PR-6104IRE Number Patterns to Algebra 5

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Number patterns 5 – Number patterns to Algebra 

Published by R.I.C. Publications ® 2012 

Copyright © Paul Swan 2012 

ISBN 978-1-922116-09-3 

RIC-6104 

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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 

Adding Consecutive Numbers..................................................................4 

Growing Squares....................................................................................5 

Visual Patterns.......................................................................................6 

Multiplication Squares.............................................................................7 

Multiplication Squares Templates.............................................................8 

Square Number Investigations............................................................9–10 

One Up, One Down................................................................................ 11 

Triangular Numbers.............................................................................. 12 

Counting Rectangles............................................................................. 13 

Intersections......................................................................................... 14 

Staircases............................................................................................ 15 

Square Numbers and Triangular Numbers............................................... 16 

Pentagonal Numbers............................................................................. 17 

ISBN............................................................................................... 18–19 

How Many Squares on a Chessboard?.................................................... 20 

Consecutive Numbers............................................................................ 21 

Even Staircase...................................................................................... 22 

Odd Staircase....................................................................................... 23 


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

Adding Consecutive Numbers 

1 What happens when you add consecutive odd numbers, 

starting at 1? 

a Try some. 

1 + 3 = 

1 + 3 + 5 = 

1 + 3 + 5 + 7 = 

1 + 3 + 5 + 7 + 9 = 

1 + 3 + 5 + 7 + 9 + 11 = 

When the information is placed in a table, 

a pattern emerges. 

b Complete the table. 

2 One step further. 

Number of odd Consecutive odd numbers Total 

numbers 

1 1 1 

2 1 + 3 4 

3 1 + 3 + 5 

4 1 + 3 + 5 + 7 

5 1 + 3 + 5 + 7 + 9 

6 

7 

8 

9 

10 

Consecutive numbers 

means following one 

after another. 

Do you recognise 

these numbers? 


a Predict the total for adding the first 20 consecutive odd numbers. 

b Explain how you made your prediction. 

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

Growing Squares 

1 A pattern with squares. 

a Continue this pattern. Note: Only count small squares. 

1 2 3 4 5 6 

b Enter your data into the table below. 

Length of side 1 2 3 4 5 6 7 10 

Number of squares 1 4 9 

c Explain how you worked out the number of squares for a side length of 10. 

2 A large square! 

a How many squares wide (and long) is a field of 225 squares? 

b Explain how you worked out your answer. 


c If you had 150 small squares, what would be the side length of the largest square 

you could make? Work with a friend to find the answer. 


0 . + 

Visual Patterns 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Consider the following sequence of patterns. 

8 pegs on the outside. 

12 pegs 

16 pegs 

1 hole in the middle. 

a Draw the next pattern in the sequence. 

b With a partner, complete the table and predict the next two numbers. 

c Check your table by drawing the next two patterns in the sequence. 

d Write about any patterns you notice. 

4 holes 

9 holes 

Pegs on the outside 8 12 16 20 24 28 

Holes in the middle 1 4 9 


2 Let me think. 

a If there were forty pegs on the outside, how many holes would you have? 

b Explain how you found your answer. 


Multiplication Squares 

1 Squares. 

Note: A square number is the answer you get when you multiply a number by 

itself; for example, 2 x 2 = 4. 

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 

4 4 8 12 16 20 24 28 32 36 

5 5 10 15 20 25 30 35 40 45 

6 6 12 18 24 30 36 42 48 54 

7 7 14 21 28 35 42 49 56 63 

8 8 16 24 32 40 48 56 64 72 

9 9 18 27 36 45 54 63 72 81 

c What do you notice? 

2 See a pattern? 

a Starting at the top left corner of 

the chart, draw successively 

larger squares. 

b Find the total of all the numbers 

in each of the first four squares. 

a Predict what you think will happen for the next three square shapes. 

b Check your predictions. 


Total Total Total 

3 Find a quick way. 

Try to find a quick way of discovering the total for the 8th, 9th and 10th squares. 

Check your predictions. Explain how you worked out the answer. Use the Multiplication 

Squares Templates on page 8. 


Multiplication Squares Templates 

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 10 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 



0 . + 

Square Number Investigations – 1 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 A square number is made when a number is multiplied by itself. 

For example, 3 x 3 = 9; so 9 is a square number. 

See how square numbers grow. 

1 

1 

1 

( 1 x 1 = 1 ) 

1 2 = 

3 2 = 

5 2 = 

7 2 = 

9 2 = 

11 2 = 

13 2 = 

15 2 = 

17 2 = 

19 2 = 

21 2 = 

23 2 = 

1 

9 

5 

2 

4 

( 2 x 2 = 4 ) 

3 4 

9 

16 

( 3 x 3 = 9 ) 

( 4 x 4 = 16 ) 

a Investigate what happens when you square odd numbers and even numbers. 

1 

9 

25 

2 

Write the last 

digit of each 

answer in 

the boxes. 

4 2 is read as four 

squared and means 

4 x 4 (=16). 

There is an interesting 

pattern to be found 

in the last digits. 

25 2 = 

2 Complete the following statements. 

a If you square an odd number, the result will be odd 

b If you square an even number, the result will be odd 

c What do you notice about the final digits of odd square numbers and 

even square numbers? 

odd: 

even: 

3 

2 2 = 

4 2 = 

6 2 = 

8 2 = 

1 0 2 = 

12 2 = 

14 2 = 

16 2 = 

1 8 2 = 

2 0 2 = 

22 2 = 

24 2 = 

26 2 = 

even 

even 


4 


0 . + 

Square Number Investigations – 2 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

Many interesting patterns may be found within square numbers. For example: 

3 2 is read as 

‘three squared’ and 

means 3 x 3. 

3 2 - 2 2 = 5 

(3 x 3) - ( 2 x 2 ) 

9 - 4 = 5 

1 Try subtracting the totals of more square numbers to see if the 

pattern continues. 

a 4 2 – 3 2 g 16 – 9 = 7 g 3 + 4 = 7 

b 5 2 – 4 2 g 25 – 16 = 9 g 4 + 5 = 9 

c 6 2 – 5 2 g g 

d 7 2 – 6 2 g g 

e 8 2 – 7 2 g g 

f 9 2 – 8 2 g g 

g 10 2 – 9 2 g g 

2 Predict the results of: 

a 20 2 – 19 2 = 

b 25 2 – 24 2 = 

c 100 2 – 99 2 = 

d Explain how you worked out the answers. 

3 Look at the pattern that results when you find the difference 

between alternate square numbers. 

3 2 – 2 2 = 5 and 2 + 3 = 5. 

I wonder if 4 2 – 3 2 gives 

the same result as 3 + 4? 

a 4 2 – 2 2 g g . 


b 5 2 – 3 2 g g 

c 6 2 – 4 2 g g 

d 7 2 – 5 2 g g 

e 8 2 – 6 2 g g 

f Predict what the difference between 9 2 and 7 2 will be. 

g Explain how you worked out the difference. 


0 . + 

One Up, One Down 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

8 2 is the same as 8 x 8 or 64. 

If you add one and subtract one, the multiplication created is 9 x 7. 

1 How do the answers relate to each other? 

a Multiply nine by seven. (Remember 8 x 8 = 64) 

b What do you notice about the two answers? 

2 Try some others. 

a 12 x 12 = 14 x 14 = 20 2 = 

13 x 11 = 15 x 13 = 21 x 19 = 

b What seems to be happening? 

c Write your own set of questions that follow the same pattern. 

x = x = x = 

x = x = x = 

d What happens every time? 

3 Do you know this too? 

a If 26 2 = 676, what does 27 x 25 produce? 

b Explain how you worked out your answer. 

4 Investigate further. 

a What happens if you add two and 

subtract two? e.g. 8 2 is 8 x 8 = 64 

Investigate 

10 x 6 = 60 

b What do you think will happen if you 

add three and subtract three? 


Test your prediction. 


Triangular Numbers 

1 Triangular pattern 

The set of numbers shown below form a triangular pattern. 

Continue the drawings to show the fifth and sixth triangular numbers. 

a Write down the number of dots in each triangle. 

1 3 6 

b If the third triangular number is six, what is the fifth triangular number? 

2 Continue the pattern to show the first ten triangular numbers. 

1, 3, 6, , , , , , , 

3 Fill in the series. 

What do you notice about the triangular numbers? 

Complete the boxes to show how the numbers are increasing. 

+2 +3 

1 3 6 

What happens when you add two consecutive triangular numbers? 

a 1 + 3 = + = 

3 + 6 = + = 

6 + = + = 

b What patterns do you notice? 

4 The triangular numbers may be redrawn to look like this: 

Hey! That 

pattern looks 

like a square. 

a Continue the drawing. When two consecutive triangular numbers are 

joined it looks like this: 

b List the next six square numbers. 

1, 4, 9, 16, , , , , , 

What does 

‘consecutive‛ 

mean? 

It means one after 

the other, e.g. the 3rd 

and 4th triangular number 

or the 7th and 8th. 


The pattern 1, 4, 9, 16, … is known as the 

square numbers because the dots may be 

drawn in the shape of a square. 


0 . + 

Counting Rectangles 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

How many rectangles can you find in the figure? 

It may help to start 

with some simpler 

examples and build up. 

Did you know a square 

is a rectangle? 

It meets all the criteria 

of a rectangle. 

1 Count the rectangles. 

a Continue the table until you reach the figure above. 

Figures Squares Rectangles Total 

3 rectangles! 

1 0 1 

2 1 3 

3 3 6 


b What patterns do you notice? 

I think I recognise 

these numbers … 

c How many rectangles, in total, were there in the original figure? 


Intersections 

An intersection is the place where two lines meet. 

1 What is the maximum number of intersections that you get with: 

a four lines? 

b five lines? 

c six lines? 

d seven lines? 

2 Place your data in a table. 

a 

The maximum number of 

intersections produced from 

two straight lines is one. 

b Do you recognise this pattern? These are the 

3 Predict how many intersections there will be for: 

The maximum number of 

intersections produced 

from three straight 

lines is three. 

Number of lines 2 3 4 5 6 7 

Maximum number of intersections 1 3 

Before you write your answer, 

try drawing the lines with as many 

intersections as you can on a 

separate sheet of paper. 


numbers. 

a six lines? 

b seven lines? 

c ten lines? 


Staircases 

A staircase is to be built using blocks. 

1 2 3 4 5 

The first step requires one block. 

Two more blocks are required to build step two and so on. 

1 Look at the staircases above. 

a Draw the fifth step on the dotted line above. 

b How many blocks are used? 

c Continue the table. 

Number of Steps 1 2 3 4 5 6 7 10 

Total Number of Blocks 1 3 6 

2 A podium. 

A podium is also composed of cubes. 

1 2 3 4 5 

step up step up 

a How many will you need? Complete the table. 

Number of Steps 1 2 3 4 5 6 7 8 9 10 

Total Number of Blocks 1 4 9 

Predict the 

tenth step. 

I recognise 

this pattern! 


b What do you notice? 


0 . + 

Square Numbers and Triangular Numbers 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Triangular numbers and square numbers are related. If you add two 

consecutive triangular numbers you end up with a square number. 

Complete the following chart, showing how the triangular numbers are related to 

the square numbers. 

▲ Number ▲ Number ■ Square number Total 

3 6 9 3 + 6 = 9 

6 10 16 6 + 10 = 16 

10 15 25 

15 21 15 + 21 = 

21 49 

2 Interesting patterns. 

Sometimes real-life situations lead to interesting number patterns. 

Consider the following question. How many handshakes would eight people 

make if they all shook hands with one another? Use a separate sheet 

for your calculations. 

64 

81 

100 


I find a diagram helps me to think. 

This pattern 

looks familiar. 

Work with a partner 

to solve this puzzle. 

handshakes. (You can test your answer at lunchtime.) 


0 . + 

Pentagonal Numbers 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 A pentagon is a five-sided shape. 

When dots are arranged into a pentagonal shape a pattern is formed. 

a Draw the next two pentagonal numbers above. 

b List the first five pentagonal numbers by adding 

the dots. 

, , , , 

c Name the next two pentagonal numbers 

, 

Draw them on a separate sheet. Can you see a pattern? 

+4 +7 +10 +13 

1 5 12 22 35… 

2 Complete the following chart. 

A pentagonal number is really made up of a square number and a triangular number. 

Draw the third and fourth pentagonal numbers showing the square number part and the 

triangular number part. (Use • for the square part and for the triangular part.) 

3 Calculate the triangular and square numbers that make up the following 

pentagonal numbers. 

a 51 = + 

n 

b 70 = + 

n 

c 92 = + 

n 

 

 

 

d 117 = + 

n 

e 145 = + 

n 

 

 

There is a building in the 

United States called 

the Pentagon. 

It has five sides. 

second third fourth 

I thought it 

was a building 

in the USA. 


Triangular numbers, 

square numbers and 

pentagonal numbers 

are part of a family 

of numbers called 

'figurative numbers‛ 

because the patterns 

look like geometric 

shapes. 


0 . + 

ISBN – 1 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 International Standard 

Book Number. 

2 Make sure the ISBN here 

is correct. 

The ISBN is a unique code found on most 

books. This makes it easy to identify a 

book. Here is an example of an ISBN: 

9 789086 640737 

The ISBN consists of 13 digits, divided into 

five groups: 

• The first three digits form a separate 

code indicating that it is a book. 

• The next group of digits indicates the 

language-sharing country group. Books 

in countries where English is spoken for 

example get a 0 or 1, where French is 

spoken a 2, and German a 3. 

• The third group identifies the publisher. 

The number of digits can be different. 

• The fourth group gives the title of the 

book. The number of digits can also vary in 

this group. 

• The last digit is the check digit. 

This last figure is put there to identify any 

errors in the ISBN. The first twelve numbers 

are used to calculate the check digit. 

• Each number in an odd position, in the 

ISBN is multiplied by 1. 

• Each number in an even position is 

multiplied by 3. 

• These two products are added. 

• The final digit of the sum is subtracted 

from 10 to give the check digit. If the 

remainder is 0, then the check digit is 0. 

Example: 

What is the check digit for this ISBN ? 

978-90-8664-073-X 

Add the numbers in the odd positions and 

multiply by 1. 

9 + 8 + 0 + 6 + 4 + 7 = 34 34 x 1 = 34 

Add the numbers in the even positions and 

multiply by 3. 

7 + 9 + 8 + 6 + 0 + 3 = 33 33 x 3 = 99 

Add the products. 34 + 99 = 133 

Subtract the final digit from 10. 

10 – 3 = 7 

So the check digit is 7. 

The full ISBN is 978-90-8664-073-7. 

9 789086 640560 

Step 1 Add the odd-positioned numbers 

(first 12 digits only). 

Step 2 Multiply this number by 1. 

Step 3 Add the even-positioned numbers 

(first 12 digits only). 

Step 4 Multiply this number by 3. 

Step 5 Add the resultant numbers of 

steps 2 and 4. 

Step 6 Subtract the last digit of this 

number from 10. If it is 0, leave it 

as 0. 

Step 7 This is your check digit! 

+ + + + + = 

Is the ISBN good? 

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

x 1 = 

+ + + + + = 

Yes 

x 3 = 

+ = 

– = 

Check digit is.... 


No

0 . + 

ISBN –2 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Follow the steps on the previous page to see if the following 

ISBNs are correct or not. 

a 

9 789059 323438 b 9 789059 323568 

+ + + + + = 

+ + + + + = 


Yes 

c 978 90 76233 17 8 

+ + + + + = 

+ + + + + = 


Yes 

No 

No 

2 Use a separate sheet to 

check the following ISBNs: 

a 978 1 74126 797 8 

b 978 1 86400 254 6 

c 978 1 74126 253 7 

d 978 1 86311 659 2 

e 978 1 86311 514 8 

f 978 1 92175 086 5 

x 3 = 

+ = 

– = 


x 3 = 

+ = 

– = 


+ + + + + = 

+ + + + + = 


Yes 

d 978 90 8664 135 7 

+ + + + + = 

+ + + + + = 


Yes 

No 

No 

x 3 = 

+ = 

– = 


x 3 = 

+ = 

– = 



3 Choose four books. Write 

the title and ISBN and 

check the ISBN, following 

the above steps. 


0 . + 

How Many Squares on a Chessboard? 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 How many squares on a chessboard? 

The answer is not 64. 

Consider; for example, that in a 2 x 2 board there would be five squares. 

+ 

4 squares + the large square 

What about a 3 x 3 board? 

If you can spot a pattern it may save you time. 


There are 

squares on a chessboard. 

I‛m going to write my 

results in a table so 

I can spot any 

patterns. 


0 . + 

Consecutive Numbers 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Consecutive numbers. 

a Look at these three consecutive numbers: 

12 , 

13 , 

14 

b Multiply the smallest number by the largest number. 

12 x 14 = 

c Square the middle number. 

13 2 = 13 x 13 or 

d Compare your results. What do you notice? 

2 Choose some more consecutive numbers. 

a , , , , , , 

x = x = x = 

= x or = x or = x or 

b What happened? 

3 Odd and even consecutive numbers. 

a Try using three consecutive 

ODD numbers. 

11 , 

13 , 

15 

x = 

b Try using three consecutive 

EVEN numbers. 

12 , 

14 , 

16 

x = 

For brainiacs only! 

Imagine the three 

consecutive numbers were: 

a, a + 1, a + 2 


= x or 

= x or 

c Write about what you notice. 


0 . + 

Even Staircase 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Even staircase. 

a Continue this staircase. 

1st 0 

2nd 2 4 

3rd 6 8 10 

4th 

5th 

6th 

7th 

b Add the numbers in each row. 

c Look for any patterns. 

They all seem to relate to 6 somehow. 

2 Making predictions. 

a Predict what the total for the 7th row will be. 

b Try it and check. 

0 6 24 

6 18 36 

I wonder what happens 

if we look at the 

difference between the 

totals? 

c Explain how you would work out the total for the tenth row. 

60 

Total 

0 

2 + 4 = 6 



0 . + 

Odd Staircase 

1 2 3 4 5 6 7 8 

AC C % √ 

9 x 

4 5 6 – 

1 2 3 = 

1 Odd staircase. 

a Continue this staircase. 

1st 1 

2nd 3 5 

3rd 7 9 11 

4th 

5th 

6th 

7th 

b Add the numbers in each row. 

c Does there appear to be a pattern? 

d Try cubing these numbers. 

1 3 2 3 3 3 

e What do you notice? 

2 Making predictions. 

Total 

1 

3 + 5 = 8 

7 + 9 + 11 = 27 

a Predict what the total would be if you added and extra row to the staircase. 

Yes 

b Explain how you would work out the total for the tenth row. 

No 

Remember, when you cube a 

number, it means to multiply 

it by itself three times; 

e.g. 4 3 = 4 x 4 x 4 or 64 


c Try to write a rule to describe how to predict the total for any row. 


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PR-6104IRE Number Patterns to Algebra 5

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