PR-6103IRE Number Patterns to Algebra 4

Viewing sample

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
Viewing sample
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
Viewing sample
• 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.
Viewing sample
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.
Viewing sample
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
÷ ÷ ÷
Viewing sample
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
Viewing sample
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
Viewing sample
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
Viewing sample
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
Viewing sample
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
Viewing sample
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
Viewing sample
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 =
Viewing sample
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
Viewing sample
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.
Viewing sample
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
Viewing sample
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.
Viewing sample
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
Viewing sample
• www.ricpublications.com.au• © R.I.C. Publications ® • Number patterns to Algebra • 19

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.
Viewing sample
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 = ➜
Viewing sample
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 =
+ =
, , , , ,
, , , , ,
, , , , ,
Viewing sample
, , , , ,
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
Viewing sample
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

Viewing sample