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PR-6103UK 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

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• 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?

3 Multiples of 4.

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.

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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

1 Multiples of 6.

a Complete this table.

2 Multiples of 7.

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.

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Yes

No

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 5


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

÷ ÷ ÷

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What

do you

notice

about

the answers?

b

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


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

b What do you notice?

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 7


Marking Multiples – 1

1 Multiples of 5.

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

2 Multiples of 7.

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

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 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

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b Test your grid by marking the

multiples of three and five.

c How wide is your grid?

columns wide

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


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

1 Multiples of 3.

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

b Describe the pattern that is formed.

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.

3 Multiples of 6.

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

4 Multiples of 8.

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

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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

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 9


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.

c What do you notice?

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.

b What do you notice?

Challenge!

3 6 9 12 15 18

4 8 12

21

16 20 24 28

2 1

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Use an empty table on page 11

and write down what happens when

you change the direction of the diagonals.

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


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

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6

7

8

9

6

7

8

9

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 11


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 =

c What do you notice?

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

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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

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


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

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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

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 13


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 =

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of the nine numbers will be.

c Explain how you made your prediction.

Have you found

the shortcut?

=

d Test your prediction.

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


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

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5 What do you notice about cubic numbers?

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 15


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

• Turn right

• Forward four steps

• Turn right

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

• Turn right

• Forward one step

• Turn right

• Forward two steps

• Turn right

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.

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


Robots – 2

1 Draw the robot's path.

a Follow these instructions to produce

the robot’s path on the grid.

• Forward two steps

• Turn right

• Forward three steps

• Turn right

• Forward one step

• Turn right

• 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

• Turn right

• Forward

• Turn right

• Forward

• Turn right

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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.

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 17


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

• Turn right

• Forward 3 steps

• Turn right

• Forward 4 steps

• Turn right

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

• Forward 3 steps

• Turn right

• Forward 4 steps

• Turn right

• Forward 2 steps

• Turn right

c Robot 3

• Forward 4 steps

• Turn right

• Forward 2 steps

• Turn right

• Forward 3 steps

• Turn right

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.

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

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


Blank Robot Grids

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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.

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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?

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


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 = ➜

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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

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 21


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 =

+ =

, , , , ,

, , , , ,

, , , , ,

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, , , , ,

f Did your numbers start to repeat themselves?

When I tried it

I kept going to

one.

Yes

No

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


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

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1 2

2 4 2 2

b What pattern is emerging?

c Predict what would happen if you folded once more.

• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 23


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