanding of the other two situations. This iswhy it is important to read a subtraction sentencesuch as 8 � 3 � 5 as “8 minus 3 equals 5” ratherthan “8 take away 3 equals 5.” Take away is justone of the three types of subtraction situations.

Peggy had 7 balloons. She gave 4 to other chil-dren. How many did she have left?

FIGURE 9-3 Some models for addition.

5 + 2 = 7

Disks orcounters Number line

0 1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 956789 1234

Balance

Linking cubes

0 1 2 3 4 5 6 7

7 – 4 =

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Developing Meanings for the Operations 201

2. Comparison, or finding the difference, involveshaving two quantities, matching them one-to-one, and noting the quantity that is the differ-ence between them. Problems of this type are alsosolved by subtraction, even though nothing is be-ing taken away.

Peggy had 7 balloons. Richard had 4 balloons.How many more balloons did Peggy have thanRichard?

learning about multiplication. Writing symbols foreach action is not essential at this early stage. Mov-ing, counting, and questioning are the importantcomponents.

Lesson Idea 9-1 illustrates a slightly more sym-bolic activity. The children solve problems in avariety of ways using the dot sticks, followed by in-dividual practice. Similar activities can be donewith counters, linking cubes, or other objects.

Initially, you should use symbols as a comple-ment to the physical manipulation of objects. Theyare a way of showing the action with the materialsand should always be introduced in conjunctionwith a concrete material or model. As the workprogresses, the amount of symbolization you offerand encourage should increase.

The number line is used in some textbooks asa model for addition and subtraction, but it mustbe used with caution. Over the years, research has

7 – 4 =

3. The final type of subtraction situation isknown as part-whole. In this type of problem, aset of objects can logically be separated into twoparts. You know how many are in the entireset and you know how many are in one of theparts. Find out how many must be in the remain-ing part.

Peggy had 7 balloons. Four of them werered and the rest were blue. How many wereblue?

1 2 3 4 5 6 7 85678 1234

7 – = 4

Some educators refer to this sort of subtractionsituation as a “missing-addend” problem because itmay be helpful to “think addition” to find the an-swer. For example, in the preceding problem, youcould ask yourself: 4 plus what equals 7?

The importance of providing many varied expe-riences in which children use physical objects tomodel or act out examples of each operation can-not be overemphasized. The Snapshot of a Lessonat the beginning of this chapter provided one suchexperience. It illustrates a way to involve eachchild in forming equal-sized groups as a lead-in to

Lesson Idea 9-1DOT STICKS

Objective: Children model and solve additionproblems using dot sticks.

Grade Level : 1

� Ask children to pick up 4 dots. (Most children willpick up the 4 stick.)• Is there another way to pick up 4 dots?

� Now ask them to pick up 6 dots.• How many ways can you do it?• Can you pick up 6 dots using 3 sticks?

� Ask for 11 dots.• Can you do it with 2 sticks?• 3 sticks?• 4 sticks?

� Ask for 8 dots using only 3 sticks.

� Ask how many different ways you can pick up 10dots: List them on the chalkboard.

� Follow up with individual worksheets.

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202 Chapter 9 • Operations: Meanings and Basic Facts

shown that children often find number lines diffi-cult to use and interpret. Sometimes they are notsure whether they should count spaces betweentic-marks or the tic-marks themselves whenattempting to model a problem, or they may not besure where on the number line to begin counting.(For example, to find 5 � 7 on a number line, theymay put their finger on 5 as they begin countingfrom 1 up to seven, whereas they actually shouldbegin counting when pointing at 6.) In other cases,children seem unsure how to relate the numberline to a problem situation. For example, considerthis number line:

sorts of things. For example, consider a problemsuch as “Andrew has two boxes of trading cards.Each box holds 24 cards. How many cards doeshe have altogether?” This problem may be written2 � 24 � 48, where the first factor (2) tells ushow many groups or sets of equal size are beingconsidered, while the second factor (24) tells usthe size of each set. The third number (48),known as the product, indicates the total of all theparts (here, the total number of cards). By con-trast, the old saying “you can’t add apples and or-anges” points out the fact that in addition andsubtraction problems, a common label must be at-tached to all the numbers involved. Apples andoranges can be added only if we relabel all thenumbers in the problem with a common label(such as “fruit”). In the trading card problem, thelabels for the numbers would be, respectively,boxes (2), cards per box (24), and cards (48).

Figure 9-4 illustrates some of the most com-monly used models for illustrating multiplicationsituations: sets of objects, arrays, and the numberline. Research indicates that children do best whenthey can use various representations for multiplica-tion and division situations and can explain the re-lationships among those representations (Koubaand Franklin, 1995).

Researchers have identified four distinct sortsof multiplicative structures: equal groups, multi-plicative comparisons, combinations, and areas/ar-rays (Greer, 1992). Problems involving the first twoof these structures are most common in elementaryschool, although students should eventually be-come familiar with all four. The four multiplicativestructures are described here to help you, as theteacher, understand and recognize their variety.You are not expected to teach these labels tochildren, but you should try to ensure that they

0 1 2 3 4 5 6 7 8

In the second national mathematics assess-ment, about twice as many nine-year-oldsresponded that “5 � 7” was pictured as did thoseresponding with the correct answer, “5 � 2 � 7.”The same error was made by 39 percent of the 13-year-olds. Carpenter et al. (1981, p. 19) suggested:

Since the model does not seem to clearly suggestthe operation, the meaning must be developed ormisunderstandings may occur. The mathematicscurriculum should be constructed to ensure thatstudents have a meaningful development of thebasic operations. Certainly, many types of mod-els can help this development, but they must becarefully selected and meaningfully taught.

Multiplication and Division

The same sequence of experiences usedfor developing understanding of addition andsubtraction—moving from concrete, to pictorial,to symbolic—should also be followed for multi-plication and division; however, one importantway that multiplication and division problems dif-fer from addition and subtraction problems is thatthe numbers in the problems represent different

Math Links 9.4If you would like to see a variety of lesson plans for

developing addition and subtraction meaningfully, Do Itwith Dominoes and Links Away may be found at theNCTM’s Illuminations web site, which you can accessfrom this book’s web site.

FIGURE 9-4 Commonly used models for multiplication.

Sets of objects

3 × 5

Arrays

4 × 6 2 × 3

Number line

0 1 2 3 4 5 6 7 8 9 10 11 12 13

4 × 2

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Developing Meanings for the Operations 203

encounter a broad range of problem situations in-volving multiplication and division.

1. Equal-groups problems involve the most com-mon type of multiplicative structure. When boththe number and size of the groups are known (butthe total is unknown), the problem can be solvedby multiplication. The problem given earlier, aboutAndrew’s trading card collection, is an example ofan equal-groups problem. When the total in anequal-groups problem is known, but either thenumber of groups or the size of the group is un-known, the problem can be solved by division. Thetwo distinct types of division situations that canarise (depending on which part is unknown) aremeasurement or partition division (described later).

2. Comparison problems involve another commonmultiplicative structure. With comparison prob-lems for subtraction, there were two different setsthat needed to be matched one-to-one to decidehow much larger one was than the other. In simi-lar fashion, comparision problems with multiplica-tive structures involve two different sets, but therelationship is not one-to-one. Rather, in multi-plicative situations, one set involves multiplecopies of the other. An example of a multiplicativecomparision situation might be: “Hilary spent $35on Christmas gifts for her family. Geoff spent 3times as much. How much did Geoff spend?” Inthis case, Hilary’s expenditures are being comparedwith Geoff’s, and the problem is solved by multipli-cation. The question does not involve “How muchmore?” (as it would if the problem involved addi-tive/subtractive comparision). Instead, the struc-ture of the problem involves “How many times asmuch?” If the problem is changed slightly to in-clude information about how much Geoff spent butto make either Hilary’s expenditures unknown orthe comparison multiplier unknown, then theproblem could be solved by division. Examples ofthese problem structures are: (a) Hilary spent acertain amount on Christmas gifts for her family,and Geoff spent 3 times as much. If Geoff spent$105, how much did Hilary spend? (b) Hilary spent$35 on Christmas gifts for her family and Geoffspent $105. How many times as much money asHilary spent did Geoff spend?

3. Combinations problems involve still anothersort of multiplicative structure. Here the two fac-tors represent the sizes of two different sets and theproduct indicates how many different pairs ofthings can be formed, with one member of eachpair taken from each of the two sets. Combinationproblems are also sometimes known as Cartesianproduct problems (Quintero, 1985). For example,consider the number of different sundaes possible

with four different ice cream flavors and two top-pings, if each sundae can have exactly one icecream flavor and one topping:

Vanila

Pineapple

Butterscotch

Cherry MintIce cream flavors

Toppings

Chocolate

Other examples of combination problems arethe number of choices for outfits given 5 T-shirtsand 4 shorts or the number of different sandwichespossible with 3 choices of meat, 2 choices ofcheese, and 2 kinds of bread.

4. Finally, area and array problems also are typi-cal examples of multiplicative structure. The areaof any rectangle (in square units) can be found ei-ther by covering the rectangle with unit squaresand counting them all individually or by multiply-ing the width of the rectangle (number of rows ofunit squares) by the length (number of unitsquares in each row). Similarly, in a rectangulararray—an arrangement of discrete, countable ob-jects (such as chairs in an auditorium)—the totalnumber of objects can be found by multiplying thenumber of rows by the number of objects in eachrow. The array model for multiplication can be es-pecially effective in helping children visualize mul-tiplication. It may serve as a natural extension ofchildren’s prior work in making and naming rec-tangles using tiles, geoboards, or graph paper:

Graph paperGeoboardTiles

These illustrations show a 2-by-3 or 3-by-2 rec-tangle. Thus, each rectangle contains six smallsquares. Asking children to build and name nu-merous rectangles with various numbers is a goodreadiness experience for the concept of multiplica-tion. In the Classroom 9-1 through 9-3 illustrateseveral experiences designed for this purpose.

Just as sets of objects, the number line, and ar-rays are useful in presenting multiplication, theycan also be useful in presenting division, with the

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204 Chapter 9 • Operations: Meanings and Basic Facts

relationship to repeated subtraction frequentlyshown. For division, however, two different typesof situations must be considered: measurement andpartition.

1. In measurement (or repeated subtraction) situ-ations, you know how many objects are in eachgroup and must determine the number of groups.

Jenny had 12 candies. She gave 3 to each per-son. How many people got candies?

Here, you can imagine Jenny beginning with 12candies and making piles of 3 repeatedly until allthe candies are gone. She is measuring how

In the Classroom 9-1CONSTRUCT A RECTANGLE!

OBJECTIVE: Building rectangles with tiles todevelop visual representations for multiplicationfacts.

GRADE LEVEL: 3

� Use 12 tiles.

� Make a rectangle using all 12 tiles.

� Write a multiplication number sentence to describeyour rectangle:

� � 12.

� Use 12 tiles over and over again to make differentrectangles. Draw a picture and write a number sen-tence for each.

� � 12

� � 12

In the Classroom 9-2RECTANGLES AND MORE

RECTANGLES!

OBJECTIVE: Building rectangles with tiles todevelop visual representations for multiplicationfacts.

GRADE LEVEL: 3

� How many ways can you � List the ways:make a rectangle with this many tiles? Draw pictures.The first one is done for you.

Drawthe rectangle you

made here.

How manydifferent wayscan you do it?

Have youtried a 6-by-2

rectangle?How about a

12-by-1?

6 tiles1 × 6 or 6 × 1

2 × 3 or 3 × 2

4 tiles or

3 tiles or

8 tiles or

or

9 tiles or

2 tiles or

10 tiles or

or

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Developing Meanings for the Operations 205

many groups of 3 she can make from the originalpile of 12.

Another example might be measuring howmany 2-foot hair ribbons can be made from a 10-foot roll of ribbon. Imagine repeatedly stretchingout 2 feet and cutting it off, thus measuring howmany hair ribbons you can make.

2. Partition, or sharing, situations are those inwhich a collection of objects is separated into agiven number of equivalent groups and you seekthe number in each group. By contrast with meas-urement situations, here you already know howmany groups you want to make, but you don’tknow how many objects must be put in eachgroup.

Gil had 15 shells. If he wanted to share themequally among 5 friends, how many should hegive to each?

Here imagine Gil passing out the shells to his fivefriends (one for you, one for you, one for you, thena second to each person, and so on) until they areall distributed, and then checking to see how manyeach person got.

In the Classroom 9-3HOW MANY SQUARES IN A

RECTANGLE?

OBJECTIVE: Drawing rectangles on grid paperto develop visual representations for multiplica-tion facts.

GRADE LEVEL: 3

� Draw a 3 � 4 rectangle:

� Your turn:Draw each rectangle, color it in, then tell how manysquares.

That's3 × 4 or 4 × 3.

It's a rectangle with12 squares.

5 × 3 = ____ 4 × 4 = ____

4 × 1 = ____ 2 × 3 = ____

Person 1 Person 2 Person 3 Person 4

Friend 1 Friend 2 Friend 3 Friend 4 Friend 5

Partitioning (or sharing) is difficult to show ina diagram, but it is relatively easy to have childrenact out. Dealing cards for a game is another in-stance of a partition situation.

It is certainly not necessary for children tolearn these terms or to name problems as meas-urement or partition situations, but it is importantfor you as a teacher to know about the two typesof division situations so you can ensure that yourstudents have opportunities to work with exam-ples of each. It is vital that students be able toidentify when a problem situation requires divi-sion, and that means being able to recognize bothtypes of situations as involving division. LessonIdea 9-2 illustrates an activity that can be used tointroduce the idea of division to children in ameaningful way.

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206 Chapter 9 • Operations: Meanings and Basic Facts

Mathematical Properties

An understanding of the mathematical proper-ties that pertain to each operation (Table 9-1) is vi-tal to children’s understanding of the operation andhow to use it. This understanding is not a prerequi-site to work with operations, but it must be devel-oped as part of understanding operations.

In elementary school, children are not expectedto state these properties precisely or identify themby name. Rather, the instructional goal is to helpchildren understand the commutative, associative,

distributive, and identity properties and to usethem when it is efficient. Table 9-1 gives the mean-ing of each property, states what children shouldunderstand, and provides examples to illustratehow the property can make learning and using thebasic facts easier.

Understanding these properties implies know-ing when they apply. For example, both additionand multiplication are commutative, but neithersubtraction nor division is:

7 � 3 is not equal to 3 � 7

28 � 7 is not equal to 7 � 28

Many children have difficulty with the idea ofcommutativity. They tend to “subtract the smallernumber from the larger” or to “divide the largernumber by the smaller” regardless of their order.Care needs to be taken to ensure that they con-struct correct notions.

Overview of Basic Fact Instruction

As children develop concepts of meanings ofoperations, instruction begins to focus on certainnumber combinations. These are generally referredto as the basic facts:

• Basic addition facts each involve two one-digitaddends and their sum. There are 100 basic ad-dition facts (from 0 � 0 up to 9 � 9; see Figure9-5). To read off a fact (say, 4 � 9 � 13), findthe first addend (4) along the left side and thesecond addend (9) along the top. By readinghorizontally across the 4-row from the left andvertically down the 9-column from the top, youfind the sum (13).

• Basic subtraction facts rely on the inverse rela-tionship of addition and subtraction for theirdefinition. The 100 basic subtraction facts re-sult from the difference between one addendand the sum for all one-digit addends. Thus the100 subtraction facts are also pictured in Figure9-5, the same table that pictures the 100 basicaddition facts. You read off a basic subtractionfact (say, 13 � 4 = 9) by finding the box in the4-row that contains the sum 13, then readingup that column to find the difference (9) at thetop of the table. Note that a sentence such as13 � 2 � 11 is neither represented in Figure 9-5nor considered a basic subtraction fact because2 � 11 � 13 is not a basic addition fact (since11 is not a one-digit addend).

Lesson Idea 9-2LOTS OF LINKS

Objective: Children model division with remaindersusing links to make equal-sized groups.

Grade Level : 3–4

Materials : 20 links (or other counters) for each child;an overhead projector

Activity: Discuss with children different ways to sep-arate the links into equal-sized groups.

� Scatter 12 links on an overhead projector.• How can we place these links into equal-sized

groups? (6 groups of 2, 4 groups of 3, 1 group of12, etc.)

• How many different ways can we do it?� Scatter 15 links on the projector.

• How many different ways can these links be splitinto equal-sized groups?

• What happens if we try dividing 13 links intogroups of 5?

2 groups and 3 leftovers

� Focus on numbers that can be divided into equal-sized groups without “leftovers”:• Which of these numbers can be divided into

groups of 3 without having leftovers?

8 4 7 3 18Try each one.

• Name a number that will have no leftovers if we di-vide it into groups of 5. How many numbers likethis can you name?

• Name a number that will have 1 leftover if we di-vide it into groups of 5. Can you name 3 numberslike this?

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Overview of Basic Fact Instruction 207

• Basic multiplication facts each involve two one-digit factors and their product. There are 100 ba-sic multiplication facts (from 0 � 0 up to 9 � 9).

• Basic division facts rely on the inverse relation-ship of multiplication and division, but thereare only 90 basic division facts. Because

division by zero is not possible, there are nofacts with zero as the divisor.

Development and mastery of the addition andsubtraction facts begins in kindergarten or firstgrade and continues as multiplication and divisionfacts are developed and practiced in third andfourth grades. Some children, however, have notmastered the facts several years later. Sometimesthe problem may be a learning disability thatmakes it virtually impossible for a child to memo-rize the facts. Use of a calculator may allow such achild to proceed with learning mathematics. Or, asresearch by Clark and Kamii (1996) indicates formultiplication, a child may have trouble with mul-tiplication facts because he or she has not devel-oped the ability to think multiplicatively.

More commonly, children’s difficulties in mas-tering basic facts may stem from one (or both) ofthe following two causes, and in these cases teach-ers can definitely provide help. First, the underlyingnumerical understandings may not have been de-veloped. Thus the process of remembering the factsquickly and accurately becomes no more than rotememorization or meaningless manipulation of sym-bols. As a result, the child has trouble remembering

Table 9-1 • Mathematical Properties for Elementary-School Children

Property Mathematical Language Child’s Language How It Helps

Commutative For all numbers a and b: If 4 � 7 � 11, then 7 � 4 must The number of addition ora � b � b � a equal 11, too. If I know 4 � 7, multiplication facts to be and I also know 7 � 4. memorized is reduced from a � b � b � a 100 to 55.

Associative For all numbers a, b, and c: When I’m adding (or When more than two numbers(a � b) � c � a � (b � c) multiplying) three or more are being added (or multiplied),and numbers, it doesn’t matter combinations that make the task(ab)c � a(bc) where I start. easier can be chosen.

For example,37 � 5 � 2 can be done as37 � (5 � 2) or 37 � 10rather than (37 � 5) � 2.

Distributive For all numbers a, b, and c: 8 � (5 � 2) is the same as Some of the more difficult basica(b � c) � ab � ac (8 � 5) � (8 � 2). facts can be split into smaller,

easier-to-remember parts.For example, 8 � 7 is the sameas (8 � 5) � (8 � 2) or 40 � 16.

Identity For any whole number a: 0 added to any number is The 19 addition facts involvinga � 0 � a and a � 1 � a easy; it’s just that number. 0 and the 19 multiplication

1 times any number is just facts involving 1 can be easilythat number. remembered once this property

is understood and established.

FIGURE 9-5 The 100 basic facts for addition.

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208 Chapter 9 • Operations: Meanings and Basic Facts

the facts. Second, the skill of fact retrieval itselfmay not be taught by teachers or understood bychildren, resulting in inefficient strategies. Teacherscan do something about both of these problems byusing a three-phase process for helping childrenlearn basic facts:

• Get ready: Start from where the children are.• Get set: Build understandings.• Go: Focus on how to remember facts.

Get Ready: Starting Where Children Are

Many children come to school knowing somebasic facts. For instance, the chances are great thatthey can say “one and one are two,” “two and twoare four,” and maybe even “five and five are ten.”They may know that 2 and 1 more is 3, and that 6and “nothing more” is still 6. But they probablydon’t know that 6 � 7 � 13, nor do they have aclear concept of the meanings of symbols suchas + and =.

Similarly, they may know that if you have 3and take away 2, you have only 1 left. But theyprobably won’t know 3 � 2 � 1 (or “three minustwo equals one”). They may know that buyingthree pieces of gum at 5¢ each will cost 15¢,but they won’t know that 3 � 5 � 15. They mayknow that eight cookies shared among four chil-dren means that each child gets two cookies, butthey won’t know that 8 � 4 � 2.

In other words, they can solve simple problemsinvolving facts, but they are not likely to be able toeither recognize or write the facts. Nor do manychildren understand why 5 � 5 � 10 or realize that4 � 2 � ❐ asks the same question as:

4� 2

It is our task to help children organize whatthey know, construct more learning to fill in thegaps, and, in the process, develop meaning.

You need to begin by determining what eachchild knows, using responses from group discus-sions, observations of how each child works withmaterials and with paper-and-pencil activities, andindividual interviews. Many teachers use an inven-tory at the beginning of the year, administeredindividually to younger children and in a question-naire format in later grades. The purpose of suchan inventory is to discover:

• Whether the children have the concept of anoperation: “What does it mean to add?” “Whydid you subtract?” “When can you multiply?”

• What basic facts they understand (demon-strated by drawing a picture to illustrate a writ-ten fact or, conversely, by writing the fact for agiven illustration)

• What strategies they use to find the solution tocombinations: “How did you know 7 � 9 � 16?”

• What basic facts they know fluently (can an-swer within about 3 seconds, without stoppingto figure them out)

Teachers use such information to plan instruc-tion. Do some children need more work withmanipulative materials to understand what multi-plication means? Do some children need help inseeing the relationship of 17 � 8 and 8 � 9? Dosome children need to be taught that counting onfrom a number is quicker than counting each num-ber? Which children need regular practice in orderto master the facts? You can group children to meetindividual needs (as suggested in Chapter 3) andprovide activities and direct instruction to fill in themissing links and strengthen understanding andcompetency. The calculator can be one of the tools.Research has indicated that the development of ba-sic facts is enhanced through calculator use.

Get Set: Building Understanding of the Basic Facts

The emphasis in helping children learn the ba-sic facts is on aiding them in organizing theirthinking and seeing relationships among the facts.Children should learn to use strategies for remem-bering the facts before drill to develop fluency.

Generally, the facts with both addends or bothfactors greater than 5 are more difficult for mostchildren, but that is relatively difficult to determinewith accuracy. What is difficult for an individualchild is really the important point. Although manytextbooks, workbooks, and computer programsemphasize practice on the generally difficult facts,many also encourage the child to keep a record ofthose facts that are difficult for him or her and sug-gest extra practice on those. The teacher shouldsuggest or reinforce this idea.

How can the basic facts for an operation beorganized meaningfully? Many textbooks presentfacts in small groups (e.g., facts with sums to 6: 0 �6 � 6, 1 � 5 � 6, 2 � 4 � 6, and so forth). Othertextbooks organize the facts in “families” (for exam-ple, facts in the “2–3–5 family” are 3 � 2 � 5, 2 �3 � 5, 5 � 3 � 2, and 5 � 2 � 3). Still other text-books organize the facts by “thinking strategies”(e.g., all facts where 1 is added, or containing “dou-bles” such as 7 � 7). No one order for teaching thebasic facts has been shown to be superior to any

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Overview of Basic Fact Instruction 209

other order. Thus the teacher can use professionaljudgment about what each group of children needsand choose whether to use the sequence in a giventextbook.

A variety of thinking strategies can be used torecall the answer to any given fact. Thinkingstrategies are efficient methods for determining an-swers on the basic facts. The more efficient thestrategy, the more quickly the student will be ableto construct the correct answer for the sum, differ-ence, product, or quotient of two numbers and,eventually, develop fluency with the facts so he orshe can quickly recall them.

Research has shown that certain thinking strate-gies help children learn the basic facts (Rathmell,1978; Thornton and Smith, 1988). Understanding ofthe facts develops in a series of stages characterizedby the thinking children use. Some of these thinkingstrategies involve using concrete materials or count-ing. Others are more mature in the sense that aknown fact is used to figure out an unknown fact.Teachers want to help children develop these ma-ture, efficient strategies to help them recall facts.The next section on thinking strategies for basicfacts provides more detail on how these skills can bedeveloped.

Many children rely heavily on counting—inparticular, finger counting—and fail to developmore efficient ways of recalling basic facts. For ex-ample, a child might count 4 fingers and then 5more to solve 4 � 5. This strategy is perfectly ac-ceptable at first; however, this counting processshould not be repeated every time 4 � 5 is encoun-tered. Teachers want the child to move beyondcounting on from 4 (which is relatively slow andinefficient), to thinking “4 � 4 � 8, so 4 � 5 is 9”or using some other more efficient strategy. Even-tually, the child must be able to recall “4 + 5 = 9”quickly and effortlessly. Some children discover ef-ficient fact strategies on their own, but many needexplicit instruction. When the teacher is satisfiedthat the children are familiar with a particularstrategy (able to model it with materials and begin-ning to use it mentally), it is time to practice thestrategy.

Go: Mastering the Basic Facts

Consider this scene: Pairs of children are key-ing numbers on a calculator and passing it backand forth. Other pairs are seated at a table, someplaying a card game and others playing boardgames. Several are busily typing numbers on com-puter keyboards. Still others are working individu-ally with flashcards. What are they all doing?Probably they are practicing basic facts.

If children are to become skillful with the al-gorithms for addition, subtraction, multiplication,and division and proficient at estimation andmental computation, they must learn the basicfacts to the level of immediate, or automatic, re-call so that they gain computational fluency andefficiency in problem solution. When should thismastery level be attempted? As soon as childrenhave a good understanding of the meanings of op-erations and symbols, the process of developingfluency with basic facts can begin. That is, asAshlock and Washbon (1978) suggest, childrenshould be able to:

• State or write related facts, given one basic fact.• Explain how they got an answer, or prove that

it is correct.• Solve a fact in two or more ways.

Research has shown that drill increases speedand accuracy on tests of basic facts (Wilson, 1930).In the Classroom 9-4 contains several interestingindividual activities that provide drill practice forbasic facts; however, drill alone will not change achild’s thinking strategies so that they become effi-cient. Drill, therefore, is most effective when thechild’s thinking is already efficient.

Some principles for drill have been proposed,based on research with primary-grade children(Davis, 1978):

• Children should attempt to memorize factsonly after understanding is attained.

• Children should participate in drill with the in-tent to develop fluency. Remembering shouldbe emphasized. This is not the time for expla-nations.

• Drill lessons should be short (5 to 10 minutes)and should be given almost every day. Childrenshould work on only a few facts in a given les-son and should constantly review previouslylearned facts.

• Children should develop confidence in theirability to remember facts fluently and shouldbe praised for good efforts. Records of theirprogress should be kept.

Math Links 9.5A variety of drill activities may be found on the web.

For example you can try games such as Mathcar Racing,Soccer Shootout, Tic Tac Toe Squares, Line Jumper, andPower Football, all of which you can access from thisbook’s web site.

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210 Chapter 9 • Operations: Meanings and Basic Facts

• Drill activities should be varied, interesting,challenging, and presented with enthusiasm.

Computer software provides a natural com-plement to more traditional materials andactivities, such as flashcards, games, and audio-taped practice, for establishing the quick recall ofbasic facts. For example, children may try gamesfound in the Math Links box on p. 209 to practicebasic facts. Children must provide the correct

answer to “equalize” the spaceship before thealiens invade.

Most programs keep track of the number ofexercises attempted and the number answered cor-rectly. Some display the time taken to give correctanswers, thus encouraging students to competeagainst their own records for speed as well as mas-tery. Requiring short response time (within threeor four seconds) is important because it promotesefficient strategies and encourages children todevelop fluent recall.

Many children enjoy computer software thatdisplays a cumulative record of their individualprogress. This feature allows children to diagnosefor themselves the basic facts they know and don’t

In the Classroom 9-4TEST YOUR FACTS!

OBJECTIVE: Using a variety of activities andpuzzles to practice multiplication and divisionfacts.

GRADE LEVEL: 3–4

� Fill in the empty � Complete these five-box: facts and match them

with the clock minutes.

� Fill in each empty box to make the next number correct:

� Multiply each number in the middle ring by the number in the center:

5 7

4 36

5 8

9 81

6 48

126 18 3

33 6 24

273 12 2

2 × 5 = __

6 × 5 = __

4 × 5 = __

5 × 5 = __

7 × 5 = __

7

762

5849

3

In the Classroom 9-5MULTIG

OBJECTIVE: Using a game to practice basicmultiplication facts.

GRADE LEVEL: 3–4

� Use the playing boardhere or make a largerone on heavy con-struction paper. Eachplayer needs somebuttons, macaroni, orchips for markers.1. Take turns. Spin

twice. Multiply the2 numbers. Findthe answer on theboard. Put amarker on it.

2. Score 1 point foreach covered �that touches aside or corner ofthe � you cover.

3. If you can’t findan uncovered �to cover, youlose your turn.

4. Opponents maychallenge anytime before thenext player spins.

5. The winner is the player with the most points atthe end of 10 rounds.

Don'tforget the spinner.

You can't play this gamewithout it.

56

42

36

30

30

40

56

20

32

63

16

25

35

49

16

64

49

42

25

48

45

24

40

48

24

49

28

63

81

54

30

24

48

32

16

42

54

45

25

36

35

64

20

20

28

40

45

72

35

72

56

32

63

36

72

54

81

28

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Overview of Basic Fact Instruction 211

know. It also provides a source of motivation be-cause each student can compete against himself orherself, with the goal of complete mastery alwaysin mind.

When using flashcards, the child should gothrough the entire set and separate the cards into apack of those known and a pack of those un-known. Each time the child works with the cards,he or she should review those in each pack, mov-ing newly learned facts to the known pack. Thisapproach makes progress evident.

Another point from research also must betaken into account: the frequency with which basicaddition and multiplication facts occur in elemen-tary school textbooks is probably a source of thedifficulty. Facts with numbers larger than 5 oc-curred up to half as frequently as those in therange of 2 to 5; 0 and 1 occurred relatively infre-quently. Thus teachers need to ensure that more

practice is provided with facts involving 0, 1, andespecially 6 through 9.

Several types of drill-and-practice procedures inthe form of games are noted In the Classroom 9-5through 9-8. Children in all age groups find suchgames an enjoyable way to practice what theyknow. These activities supplement the many otherdrill-and-practice procedures that you will find intextbooks, journals, computer software, and othersources.

In the Classroom 9-6ADDITION BINGO!

OBJECTIVE: Using a bingo game to practicebasic addition facts.

GRADE LEVEL: 1–2

� Each player needs a different Bingo card and somebuttons or macaroni for markers.

� It’s easy to play:• The leader draws a card and reads the addends on it.• Each player covers the sum on his or her Bingo

card.Not all sums are given on each card.Some sums are given more than once on a Bingocard, but a player may cover only one answer foreach pair of addends.The winner is the first person with 5 markers in a row!

7+6

8+4

9+5

The leader needs a packof cards like these with all possible

combinations (basic facts).9

5

15

7

2

6

8

11

9

13

17

10

2

11

11

3

13

1

18

5

7

14

12

17

Free

In the Classroom 9-7ZERO WINS!

OBJECTIVE: Using a game to develop numbersense and practice addition and subtractionfacts.

GRADE LEVEL: 2–3

� Make two identical sets of 19 cards with a numberfrom 0 to 18 on each!

� Follow these rules:• After shuffling, the leader deals 4 cards to each

player and puts the remaining cards face down inthe center of the table.

• Players must add or subtract the numbers ontheir 4 cards so they equal 0. For example, sup-pose you had these cards:

• With these cards you could write10 � 6 � 2 � 6 � 0or6 � 6 � 10 � 2 � 0or various other number sentences.

• On each round of play, the players may exchangeone card if they wish, and each player takes a turnbeing first to exchange a card on a round.Tomake an exchange, the first player draws a cardand discards a card, face up. Other players candraw from either the face-down pile or the face-up discard pile.

The first player to get 0 on a round wins the round!

180 180

6 10 2 6

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212 Chapter 9 • Operations: Meanings and Basic Facts

Thinking Strategies for Basic Facts

In the following four sections, we discussthinking strategies for basic facts for addition, sub-traction, multiplication, and division and illustrateways to teach them.

Thinking Strategies for AdditionFacts

The 100 basic facts for addition are shown inFigure 9-5. They are not presented to children inthis completed form; rather, the children graduallyand systematically learn the facts and may fill in orcheck them off on the chart.

Lesson Idea 9-3 presents questions that ateacher might use to help children see the orderli-ness of the basic addition facts. This overview canhelp children see their goal as they begin to workon fluency with facts.

The thinking strategies that can be used whenteaching basic addition facts include commutativ-ity; adding 0, 1, and doubles; counting on; andadding to 10 and beyond. For many facts, morethan one strategy is appropriate.

1. Commutativity. The task of learning the basicaddition facts is simplified because of the commu-tative property. Changing the order of the addendsdoes not affect the sum. Children encounter thisidea when they note that putting 2 blue objectsand 3 yellow objects together gives the same quan-tity as putting 3 blue objects and 2 yellow objectstogether:

In the Classroom 9-821 OR BUST!

OBJECTIVE: Using a game to develop logicalreasoning and to practice addition.

GRADE LEVEL: 3–4

� Play this game with a partner:

• Enter 1, 2, 3, 4, or 5 in your

• Give the to your opponent, who adds

1, 2, 3, 4, or 5 to the displayed number.• Take turns adding 1, 2, 3, 4, or 5 to the total.The first player to reach 21 wins! If you go over 21,you “bust,” or lose!

What next?

987

654

321

=

OFF ÷

–

+•0

ON

987

654

321

=

OFF ÷

–

+•0

ON

I'll add a4 to that

I'll startat 3.

73

I'll add another 5.

I'll add 5.

1712

Math Links 9.6Some basic fact games and puzzles also involve us-

ing strategy and problem solving. Number Bounce,Number Puzzles, and The Product Game illustrate thistype of activity, all of which you can access from thisbook’s web site.

In work with the basic addition facts, childrenwill see or write, for example:

2 5�5 and �2

or

2 � 5 � � and 5 � 2 � �

Students must realize that the same two num-bers have the same sum, no matter which comesfirst. They need to be able to put this idea into theirown words; they do not need to know the term com-mutative property. They need to use the idea as theywork with basic facts, not merely parrot a term.

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Thinking Strategies for Basic Facts 213

Encourage the use of commutativity by usingmaterials such as a chain of loops:

Lesson Idea 9-3THE BIG PICTURE

Materials : Activity sheetfor each child; a transparencyof it for use on the overheadprojector.

Objective: Children dis-cover the patterns of the ba-sic facts using an organizedtable.

Grade Level : 2

� Ask each child to studycarefully examples on theactivity sheet.

• What is alike about theexamples?

• What patterns areapparent?

� Discuss the top row ofexamples—those involving 0. Have the children fill in each of the sums.

� Discuss the second row of examples.

0• Why isn’t �1 included? (This fact is in the top row.)

• How can each sum be found quickly? (By counting on one.)

� Look at the diagonal containing these facts:

0 1 2 3 4 5 6 7 8 9�1 �1 �2 �3 �4 �5 �6 �7 �8 �9

• Find the sums.

� On the overhead projector, quickly fill in the remain-ing sums.Then focus attention on the entire table.

• What patterns are apparent?

• What is the largest sum? What are its addends?

• What is the smallest sum? What are its addends?

• Ask children to circle all examples whose sum is 8.

• Where are they?

• What patterns do you see in the addends?

5• Why isn’t �7 in the chart?

� Continue discussing patterns as long as you feelyour children are benefiting from the experience.Encourage the children to understand that all thebasic facts to be memorized are included on thissheet.

9+9

8+8

7+7

6+6

5+5

4+3

3+3

5+4

4+4

2+2

3+2

4+2

2+1

3+1

4+1

5+1

6+1

7+1

8+1

9+1

1+1

0+0

2+0

3+0

4+0

5+0

6+0

7+0

8+0

9+0

1+0

52

5

2

5

25

2

5

2

2

5

Have the children note that 5 is followed by 2and 2 is followed by 5 all around the chain. Thechain can be turned as they read and add:

5 � 2 � 7, 2 � 5 � 7, 5 � 2 � 7, . . .

The calculator can also help children verifythat the order of the addends is irrelevant. Havethem key into their calculators:

5 � 8 � and 8 � 5 �

Use a variety of combinations, so the idea thatthe order does not affect the outcome becomes evi-dent. In the Classroom 9-9 presents another way ofhelping them develop and use the idea of commu-tativity.

The blank boxes in Figure 9-6 help you seethat the commutative property reduces the num-ber of facts to be learned by 45. Each blank boxbelow the diagonal of the table can be matchedwith a box above the diagonal with the same ad-dends and sum. (If there are 100 facts altogether,why are there 55 distinct facts to learn, ratherthan just 50?)

2. Strategies for Adding One and Adding Zero.Adding one to a number is easy for most children.In fact, most children learn this idea before theycome to school, and they only have to practice therecognition and writing of it rather than developinitial understanding. To reinforce their initial con-cept, experiences with objects come first, followedby such paper-and-pencil activities as these:

5 + 1 =

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214 Chapter 9 • Operations: Meanings and Basic Facts

Recognition of the pattern is then encouraged:

Although adding zero may seem easy to adults,this is actually one of the hardest strategies forchildren to learn. Concrete modeling of the situa-tion is tricky because it is difficult to pictureadding nothing. Therefore, explicit work on factsinvolving zero should be postponed until childrenhave mastered some of the other fact strategies.

3. Strategies for Adding Doubles and Near Dou-bles. Doubles are basic facts in which both addendsare the same number, such as 4 � 4 or 9 � 9. Mostchildren learn these facts quickly, often parrotingthem before they come to school. Connecting dou-bles facts to familiar situations often helps studentsremember them (e.g., two hands shows 5 � 5 �10, an egg carton shows 6 � 6 � 12, two weeks ona calendar shows 7 � 7 � 14) Students can profitfrom work with objects followed by drawings:

In the Classroom 9-9ARRANGING AND REARRANGING

OBJECTIVE: Developing number sense by us-ing counters to illustrate different sums to 10.

GRADE LEVEL: 2

� Use 10 counters and string for the rings to makethis arrangement:

• How many counters are in Ring A? • How many counters are in Ring B?

� Rearrange your counters and rings to show thesame numbers, and then move 1 counter from Ring A to Ring B.• How many counters are now in Ring A? • How many counters are in Ring B?

� See how many different ways you can put 10 coun-ters in the two rings.• Use your counters and rings, and list the ways

here:4 � 6

A BFIGURE 9-6 Addition facts derived by the commuta-tive thinking strategy.

1

1 + 1 =

2 + 1 =

3 + 1 =

4 + 1 =

The strategy for adding zero applies to factsthat have zero as one addend. These facts shouldbe learned, through experience, as a generaliza-tion: Zero added to any number does not changethe number. This idea follows from many concreteexamples in which children see that any time theyadd “no more” (zero) they have the same amount.Activities then focus on this pattern:

4 + 0 =

3 + 0 =

2 + 0 =

1 + 0 =

etc.

0 + 1 =

4 + 4 = 8 + 8 =

Another strategy, “near doubles,” can be usedfor the facts that are one more or one less than thedoubles:

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Thinking Strategies for Basic Facts 215

Think7 � 8 � � 7 � 7 � 14

So 7 � 8 is one more.7 � 8 � 15

Think7 � 6 � � 7 � 7 � 14

So 7 � 6 is one less.7 � 6 � 13

Lesson Idea 9-4 shows one way of presentingthis strategy.

The four thinking strategies in the precedingtwo sections can be used with the addition factsshown in Figure 9-7, parts A and B.

4. Counting On. The strategy of counting on canbe used for any addition facts but is most easilyused when one of the addends is 1, 2, or 3. To beefficient, it is important to count on from the largeraddend. For example,

Think2 � 6 � � 6 . . . 7 . . . 8

2 � 6 � 8

Initially, children will probably count all ob-jects in a group, as noted in Chapter 7.

Lesson Idea 9-4NEARLY DOUBLE

Materials : Flannel board or magnetic board anddisks for demonstration, 1 die, a pile of chips or tiles,and paper for each pair of children for seat work.

Objective: Children determine strategies for addingdoubles and near doubles using plastic disks.

Grade Level : 1

� Put a group of 6 disks on the board and have chil-dren do the same at their seats.

� Have them add a second group of 6 (and do so fordemonstration too).

• Ask: How many in each group? How many in all?� Write 6 � 6 � 12 on the board.� Have children add one more disk to the second

group:

• Ask: How many in the first group?How many in the second group?How many in all?

� Write 6 � 7 � 13 on the board.� Demonstrate how children can use their dice and

chips to build more examples like this and to writethe corresponding sentences on their papers. Eachpair of children should repeatedly roll their die.After each roll they should model, say, and write anear doubles fact. For example, if they roll 5, theyuse chips to model, say, and write 5 � 6 � 11.

FIGURE 9-7 Addition facts derived by four strategies.

B Addition facts derived by the adding doubles and near-doubles strategies

A Addition facts derived by the adding one and adding zero strategies

1 2 7 83 4 5 6 . . .

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216 Chapter 9 • Operations: Meanings and Basic Facts

They need to learn to start from the larger ad-dend, 6, and count on, 7, 8. (Notice that under-standing of the commutative property is assumed.)Research indicates that young children will counton but not necessarily from the larger addend(Ginsberg, 1977). Thus the strategy must be taughtto many children using activities such as this one:

5. Adding to 10 and Beyond. With the strategy ofadding to 10 and beyond, one addend is increasedand the other decreased, to make one of the ad-dends 10. This strategy is used most easily whenone of the addends is 8 or 9, although some chil-dren also find it useful when adding 6 or 7. Here isan example:

Think8 � 5 � � 8 � 2 � 10 and 5 � 2 � 3

So 10 � 3 � 13, so8 � 5 � 13

The child recognizes that 8 is close to 10, andthen mentally breaks 5 apart into 2 � 3. The 2 isused with the 8 to “add to ten,” and the 3 is usedto go “beyond.” A ten-frame can be helpful inteaching this strategy because it provides a visualimage of adding to 10 and going beyond:

6 + 2 =

6

How many dots?

6 . . . 7, 8

The counting-on strategy can be rather effi-ciently used with the addition facts noted in Figure9-8 (where at least one addend is 1, 2, or 3); how-ever, it is not efficient to use counting on whenboth addends are larger than 3 (e.g., for 6 � 8 � 14or 5 � 8 � 13 or 9 � 8 � 17). Counting on for factssuch as these is both slow and prone to error.Unfortunately, some children develop the habit ofusing counting on all the time and thereby fail todevelop more efficient strategies, sometimes still re-lying on counting on even after they have gradu-ated to middle school or high school. It is importantto help students move beyond counting on becausefluency (efficiency and accuracy) is extremely im-portant for work with basic facts. When studentsare slowed down by inefficient strategies for basicfact retrieval, they are often hampered in doingmore advanced work in mathematics.

FIGURE 9-8 Addition facts derived by the counting-on strategy.

8 + 5 = 10 + 3 =

5

8

The research of Funkhouser (1995) indicatesthat working with five-frames as a base and thenmoving to ten-frames may be particularly helpfulfor children with learning disabilities.

Children must know the sums to 10 well in or-der to use the adding-to-ten-and-beyond strategy.Practice with regrouping to 10 is needed to helpthem become proficient. They also need to realizehow easy it is to add any single digit to 10 to get anumber in the teens, without having to thinkabout counting on. Which is easier?

10 � 5 or 9 � 6

7 � 8 or 5 � 10

As another example, change this problem to aneasier one:

9 + 6 =

In all cases, talk the strategy through withdrawings as well as objects:

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Thinking Strategies for Basic Facts 217

The adding-to-10-and-beyond strategy can beused with the addition facts shown in Figure 9-9.

In many cases, more than one strategy can beused to aid in recalling a fact. This point should bemade with the children. It encourages them to trydifferent ways of recalling a fact, and it maystrengthen their understanding of the relationshipsinvolved. Notice from Figure 9-10 that, when thestrategies for adding one, adding zero, adding dou-bles and near doubles, counting on, and adding-to-10-and-beyond have been learned, only six basicfacts remain. These missing facts can be derived us-ing one of the strategies (and commutativity) or sim-ply taught separately. Children should be encouragedto look for patterns and relationships because almostall of the 100 basic addition facts can be developedfrom a variety of relationships with other facts.

It also should be noted that children might in-vent strategies of their own, such as:

Think6 � 7 � � 6 is 5 � 1

7 is 5 � 2So 10 � 313

Think6 � 8 � � 3 � 3 � 8

3 � 1114

Encourage their ideas!

Thinking Strategies for SubtractionFacts

For each basic addition fact, there is a relatedsubtraction fact. In some mathematics programs,the two operations are taught simultaneously. Therelationship between them is then readily empha-sized, and learning the basic facts for both opera-tions proceeds as if they were in the same family.Even when they are not taught simultaneously,however, the idea of a fact family is frequentlyused (see Figure 9-11).

“Think addition” is the major thinking strategyfor learning and recalling the subtraction facts. En-courage children to recognize, think about, and usethe relationships between addition and subtractionfacts. They can find the answers to subtractionfacts by thinking about missing addends.

8 + 4 = 10 + 2 =

FIGURE 9-9 Addition facts derived by the adding-to-10 strategy.

FIGURE 9-10 Addition facts derived by all thinkingstrategies for addition.

FIGURE 9-11 Examples of fact families.

77

–3

7

–4

3

+4

4

+3

7

4 3

7

5

5 + = 7

+ 5 = 7

7 – 5 =

7 – = 5

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218 Chapter 9 • Operations: Meanings and Basic Facts

For example,

Think15 � 7 � � 7 � 8 � 15

So 15 � 7 � 8

Other strategies for finding subtraction factsalso can be taught: using 0 and 1, doubles, count-ing back, and counting on.

1. Using 0 and 1. Once they have learned strate-gies for adding 0 and adding 1, most children findit rather easy to learn the related subtraction factsinvolving 0 and 1. They can profit from work withmaterials and from observing patterns similar tothose used for addition facts.

2. Doubles. The strategy for doubles may need tobe taught more explicitly for subtraction facts thanfor addition facts. It rests on the assumption thatchildren know the doubles for addition. Here is anexample:

Think16 � 8 � � 8 � 8 � 16

16 – 8 � 8

3. Counting Back. The strategy of counting backis related to counting on in addition, and thereforeit is most effectively used when the number to besubtracted is 1, 2, or 3:

Think9 �3 � � 9 . . . 8, 7, 6

9 – 3 � 6

As for other strategies, use problems and a va-riety of manipulative materials to help childrengain facility in counting back from given numbers.Focus especially on the numbers involved in sub-traction facts, as in the following examples.

Write the numbers in order backward:

(i.e., when it is easy to see that the two numbersinvolved in the subtraction are close together).“Think addition” by counting on:

Think8 – 6 � � 6 . . . 7, 8

8 � 6 � 2

Activities for developing this strategy include thefollowing:

Write the numbers you say when you countback.

Write the answer.

6 – 2 =

6, __, __

4. Counting On. The strategy of counting on isused most easily when the difference is 1, 2, or 3

12–9 9

Begin with 9. Count on until you reach 12.How many?

8

–5

0 1 2 3 4 5 6 7 8 9

1 2 3

Begin with 5. Count on until you reach 8.How much?

The emphasis in using the counting-on strategycan also encompass adding on: “How much morewould I need?” The child is encouraged to useknown addition facts to reach the solution. This isparticularly valuable with missing-addend situa-tions, such as 6 � ❐ � 9.

Thinking Strategies forMultiplication Facts

Multiplication is frequently viewed as a specialcase of addition in which all the addends are ofequal size. The solution to multiplication problemscan be attained by adding or counting, but multi-plication is used because it is so much quicker.

Instruction on multiplication ideas begins inkindergarten as children develop ideas aboutgroups, numbers, and addition. In grades 1 and 2,counting by twos, threes, fours, fives, tens, andpossibly other numbers should be taught. Such ex-pressions provide a basis for understanding thepatterns that occur with the basic multiplicationfacts. Use of the calculator as described in Chapter7 can aid teachers in developing ideas about thesepatterns of multiplication. Using the constant func-tion on calculators, children realize that two sixesequal 12, three sixes equal 18, and so on.

The basic multiplication facts pair two one-digitfactors with a product, as shown in Figure 9-12.The basic multiplication facts should not be givento children in the form of a table or chart of factsuntil they have been meaningfully introduced.

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Thinking Strategies for Basic Facts 219

Rather, the facts should be developed through prob-lem situations, experiences with manipulative andother materials, and various thinking strategies.The table becomes the end result of this process ofdeveloping understanding of operations and facts.

The thinking strategies for multiplication factsprovide an efficient way for a child to attain eachfact. These strategies include commutativity, using1 and 0, skip counting, repeated addition, splittinginto known parts, and patterns.

1. Commutativity. Commutativity applies to mul-tiplication just as it does to addition. It is, there-fore, a primary strategy for helping students learnthe multiplication facts. In the Classroom 9-2through 9-4 (presented earlier) emphasize thisproperty. The calculator is also useful in reinforc-ing the idea. Children can multiply 4 � 6, then 6 �4, for example, and realize that the answer to bothis 24.

Here are some other examples:

3 � 6 � 18 99: 6 � � � 18

7 � 5 � 35 99: � � 7 � 35

After they have tried many combinations, stu-dents should be able to verbalize that the order ofthe factors is irrelevant. Figure 9-13 (where eachbox in the upper right can be matched with anunshaded box in the lower left) shows that, as foraddition, there really are only 55 multiplicationfacts to be learned if you recognize the power ofcommutativity.

2. Skip Counting. The strategy of skip countingworks best for the multiples children know best,twos and fives, but it also may be applied to threes

and fours (or other numbers) if children havelearned to skip count by them.

Here is an example for 5:

Think4 � 5 � � 5, 10, 15, 20

4 � 5 � 20

Skip counting around the clock face (as you dowhen counting minutes after the hour) is a goodway to reinforce multiples of 5 by skip counting.The facts that can be established with the skip-counting strategy for twos and fives are noted inFigure 9-14.

3. Repeated Addition. The strategy of repeated ad-dition can be used most efficiently when one of

FIGURE 9-12 The 100 basic facts for multiplication. FIGURE 9-13 Multiplication facts derived by thecommutative thinking strategy.

FIGURE 9-14 Multiplication facts derived by theskip-counting strategy.

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220 Chapter 9 • Operations: Meanings and Basic Facts

the factors is less than 5. The child changes themultiplication example to an addition example:

Think3 � 6 � � 6 � 6 � 6 � 18

3 � 6 � 18

Because this strategy is based on one inter-pretation of multiplication, children should havehad many experiences with objects and materials.Drawings and the calculator can be used toprovide additional experiences to help developthis strategy as well as the concept for the opera-tion. Lesson Idea 9-5 illustrates these ideas. InFigure 9-15, note the facts that can be learnedwith this strategy.

4. Splitting the Product into Known Parts. Aschildren gain assurance with some basic facts,they can use their known facts to derive others.The strategy known as splitting the product isbased on the distributive property of multiplica-tion. It can be approached in terms of “one moreset,” “twice as much as a known fact,” or“known facts of 5.”

• The idea of one more set can be used for almostall multiplication facts. If one multiple of anumber is known, the next multiple can be de-termined by adding a single-digit number. Forexample, to find 3 � 5 if doubles are alreadyknown, you can think of 2 � 5 (10) and addone more 5 (to get 15). The computation isslightly more difficult when the addition re-quires renaming (as, for example, below, where

8 � 7 is found by adding one more set of 7 tothe known fact 7 � 7 � 49):

Think8 � 7 � � 7 � 7 � 49

8 � 7 � 49 � 78 � 7 � 56

Each fact can be used to help learn the nextmultiple of either factor. Illustrating this strat-egy using an array model can be helpful:

FIGURE 9-15 Multiplication facts derived by the re-peated addition strategy.

Lesson Idea 9-5HOW MANY?

Materials : Counting chips, calculator.

Objective: Children use repeated addition to modelmultiplication, using counting chips and calculators.

Grade Level : 3

� Look at these dots:

• How many groups of dots?• How many dots in each group?• How many dots in all?

Count them: 5 , , ,Add them: 5 � 5 � 5 � 5 � ��

Multiply them: 4 � 5 � ��

� Now look at these dots:

• Ring sets of 7. How many sevens?• How many dots in all?

Count them: 7 , , ,Add them: 7 � 7 � 7 � 7 � ��

Multiply them: 4 � 7 � ��

� Do these using a calculator:7 � 7 � 7 � 7 � 7 �

5 � 5 � 5 � 5 � 5 �

• Do you know a simpler way? Show it here:

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Thinking Strategies for Basic Facts 221

Ask children to name each part of the array andwrite the multiplication fact for the whole array.

• Twice as much as a known fact is a variation ofthe foregoing strategy. It can be applied to mul-tiples of 4, 6, and 8 because an array with oneof these numbers can be split in half. The prod-uct is twice as much as each half:

Think6 � 8 � � 3 � 8 � 24

6 � 8 is twice as much, or 24 � 246 � 8 � 48

Note that a difficulty may arise when renam-ing is needed to do the doubling by addition.

Again, using models helps provide a visualimage of this strategy:

with large factors but is most useful for multi-ples of 6 and 8 because both 5 sixes and 5eights are multiples of 10, so it is rather easy toadd on the remaining part without renaming.

For example, to figure 7 � 6, recognize that7 can be conveniently split into 5 � 2 because5 � 6 (30) is easy to work with.

Think7 � 6 � � 5 � 6 � 30

2 � 6 � 12So 7 � 6 is 30 � 12, or 42

To illustrate this strategy, the array is dividedso that 5 sixes or eights (or some other number)are separated from the remaining portion:

5 × 4 =

1 × 4 =

6 × 4 =

4 × 7 =

2 sevens is ____

2 sevens is ____

In this case, children work with already di-vided arrays.

As they progress, children can divide an ar-ray, such as the following; write about eachpart; and write the multiplication fact for thewhole array.

Some options for the array above (6 � 8)might be:

(5 � 8) � 40 and 1 � 8 � 8, so 6 � 8 � 40 � 8 � 48or 6 � 4 � 24 and 6 � 8 is twice as much, so 6 � 8 � 24 � 24 � 48.

• Working from known facts of 5 also will aidchildren. It can be helpful for any problem

5 × __ =

2 × __ =

__ × __ =

Call attention to how the array is divided.Have the children work with other arrays, de-termining when it seems reasonable to workwith particular numbers. The facts that can besolved by splitting the product into knownparts are shown in Figure 9-16.

5. Using 0 and 1. The facts with zero and one aregenerally learned from experience working withmultiplication. Children need to be able to generalizethat “multiplying with 1 does not change the othernumber” and that “multiplying by 0 results in a

FIGURE 9-16 Multiplication facts derived by thestrategy of splitting the product into known parts.

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222 Chapter 9 • Operations: Meanings and Basic Facts

product of 0.” Figure 9-17 indicates the facts that canbe learned with these strategies.

The preceding strategies account for all themultiplication facts. However, one more strategycan provide help with some difficult facts.

6. Patterns. Finding patterns is helpful with sev-eral multiplication facts. One of the most usefuland interesting patterns concerns nines. Look atthe products of the facts involving nine. The digitsof the products always sum to 9. Furthermore, thetens digit of each product is always one less thanwhatever factor was being multiplied times 9.

FIGURE 9-17 Multiplication facts derived by thestrategies for 0 and 1.

1 × 9 = 92 × 9 = 193 × 9 = 274 × 9 = 36

The tensdigit is 1 lessthan 4.

0 + 9 = 91 + 8 = 92 + 7 = 93 + 6 = 9

The sumof the digitsof 36 is 9.

So for 5 � 9,

The tens digit is one less than 5 9: 4The sum of the digits is 9, so 4 � � � 9 9: 5Thus, 5 � 9 � 45

Now try 7 � 9 � ❐Challenge children to find this and other inter-

esting patterns in a table or chart such as the onein Figure 9-12. They should note, for instance, thatthe columns (and rows) for 2, 4, 6, and 8 containall even numbers, and the columns for 1, 3, 5, 7,and 9 alternate even and odd numbers.

You will probably find that children also enjoyvarious forms of “finger multiplication” (see Figure9-18), seemingly magical tricks for using fingermanipulations to determine basic facts. Actually,

8 × 4 = 32

4 × 8 = 32

32 ÷ 8 = 4

32 ÷ 4 = 8

though, finger multiplication works because of thepatterns that are inherent in the basic fact table.

Thinking Strategies for Division Facts

Teaching division has traditionally taken alarge portion of time in elementary school . Now,with the increased use of calculators, many educa-tors advocate reducing the attention accorded to it.Nevertheless, children still need an understandingof the division process and division facts. The factshelp them to respond quickly to simple divisionsituations and to understand better the nature ofdivision and its relationship to multiplication.

Just as “think addition” is an important strategyfor subtraction, “think multiplication” is the pri-mary thinking strategy to aid children inunderstanding and recalling the division facts.Division is the inverse of multiplication; that is, in adivision problem you are seeking an unknown fac-tor when the product and some other factor areknown. The multiplication table illustrates all thedivision facts; simply read it differently. For the divi-sion fact 54 ÷ 9 � ❐, look in the 9-row of the multi-plication table for the number 54, then read up thatcolumn to find the other factor, 6. Students gener-ally do not learn division facts separately from mul-tiplication facts. Instead, they learn division factssuch as 48 � 6 � 8 by remembering (and connect-ing with) multiplication facts such as 6 � 8 � 48.

It is important to realize that most divisionproblems, in computations and in real-worldproblem situations, do not directly involve a multi-plication fact that you have learned. For example,consider the computation 49 � 6. There is no basicfact involving 49 and 6. So, what do you do? In thissituation most people quickly and automaticallymentally review the 6-facts that they know, lookingfor the facts that come closest to 49 (6 � 7 � 42—too small, 6 � 8 � 48—just a little too small, 6 � 9� 54—too big). From this mental review, you canconclude that 49 ÷ 6 = 8, with 1 left over. Chil-dren need practice in thinking this way (mentallyfinding the answers to problems involving one-digitdivisors and one-digit answers plus remainders).

Just as fact families can be developed for addi-tion and subtraction, so can they be useful formultiplication and division:

Because of its relationship to multiplication,division can be stated in terms of multiplication:

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Thinking Strategies for Basic Facts 223

5 × 9 = 40 + 5 = 45

Multiply 8 × 9 Differencebetween8 and 10 is 2(fold down2 fingers)

2 × 1 = 270 + 28 × 9 = 72

Differencebetween9 and 10 is 1(fold down1 fingers)

3 × 9 = 20 + 7 = 27

10 720 61× 3 4

5

2 3

510420 240

330

30 60

40

5020 70

10

1× 5

Finger Multiplication for9-Times Facts Only

1. Hold both hands in front of you with palms facingaway.

2. To do 3 � 9, bend down the third finger fromthe left.

3. The fingers to the left of the bent finger representtens. The fingers to the right of the bent fingerrepresent ones. Read off the answer: 2 tens and7 ones, or 27.

4. To do 5 � 9, bend down the fifth finger fromthe left, etc. Try this method for any 9-facts:1 � 9 through 10 � 9.

Finger Multiplication for Factswith Factors 6 and Higher Only

1. Hold both hands in front of you with palmsfacing you.

2. Think of a multiplication in which both factorsare 6 or more (up to 10). For example, you cando anything from 6 � 6 up to 10 � 10.

3. For each factor, find the difference from 10.

• Let's try the example 8 � 9.

• For the factor 8, think 10 � 8 � 2, andbend down 2 fingers on the left hand.

• For the factor 9, think 10 � 9 � 1 , andbend down 1 finger on the right hand.

4. Look at all fingers that are still up, and countby ten (10, 20, 30, 40, 50, 60, 70).

5. Look at the fingers that are bent down. Multiplythe number of fingers bent down on the lefthand by the number bent down on the righthand (2 � 1 � 2).

6. So 8 � 9 � 70 � 2 or 72.

7. Try this method for any multiplication facts from6 � 6 up through 10 � 10.

FIGURE 9-18 Two forms of finger multiplication.

42 � 6 � � 99: 6 � � � 42

Thus children must search for the missing fac-tor in the multiplication problem. Because multipli-cation facts are usually encountered and learnedfirst, children can use what they know to learn themore difficult division facts. Moreover, division isrelated to subtraction, and division problems canbe solved by repeated subtraction:

12 � 3 � 12 � 3 � 3 � 3 � 3

0 1 2 3 4 5 6 7 8 9 10 11 12

Four threes

However, repeated subtraction and the relatedstrategies of counting backward or skip countingare confusing for many children. You may discuss

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224 Chapter 9 • Operations: Meanings and Basic Facts

A Glance at Where We’ve Been

Skill in computation with whole numbers is de-veloped through concrete experiences. In this chap-ter, we considered the importance of experienceswith counting, concrete experiences, problem-solv-ing contexts, and language, and we described andmodeled meanings for each operation. We also pre-sented mathematical properties to be developed aspart of the understanding of the operations.

The next part of the chapter focused on the ba-sic facts for each operation. Starting from whatchildren know, the facts are developed using expe-riences that range from concrete to pictorial tosymbolic. Thinking strategies for the basic facts foreach operation help children move from countingto more mature, efficient ways of developing thefacts. Specific suggestions for providing practiceand promoting mastery help children master thebasic facts for quick recall.

Things to Do: From What You’ve Read

1. What prerequisites must children have beforeengaging in the lesson that opens this chapter?

2. Discuss each of these statements:(a) When you teach multiplication, you begin

preparation for learning division.(b) Children should not be allowed to count on

their fingers when they start addition.(c) With the wide use of calculators, there is

little need for children to attain prompt re-call of the basic facts.

3. What properties of addition and multiplicationare especially helpful in teaching the basic facts?

4. Describe the thinking strategies a child mightuse with each of the following computations:

8 � 0 � �� 8 � 5 � �� 7 � 8 � ��

18 � 3 � �� 16 � 7 � ��

5. For each addition fact strategy (i.e., commuta-tivity, adding one, doubles and near doubles,adding zero, counting on, adding to 10 and be-yond), list three different facts that could usethat strategy and explain what a child wouldthink when using that strategy.

6. When is counting back an effective strategy forsubtraction? Give an example.

them as ideas children might like to try, particularlywhen these strategies occur spontaneously in thecourse of some students’ work, but don’t be sur-prised if only a few children actually use them.

Think15 � 3 � � 15 . . . 12, 9, 6, 3, 0

That’s 5 numbers15 � 3 � 5

Think28 � 7 � � 28 � 7

21 � 7 4 subtractions14 � 7

7 � 70

28 � 7 � 4

Splitting the product into known parts reliesheavily on knowledge of multiplication facts, aswell as on the ability to keep in mind the compo-nent parts.

Think35 � 7 � � 2 � 7 � 14

3 � 7 � 2114 � 21 � 352 � 3 � 5

So 35 � 7 � 5

As with multiplication, work with arrays helpschildren relate the symbols to the action.

In general, children have little difficulty in divid-ing by one. They need to exercise caution when zerois involved, however. Division by zero and divisionof zero present two different situations. Divide 0 by6 (0 � 6); the result is 0. Check this by multiplying:6 � 0 � 0. But division by zero is impossible.

For example, to solve 6 � 0 � ❐ would requirea solution so that 6 � ❐ � 0. However, there is novalue for ❐ that would make this sentence true.Therefore, 6 � 0 has no solution, and division byzero is undefined in mathematics. Just as you mayhave difficulty remembering which is possible, di-vision of 0 or division by 0, so will children havedifficulty in remembering and need to be givenpractice.

Thinking strategies for division are far moredifficult for children to learn than are the strategiesfor the other operations. The child must remembermore, and regrouping is often necessary. Whenskip counting, for instance, the child must keeptrack of the number of times a number is namedeven as the struggle to count backward proceeds.Therefore, the primary burden falls on the child’sfacility with the multiplication facts. Being able torecall those facts quickly will facilitate recall of thedivision facts.

14

24

3

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Things to Do: Going Beyond This Book 225

7. How can Cuisenaire rods be used to show thecommutative property of addition? Sketch sev-eral examples and describe an activity usingCuisenaire rods that would involve students ininvestigating this property.

8. Explain why division by zero is considered“undefined” by mathematicians.

9. There are two types of division that arise inreal-world problems: measurement (or re-peated subtraction) division and fair sharing(or partitive) division.(a) Write a partitive (or fair sharing) division

word problem that would correspond to 24 � 6. Draw a picture to illustrate the so-lution to your problem and in simple lan-guage explain why your numerical answermakes sense.

(b) Write a measurement (or repeated subtrac-tion) division word problem that wouldcorrespond to 4 � . Draw a picture to il-lustrate the solution to your problem andin simple language, explain why your nu-merical answer makes sense.

10. Choose a method for developing meanings andthinking strategies for teaching addition, sub-traction, multiplication, or division and plan alesson idea for teaching.

Things to Do: GoingBeyond This Book

1. Plan an interactive bulletin board to help chil-dren learn about one or more thinking strate-gies for subtraction.

2. Examine a recently published textbook forgrades 2, 3, or 4. Analyze how basic factsfor the four basic operations are presented inthat text. How are the facts grouped? Are think-ing strategies presented? If so, how do theycompare with the strategies described in thischapter? How would you use this text in teach-ing basic facts? If you think you would proba-bly want to supplement the text, explain whatsorts of supplementary materials and experi-ences you would use and why.

3. *Building on children’s literature. Find a populartrade book (perhaps from the Book Nook for

IN THE FIELD

34

12

Children) that would be useful as a motivatorfor a problem-based lesson revolving aroundone of the four basic operations (addition,subtraction, multiplication, or division). Identifythe operation that the story evokes and write aproblem that you might challenge students towork on after hearing the story. If possible, readthe story to some children, have them try solv-ing your problem, and analyze their writtenwork for evidence of understanding of the basicoperation the story involves.

4. *Timed tests. Observe children as they take atimed test on basic facts. What behaviors andemotions can you document? How are the re-sults of the test used by the teacher? Talk to atleast five children or adults about their memo-ries of timed tests on basic facts. What do theysay about their experiences?

5. *Student difficulties with basic facts. Talk to afifth- or sixth-grade teacher about what he orshe does with students who are not fluent in ba-sic facts. Do they receive special instruction infact strategies? Do they use calculators routinelyfor classwork and homework? How many in theclass have problems with basic facts? Whichfacts cause the most difficulties?

6. Use the Basic Addition and Multiplication Factscharts, found in Appendix B of your text or inTeaching Elementary Mathematics: A Resourcefor Field Experiences, to assess a child’s fluencywith the basic facts. Randomly say a fact forthe child to answer orally. Use the chart torecord his or her responses. You may want todevelop some symbols to indicate correctnessand quickness of response.

7. Read the Number Standards for Grades pre-K–2 and 3–5 in the NCTM’s Principlesand Standards for School Mathematics (www.nctm.org). What recommendations are madeconcerning teaching basic facts? Write a posi-tion paper either agreeing with the NCTM’srecommendations or taking an opposing posi-tion. Clearly explain and defend your position.

8. See the activity described in “Problem Solvingwith Combinations” (English, 1992). How is itrelated to ideas presented in this chapter?

9. Look at the book Mental Math in the PrimaryGrades by Jack Hope et al. (1987). How do thestrategies described in that book compare tothose described in this chapter? The book

WITH ADDITIONAL RESOURCES

IN YOUR JOURNAL

*Additional activities, suggestions, or questions are pro-vided in Teaching Elementary Mathematics: A Resourcefor Field Experiences (Wiley, 2004).

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226 Chapter 9 • Operations: Meanings and Basic Facts

Book Nook for ChildrenBurningham, John. Pigs Plus. New York: VikingPress, 1983.Mishaps occur when an old jalopy and its drivermake an eventful trip. Simple addition conceptsand sentences are used to describe these events.Each mishap is a separate little story told inthree foldout pages. The first page shows thedistressed riders and an appropriate numeral,the second page shows one pig to the rescuewith a “�1,” and the last page shows the pigswith the problem solved.Charosh, Mannis. Number Ideas through Pictures.New York: Thomas Y. Crowell Publishers, 1974.Colorful drawings and simple text graduallylead the child into activities and generalizationsin an exploration of odd, even, square, and tri-angular numbers. An example of this visualizedprocess is presented in the context of addingtwo odd numbers. Endelee, Judith Ross, and Tessler, StephanieGordon. Six Creepy Sheep. Hornesdale, PA:Boyds Mills Press, 1992.This counting book uses rhymes to depict howsix sheep dressed as creepy ghosts are fright-ened off, one by one, by trick-or-treaters untilonly one creepy sheep is left to knock on thedoor of a lonely barn.Hawkins, Colin. Take Away Monsters. NewYork: G. P. Putnam’s Sons, 1984.Monstrous characters, clever situations, andgood rhymes support learning mathematics.Add paper mechanics that allow subtractionproblems by using paper mechanics on eachpage to be solved as the illustration changes toaccurately reflect the answer. For example, averse states, “Five monsters living in jail. Oneescapes, and they all start to wail.” The partialnumber sentence 5 � 1 � appears with theverse. When the reader pulls a tab, the humor-ous scene depicting five monsters is trans-formed.Hulme, Joy N. Counting by Kangaroos: A Multi-plication Concept Book. New York: ScientificAmerican Books for Young Readers, 1995.Multiplication is illustrated using animals fromAustralia. Groups of three squirrel gliders, fourKoalas, five bandicoots, . . . ten wallaby joys allcrowd a house.

includes black-line masters for instruction,activity sheets for individual seatwork, and sug-gestions for using manipulatives to help chil-dren develop basic fact strategies. How mightyou integrate use of materials from that bookwith regular textbook instruction on basic facts?

10. Pick a concept or skill from this chapter andcompile a brief bibliography of recent articleswith research or teaching ideas on the topic.

11. Find the Focus issue on The Magic of Numberin the journal Mathematics Teaching in MiddleSchool. Read one of the articles and give a writ-ten or oral report to your teaching peers.

12. Find a calculator activity, software package, orgame on the web that focuses on basic facts.Share the ideas you find.

13. Find a computer program that claims to helpchildren learn basic facts and try it out. Manyof these programs are on the market. Most focus on encouraging development of quick responses, but few encourage use of thinkingstrategies. Write a half-page review of the com-puter program (similar to those published inteacher journals). In your review you shouldidentify the name and publisher of the program,the cost, and the type of hardware required torun it. Explain how the program works. Includeanswers to the following questions:• Does the program help children learn strate-

gies for basic facts?• Does the program keep track of student

progress?• Does it modify the facts presented to the stu-

dent according to which ones he or she hasgotten right or wrong in the past?

• Do you think the program would be effectivein promoting quick recall of facts? Why?

• Do you think children would enjoy usingthis program? Why?

• How would you use such a program if yourclassroom had only one or two computers?

• Would you recommend this program to otherteachers? Why?

Resources

ANNOTATED BOOKSFosnot, Catherine T., and Dolk, Maarten. Young Mathemati-

cians at Work: Constructing Number Sense, Addition, andSubtraction. Portsmouth, NH: Heinemann, 2001.

WITH TECHNOLOGY

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Resources 227

In our efforts to reform mathematics education, we’velearned a tremendous amount about young students’strategies and the ways they construct knowledge, with-out fully understanding how to support such develop-ment over time. In this book, the authors tell what theyhave learned after several years of intensive study in nu-merous urban classrooms. The first in a three-volumeset, Young Mathematicians at Work focuses on youngchildren ages four to eight as they construct a deep un-derstanding of number and the operations of additionand subtraction. Drawing from previous work, they de-fine mathematics as an activity of structuring, modeling,and interpreting one’s “lived world” mathematically.They describe teachers who use rich problematic situa-tions to promote inquiry, problem solving, and construc-tion, and children who raise and pursue their own math-ematical ideas. Sample mini-lessons on the use of theopen number line model are provided to show you howto support the development of efficient computation.

Piccirilli, Richard. Mental Math Kids Can’t Resist! (Grades2–4). New York: Scholastic Professor Book Division, 2000. Tips, shortcut strategies, and 60 practice pages that rein-force math skills. Students apply the strategies as theywork on computation, problem solving, and estimation.

Trafton, Paul, and Thiessen, Diane. Learning through Prob-lems, Number Sense and Computational Strategies, A Re-source for Primary Teachers. Portsmouth, NH: Heine-mann, 1999.A good book for K–4 teachers who are looking for a better way using problem solving. Kids can come upwith all sorts of ingenious solutions to problems, gainingconfidence in working on new problems. The problemsfocus on simple, everyday situations.

ANNOTATED ARTICLESAskew, Mike. “Mental mathematics.” Mathematics Teach-

ing, 160(6), (September 1997), pp. 3–6.Focuses on the mental methods of computation that chil-dren use. Highlighting two aspects of mental mathemat-ics used by children in the seven- to twelve-year-old agegroup; illustration of the process of computation; whatchildren are expected to know at certain stages; how tomanage mental mathematics programs.

Buckleitner, Warren. “Tech Notebook: Math Motivators.” Instructor, 109 (April 2000), pp. 66–69.A collection of computer programs for teaching elemen-tary mathematics. At the primary level, students aretaught skip counting, measurement, graphs, number pat-terns, place value, addition, and subtraction. An actiongame teaches mental math, addition, and multiplication,as well as other skills for upper elementary students.

Leutzinger, Larry P. “Developing Thinking Strategies forAddition Facts.” Teaching Children Mathematics, 6 (September 1999), pp. 14–18.Leutzinger demonstrates activities for primary-grade stu-dents to address basic facts like counting on, using dou-bles, and making ten with a focus on reasoning andcommunication. Probability, spatial sense, and moneyare also discussed.

Sowell, Evelyn J. “Multipurpose Mathematics Games FromDiscards.” Arithmetic Teacher, 23 (October 1976), pp.414–416.Sowell gives suggestions for developing effective mathe-matics games and gives rules for two specific games.

“The Magic of Number,” Focus Issue of Mathematics Teach-ing in Middle School, 7 (April 2002).

This focus issue provides several suggestions for teach-ing numbers and operations according to the Standards.

ADDITIONAL REFERENCESAshlock, Robert B., and Washbon, Carolynn A. “Games:

Practice Activities for the Basic Facts.” In DevelopingComputational Skills, 1978 Yearbook (ed. Marilyn N.Suydam). Reston, VA: NCTM, 1978, pp. 39–50.

Baroody, Arthur J., and Standifer, Dorothy J. “Addition andSubtraction in the Primary Grades.” In Research Ideasfor the Classroom: Early Childhood Mathematics (ed.Robert J. Jensen). Reston, VA: NCTM, and New York:Macmillan, 1993, pp. 72–102.

Carpenter, Thomas P.; Corbitt, Mary Kay; Kepner, Henry S.Jr.; Lindquist, Mary Montgomery; and Reys, Robert E.Results from the Second Mathematics Assessment of theNational Assessment of Educational Progress. Reston, VA:NCTM, 1981.

Clark, Faye B., and Kamii, Constance. “Identification ofMultiplicative Thinking in Children in Grades 1–5.”Journal for Research in Mathematics Education, 27 (January 1996), pp. 41–51.

Davis, Edward J. “Suggestions for Teaching the Basic Factsof Arithmetic.” In Developing Computational Skills, 1978Yearbook (ed. Marilyn N. Suydam). Reston, VA: NCTM,1978, pp. 51–60.

English, Lyn. “Problem Solving with Combinations.”Arithmetic Teacher, 40 (October 1992), pp. 72–77.

Funkhouser, Charles. “Developing Number Sense andBasic Computational Skills in Students with SpecialNeeds.” School Science and Mathematics, 95 (May 1995),pp. 236–239.

Ginsberg, Herbert. Children’s Arithmetic: The LearningProcess. New York: Van Nostrand Reinhold, 1977.

Greer, Brian. Multiplication and Division as Models of Situ-ations. In Handbook of Research on Mathematics Teach-ing and Learning (ed. Douglas A. Grouws). New York:Macmillan, 1992, pp. 276–295.

Hope, Jack A.; Leutzinger, Larry; Reys, Barbara J.; andReys, Robert E. Mental Math in the Primary Grades. PaloAlto, CA: Dale Seymour, 1987.

Kamii, Constance; Lewis, Barbara A.; and Booker, Bobby M.“Instead of Teaching Missing Addends.” Teaching Chil-dren Mathematics, 4 (April 1998), pp. 458–461.

Kouba, Vicky L., and Franklin, Kathy. “Multiplication andDivision: Sense Making and Meaning.” In Research Ideasfor the Classroom: Early Childhood Mathematics (ed.Robert J. Jensen). Reston, VA: NCTM, and New York:Macmillan, 1993, pp. 103–126.

Kouba, Vicky L., and Franklin, Kathy. “Research into Prac-tice: Multiplication and Division: Sense Making andMeaning.” Teaching Children Mathematics, 1 (May1995), pp. 574–577.

Quintero, Ana Helvia. “Conceptual Understanding of Multi-plication: Problems Involving Combination.” ArithmeticTeacher, 33 (November 1985), pp. 36–39.

Rathmell, Edward C. “Using Thinking Strategies to Teachthe Basic Facts.” In Developing Computational Skills,1978 Yearbook (ed. Marilyn N. Suydam). Reston, VA:NCTM, 1978, pp. 13–38.

Sowell, Evelyn J. “Effects of Manipulative Materials inMathematics Instruction.” Journal for Research in Mathe-matics Education, 20 (November 1989), pp. 498–505.

Thornton, Carol A., and Smith, Paula J. “Action Research:Strategies for Learning Subtraction Facts.” ArithmeticTeacher, 35 (April 1988), pp. 8–12.

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