PR-6103IRE Number Patterns to Algebra 4

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Number patterns 4 – Number patterns to Algebra 

Published by R.I.C. Publications ® 2012 

Copyright © Paul Swan 2012 

ISBN 978-1-922116-08-6 

RIC-6103 

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Published by 

R.I.C. Publications ® Pty Ltd 

PO Box 332, Greenwood 

Western Australia 6924 

Copyright Notice 

No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying or 

recording, or by an information retrieval system without written permission from the publisher.

CONTENTS 

Graphing Multiples.............................................................................. 4–5 

Exchange Rates ................................................................................. 6–7 

Marking Multiples................................................................................ 8–9 

Diagonal Dilemmas............................................................................... 10 

Empty Tables........................................................................................ 11 

Multiplication Patterns.......................................................................... 12 

Multiplication Tables............................................................................. 13 

In the Middle......................................................................................... 14 

The Boomerang.................................................................................... 15 

Robots............................................................................................ 16–17 

Even More Robots................................................................................. 18 

Blank Robot Grids................................................................................. 19 

Circle Patterns...................................................................................... 20 

Digit Sums............................................................................................ 21 

Square the Digits.................................................................................. 22 

Paper Folding....................................................................................... 23 


• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 3

Graphing Multiples – 1 

Ruler 

1 2 3 4 5 6 7 8 9 10 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 

Multiples of two are 

2, 4, 6, 8, 10, 12, … 

The first multiple is 2, 

the second is 4, the third, 

6 and so on. 

Multiples of two may 

be graphed as shown. 

1 Mark in the 7th, 8th and 9th multiples 

of two on the graph and join the points. 

2 Multiples of 3. 

a List the first ten multiples of three. 

b Graph the first ten multiples of three. Use a 

different colour pencil. (Make sure it is sharp.) 

c Compare the graph of the multiples of two with 

the multiples of three graph. What do you 

notice? 


a Describe what you think the graph of the 

multiples of four will look like. 

b List the first ten multiples of four. 

c Graph the multiples of four. Use a different 

colour pencil. 


4 Predict what you think would happen if 

you graphed the multiples of five. 

Graphs are often a 

good way to find patterns 

and relationships. 

4 • Number patterns to Algebra • © R.I.C. Publications ® • www.ricpublications.com.au •

Graphing Multiples – 2 

Ruler 

1 2 3 4 5 6 7 8 9 10 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 


a Complete this table. 


a Complete this table. 

b Graph these points 

onto the grid. 

3 Which line has the steeper slope? 

4 Let me think ... 

a What would the graph of the 1 times table 

look like? 

b Draw the graph to check your prediction. 

c Were you correct? 

b Graph these points 

onto the grid in a 

different colour. 


Yes 

No 


Exchange Rates 

1 Do this with three other numbers. 

a 

This is fun! 

b What pattern do you see? 

2 Further patterns can be found if you use numbers in sequences; for example; 90, 

91, 92, 93, 94 etc. There are nine two-digit numbers in the nineties where the 

first digit is larger than the second. 

a Write them all down, and calculate the difference. 

90 91 92 93 

- 9 - 19 - 29 - 39 

• Choose a two-digit number. The digit for 

the tens must be greater than the 

number of units: 42 

• Reverse the digits: 24 

• Subtract them: 42 – 24 = 18 

• Divide the answer by 9: 18 ÷ 9 = 2 

÷ ÷ ÷ 


What 

do you 

notice 

about 

the answers? 

b 


Exchange Rates (continued) 

3 There are only eight two-digit numbers in the eighties where the first digit 

is greater than the second. 

a Write them all down, and calculate the difference. 

b What do you notice? 

c Calculate the difference between the first and second digit; 

8 - 0 =, 8 - 1 =, 8 - 2 = etc. 

What do you notice? 

4 Investigate what happens when you work with two-digit numbers in the 

seventies, sixties and fifties. 

a Write the numbers and calculate the difference between the first and second 

digits. 

Challenge!Viewing sample 



Marking Multiples – 1 


a Colour all the multiples of five on the grid. 

b Describe the pattern that is formed. 

1 2 3 4 5 6 

7 8 9 10 11 12 

13 14 15 16 17 18 


a Use a different colour and mark all the multiples of seven 

on the grid. 


1 2 3 4 5 6 7 8 

9 10 11 12 13 14 15 16 

17 18 19 20 21 22 23 24 

25 26 27 28 29 30 31 32 

33 34 35 36 37 38 39 40 

41 42 43 44 45 46 47 48 

49 50 51 52 53 54 55 56 

57 58 59 60 61 62 63 64 

65 66 67 68 69 70 71 72 

73 74 75 76 77 78 79 80 

4 Can you do this? 

a Design a grid that will produce a 

diagonal pattern when the 

multiples of three and five are 

coloured. 

You don‛t need to 

use all of the grid. 

3 Equally puzzling. 

a Which multiples do you think will form a diagonal 

pattern on this 1–80 grid? 

b Colour these multiples on the grid. 

19 20 21 22 23 24 

25 26 27 28 29 30 

31 32 33 34 35 36 

37 38 39 40 41 42 

43 44 45 46 47 48 

49 50 51 52 53 54 

55 56 57 58 59 60 


b Test your grid by marking the 

multiples of three and five. 

c How wide is your grid? 

columns wide 


Marking Multiples – 2 

Ruler 

1 2 3 4 5 6 7 8 9 10 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 


a Colour all the multiples of three onto the 1–60 grid. 


2 Find multiples. 

a Which multiples do you think will produce the same pattern 

on the 1–80 grid below? 

b Colour these multiples on the 1–80 grid. 


a List the multiples of six. 

b On page 11, design a grid that will produce the same 

pattern when the multiples of six are coloured. 

c How wide is your grid? 

columns wide 


a Go back to your original 1-60 grid and mark in the 

multiples of eight using a different colour. 

b Which multiples would you need to mark on the 1–80 

grid to produce a similar pattern? 

1 – 60 grid 

1 2 3 4 5 6 

7 8 9 10 11 12 

13 14 15 16 17 18 

19 20 21 22 23 24 

25 26 27 28 29 30 

31 32 33 34 35 36 

37 38 39 40 41 42 

43 44 45 46 47 48 

49 50 51 52 53 54 

55 56 57 58 59 60 

1 – 80 grid 

1 2 3 4 5 6 7 8 

9 10 11 12 13 14 15 16 

17 18 19 20 21 22 23 24 

25 26 27 28 29 30 31 32 

33 34 35 36 37 38 39 40 

41 42 43 44 45 46 47 48 


c List the multiples that you would need to colour on 

your own grid (page 11), to produce a similar pattern. 

49 50 51 52 53 54 55 56 

57 58 59 60 61 62 63 64 

65 66 67 68 69 70 71 72 

73 74 75 76 77 78 79 80 


Diagonal Dilemmas 

2 More diagonals. 

3 Make your own tables. 

1 Complete this multiplication table. 

a Shade the numbers in the third and 

fourth rows. 

b Can you see the lines joining numbers in 

each row along a diagonal? Find the 

differences between the two numbers on 

each of the diagonals. 

, , , , , , , 

c What do you notice? 

a Shade the numbers in the seventh and eighth rows. Mark the diagonals. 

b Find the differences between each pair of numbers on the diagonals. 


a Use an empty table on page 11. Choose two more rows that are next to each other 

and mark in the diagonals. Write the differences between each pair of numbers 

along the diagonals. 


Challenge! 

3 6 9 12 15 18 

4 8 12 

21 

16 20 24 28 

2 1 


Use an empty table on page 11 

and write down what happens when 

you change the direction of the diagonals. 


Empty Tables 

a Empty tables for Marking Multiples – 2 

b Empty tables for Diagonal Dilemmas 

X 1 2 3 4 5 6 7 8 9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

X 1 2 3 4 5 6 7 8 9 

1 

1 

2 

2 

3 

3 

4 

4 

5 

5 

X 1 2 3 4 5 6 7 8 9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

X 1 2 3 4 5 6 7 8 9 


6 

7 

8 

9 

6 

7 

8 

9 


Multiplication Patterns 

1 Look at the following table. 

2 What is happening? 

a Draw a 3 x 2 rectangle around six numbers 

in the grid. 

e.g. 

b Multiply the two numbers in the opposite 

corners. 

x = 

x = 


a Draw more 3 x 2 rectangles in the grids on page 13. Multiply the numbers in the 

corners opposite each other. 

x = x = 

x = x = 

b What happens each time? 

3 Draw different sized rectangles on the grids and see what happens. 

a 4 x 2 rectangles 

x = x = 

x = x = 

b 3 x 3 rectangles 


x = x = 

x = x = 

4 Think for a moment! 

Predict what will 

happen if you have 

a 4 x 3 rectangle. 

Now check and see. 

Correct? Yes 

No 


Multiplication Tables 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 

2 2 4 6 8 10 12 14 16 18 

3 3 6 9 12 15 18 

4 4 8 12 16 20 24 

5 

6 

7 

8 

9 

5 10 15 20 25 30 

6 12 18 24 30 36 

7 14 21 28 35 42 

8 16 24 32 40 48 

9 18 27 36 45 54 

21 24 27 

28 32 36 

35 40 45 

42 48 54 

49 56 63 

56 64 72 

63 72 81 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 

2 2 4 6 8 10 12 14 16 18 

3 3 6 9 12 15 18 

4 4 8 12 16 20 24 

5 

6 

7 

8 

9 

5 10 15 20 25 30 

6 12 18 24 30 36 

7 14 21 28 35 42 

8 16 24 32 40 48 

9 18 27 36 45 54 

21 24 27 

28 32 36 

35 40 45 

42 48 54 

49 56 63 

56 64 72 

63 72 81 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 

2 2 4 6 8 10 12 14 16 18 

2 2 4 6 8 10 12 14 16 18 

3 3 6 9 12 15 18 

4 4 8 12 16 20 24 

5 

6 

7 

8 

9 

5 

6 

7 

8 

9 

5 10 15 20 25 30 

6 12 18 24 30 36 

7 14 21 28 35 42 

8 16 24 32 40 48 

9 18 27 36 45 54 

3 3 6 9 12 15 18 

4 4 8 12 16 20 24 

5 10 15 20 25 30 

6 12 18 24 30 36 

7 14 21 28 35 42 

8 16 24 32 40 48 

9 18 27 36 45 54 

21 24 27 

28 32 36 

35 40 45 

42 48 54 

49 56 63 

56 64 72 

63 72 81 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 

2 2 4 6 8 10 12 14 16 18 

21 24 27 

28 32 36 

35 40 45 

42 48 54 

49 56 63 

56 64 72 

63 72 81 

X 1 2 3 4 5 6 7 8 9 

1 1 2 3 4 5 6 7 8 9 


2 2 4 6 8 10 12 14 16 18 

3 3 6 9 12 15 18 21 24 27 

3 3 6 9 12 15 18 21 24 27 

4 4 8 12 16 20 24 28 32 36 

4 4 8 12 16 20 24 28 32 36 

5 5 10 15 20 25 30 35 40 45 

5 5 10 15 20 25 30 35 40 45 

6 

6 12 18 24 30 36 

42 48 54 

6 

6 12 18 24 30 36 

42 48 54 

7 

7 14 21 28 35 42 

49 56 63 

7 

7 14 21 28 35 42 

49 56 63 

8 

8 16 24 32 40 48 

56 64 72 

8 

8 16 24 32 40 48 

56 64 72 

9 

9 18 27 36 45 54 

63 72 81 

9 

9 18 27 36 45 54 

63 72 81 


In the Middle 

1 Three times three. 

a Step 1: 

b Step 2: 

2 There is a quicker way to find the answer … 

Add the nine numbers in the first 3 x 3 

block of numbers. 

Divide the total by the number in the middle of the box. ÷ 6 = 

c Repeat steps 1 and 2 for the other four 3 x 3 blocks. 

d What do you notice? 

a Make your own 3 x 3 block. 

b Predict what you think the total 

= ÷ 14 = 

= ÷ 25 = 

= ÷ 16 = 

= ÷ 64 = 


of the nine numbers will be. 

c Explain how you made your prediction. 

Have you found 

the shortcut? 

= 

d Test your prediction. 


The Boomerang 

2 Calculate the answers to each of the following multiplications. 

1 Add the numbers in each of the first 

four boomerangs together. 

1 x 1 x 1 = 2 x 2 x 2 = 3 x 3 x 3 = 4 x 4 x 4 = 

3 What do you notice about the total for the first four boomerangs and the first 

four cubic numbers? 

4 Further boomerangs. 

a Predict the sums for the next three boomerangs. 

A number which is multiplied by 

itself is called a ‘square number‛. 

A number which is multiplied by itself 

and then multiplied by itself again is 

called a ‘cubic number‛. 

You can colour 

the boomerangs 

b Explain how you made your prediction. 

different colours 

to help you. 

c Check your predictions by finding totals for the 5th, 6th and 7th 

boomerangs. 

5 th 6 th 7 th 


5 What do you notice about cubic numbers? 


Robots – 1 

1 Toy Robot 

A toy robot may only be given 

two instructions. 

• Turn right; and 

• Forward steps. 

The following instructions were 

given to a robot. 

• Forward one step 

• Turn right 

• Forward two steps 


• Forward four steps 


These instructions are repeated 

until the robot gets back to the 

start. 

a Draw the robot’s path. 

b How many times did you 

repeat the pattern? 

2 A different pattern 

a Follow these instructions to show 

the robot’s path on the grid. 

• Forward four steps 






b Continue the pattern until the robot 

gets back to the start. 

c How many times did 

you repeat the pattern? 

3 What do you notice 

about the patterns? 

A right turn is 

the same as a 

quarter turn 

or a 90° turn. 



Robots – 2 

1 Draw the robot's path. 

a Follow these instructions to produce 

the robot’s path on the grid. 



• Forward three steps 




• Repeat four times. 

b Where do you finish? 

c Describe the pattern. 

2 Make one yourself. 

a Write your own set of instructions 

to produce a similar pattern. 

• Forward 


• Forward 


• Forward 



b Try your set of instructions on the 

grid to check whether your pattern 

is similar. Watch where you start! 

Write a set of 

robot instructions for 

a friend to draw. 

You can use the grids on 

page 19. 


Even More Robots 

1 Use the blank grids on page 19 to draw the paths that 

these robots would travel. 

a Robot 1 

• Forward 2 steps 






d What do you notice about the paths travelled by the robots? 

e What do you notice about the instructions? 

2 What happens if you change the order; for example, 3 steps, 2 steps and 

then 4 steps? Test what happens when the order is changed. 

3 What happens if the robots are instructed to walk 

2 steps, 3 steps and then 6 steps? 

4 A different pattern. 

b Robot 2 







c Robot 3 







a Write your instructions and draw the robot’s walk on the grid on page 19. 

b What happens? 

a Write your instructions and draw the robot’s walk on the grid on page 19. 

b What happens? 

a Now try 2 steps, 1 step and 5 steps. 

You may have noticed that the 

order of the instructions is the 

same in each case. 


b What do you notice about the 2, 3, 6 pattern and the 2, 1, 5 pattern? 


Blank Robot Grids 



Circle Patterns 

1 2 3 4 5 6 7 8 9 10 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 

1 Connect the dots. 

Joining the points according to certain rules will produce some interesting patterns. 

The first rule is n ➜ 2n which means you will need to join each number to its double. 

e.g. 1 ➜ 2, 2 ➜ 4, 3 ➜ 6 and so on. 

The following circle has been 

divided into 36 sections. The numbers 

closest to the circle go from 1–36 in a 

clockwise direction. The numbers on the 

outside go from 1–36 in an 

anticlockwise direction. 

Join all the points to their double in a clockwise direction using a ruler and sharp pencil 

until you reach 18. 


Now starting from 1 and working in an anticlockwise direction, join each point to its 

double until you reach 18. 

The shape you have drawn 

is called a 'cardioid' 

or heart shape. 

Look in your dictionary 

for words beginning 

with 'cardio'. 

What are they 

related to? 


Digit Sums 

1 Calculate the digit sums for these numbers. 

Example: 38 ➜ 3 + 8 ➜ 11 ➜ 1 + 1 = 2 

a 41 ➜ + = 

b 382 ➜ + + ➜ ➜ + = 

c 4886 ➜ + + + ➜ ➜ + = 

2 Many interesting patterns may be found by looking at the 

digit sums found in the tables. 

a Calculate the digit sum for the six-times table. 

1 x 6 = 6 ➜ 6 ➜ 6 

2 x 6 = 12 ➜ 1 + 2 ➜ 3 

3 x 6 = ➜ 

4 x 6 = ➜ 

5 x 6 = ➜ 

6 x 6 = ➜ 

7 x 6 = ➜ 

8 x 6 = ➜ 

9 x 6 = ➜ 

10 x 6 = ➜ 

Digit Sum 

b Write about any patterns you notice. 

Every number has a digit sum. The digit sum of 7 

is seven. The digit sum of 62 is eight (6 + 2). 

The digit sum of 728 is also eight (7 + 2 + 8 = 17, 1 + 7 = 8). 

The digit sum of a number is found by adding all the 

digits in the number until a single digit is left. 

Digit Sum 

11 x 6 = ➜ 

12 x 6 = ➜ 

13 x 6 = ➜ 

14 x 6 = ➜ 

15 x 6 = ➜ 

16 x 6 = ➜ 

17 x 6 = ➜ 

18 x 6 = ➜ 


19 x 6 = ➜ 

20 x 6 = ➜ 

c Predict the digit sums for the: 

21st multiple of six 

31st multiple of six 

25th multiple of six 

35th multiple of six 


Square the Digits 

1 Try beginning with a different starting number and see 

what happens. 

a Choose a two-digit number: 

• Choose a two-digit number. e.g. 35 

• Square its digits. 3 2 = 3 x 3 or 9 

5 2 = 5 x 5 or 25 

• Add the two numbers. 9 + 25 = 34 

• Now square the digits of this 

new number and add them. 3 2 + 4 2 

9 + 16 = 25 

• Continue the process. 35, 34, 25, 29, 

85, 89, 145, 42, 

20, 4, 16, 37, 

58, 89, 145 ... 

You will have noticed that the digits start to repeat 

themselves. 

b Square its digits. x = 

x = 

c Add the two numbers. + = 

d Now square the digits of this new number and add them. 

e Continue the process and list all your numbers below. 

x = 

x = 

+ = 

, , , , , 

, , , , , 

, , , , , 


, , , , , 

f Did your numbers start to repeat themselves? 

When I tried it 

I kept going to 

one. 

Yes 

No 


Paper Folding 

1 Fold the paper. 

a Start with a sheet of A4 paper. This 

sheet represents one rectangle. 

d Fold again. 

b Fold the paper once across the middle. 

You will have created two congruent 

rectangles (same size and shape). 

c Fold the paper again (lengthways). 

2 Collect data. 

a Write your data into a table. 

e And again. 

No of folds No of Rectangles Other expression 

0 1 


1 2 

2 4 2 2 

b What pattern is emerging? 

c Predict what would happen if you folded once more. 


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PR-6103IRE Number Patterns to Algebra 4

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