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Arrays, multiplication and division

Jennie Pennant, with the help of Jenni Way and Mike Askew, explores how the array can be used as a thinking

tool to help children develop an in-depth understanding of multiplication and division.

Using Arrays to Explore Numbers

Arrays are useful models for multiplication which can be used in a variety of ways, ranging from highly

structured lessons to games and open investigations.

An array is formed by arranging a set of objects into rows and columns. Each column must contain the same

number of objects as the other columns, and each row must have the same number as the other rows.

The following array, consisting of four columns and three rows, could be used to represent the number sentence

3 x 4 = 12, 4 x 3 =12, 3 + 3 + 3 + 3 = 12 and 4 + 4 + 4 =12.

Building Multiplication Facts and Tables

Arrays can be used for building multiplication facts in a meaningful way. Before drilling and memorising

tables, children must understand how these facts are derived. For example, by progressively adding another

column of three objects, children can build the three-times tables for themselves. This representation not only

assists in understanding the process, but provides a visual image for children to draw upon as they begin to use

and memorise the basic number facts.

Using arrays to explore larger numbers

Arrays can be helpfully used to explore calculations such as 13 x 5 where the array can be split into useful

chunks such as 10 and 3. This means that children can use their known number facts to work out calculations.

Here 13 x 5 = (10 x 5) + (3 x 5).

After a while drawing all the dots can get very tedious! The blank array then becomes a very useful tool for

helping children model their thinking and work out how to do more complex multiplications in an informal

way.

Here's a child using the blank array, as a thinking tool, to help them work out 15 x 14.

The blank array helps children to use other strategies, such as compensating, when carrying out multiplication.

Here, to work out 34 x 9, the child has decided to do 34 x 10 and then take off the 34 x 1.

Beyond the blank array this 'dividing the multiplication into easy parts' strategy can be formalised into the grid

method. The children can see how the 'abstract' grid method overlays the array and formalises the blank array

into a standard form.

Division as the Inverse Operation of Multiplication

Of the four operations, division is the most troublesome for young students. Full understanding of division tends

to lag well behind the other operations. For many children opportunities to explore the concept with concrete

materials are curtailed well before they perceive the relationships between division and the other four

operations. One such relationship, the inverse relationship between division and multiplication, can be

effectively illustrated using arrays.

For example; 3×5=15 or 3 rows of 5 make 15, can be represented by the following array.

Looking at the array differently reveals the inverse, that is

15÷3=5 or 15 put into 3 rows makes 5 columns - or 5 in each row.

Language clearly plays an important role in being able to express the mathematical relationships and the

physical array supports this aspect of understanding by giving the children a concrete image to talk about.

Placing the mathematics into a real-life context through word problems can facilitate both understanding of the

relationship and its expression through words.

For example, "The gardener planted 3 rows of 5 seeds. How many seeds did she plant?" poses quite a different

problem to "The gardener planted 15 seeds in 3 equal rows. How many seeds in each row?" yet both these word

problems can be modelled using the same array.

Further exploration of the array reveals two more ways of expressing inverse relationships: 5×3=15 and 15÷3=5

.

The word problems can be adapted to describe these operations and highlight the similarities and differences

between the four expressions modelled by the one array.

Using the blank array

Suppose you want to figure out 176 ÷ 8. We can set this up as an array with the value of one side missing.

Using known multiplication facts the value of the missing side can be built up.

So the child can see that 22 lots of 8 is the same as 176.

The array is a very powerful tool for suporting the development of children's thinking around both

multiplication and division.

Further reading

To read about the use of the arrays to illustrate number properties go here to read Jenni Way's article entitled

Illustrating Number Properties with Arrays.

Use and Connect Mathematical Representations Effective teaching of mathematics engages students in making connections among mathematical

representations to deepen understanding of mathematical concepts and procedures as tools for problem

solving

Tape Diagrams

Addition & Subtraction

Part-Whole ModelAddition & Subtraction: Variation 1

Given 2 parts, find the whole

4 5?

Jason went on 4 rides at the carnival on Saturday. On sunday,

he went on 5 rides. How many rides did he go on altogether?

Part-Whole ModelAddition & Subtraction: Variation 2

Find the missing part when given 1 part and the whole.

4 ?9

Last weekend, Ryan ran 9 miles. On Saturday, he ran 4 miles. How many miles did he run on Sunday?

The Comparison Model

Larger Quantity

Smaller Quantity

larger quantity - smaller quantity = difference !

smaller quantity + difference = larger quantity

difference

Addition & Subtraction

The Comparison Model

6

?

There were 6 dogs at the pet store. There were 2 more dogs than cats. How many

cats were there in the pet store?

2

Addition & Subtraction

The Comparison Model

6

4

There were 6 dogs at the pet store. If there were 4 cats. How many more dogs than cats were there?

?

Addition & Subtraction

The Comparison Model

?

4

There were 4 cats in the pet store. If there were 2 more dogs than cats. How many dogs were

there altogether?

2

Addition & Subtraction

Multiplication & Division

Part - Whole ModelMultiplication & Division

whole

part

one part x number of parts = whole whole ÷ number of parts = one part whole ÷ one part = number of parts

Part - Whole ModelMultiplication & Division: Variation 1

5

whole

part!!

There were 5 goldfish in each fish tank. If there were 3 fish tanks, how many goldfish were there altogether?

Part - Whole ModelMultiplication & Division: Variation 2

15

part

If there were 15 goldfish in the pet store and there were 3 tanks, how many goldfish were in each tank?

Part - Whole ModelMultiplication & Division

5

15

part

There were 15 goldfish in the pet store. There were 5 in each tank. How many tanks

were there altogether?

The Comparison Model Multiplication & Division

multiple

larger quantity ÷ smaller quantity = multiple

smaller quantity x multiple = larger quantity

larger quantity ÷ multiple = smaller quantity

larger quantity

smaller quantity

The Comparison Model Multiplication & Division: Variation 1

multiple

larger quantity ÷ smaller quantity = multiple

35

7

A farmer has 35 horses. He has 7 cows. How many times more horses than cows does he have?

The Comparison Model Multiplication & Division: Variation 2

5

smaller quantity x multiple = larger quantity

?

7A farmer has 7 cows. He has 5

times as many horses. How many horses does he have?

The Comparison Model Multiplication & Division

7 7 7 7 7

35

?

larger quantity ÷ multiple = smaller quantity

A farmer has 35 horses. He has 5 times as many horses as cows. How many

cows does he have?

Fractions

Part-Whole ModelShowing Fractions

Part

whole

The part is 1/3 of the whole.

Part-Whole Model

1/4 unit1 unit

3/4 unit

3/4 means 1/4 + 1/4 + 1/4 !

or !

3/4 means 3 x 1/4

Showing Fractions of 1

Part-Whole ModelFinding Part of a Whole Quantity

12

?

1 unit = 12/4 = 3 3 units = 3 x 3 = 9

The blue portion of the tape diagram is 9.

Part-Whole ModelFinding the Whole from a Partial Quantity

?

9

1 unit = 9/3 = 3 4 units = 4 x 3 = 12

There are 12 altogether.

The Comparison ModelFractions

A

B

A is 4 times as much as B. So, B is 1/4 as much as A.

The Comparison ModelFractions

Dogs

Cats

?

40

There are 3/4 as many dogs as cats. If there are 40 cats, how many dogs are there?

The Comparison ModelFractions

Dogs

Cats?

40

There are 3/4 as many dogs as cats. If there are 40 cats,

how many more cats than dogs are there?

The Comparison ModelFractions

Dogs

Cats

There are 3/4 as many dogs as cats. If there are 70 animals,

how many cats are there?

70

Using Visual Representations Notetaking Guide

Visual Models for Thinking

Dot cards

Number Lines K-1 Counting On Gr 2 and beyond

5 Frames/ 10 Frames

Number Bonds (number paths)

Place value

L. Pruske 2016

Visual Models for Thinking Arrow Notation

Tape Diagram

Using Visual Representations Notetaking Guide

Double Number Lines:

Additional Notes:

Using multiple visual models together helps students develop fluency and move between models, securing their understanding of math concepts.