malaysia operations framework

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SunsetMaths Project: Operations, basic facts and algorithms Frameworks for teaching primary mathematics

| AND SUBTRACTION

|

| MENTAL COMPUTATION AND ALGORITHMS FOR MULTIPLICATION

| AND DIVISION

|

| COMPUTATIONAL ESTIMATION|

V CALCULATORS

This sequence is not pure. The steps intermingle. You are teaching basic facts for addition

and subtraction while you are introducing concepts of multiplication and division. The crucial thing is

that multiplication and division follow after addition and subtraction, that basic facts follow concepts,

and that computation follows basic facts. As well, it is common for mental computation to begin

about the same time as the algorithms (if both are attempted) and for these to be followed by

computational estimation and calculator use for algorithms. A major reason for this is that

computational estimation and calculator algorithms are used on numbers greater than 1 000, whilemental computation and pen-and-paper algorithms are normally restricted to numbers less than 1 000.

There are a lot of interrelationships in this sequence. Multiplication can be considered as

repeated addition. Similarly, one way to think of division is as repeated subtraction. The

multiplication algorithm for two digit numbers involves the addition algorithm and long division

involves subtraction. For example:

3x4 15/5 3 4 2 3 4

x 2 5 4 ) 9 3 6

4+4+4=12 15-5-5-5=0 1 7 0 86 8 0 1 3

8 5 0 1 2

1 6

One major issue in sequencing is the relationship between numeration and computation.

Many of the procedures rely on aligning place values and calculating each position separately.

Obviously many procedures use renaming and regrouping.

There is also a need for students to learn multiple of tens facts (e.g., 20x40=800) and to

understand the operation properties and principles of how computation changes as the numberchanges, for example, the inverse proportion principle (36/12=3, 36/6=6, 36/4=9) - the smaller the

divisor the larger the quotient.

CONCEPTS

The concepts of addition, subtraction, multiplication and division are the meanings that lie

behind the operations, not the ways of getting answers. In this section, we will look at: (a) all the

meanings for addition, subtraction, multiplication and division; (b) the instructional sequence to

introduce symbols ; and (c) equals sign, dynamic arithmetic and operation principles.

Meanings of the operations

Tom Cooper, School of Mathematics, Science and Technology Education, QUT, 1999 Page 2


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Addition and subtraction

The various meanings for addition and subtraction are as below. They are based on the

following:

(1) Situations. There are four situations -joining, separating, comparing and inaction. Join and

separate are two inverse actions - addition and subtraction are in both. Comparing involves twogroups but they are not joined, the sum is one of the two groups. Inaction is where two parts are

considered in terms of a whole by including them in a wider set (i.e., apples and bananas become fruit,

Fords and Holdens become cars).

(2) Models. For teaching purposes, the operations of addition and subtraction need to be modelled

with 2 types of materials - set, those that involve sets of discrete objects (e.g., unifix or counters), and

length, those that involve distance (unifix stuck together or number lines).

(3) Overall meaning. The situations can be combined under part-part-total. Addition is when

know the parts and want the total, while subtraction is when know the total and one part and want theother part.

OPERATION MEANING REAL-WORLD PROBLEM

Addition Join There were 3 cars in the park, 2 drove in, how many cars in the park?

[set]

The building had 3 storeys, the crane lifted on another 2 storeys, how

high is the building? [length]

Take-away There were cars in the park, 2 drove out, this left 3, how many carswere in the park to begin with? [set]

The crane knocked off the top 2 storeys, this left 3 storeys, how many

storeys high was the building? [length]

Compariso

n

Fred had 3 cars, Jack has 2 more cars than Fred, how many cars does

Jack have? [set]

The Fox building is 3 storeys, the Jed building has 2 more storeys

than the Fox building, how many storeys does the Jed building have?

[length]

Inaction There were 3 holdens and 2 fords, how many cars? [set]

The building had 3 blue storeys and 2 white storeys, how many

storeys high was the building? [length]

Subtraction Join There were 3 cars in the park, some cars drove in, there are now 5

cars in the park, how many cars drove in? / There were some cars in

the park, 2 cars drove in, there are now 5 cars in the park, how many

cars were in the park at the start? [set]

The crane lifted some extra storeys onto the top of the 2 storey

building, the building is now 5 storeys high, how many extra storeyswere lifted by the crane? / The crane lifted 3 extra storeys onto the

top of the building, the building is now 5 storeys high, how high was



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the building to start with? [length]

Take-away There were 5 cars in the park, 2 drove out, how many cars are left? /

There were 5 cars in the park, some drove out, 3 were left, how many

cars are drove out? [set]

The building had 5 storeys, the crane knocked off the 2 top storeys,

how many storeys are left? / The building had 5 storeys, the crane

knocked off some of the top storeys, 3 storeys are left, how many

storeys were knocked off? [length]

Compariso

n

Fred had some cars, Jack has 2 more cars than Fred, Jack has 5 cars,

how many cars does Fred have? / Fred has 3 cars, Jack has 5 cars,

how many more cars does Jack have than Fred? [set]

The Jed building has 2 more storeys than the Fox building, the Fox

building has 5 storeys, how many storeys does the Jed building

have? / The Fox building is 3 storeys, the Jed building has 5 storeys,how many more storeys does the Jed building have than the Fox

building? [length]

Inaction There were 5 cars and 2 fords, how many holdens? [set]

The blue and white building was 5 storeys high, 2 were white, how

many storeys were blue? [length]

As we said earlier, the meanings can be combined underpart-part-total. This allows all

meanings to be integrated and a single method to be used to determine whether a problem is addition

or subtraction. (Note: There are some difficulties in integrating comparison but it is worth a little

sleight of hand to get a single integrating idea.)

OPERATION MEANING PROBLEM THINKING

Addition Know parts

want total

I took $5 238 from my

account, this left $11 892,

what was in the account to

start with?

The $5 238 & $11 892 are parts. The

wanted amount is the total. So, the

operation is addition.

Subtraction Know total

want a part

I added the grant to my

$7 832 account, this gave

me $9 561, how much was

in the grant?

The $7 832 is a part, the $9 561 is the

total. The wanted amount is a part. So,

the operation is subtraction.

Multiplication and division situations

The various situations for multiplication and division are as below.

(1) Situations. There are five situations - combining, partitioning, comparing, combinations

and inaction (set inclusion). Once again, combine and partition are inverse actions (which

contain both multiplication and division), comparison has two sets (one of which is the

product), combinations is a new situation based on counting when two or more possibilities are



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available for different combining attributes (e.g., dice and spinner, shirt and pants), and

inaction is set inclusion - taking a wider view of the objects. (Note: Some educators see

inaction as another form of combining and do not give it a separate meaning.)

For division, the unknown may be the number of groups (called grouping or repeated

subtraction) or the number in each group (called sharing)

(2) Models. There are three models - similar to addition and subtraction, there are set (e.g., unifix

and counters) and length models (e.g., unifix stuck together or number lines) and there is also

array or area model (e.g., counters, unifix, dot paper or graph paper), a new one for

multiplication and division.

(3) Overall meaning. The situations can be combined again under part-part-product, where each

part is a factor. Multiplication is when know the parts and want the product, while division is

when know the product and one part and want the other part. (Note: Here, a part refers to

either the number multiplying or the number being multiplied.)

OPERATION MEANING REAL-WORLD PROBLEM

Multiplication Combining

(equal

groups

)

There were 3 cars with 4 people in each, how many people? [set]

There were 3 trains each with 4 carriages, how many carriages?

[length]

There are 3 rows and 4 trees in each row, how many trees

[array]

Partitioning People were put into groups of 4, there were three groups, howmany people? [set]

The carriages were divided into 3 trains each of 4 carriages, how

many carriages? [length]

The trees were planted into 3 rows of 4, how many trees? [array]

Comparing Jack has 3 times as many cars as Fred, Jack has 12 cars, how

many cars does Fred have? [set]

The steam train has 4 carriages, the electric train has 12, how

many times as many carriages does the electric have than the

steam? [length]

Combinations Jane has 3 tops and 4 skirts, how many outfits?

Inaction 4 apples, 4 pears and 4 bananas, how many pieces of fruit?

Division Combining Groups of 4 people joined for a dinner of 12, how many groups?

[set - grouping] / 3 groups of people joined for a dinner of 12,

each group was the same size, how many in each group? [set -

sharing]

Trains of 4 carriages were shunted together to form a train of 12

carriages, how many trains? [length - grouping] / 3 trains wereshunted together to form a train of 12 carriages, how many

carriages in each train if the trains were the same length? [length -



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sharing]

I planted rows of trees with 4 trees in each row, I used 12 trees,

how many rows? [array - grouping] / I planted 3 rows of tress, I

used 12 trees, how many in each row? [array sharing]

Partitioning 12 people put into groups of 4, how many groups? [set -

grouping] / 12 people were shared amongst 3 cars, how many

people in each car? [set - sharing]

The 12 carriages were divided into trains of 4 carriages, how

many trains? [length grouping] / The 12 carriages were shared

amongst 3 trains, how many carriages in each train? [length -

sharing]

12 trees were divided into rows of 4, how many rows? [array -

grouping] / 12 trees were divided into 3 rows, how many trees per

row? [array - sharing]Comparing Jack has 3 times as many cars as Fred, Jack has 12 cars, how

many cars does Fred have? [set] / Jack has 12 cars, Fred has 4,

how many times as many cars has Jack as Fred? [set]

The electric train has 12 carriages, the electric train has 3 times as

many carriages as the steam, how many carriages does the steam

train have? [length] / The steam train has 4 carriages, the electric

train has 12, how many times as many carriages does the electric

have than the steam? [length]

Combinations Jane has 12 outfits, There are3 tops, how many skirts?

Inaction 4 of each fruit on the table, 12 pieces of fruit in all, how many

different types of fruit? [inaction grouping] / The same number

of apples, oranges and bananas, 12 pieces of fruit, how many of

each fruit? [inaction sharing]

As we said earlier, similar to addition and subtraction, the meanings of multiplication anddivision can be combined underpart-part-product. This allows all meanings to be integrated and a

single method to be used to determine whether a problem is addition or subtraction. (Note: There are

also some difficulties in integrating comparison as there were for addition and subtraction.)

OPERATION MEANING PROBLEM THINKING

Multiplication Know parts

want

product

The money was divided amongst

the employees, each received

$436, there were 57 employees,how much money was divided?

The $436 is a part. The 57 is a

part. The wanted amount is the

product. So, the operation ismultiplication.



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Division Know

product

want a part

The number of apples is 8

times the number of oranges,

there are 56 apples, how many

oranges?

The 8 is a part. The 56 is the

product. The wanted amount is

also a part. So, the operation is

division.

Introducing symbols

A crucial part of developing meaning is to relate symbols to models and real-world situations.

The approach advocated is to have the children act out real-world situations with materials while using

the language of the operations. Below is one way it could be done. It is also possible to be more open

and let the children explore things in their own way (model explore share).

The attaching of symbols is contentious. A particular setting out can be encouraged or the

students could be allowed to develop their own setting out. Below is one way it could be done. It is

also possible to be more open and let the children explore things in their own way and to develop their

own setting out (model explore share). The pendulum is moving to child developed setting out -

to sharing a variety of methods with the class and letting the children choose their own. However, it is

useful to see some ways in which a setting out can be directly developed.

Addition

Materials and language

PROBLEM Real world There are two cats on a fence, three more jumped up to join

AND MODEL problem them. How many cats are now on the fence?

Modelling Show me the two cats on the fence! O O

Show me the three cats on the ground! O O O O O

Show the three joining the two! O O O O O

How many cats on the fence? [5] O O O O O

REFLECTION Questions Show me the part that was the cats that were on the fence?

How many cats in this part? [2]

Show me the part that was the cats that were on the ground?


Show me the total which is all the cats together?How many cats in the total? [5]

Repeat qus How many on fence? How many on ground? How many in

total?

Materials, language and symbols

MODEL Materials There are two cats on a fence, O O O O O

three more jumped up to join them.

How many cats are now on the fence?

INTRODUCTION Symbols How many cats on the fence? [2] 2 2

OF SYMBOLS Write this numeral down!

How many cats on the ground? [3] 2 2 3



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Write this numeral under the 2 (beside the 2)! 3

What are we trying to find? [The total] 2 2+3=

Write the addition symbol as shown and the + 3

line (equals sign) to show we want an answer!

What is the total? [5] 2 2+3=5

Write this under the line (beside the equals)! + 3

5

CONTINUED Action with Two cats on fence! O O 2

RELATION OF materials and Three cats on ground! O O O O O 2

SYMBOLS TO symbols 3

MATERIALS AND Three cats join the two! O O O O O 2

REAL WORLD + 3

SITUATION How many cats on the fence? 2+ 3

5

Subtraction


PROBLEM Real world There are six cats on a fence, two jumped down.

AND MODEL problem How many cats are now on the fence?

Modelling Show me the six cats on the fence! O O O O O OShow me the two cats jumping down! O O O O O O

How many cats on the fence? [4] O O O O

REFLECTION Questions Show me the total that was the cats that were on the fence?

How many cats in this total? [6]

Show me the part that was the cats that jumped to the ground?


Show me the part which is the cats left on the fence?


Repeat qus How many on fence? How many jumped down? How many

left?


MODEL Materials There are six cats on a fence, O O O O O O

two jumped down to the ground.

How many cats are now on the fence?

INTRODUCTION Symbols How many cats on the fence? [6] 6 6


How many cats jumped down? [2] 6 6 2Write this numeral under the 6 (beside the 6)! 2

What are we trying to find? [The part] 6 6-2=



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Write the subtraction symbol as shown and the - 2


What is the part (the cats left)? [4] 6 6-2=4

Write this under the line (beside the equals)! - 2

4

CONTINUED Action with Six cats on fence! O O O O O O 6

RELATION OF materials and Two cats jump down! O O O O O O 6

SYMBOLS TO symbols - 2

MATERIALS AND How many cats on the fence? O O O O O O 6

REAL WORLD - 2

SITUATION 4

Please note: When acting out subtraction, it is useful not to remove the subtracted material

completely, but to move aside or down. Then, the subtraction can be checked by moving thesubtracted part back, that is, rejoining. Some educators also recommend covering the subtracted part

(with hand or etc.) instead of moving it.

Multiplication


PROBLEM Real world There were 3 bags of lollies, 4 lollies in each bag,

AND MODEL problem how many lollies in all?

Modelling Show me the first bag of lollies! oooo

Show me the second and third bag! oooo oooo ooooHow many lollies in all? [12] oooooooooooo

REFLECTION Questions Show me the separate bags of lollies?

How many bags? [3]

Show me one of the bags? How many in each bag? [4]

Show me all the lollies. How many lollies? [12]

Repeat qus How many groups? How many in each group? How many in

all?


MODEL Materials There were 3 bags of lollies, oooo

4 lollies in each bag, oooo

how many lollies in all? oooo

INTRODUCTION Symbols How many groups [3]

OF SYMBOLS Write this numeral down 3

(second position for vertical)! 3

How many in each group? [4] 4 3 4

Write this numeral above the 3 (beside the 3)! 3

What are we trying to find? [The product] 4 3x4=Write the multiplication symbol as shown and the x 3




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What is the product (h.m. cats in all?)? [12] 4 3x4=12

Write this under the line (beside the equals)! x 3

1 2

CONTINUED Action with Three bags! 3

RELATION OF materials and Four lollies in each bag! [oooo] [oooo] [oooo] 4

SYMBOLS TO symbols x 3

MATERIALS AND

REAL WORLD How many lollies in all? oooooooooooo 4

SITUATION x 3

1 2

Division


PROBLEM Real world $8 shared amongst 2 people, how much money

AND MODEL problem does each person get?

Modelling Show me $8! Show me the 2 people! O O O O O O O

Share the dollars! ( ) ( )

How many dollars to each person? [4] (OOOO) (OOOO)

REFLECTION Questions Show me the money we started with? How much? [8]

Show me the people sharing. How many people? [2]

Show me what each person got after sharing. How much? [4]

Repeat qus How much to share? How many people sharing?How much to each person?


MODEL Materials There is $8 to share, 2 people sharing, O O O O O O O O

how much does each get? ( ) ( )

INTRODUCTION Symbols How much money is there to share? [8] 8 8


How many people sharing? [2] What are 8/2 2 ) 8

we trying to find? [How much each

gets.] Write the number 2 beside the 8 withappropriate sign for sharing!

Share the money. How much did each 8/2=4 2 ) 8

person get? [4] Write this appropriately. 4

CONTINUED Action with $8 to share! O O O O O O O O 8

RELATION OF materials and

SYMBOLS TO symbols 2 people to share! ( ) ( ) 2 ) 8

MATERIALS AND

REAL WORLD How much to each person? (OOOO) (OOOO) 2 ) 8

SITUATION 4



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Note: It is important to reverse these procedures - to start, say, with 5+8 and move from symbols to

modelling and real-world situations. So:

real world model symbols

and symbols model real world

The equals sign, dynamic arithmetic and operation principles

This section looks at three other issues important in operations. The first is the equals sign.

The second is a way of looking at operations that considers them as changesor transformations not

relationships. The third is the properties of the operations that are useful in understanding

computation and estimation.

Equals sign

Equations involve the equals sign. Children come to believe that the equals sign shows wherethe answer should go or that it is an indication to do something. Both of these are inadequate.

Children need to taught that equals means the same as, that is, that the right hand side is the same

value as the left hand side. Equals is therefore equivalence.

It is also important children learn: (a) that the line in the vertical setting out of the algorithm is

somewhat similar to the equals sign; and (b) to relate the use of the equals sign to real world

situations.

SITUATION REPRESENTATION

I had $5 and my father gave me $3. 5+3

I had $5 and my father gave me $3. How much

money do I have? 5+3=

I had $5 and my father gave me $3. I ended up

with the same money as my friend, $8. 5+3=8

I had $5 and my father gave me $3. I ended up

with the same as Joe who had $10 and lost $2. 5+3=10-2

Children have to have the experience to see that the equals sign means the same thing in each

of the following examples:

(a) 2+5= (b) 3x7=21 (c) 45=9x5 (d) 18-2=4x4

It is important to build the idea that as long as you do the same to both sides, the equation

remains correct. For example:

If 7x8=56 then 7x8+42=56+42 is true since adding the same 42 to both sides

If 45+76=11x11 then (45+76)x53=(11x11)+53 is incorrect since x53 & +53 are different

If 145-78=9x7+4 then 145-78+237=9x7+4-237 is incorrect since+237 & -237 are different

Dynamic arithmetic



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Arithmetic can be thought of as relationship, e.g., 2+3=5. However, arithmetic can also be

seen as dynamic, that is, as changes. In this view,

+3 -7

2+3=5 can be considered as 2 5 and 16-7=9 can be considered as 16 9

and an arithmetical excursion can be represented as:

x8 +15 +46 /11 x56 -168

2 16 31 77 7 392 224

Children can make up their own excursions:

(a) 56 579

(b) 1 201 43

(c) 78 654 468

Operation principles (or properties)

The following principles are important for mathematics across years 1-10. They are

particularly important for estimation.

These principles are really patterns or properties that recur for different numbers. Therefore,

the way of teaching them is to use reflection on examples. Putting examples like below together will

enable children to see that increasing the minuend or the divisor actually decreases the subtrahend

(difference) or the quotient.

12-4=8 24/12=2

12-5=7 24/8=3

12-6=6 24/6=4

12-7=5 24/4=6 and so on

(1) Addition-increase principle. If we take an addition example, say a+b=c, then c increases as a

or b increases (and vice versa), and c stays the same if a and b vary in the opposite direction.

(2) Inverse subtraction principle. If we take a subtraction example, say c-a=b, then b increases as

c increases, but b decreases as a increases. There is the inverse relation between total and

minuend.

(3) Difference principle. For a subtraction example, say, c-a=b, if c and a are changed the same

(the same number is added to or subtracted from c and a), then b will stay the same. This is the

difference principle. It means that if we have to calculate 428-195, we can change this to 433-

200 by adding 5 to both sides. 433-200 is easier to solve than 428-195 and gives the same

answer.

(4) Multiplication-increase principle. Similar to addition, a multiplication example, say, pxq=r, is

such that r increases if p and q increase (and vice versa), and r stays the same if p and q vary in

the opposite way.

(5) Inverse proportion principle. For division example r/p=q, q increases if r increases and q

decreases if p increases (inverse proportion).



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(6) Quotient principle. Similar to subtraction, a division example, say r/p=q, q stays the same if r

and p vary in the same way (r and p are multiplied and divided by the same number).

The principles above are related to how operations are effected by number changes. There are

other principles that relate to properties of the operations. Some of these are given below. They can

be developed using counters and, for multiplication, using arrays.

(1) Commutative principle. This is turn arounds, for + and x only - 3+7=7+3 and 5x8=8x5.

(2) Associative principle. This is that numbers can be reassociated for + and x. For example:

6+5=(2+4)+5=2+9=2+(4+5) 12x5=(3x4)x5=3x20=3x(4x5)

(3) Identity principle. For + and x, there are numbers that leave things unchanged - +0 and x1.

(4) Inverse principle. - and / are the inverses of + and x, for example, +3, -3 and x5, /5.

BASIC FACTS

Once the concepts of the operations are introduced, it is time to teach ways to calculate the

answers quicker than representing the operation with counters and counting to get the answer. The

first of the calculations to teach are those that form the basis of the later algorithms and estimation -

the basic facts. It is still accepted that these facts have to be learnt off by heart, that is, automated by

practice (drill). This is because that any knowledge that is automated is available for problem solving

without taking any thinking power from the problems.

The basic facts are all the calculations with numbers less than 10 for addition and

multiplication and the inverse operations for subtraction and division, that is:

0+0, 0+1, 0+2, ....., 1+0, 1+1. 1+2, ...., 2+0, 2+1, ...., 9+0, 9+1, 9+2, ...., 9+9

0-0, 1-0, 2-0, ...., 9-0, 1-1, 2-1, ...., 10-1, 2-2, ...., 11-2, ...., 9-9, 10-9, ...., 18-9

0x0, 1x0, ...., 9x0, 0x1, 1x1, ...., 9x1, 2x0, 2x1, ...., 9x0, 9x1, 9x2, ...., 9x9

0/1, 1/1, ...., 9/1, ...., 0/2, 2/2, ...., 18/2, 0/3, 3/3, ...., 27/3, ...., 9/9, ...., 81/9

For algorithms and estimation, it is important that these facts are extended to 30+50, 20x30,

800/20, and so on. These are multiples of tens facts.

Although in the end the basic facts are automated, on the way to this end it is important to

develop some way in which students can work out answers so that drills can improve speed to

automaticity. This is done by teaching thinking strategies. These thinking strategies are ways in

which answers can be got for a set of basic facts.

This section on basic facts will, therefore, cover: (a) thinking strategies for addition and

subtraction; (b) thinking strategies for multiplication and division; (c) multiples of tens facts; and (d)

diagnosis and practice activities.

Addition and subtraction thinking strategies



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The strategies for addition and subtraction can be categorised as six types: counting (counting

on and counting back), turn arounds, near doubles (doubles+1, doubles+2, levelling pairs),

near tens (build to 10 and add 10), think addition, andfamilies. The material that can be used to

introduce them is counters and unifix cubes, 2 cm graph paper, addition grids, paper, and coloured

pens.

Counting on and counting back

The strategy of counting on is used for addition facts where one number is 0, 1, 2 and 3. The

idea is to change counting from both numbers all together (called SUM) to where only the 0, 1, 2 and

3 are counted and the other number is the start. For example, 6+2 is six, seven, eight. One way to

do this is to cover the larger number, recall its number and then count on the 0, 1, 2 or 3. This can be

done with a hand, or a container. For example: Put 4 counters into your left hand. Put 2 counters in

your right hand. Say four showing the left hand and then drop in the counters one at a time from

the right to the left hand, saying five, six.

The strategy is also used when subtracting 0, 1, 2 and 3 (counting back). For example, 7-2 is

seven, six, five. This can be taught by dropping counters out of a hand or a container: Put 5

counters in the left hand, show hand and say five, drop out 3 counters one at a time into the right

hand saying four, three, two.

This 3D activitiy can be reinforced with an activity card that relates +2 or -3 to counters or

dots. For example (counting on): Take an A4 sheet of paper. Fold it back on itself half way. Place 4

+ 2 vertically on the top half and the bottom folded sheet. Open the fold and place two circles behind

the fold. Show the folded page, say 4+2, open the fold, and say four, five, six. Similarly, another

example (counting back): Take an A4 sheet of paper. Fold it back on itself half way. Place 7 - 3vertically on the top half and the bottom folded sheet. Open the fold and place three circles behind

the fold. Show the folded page, say 7-3, open the fold, and say seven, six, five, four. The two

examples are drawn below:

Turn arounds



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This strategy is for all facts. It nearly halves the number of facts to be learnt by showing that

bigger + smaller (e.g., 5+2) is the same as smaller + bigger (e.g., 2+5). For example, 4+7 is 7+4

equals 11. This is done by showing that the counters can be joined either way. For example: Put out

6 counters, add 3 counters to it. Put out 3 counters, add 6 counters to it. Are the final amounts the

same? Put out 9 counters. Separate into 6 and 3. Remove and add the 6. Repeat for the 3. Say

3+6 is the same as 6+3.

A way to reinforce this would be a sheet with, for example, 6 circles in the top half with the

numeral 6 on its left side and an upside down +6 on its right side and 3 circles on the bottom half with

the symbol and numeral +3 on its left side and an upside down 3 on its right side. This can be shown

to children the right way up (and they will see 6+3 vertically) and then upside down (and they will see

3+6).

Near doubles

This strategy is for doubles and for facts that are 1 or 2 from doubles (e.g., 4+5 is double four,

eight, plus one, eight, nine, and 6+8 is double 6, twelve, plus two, twelve, thirteen, fourteen). The

first step is to learn the doubles. This can be done as follows:

NUMBER TO DOUBLE THING TO THINK OF

1 The 2 feet or 2 hands on a person

2 The 4 tyres on a car

3 The 6 wickets in cricket

4 The 8 legs of a spider

5 The 10 fingers on our hands

6 The 12 eggs in an egg carton7 The 14 days in a fortnight

8 The 16 legs in 2 octapuses

9 The 18 dots in 2 Channel 9 symbols

Then doubles+1 and doubles+2 are introduced. This can be taught by putting counters for

the two numbers in rows in 1:1 correspondence and covering the extras. Then the children can see a

double. Lifting the hand will enable the extras to be counted. For example (double+1): Use sheet

with 2 rows of ten 2cm squares. Place 5 unifix on the top row. Place 6 on the bottom row. Cover the

extra unifix so double 5 is showing, say double five is 10, reveal extra cube, and say count on, ten,

eleven. Similarly, another example (double+2): Use sheet with 2 rows of ten 2cm squares. Place 6unifix on the top row. Place 8 on the bottom row. Cover the extra unifix so double 6 is showing, say

double six is 12, reveal extra 2 cubes, and say count on, twelve, thirteen, fourteen. A special

diagram can be used as below (using example, 7+5):

diagram |---|---|---|---|---|---|---|---|---|---| 7+5 |---|---|---|---|---|---|---|---|---|---|

|---|---|---|---|---|---|---|---|---|---| |---|---|---|---|---|---|---|---|---|---|

|---|---|---|---|---|---|---|---|---|---| |---|---|---|---|---|---|---|---|---|---|

5+5 +2

There is also a special sheet for reinforcing this strategy: Take an A4 sheet, fold both endsback 5 cm. On the folded left end, write 5+5 vertically. On back of the folded left end write 5+6

vertically. Put two rows of 5 circles on unfolded section and one extra circle under right fold on



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bottom row. Fold over both ends, show and say 5+5, double 5, ten, unfold both ends and say

5+6, double 5 and 1, ten, eleven. The diagram below shows this aid for 5+6 and another for 6+8.

It should be noted that counting on is not the only way to do this strategy. Some children

count back(e.g., 6+8 is double eight, sixteen, back two, sixteen, fifteen, fourteen), while some

children level pairs, for the two case (e.g., 6+8 is 7+7 by adding 1 to 6 and taking 1 from 8, is

double 7, 14).

Near tens

This strategy is for all the remaining addition facts - where one number is not 0, 1,2 or 3 and

not a near double (e.g., 8+5 and 9+4). Usually, one of the numbers is 7, 8 or 9. The first thing to betaught is the difference between each number to 9 and 10. This can be done on the fingers: Show 10

fingers on your 2 hands. Drop your first 7 fingers. How many left? How many to the ten? Repeat

for 4, 6 and 8 fingers.

Once this is known, the strategy can be used in build to 10 mode. For example, 9+5 is 9+1

to make 10 plus another 4, 14. This can be taught on a sheet with two 2x5 arrays of 2cm squares:

Place 8 unifix on the first array and 5 unifix on the second array. Say 5+8. Move counters from

second to first array until all ten squares are covered. Say 5+8 is 10+3 is 13. Repeat this for 7+4

and 9+6. This can be shown diagrammatically as below (for example 8+5):

diagram |---|---|---|---|---| |---|---|---|---|---|

|---|---|---|---|---| |---|---|---|---|---|

|---|---|---|---|---| |---|---|---|---|---|

8+5 |---|---|---|---|---| |---|---|---|---|---|

|---|---|---|---|---| |---|---|---|---|---|

|---|---|---|---|---| |---|---|---|---|---|

is the same as 10+3 |---|---|---|---|---| |---|---|---|---|---|

if 2 is moved from |---|---|---|---|---| |---|---|---|---|---|

the 5 to the 8 |---|---|---|---|---| |---|---|---|---|---|



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There is another near ten strategy called add ten in which 9+4 is considered as 4+9 and

thought of as 4+10=14-1. This could be introduced with MAB by putting out 4, adding 9, and then

trading to 10 and 3. This could be followed by discussion of how this could be short circuited by

directly taking the 10 and handing in a 1 (instead of adding the 9 first). A 99 board can also help,

adding 9 can be seen as adding 10 and going back 1, while adding 8 can be seen as adding 10 and

going back 2.

Think addition

This strategy is used for subtraction facts. The idea is not to do subtraction but to think of the

facts in addition terms. For example 8=3 is thought of as what is added to 3 to make 8. To teach

this, need to show that subtraction and addition are inverses of each other: Take 7 counters and 4

counters. Combine them to make 11. Separate them back to 7 and 4. Repeat this for 3 and 6

counters and 5 and 8 counters. The notion of adding on to get a subtraction can also be directly

modelled: Put out 11 counters. Below them put 7 counters. Add counters to the bottom group until

both groups are the same. Repeat for 8 and 13.

There is also a sheet to reinforce this (for example 9-6): Take an A4 sheet of paper. Fold in

half length wise. Split the top fold in half again so it covers two half areas. Put 6 circles under the

left fold and 3 under the right. Put 9 on both right and left fold. Lift up left fold, say I have 6 and I

want to get to 9, lift up other side, say six, seven, eight, nine - three more. This aid can be shown

diagrammatically below (for examples 9-6 and 13-8):

Families

This strategy is to reinforce think addition and to relate + and -. For each

addition/subtraction fact, there are 4 members of the family - 3+5=8, 5+3=8, 8-5=3, and 8-3=5.

Families for 4+7 and 15-9 are:

4+7=11, 7+4=11, 11-4=7, 11-7=4 9+6=15, 6+9=15, 15-9=6, 15-6=9

Multiplication and division thinking strategies

The strategies for multiplication and division can be categorised as five types: turn arounds,patterns, connections, think multiplication, andfamilies. Strategies are used differently for

multiplication and division than they are for addition and subtraction. In addition and subtraction,



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strategies covered a variety of facts and there was no need to focus on tables. In multiplication and

division, there is more focus on tables (e.g., 4x, 7x tables).

Turn arounds

This strategy is applied to all facts. It means that larger x smaller is th same as smaller x

larger, that is, 7x3 = 3x7. It can be taught by comparing groups of counters and by rotating the arraymodel of multiplication a right angle: Put out 3 groups of 5 counters, and put out 5 groups of 3

counters. Which is larger? Are they the same? Construct an array for 4x7. Turn this array 90

degrees. Has the amount changed? What does this mean?

Patterns

One of the two major strategies for multiplication is patterns. This strategy applies to any of

the tables for which there is a pattern that could help children remember the facts. The following

tables have patterns:

TABLE PATTERN2x Doubles - 2, 4, 6, 8, 0, .... and so on.

5x Fives - 5, 0, 5, 0 , 5, .... and so on; 5, 10, 15, ...; half the 10x

tables; hands; clockface (minutes in one hour).

9x Nines - tens are one less than number to be multiplied by 9, ones

are such that tens & ones digits add to 9; 9, 18, 27, ....

and so on.

4x Fours - 4, 8, 2, 6, 0 , 4, .... and so on; odd tens is 2 and 6 for

ones and even tens is 0, 4 and 8 for ones.

0 4 812 16

20 24 28

32 36

3x Threes - the one back pattern

0 3 6 9

12 15 18

21 24 27

These patterns can be most easily seen with a calculator, unifix, and large and small 99 boards.

The table for the pattern is chosen (e.g., 4x). The number of the table is entered on the calculator and

[+] [=] pressed (e.g., [4] [+] [=]). The result (4) is covered on the 99 board with a unifix. From there,

[=] is continually pressed (adding 4) and the number shown is covered. Once sufficient numbers are

covered to see the visual pattern on the 99 board, this pattern is transferred to the small 99 board by

colouring squares. The numbers coloured are discussed to arrive at the pattern. If more reinforcement

is needed, [number] [+] [=] [=] [=] [=] [=] ... is pressed on the calculator and the ones or tens called

out at each [=] press (see below). This enables children to verbally hear patterns. The numbers could

also be written down for inspection for pattern.

Press [5] [+] [=] [=] [=] ... stating the ones positionPress [9] [+] [=] [=] [=] ... stating the ones position, then repeat, stating the tens position

Press [4] [+] [=] [=] [=] ... stating the ones position



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Connections

The other major strategy is for all the tables not covered by patterns. Here, the unknown table

is connected to a known table using the distributive principle. Here are some examples. We can use

counters, unifix, dot paper and graph paper for the models. The idea is that the answers to the

unknown table are found from the known tables.

KNOWN UNKNOWN CONNECTION

TABLE(S) TABLE 5

2x 3x 3x5 o o o o o same as 2x5 + 1x5 = 2x5+5

2 o o o o o i.e., 3x tables is 2x table

---------- plus the number

1 o o o o o

2x 4x 4x6 6 same as 2x6 + 2x6,

o o o o o o i.e., 4x6 is double 2x6,

2 o o o o o o i.e., 4x6 is double doubles-------------

2 o o o o o o

o o o o o o

2x, 4x 8x 8x6 is the same as 4x6 + 4x6, i.e. double double doubles

2x, 3x, 5x 6x 6x7 is the same as 3x6 + 3x6, i.e., double 3x (see below)

or 6x7 is the same as 5x6 + 1x6, i.e., 5x plus number

2x, 5x 7x 7x8 is the same as 5x8+2x8, i.e., 5x plus 2x (see below)

7 8

o o o o o o o o o o o o o o o o

{ 3 o o o o o o o o o o o o o o o o

{ o o o o o o o { 5 o o o o o o o o o

6 { --------------- { o o o o o o o o o

{ o o o o o o o 7 { o o o o o o o o o

{ 3 o o o o o o o { -------------------

o o o o o o o { 2 o o o o o o o o o

o o o o o o o o o

An excellent sequence, taking into account patterns and connections, is 2x, 5x, 9x, 4x, 8x, 3x,6x, and 7x. This, of course, is not the only correct or appropriate sequence. (For example, 2x, 4x, 8x,

5x, 3x, 6x, 9x, 7x is also interesting.)

Think multiplication

This strategy is for all division facts. The division facts are reversed in thinking to

multiplication form, for example, 36/9 is rethought as what time 9 equals 36. It can be taught by

looking at combining and partitioning: Take 3 groups of 5 counters and combine. Partition 15 into

groups of 5. Repeat. State 15 divided by 5 is the same as 5 multiplied by what is 15. Do the same

for 4x7=28.Families



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This strategy reinforces think multiplication and relate multiplication to division. For each

multiplication/division fact, there are 4 members of the family, for example, 3x5=15, 5x3=15, 15/5=3,

and 15/3=5. Families for 4x7 and 36/9 are:

4x7=28, 7x4=28, 28/7=4. 28/7=4 4x9=36, 9x4=36, 36/9=4, 36/4=9

Multiples of tens facts

Once basic facts are known, it is important to extend them to multiples of ten facts. Multiples

of tens facts are the basic facts applied to tens, hundreds, thousands, etc.. For example, 3+5=8 can be

extended to 300+500, while 3x4=12 can be extended to 30x40=1200. materials that can be used

include MAB and calculators.

Addition and subtraction multiples of tens facts

It appears fairly simple to extend basic addition and subtraction facts to multiples of tens facts.

One approach would be to use MAB. The basic fact 2+3=5 can be related to MAB - 2 longs + 3 longs

= 5 longs and 2 flats + 3 flats = 5 flats. From this, it can be seen that 20+30=50 and 200+300=500.Similarly, subtraction multiples of tens facts can be developed, that is, 6-2=4 can lead to 6 flats - 2

flats which in turn can lead to 600-200=400.

Another approach could be to simply consider that you are adding/subtracting tens and

hundreds like you would add and subtract apples, cars, etc.: 3 apples + 5 apples = 8 apples,

therefore 3 tens + 5 tens = 8 tens and 3 hundreds + 5 hundreds = 8 hundreds, and, thus, 30+50=8

and 300+500=800. Similarly, 3 apples and 5 trucks is neither 8 apples or 8 trucks, thus 30+500 is

not 800 or 80. In the same way, we can relate 13 pears - 5 pears = 8 pears to 130-50=80 and 1300-

500=800.

Multiplication and division multiples of tens facts

These facts are not as simple as addition and subtraction. They are based on the facts:

10x10=100 10x100=1 000 100x100=10 000

One way to learn the pattern with respect to multiples of tens multiplication facts is to use

calculators with respect to examples like those below:

2x4= 5x3= 8x7=

20x4= 50x3= 80x7=

2x40= 5x30= 8x70=

20x40= 50x30= 80x70=

200x40= 500x30= 800x70= and so on ...

The students are asked to complete the examples and then to look at them for any patterns that

emerge that enable the multiples of tens facts to be calculated from the basic facts. The pattern that

should emerge is that the zeros are combined, that is, 4x6=24 means that 40x600 is 24 with 3 zeros,

i.e., 24 000. To check that the students understand the patterns, they are asked to complete the

following without a calculator:

7x6 70x6 70x600 7000x60

In a similar manner, examples such as those below could be completed with a calculator:



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12/6= 21/7= 56/8=

120/6= 210/7= 560/8=

120/60= 210/70= 560/80=

1200/60= 2100/70= 5600/80= and so on ...

Studying these examples can lead to the pattern that the zeros are subtracted, that is, 56000/80

is 700 because 56/8=7 and 3 zeros - 1 zero = 2 zeros. Once again, students patterns can be checked

by asking them to complete the following without a calculator:

45/9 450/9 45 000/90 450 000/9 000

Diagnosis, practice activities, patterns and oddities

There are a lot of interesting activities that reinforce basic facts.

Diagnosis

Addition and subtraction grids can be used to determine the strategies needed by children

making errors. For instance, the count, near doubles and near ten facts can be placed on an addition

grid using different colours. The grid would look like that below. Then, if a childs errors are placed

on the grid, the position of the errors will determine which strategy or strategies are needed. The case

for multiplication is similar, but here the errors just show the unknown x tables and the strategy would

have to be determined from that which is needed for the x table. Thus, the multiplication grid is

broken into two sections - those that show the need for patterns and those that show the need for

connections (see below).



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Practice activities

After strategies, there is drill and drill and drill. It should be motivating.

Basic fact practice games

1. Bingo game. Pick a set of facts to practice, say near doubles. Write down all the facts to be

practiced. Place these on a deck of cards. Take all the answers and place them randomly on 1

cm graph paper. Students take a sheet and for each game they put a line around a square of 9

answers anywhere on sheet. One student calls out the facts by drawing a card from the deck.

The other students circle any answer in their 3x3 square. First person who circles a row or

column of answers and calls bingo wins.

2. Race track game. Pick a set of facts to practice, say think addition. Write down all the facts

to be practiced. Place these on a deck of cards. Take a coloured manilla folder. Place start

on one corner and end of another. Join start to end with a series of stick-on dots. Some dots

can be made special with stars. There can be special dots that allow you to jump forward orfall back. Children throw die, move this distance if correctly answer a fact card taken

randomly from the deck of cards. Special spots require special cards. Different children can

have different card decks.

Basic facts practice sheets

1. Many ways worksheets

(a) Count ons - 3 columns - fact (5+2) on left, cup with 5 on it and two counters above it

in middle, and 7 on right. Leave 2 of the 3 columns empty for each example.

(b) Addition turn arounds - 3 columns - fact (8+5) on left, turn around (5+8) next, and 13

on right. Leave 2 out of 3 columns empty for each example.

(c) Near doubles - 4 columns - fact (6+8) on left, double (6+6) next, 12+2 next, and 14

on right. Leave 3 out of 4 columns empty for each example.

(d) Build to 10s - 4 columns - fact (8+5) on left, 8+2=10 next, 10+3 next, and 13 on

right. Leave 3 out of 4 columns empty for each example.

(e) Think additions - 3 columns - fact (13-8 vertically) on left, think addition fact next(8+?=13 vertically), and 5 on right. Leave 2 out of 3 columns empty for each example.

(f) Multiplication turn arounds - 3 columns - fact (5x3) on left, turn around (3x5) next,

and 15 on right. Leave 2 out of 3 columns empty for each example.

(g) Think multiplications - 3 columns - fact (27/3) on left, think multiplication fact next

(3x?=27), and 9 on right. Leave 2 out of 3 columns empty for each example.

(h) Multiples of tens addition/subtraction - 4 columns - fact (2+7 vertically) on left,

multiple of tens facts in next 3 columns (20+70, 200+700, 2000+7000). Leave 3 out of

4 columns empty for each example.



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2. 3 minute kilometre activity. Choose a set of facts to practice - say 2, 3, 4 and 5 multiplication

facts. Place all these facts randomly on a sheet. Students are given 3 minutes to do as many

examples on the sheet as they can. The same csheet is used over a 2 week period. Children

try to improve their score each day. Children record errors on a small card and practice these

errors.

3. Beat the audio tape. Read facts randomly onto a tape - 10 to a set. Make up more than one

set. Read at different speeds - slow to fast. Children work through the sets - seeing if they can

continue to get them all right as the speed gets faster.

Drills should be motivating and effective and efficient. The following is an example of a non

efficient and non effective drill: The teacher puts the children in a circle and asked facts, in turn,

around the circle. If a child is incorrect they sit down and take no further part in the game. The

winner is the last standing. The game takes 30 minutes.

Patterns and oddities

Hands can be used for the 9x tables and for all multiplication facts above 5x5 (the Russian

Peasant method). These two methods are shown below.

COMPUTATION

Computation covers the procedures for operations when both numbers are 10 or over.

Computation can be accurate, give the correct answer, or it can be an estimate, get close to the correct

answer (as close as the accuracy required). Computation can be achieved through:



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(a) algorithms - procedures that will always give the correct answer if followed; and

(b) heuristics or strategies - rules of thumb that point the direction to an answer.

Whether by algorithms or strategies, there are four types of computation to now be considered

in computation. They are

(a) mental computation - accurately finding an answer without a calculating aid or using a

pen-and-paper procedure;

(b) pen-and-paper algorithms - a pen-and-paper procedure for accurately finding an

answer (this used to be the manstay of the primary syllabus);

(c) computational estimation - a way of getting estimates of large number calculations

(most useful in checking the use of a calculating device); and

(d) calculator algorithms - how to use a calculator to correctly compute the answer (used

mostly in problem situations.

At the present time, the conventional wisdom is to teach the following with respect to addition

and subtraction computation:

(a) accurate computation less than 1000 - have a procedure for accurately calculating for

numbers less than 1000 without aids (either mental computation, a traditional pen-and-

paper algorithm, or a child-developed pen-and-paper version of a non-traditional

mental strategy);

(b) accurate computation greater than 1000 - use calculators when both numbers become

larger than 1000; and

(c) estimates - have techniques to estimate (without using a calculating device) for any size

numbers.

As all algorithms are based on a strategy, we will look at: (a) strategies for addition and

subtraction and multiplication and division computation; (b) mental computation for all operations; (c)

pen-and-paper algorithms for all operations; (d) computational estimation for all operations; (e)

calculators and how they can be used; and (f) practice activities for all the forms of computation.

Strategies

There are two sets of strategies, one set for addition and subtraction, and a second set formultiplication and division.

Addition and subtraction strategies

The strategies for addition are as follows:

(i) Counting on (start with the first number and count on the second)

(ii) Separation R-L (separate into place values and add the ones first)

(iii) Separation L-R (separate into place values and add the tens first)

(iv) Aggregation (start with the first number, separate the second into place values and add

it cumulatively to the first - starting with the ones and starting with the tens)



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(vi) Wholistic(change one number into an easily added number, add it and then compensate

the answer or change one number and do the opposite to the other until one number is

an easily added number, then add)

When two numbers are being subtracted, the second can be removed from the first (this is

called subtractive) or the second can be built up to the first (this is called additive). That is, 74-32

can be calculated by subtracting two ones and then three tens from 74 to give 42, or by adding two

ones and four tens to 32 to give 74. In either case, the answer is 42. Thus, the strategies for

subtraction are the same as addition, but with the added factor that they can be subtractive or additive:

(i) Counting back(subtractive - start with the first number and count back the second;

additive - start with second number and count on to the first)

(ii) Separation R-L (subtractive - separate into place values and subtract the ones first;

additive - separate into place values and add forward smaller to larger, ones first)

(iii) Separation L-R (subtractive - separate into place values and subtract the tens first;additive - separate into place values and add forward smaller to larger, tens first)

(iv) Aggregation R-L (subtractive - start with the first number, separate the second into

place values and subtract it cumulatively to the first; additive - start with the second

number and add cumulatively forward to the first number)

(vi) Wholistic (subtractive - change second number into an easily subtracted number,

subtract it and then compensate the answer, or change one number and do the opposite

to the other until one number is an easily subtracted number; additive change second

number until can easily add to get first number then compensate, or change both

numbers the same until it easy to see what to add to the second number to get the first)

The computation strategies are described in the table below. Examples for both addition and

subtraction are given, as well as examples for both subtractive and additive subtraction.

Strategy Example

Counting 28+35: 28, 29, 30, ... (count on by 1)

52-24: 52, 51, 50, (count back by 1 - subtractive)

: 24, 25, 26, ... 52 (count on by 1 - additive)

Separation right to left

(u-1010)

left to right

(1010)

28+35: 8+5=13, 20+30=50, 63

52-24: 12-4=8, 40-20=20, 28 (subtractive)

: 4+8=12, 20+20=40, 28 (additive)

28+35: 20+30=50, 8+5=13, 63

52-24: 40-20=20, 12-4=8, 28 (subtractive)

: 20+20=40, 4+8=12, 28 (additive)

Aggregation right to left

(u-N10)

28+35: 28+5=33, 33+30=63

52-24: 52-4=48, 48-20=28 (subtractive)

:24+8=32, 32+ 20=52, 28 (additive)



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left to right

(N10)

28+35: 28+30=58, 58+5=63

52-24: 52-20=32, 32-4=28 (subtractive)

: 24+20=44, 44+8=52, 28 (additive)

Wholistic compensation

levelling

28+35: (28+2)+35=30+35=65-2=63

52-24: 52-(24+6)=52-30=22, 22+6=28 (subtractive)

: 24+26=50, 50+2=52, 26+2=28 (additive)

28+35: 30+33=63

52-24: 58-30=28 (subtractive)

: 22+28=50, 28 (additive)

The separation R-L strategy is, of course, very similar to the traditional pen-and-paper procedure

taught for many years in schools. Thus, it is called the traditional strategy. The other strategies are

called non-traditional. Counting is an ineffective and inefficient strategy and should be extended into

one of the others.

The multiplication and division strategies

The multiplication strategies are as follows:

(i) Counting (repeated addition or skip counting of the second number for the first

numbers number of times);

(ii) R-L multiplication (the standard pen-and-paper multiplication algorithm procedure -

separating both numbers into place values and then multiplying components, ones x

ones, tens x ones, hundreds x ones, ..., ones x tens, tens x tens, ..., ones x hundreds, ...and so on);

(iii) L-R multiplication (the standard pen-and-paper algorithm starting with the larger place

values - separating both numbers into place values and then multiplying

components, ..., hundreds x hundreds, tens x hundreds, ones x hundreds, ..., tens x tens,

ones x tens, ..., tens x ones, ones x ones);

(iv) Standard division (the reverse of the standard sharing pen-and-paper algorithm - asking

what number shared amongst the first number (say, of people) will give the second

number - this strategy is unlikely to be used in real life); and

(v) Wholistic (not separating numbers into place values and trying to relate the example to

another example for which the answer is known).

The division strategies are the same:

(i) Counting (repeated subtraction of the second number from the first number until zero is

reached, or repeated addition of the second number until the first is reached);

(ii) R-L multiplication (separating the numbers into their place values, thinking of the

division as a multiplication and using the standard R-L algorithm);

(iii) L-R multiplication (separating the numbers into their place values, thinking of the

division as a multiplication and using the non-standard L-R algorithm);



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(iv) Standard division (separating the numbers into their place values, sharing the place

values L-R amongst the second number of people); and

(v) Wholistic (not separating numbers into place values and trying to relate the example to

another example for which the answer is known).

The computation strategies are described in the table below. Examples for both multiplicationand division are given.

Strategy Description Example

Counting count all, modelling,

transitional count (with

modelling), double count

(without modelling), repeated

addition, doubles/near doubles,halves/near halves, repeated

subtraction.

5x39: 5, 10, 15, 20, .... [39 times] ...., 135, 140

192/6: 192, 186, 180, ...., 6, 0 [32 times]

192/6: 6, 12, ...., 192 [32 times]

Standard R-L

multiplication

right to left separation 5x39: 5x9=45, 5x30=150, 150+45=195

192/6: 6x2=12, leaves 192-12=180, 6x30=180,

30+2=32

Non standard

L-R

multiplication

left to right separation 5x39: 5x30=150, 5x9=45, 150+45=195

192/6: 5x10=50, 5x20=100, 5x30=180, 5x2=12,

18+12=192, 30+2=32

Standard

division

left to right separation 5x39: what number is such that shared among it

gives 39, 15 shared amongst 5 is 3, 45 shared

amongst 5 is 9, number is 150+45=195

192/6: 19 tens shared amongst 6 is 3 tens, 1 ten

left over means 12 ones shared amongst 6 to give

2 ones, answer is 32.

Wholistic direct compensation, inverse

compensation, left to right and

add 0.

5x39: 5x40=200 less 5= 195,

192/6: 600/6=100, 300/6=50, 150/6=25, 42/6=7,

answer=25+7=32

Mental computation

Mental computation is best based on the strategy that is natural for the child. Hence, children

should be taught diagnostically, and not directly taught a method unless they do not acquire their own

efficient and effective method.

The separation strategies have to be taught by materials that show place value (i.e., base and

position). Hence, the material for the separation strategies is MAB and place value charts. This is not

the case for aggregation and wholistic strategies. We need a material that does not separate at least



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one of the numbers into place value. The two best options for teaching the aggregation and wholistic

strategies appear to be the number line and the 99 board.

(1) Number line . Free number lines are drawn. If two numbers are to be added or subtracted, the

first number is written on the line and the student is encouraged to hop along the line (by tens,

by ones, by other numbers) the amount of the second number (either forward or back

depending on the operation). Additive subtraction can also be encouraged by getting the

students to put both numbers on the line and jumping along the line from the smaller to the

bigger. Numbers like 8 and 9 can be added by a 10 hop and a back 1 hop. Similar for

subtracting 8 or 9.

(2) 99 boards. With this material, the students are taught how to find numbers from zero - down

to increase tens, up to decrease tens, across to increase ones, and back to decrease ones. Then,

addition is done by finding the first number and then adding the tens (going down) and adding

the ones (going across) of the second number. Subtraction is done by finding the first number

and then subtracting the tens (going up) and subtracting the ones (going back) of the secondnumber. One can also add numbers like 9 and 8 by adding an extra 10 and the going back.

Similarly for subtracting 9 or 8.

The number lines and the 99 boards tend to focus initially on the aggregation strategy. The

wholistic strategy comes in with practice as the students seek a quicker way for adding on 9 or

subtracting 9. The two materials also tend to focus students on the L-R strategy (not the R-L) because

it is just common sense efficient to consider the tens first.

The most effective strategy for both multiplication and division mental computation is L-R

multiplication. This can be taught with the aid of a game, Two-step Target. The two-step targetgame is based on the game Target. Target is a game in which you enter a starting number and x on a

calculator and, then, you enter guesses and = in an attempt to reach a target number. As long as

clear and x are not pressed again, each of your guesses will be multiplied by the starting number and

you can keep doing better guesses until you reach the target number. We are adapting the game here

to teach mental division by L-R multiplication. In the two-step target game, the aim is to find the

number in two steps, the first finding the ten and the second finding the one. For example: Starting

number is 7 and target is 679. The aim of the game is to find in two steps the number which

multiplied by 6 gives 679 and on the calculator and press x, the two steps being finding the ten and

then the one. 7x90=630 so the ten is 90. 7x7=49 and 630+49=679. Hence, the one is 7 and theguess is 97.

Traditional algorithms

The modern approach to written algorithms is that students should be allowed to develop their

own strategies and their own setting out - as they do for mental computation. However, many schools

still require the traditional algorithms to be taught to all children. Therefore, we include what we

think is the best way to do this. It is based on real world problem model language

symbol. Materials used are MAB, bundling sticks, place value charts, dot paper, graph paper, and

coloured pens.Addition



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REAL WORLD PROB John had $47, Frank gave him $38, how much money does he have?

MODEL AND Show the 47 Tens | Ones

LANGUAGE | | | | | . . . . . . .

Show the 38 Tens | Ones

| | | | | . . . . . . .

| | | | . . . . . . . .

What do you want? [the total] Tens | Ones

Join the ones. How many ones? [15] | | | | |

| | | |

| . . . . . . . . . . . . . . .

Is there enough ones to make a 10? [Yes] Tens | Ones

Make the 10, put it above the other tens, | |

how many ones are left? [5] | | | | |

| | | || . . . . .

Join the tens. Tens | Ones

How many tens are there? [8] | | | | | | | | | . . . . .

What is the answer? [8 tens & 7 ones - 87]

MODEL, Show the 47. Write 47. Tens | Ones 4 7

LANGUAGE | | | | | . . . . . . .

AND SYMBOLS Show the 38. Write 38 underneath Tens | Ones 4 7

the 47 with place values aligned. | | | | | . . . . . . . 3 8

| | | | . . . . . . . .

What do you want? [the total] Tens | Ones 4 7

Draw a line underneath the 38 and put the | | | | | + 3 8

plus sign beside the 38. | | | |

Join the ones. How many ones? [15] | . . . . . . . . . .

| . . . . .

Is there enough ones to make a 10? [Yes] Tens | Ones 1

Make the 10, put it above the other tens, | | 4 7

how many ones are left? [5] | | | | | + 3 8

Write the 5 under the line in the ones | | | | 5

position | . . . . .

Join the tens. Tens | Ones 1

How many tens are there? [8] | | | | | | | | | . . . . . 4 7

Write the 8 in the tens position under the line. + 3 8

What is the answer? [8 tens & 5 ones - 85] 8 5

Subtraction

REAL WORLD PROB John had $72, he gave Frank $35, how much money does he have?

MODEL AND Show the 72. Tens | Ones



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LANGUAGE | | | | | | | | . .

What do you want? [to subtract 35] Tens | Ones

Do you have enough loose ones to | | | | | | | | . .

give 5 ones? [No]

Trade a ten for 10 ones. How many Tens | Ones

ones now? [12]. How many tens? [6] | | | | | | | . .

give 5 ones? [No] | . . . . . . . . . .

Remove the 5 ones. How many ones Tens | Ones

are left? [7] | | | | | | | . . . . . . .

|

| . . . . .

Remove the 3 tens. How many tens Tens | Ones

are left? [3] | | | | . . . . . . .

What is the answer? [3 tens & 7 ones - 37] || | | | . . . . .

MODEL Show the 72. Write 72. Tens | Ones 7 2

LANGUAGE | | | | | | | | . .

AND SYMBOLS What do you want? [to subtract 35] Tens | Ones 7 2

Write 35, minus sign and line under 72. | | | | | | | | . . - 3 5

Do you have enough loose ones to

give 5 ones? [No]

Trade a ten for 10 ones. How many Tens | Ones 6 12

ones now? [12]. How many tens? [6] | | | | | | | . . 7 2give 5 ones? [No] | . . . . . . . . . .- 3 5

Write 6 and 12 above the 7 and the 2.

Remove the 5 ones. How many ones Tens | Ones 6 12

are left? [7] | | | | | | | . . . . . . . 7 2

Write 7 in the ones under the line. | - 3 5

| . . . . . 7

Remove the 3 tens. How many tens Tens | Ones 6 12

are left? [3] | | | | . . . . . . . 7 2

Write the 3 in the tens under the line. | - 3 5

What is the answer? [3 tens & 7 ones - 37] | | | | . . . . . 3 7

Multiplication using MAB

REAL WORLD PROB John bought 3 radios for $46, how much did this cost?

MODEL AND Show the first 46. Tens | Ones

LANGUAGE | | | | | . . . . . .

Show me the second and third 46. Tens | OnesHow many 36s? [3] So 3 lots of 36. | | | | | . . . . . .

What do you want? [the total] | | | | | . . . . . .



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Look at the ones. How many groups are | | | | | . . . . . .

there? [3] How many in each group? [6]

So there are 3 lots of 6 or 3x56 ones.

Therefore, how many ones are there? [18] Tens | Ones

Join the ones together and check this. | | | | |

Is it so? [Yes] | | | | |

| | | | |

| . . . . . . . . . .

| . . . . . . . .

Is there enough ones to make a 10? [Yes] Tens | Ones

Make the 10, put it above the other tens, | |

how many ones are left? [8] | | | | |

Ignoring the carried 10, how many groups | | | | |

of ten? [3] How many tens in each | | | | |

group? [4] So, there are 3 lots of 4 tens | . . . . . . . .

or 3x4 tens plus the extra ten

Therefore, how many tens? [13] Tens | Ones

Join the tens. Form hundreds if | | | | | | | | | | | | | | . . . . . . . .

necessary.

What is the answer? [138] Hundreds | Tens | Ones

[] | | | | | . . . . . . . .

MODEL, Show the first 46. Write down 46. Tens | Ones 4 6

LANGUAGE | | | | | . . . . . .ANDSYMBOLS Show me the second and third 46 Tens | Ones 4 6

How many 36s? [3] So 3 lots of 36. | | | | | . . . . . . x 3

What do you want? [the total] | | | | | . . . . . .

Write down 3 under the 6. Put x beside | | | | | . . . . . .

the 3 and draw a line underneath.

Look at the ones. How many groups are there? [3] How many

in each group? [6] So there are 3 lots of 6 or 3x56 ones.

Therefore, how many ones are there? [18] Tens | Ones

Join the ones together and check this. | | | | |

Is it so? [Yes] | | | | |

| | | | |

| . . . . . . . . . .

| . . . . . . .

Is there enough ones to make a 10? [Yes] Tens | Ones 1

Make the 10, put it above the other tens, | | 4.6

how many ones are left? [8] | | | | | x 3

Ignoring the carried 10, how many groups | | | | | 8

of ten? [3] How many tens in each | | | | |

group? [4] So, there are 3 lots of 4 tens | . . . . . . . .



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or 3x4 tens plus the extra ten

1

Therefore, how many tens? [13] Tens | Ones 4 6

Join the tens. Form hundreds if | | | | | | | | | | | | | | . . . . . . . . x 3

necessary. 1 3 8

What is the answer? [138] Hundreds | Tens | Ones

[] | | | | | . . . . . . . .

Multiplication using arrays

(a) 2x1 algorithms as ones by ones and tens by tens .

REAL WORLD PROB A tiler constructed 3 rows of tiles. Each row has 24 tiles. How many

tiles?

24

MODEL AND Show me the 3 rows of tiles ........................

LANGUAGE by putting a rough rectangle 3 ........................

around 3 rows of 24 dots. ........................

Consider the rows in terms of 20 4

20 tiles and another 4 tiles. .................... ....

Then, 3 rows of 24 is 3 rows 3 .................... ....

of 20 and 3 rows of 4. .................... ....

MODEL, Show me the 3 rows of tiles 24 2 4

LANGUAGE by putting a rough rectangle ........................ x 3AND around 3 rows of 24 dots. 3 ........................

SYMBOLS Write 24 multiplied by 3. ........................

Consider the rows in terms of 20 4 2 0 4

20 tiles and another 4 tiles. .................... .... x 3 x 3

Then, 3 rows of 24 is 3 rows 3 .................... .... 6 0 1 2

of 20 and 3 rows of 4 .................... ....

3x24 can be considered as 3x20 24

and 3x4. That is, 60+12=72 x 3

7 2

(b) 2x2 algorithms as tens and ones by ones and tens and ones by tens .

REAL WORLD PROB A tiler constructed 32 rows of tiles. Each row has 24 tiles. How many

tiles?

24

MODEL AND Show me the 32 rows of tiles |-------------------------- |

LANGUAGE by putting a rectangle 32 | |

around 32 rows of 24 dots. |-------------------------- |

Consider the columns in terms of 2430 tiles and another 2 tiles. |-------------------------- |

Then, 32 rows of 24 is 30 30 | |



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rows of 24 & 2 rows of 24. | |

|-------------------------- |

2 | |

|-------------------------- |

MODEL, Show me the 32 rows of tiles 24 2 4

LANGUAGE by putting a rectangle |-------------------------- | x 3 2

AND around 32 rows of 24 dots. 32 | |

SYMBOLS Write 24 multiplied by 32. |-------------------------- |



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Consider the columns in terms of 24 2 4 2 4

30 tiles and another 2 tiles. |-------------------------- | x 3 0| x 2

Then, 32 rows of 24 is 30 30 | | 7 2 0 4 8

rows of 24 & 2 rows of 24. | | 2 4

32x24 can be considered as |-------------------------- | x 3 2

30x24 and 2x24. That is, 2 | | 4 8720+48=768. |-------------------------- | 7 2 0

7 6 8

Division

REAL WORLDPROB John had $72, he shared this amongst his 3 nephews, how much did

each get?

Tens | Ones

MODEL AND Show the 72. Show the 3 nephews. | | | | | | | | . .LANGUAGE ( ) ( ) ( )

What do we have to do? [Share the money] Tens | Ones

What shall we share first? [tens] | | | | | | | | . .

Are there enough tens for one ten to each ( ) ( ) ( )

nephew, for 2 tens to each nephew? [Yes]

Share out the tens. How many tens to Tens | Ones

each nephew? [2] How many tens left | | . .

over? [1] How many tens used? [6] (| |) (| |) (| |)

Trade the ten for 10 ones. How many Tens | Ones

ones now? [12]. | . . . . . . . . . . . .

(| |) (| |) (| |)

Share out the ones. How many ones to Tens | Ones

each nephew? [4] How many ones |

used? [12] How many ones left? [0] (| | ....) (| | ....) (| | ....)

What did each nephew get? [$24]

MODEL, Show the 72. Show the 3 nephews. Tens | Ones

LANGUAGE AND Write down the 72.| | | | | | | | . . 7 2

SYMBOLS ( ) ( ) ( )

What do we have to do? [Share the money] Tens | Ones

Write down the 3 & the symbols for divide. | | | | | | | | . . 3) 7 2

What shall we share first? [tens]

Are there enough tens for one ten to each ( ) ( ) ( )

nephew, for 2 tens to each nephew? [Yes]

Share out the tens. How many tens to Tens | Ones 2

each nephew? [2] How many tens left | | . . 3) 7 2

over? [1] How many tens used? [6] 6Put 6 under 7, 1 below this and 2 above 7. (| |) (| |) (| |) 1

Trade the ten for 10 ones. How many Tens | Ones 2



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ones now? [12]. | ............ 3) 7 2

Write 2 beside 1 to show 12. (| |) (| |) (| |) 6

1 2

Share out the ones. How many ones to Tens | Ones 2 4

each nephew? [4] How many ones | 3) 7 2

used? [12] How many ones left? [0] (| | ....) (| | ....) (| | ....) 6Write 12 below 12, 0 below this and 1 2

4 above 2 in 72. 1 2

What did each nephew get? [$24] 0

Computational estimation

Computational estimation is based on

(i) strategies (front end, rounding, straddling, nice numbers, and getting closer);

(ii) place value;

(iii) basic facts and multiples of tens facts; and

(iv) operation principles (addition-increase, inverse subtraction, multiplication-increase,

and inverse proportion).

Computational-estimation strategies

Examples of the computational-estimation strategies are given in the table below. Examples

for addition, subtraction, multiplication and division are given. In particular, many examples are

given for the getting closer strategy. Each of the results for each of the strategies is taken and looked

at in terms of getting closer.

Strategy Description and example

Front end Cover all digits but the highest place value digits (the front digits), computing with

these to give the estimate. For example:

4 567 + 8 329 = 4 000 + 8 000 = 12 000

55 181 - 27 988 = 50 000 - 20 000 = 30 000

37 x 456 = 30 x 400 = 12 000

87 567 / 45 = 80 000 / 40 = 2 000

Rounding Round the two numbers to the place value that gives the required accuracy and then

compute. For example:

4 567 + 8 329 = 5 000 + 8 000 = 13 000

55 181 - 27 988 = 60 000 - 30 000 = 30 000 or = 55 000 - 28 000 = 27 000

37 x 456 = 40 x 500 = 20 000

* 87 567 / 45 = 90 000 / 50 = 1 800

Straddling Rounding up and down the numbers so that a larger and a smaller numbers can be

calculated between which the answer is. For example:

4 567 + 8 329 is between 5 000 + 9 000 = 14 000 and 4 000 + 8 000 = 12 000 56 181 - 27 988 is between 60 000 - 20 000 = 40 000 and 50 000 - 30 000 = 20 000



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37 x 456 is between 30 x 400 = 12 000 and 40 x 500 = 20 000

87 567 / 45 is between 80 000 / 50 = 1 600 and 90 000 / 40 = 2 250

Nice

numbers

The numbers are rounded to numbers which easily compute. For example:

4 567 + 8 329 = 4 500 + 8 500 = 13 000

55 181 - 27 988 = 56 000 - 26 000 = 30 000

37 x 456 = 40 x 450 = 1 800

87 567 / 45 = 88 000 / 44 = 2 000

Getting

closer

After doing one of the strategies above, the answer is looked at in terms of whether it

should be increased/decreased, and by how much, for a more accurate estimate. For

example:

4 567 + 8 329 has estimate of 12 000 by front end - this is too low as both

numbers have been reduced and should be increased about another 900

4 567 + 8 329 has an estimate of 13 000 by rounding - this is a little too high (by

about 100) as the 4 567 has gone up more than the 8 329 has gone down

4 567 + 8 329 is between 14 000 and 12 000 by straddling - a better estimate is a

little under half way

malaysia operations framework

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