PR-6102IRE Investigating Number Patterns 3

Viewing sample

Number patterns 3 – Investigating Number patterns
Published by R.I.C. Publications ® 2012
Copyright © Paul Swan 2012
ISBN 978-1-922116-07-9
RIC-6102
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
Addition Patterns..............................................................................................4–7
Patterns Within the Addition Table...............................................................8–10
Place Value Patterns.................................................................................... 11–13
Noticing Nines............................................................................................... 14–15
The 99 Times Table.............................................................................................16
Eleven Times.......................................................................................................17
Puzzling Patterns................................................................................................18
Function Machines....................................................................................... 19–21
Patterns in Tables Charts............................................................................ 22–23
Viewing sample
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 3

Addition Patterns
1 Add together.
+
0 1 2 3 4
5 6 7
8 9
0 0
7
1
4
2 2
3
4
5
6
7
8
9
2 Refer to the table above.
3 Predict the results.
= = 1ste row rij
= = 2nd 2de rij row
= = 3de 3rd rij row
= = 4th row
4de rij
a Add all the numbers in the 1st, 2nd, 3rd and 4th rows together.
b Write down any patterns that you notice.
a Predict the results of adding the
numbers in the:
5th row
6th row
7th row
8th row
b In the box, add the numbers in each
row to check your predictions.
Viewing sample
Tick the answers if you
predicted correctly.
I could use the answer from adding the numbers
in the first row to help me work this out.
4 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Addition Patterns (continued)
4 Predict the results for adding:
a 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 =
b 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 =
c 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 =
d 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 =
e 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100 + 110 =
5 Create even more new rows!
f + + + + + + + + + =
g + + + + + + + + + =
h + + + + + + + + + =
Viewing sample
i + + + + + + + + + =
j + + + + + + + + + =
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 5

You'll need these
diagonals for the
questions on the
following page.
0 . +
More Addition Patterns
1 2 3 4 5 6 7 8
AC C % ÷
9 x
4 5 6 –
1 2 3 =
1 Complete this table of addition facts.
2 Add diagonally.
7th diagonal
Viewing sample
8th diagonal
9th diagonal
10th diagonal
a Add the numbers in the first 6 diagonals, beginning at the top right corner, and put the answers in the diagram at right top
of the page. The first answer (9) and the last (54), have already been done for you.
b Describe any patterns you notice.
6 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

More Addition Patterns (continued)
3 Complete the following diagonals.
a Estimate the answers of the:
• 7th diagonal
b Add the numbers along each diagonal.
Write the answers in the boxes.
• 8th diagonal
• 9th diagonal
• 10th diagonal
4 Go back further!
a What do you think will happen if you continue to add the numbers along each of the
remaining diagonals? (The next diagonal of numbers starts at the .)
b Calculate the diagonal at the star to see if you're right
Total : Were you correct?
5 Now complete the diagonals from right to left, down
from the top left corner.
a Examine the diagonals; the first is listed below:
Yes
Tick the answer if you
predicted correctly.
Viewing sample
No
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 7

Patterns Within the Addition Table
1 Look at the following table of addition facts.
d Try the same thing with other blocks of four
numbers.
e What do you notice?
2 Use the space below.
a Draw a rectangle around
a block of four numbers;
e.g.
b Add all the numbers in
the box;
e.g. 5 + 6 + 6 + 7 =
c Divide the total by 4;
e.g. ÷ 4 =
a Look for a relationship that will allow you to work out the total of the four
Viewing sample
numbers in the box.
b Explain how your method will allow you to quickly work out the total of the four
numbers mentally.
8 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Patterns Within the Addition Table (cont.)
3 Even more patterns.
a
b What other rules or patterns have you discovered? Write them down.
ChallengeViewing sample
Work out the total of the 100
numbers in the addition table.
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 9

More Patterns in the Addition Table
1 Look at the following addition table.
2 Refer to the table above.
a Draw some crosses of your own on the addition table.
3 Make it bigger.
a Step 1:
Look at the cross shown on the
table and write down the numbers in
the top left corner and the bottom
right corner and multiply them.
x =
b Step 2:
Multiply the numbers in the top
right and bottom left corners of
the cross.
x =
c Step 3:
Subtract the smaller answer from
the larger.
– =
b Multiply the numbers in opposite corners of the cross and then subtract the smaller
from the larger number.
c What did you notice?
x =
x =
- =
a Try drawing a larger cross on the addition table.
For example, a 4 x 4 cross would look like this:
Repeat the steps above for two different 4 x 4 crosses.
x =
x =
- =
Viewing sample
x =
x =
x =
- =
x =
- =
b What did you notice about the answers?
10 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Place Value Patterns – 1
1 Try the following multiplications.
a 26 x 1 =
A calculator
will help.
b 26 x 10 =
c 26 x 100 =
d 26 x 1000 =
e 26 x 0.1 =
f 26 x 0.01 =
g 26 x 0.001 =
h 26 x 0.0001 =
2 See a pattern? 3 How is that?
a Describe the pattern you can see.
b In what ways are the answers the
same?
c In what ways are the answers
different?
4 Describe a method for doing calculations, like those above, in your head.
5 Try these without a calculator first.
a Describe what happens when you
multiply by a number greater than
one.
a 39 x 10 =
b Describe what happens when you
multiply by a number less than one.
Viewing sample
b 39 x 100 =
Then check your answers
with a calculator.
c 39 x 0.01 =
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 11

0 . +
Place Value Patterns – 2
1 2 3 4 5 6 7 8
AC C % ∏
9 x
4 5 6 –
1 2 3 =
1 Try the following divisions.
a 43 ÷ 1 =
b 43 ÷ 10 =
c 43 ÷ 100 =
d 43 ÷ 1 000 =
e 43 ÷ 10 000 =
2 Patterns.
a Describe any patterns you notice.
3 Try these without a calculator first.
f 43 ÷ 0.1 =
g 43 ÷ 0.01 =
h 43 ÷ 0.001 =
i 43 ÷ 0.0001 =
j 43 ÷ 0.00001 =
b Describe a method for doing calculations, like those above, in your head.
a 79 ÷ 10 =
b 79 ÷ 1000 =
c 79 ÷ 10 000 =
d 79 ÷ 0.1 =
e 79 ÷ 0.01 =
f 79 ÷ 0.0001 =
A calculator
will help.
Viewing sample
Check your answers with
a calculator.
12 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Place Value Patterns – 3
1 Try the following multiplications.
a 173 x 27 =
b 173 x 2.7 =
c 173 x 0.27 =
d 173 x 0.027 =
2 Determine.
a Write about any patterns you notice.
b In what ways are the answers the same?
c In what ways are the answers different?
e 17.3 x 27 =
f 1.73 x 27 =
g 17.3 x 2.7 =
h 1.73 x 2.7 =
3 237 x 16 = 3792. Use this information to help work out the answers to:
a 23.7 x 16 =
b 2.37 x 16 =
c 23.7 x 1.6 =
d 237 x 0.16 =
A calculator
will help.
d How does the position of the decimal point change the answer?
e 237 x 160 =
Viewing sample
f 2370 x 16 =
g 2370 x 1.6 =
h 0.237 x 160 =
4 Explain how patterns can help to work out answers to questions like those above.
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 13

Noticing Nines – 1
1 You'll be familiar with this?
a Complete the nine times table.
1 x 9 =
2 x 9 =
3 x 9 =
4 x 9 =
5 x 9 =
6 x 9 =
7 x 9 =
8 x 9 =
9 x 9 =
10 x 9 =
What do you notice
about the tens?
b Record your observations.
What happens when
you add the digits?
e.g. 2 x 9 = 18
1 + 8 = 9
What do you notice
about the ones?
What do you notice
about the answers?
9 = 9
1 + 8 = 9
Viewing sample
2 What happens if, instead of adding the digits in the answer, you subtract the
smaller number from the larger? e.g. 4 x 9 = 36, 6 – 3 = 3.
Describe the pattern that is formed.
=
=
=
=
=
=
=
=
14 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

0 . +
Noticing Nines – 2
1 2 3 4 5 6 7 8
AC C % ∏
9 x
4 5 6 –
1 2 3 =
1 List the answers to the nine times table.
1 x 9 =
9
(
odd
)
2 x 9 = ( )
3 x 9 = ( )
4 x 9 = ( )
5 x 9 = ( )
1 8
2 7
3 6
4 5
5 4
6 3
7 2
8 1
9 0
2 When the digits in the answer are split, more
patterns may be found.
3 What happens when the numbers along each of the diagonals are
subtracted? (Always take the smaller number away from the bigger.)
4 Complete the following multiplications.
Write down
whether the answer
is odd or even.
6 x 9 = ( )
7 x 9 = ( )
8 x 9 = ( )
9 x 9 = ( )
10 x 9 = ( )
a What happens when you add the digits along the diagonal?
b Try adding the digits along the other diagonal; i.e.
Write about what you notice.
a 736 x 9 =
6624 6 + 6 + 2 + 4
=
18
=
9
b 437 x 9 = = =
c 615 x 9 = = =
d 336 x 9 = = =
e 167 x 99 = = =
2
8
Add the digits in the
answer and keep
adding until a single
digit is found.
Viewing sample
5 What do you notice when the digits are added?
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 15

The 99 Times Table
1 2 3 4 5 6 7 8
AC C % ÷
7 8 9 x
4 5 6 –
1 2 3 =
0 . +
2 You have discovered patterns in the table of 9.
You can also see patterns in the table of 99.
a What do you notice about the units, tens and hundreds?
1 Complete the 99
times table.
b What happens when you add the digits in the answer?
1 x 99 = 99
2 x 99 =
3 x 99 =
4 x 99 =
5 x 99 =
6 x 99 =
7 x 99 =
8 x 99 =
9 x 99 =
10 x 99 =
11 x 99 =
12 x 99 =
13 x 99 =
14 x 99 =
15 x 99 =
16 x 99 =
17 x 99 =
18 x 99 =
19 x 99 =
20 x 99 =
21 x 99 =
198
297
3 Try to write a rule to determine whether a number
is divisible by nine without leaving a remainder. Share
and discuss your ideas with a friend.
Viewing sample
16 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Eleven Times
1 2 3 4 5 6 7 8
AC C % ÷
7 8 9 x
4 5 6 –
1 2 3 =
0 . +
1 Notice what happens when the
table is extended further.
a 11 x 11 =
Do you know the pattern in
the eleven times table?
11, 22, 33, 44, 55, 66, 77, 88, 99
b 12 x 11 =
c 13 x 11 =
d 14 x 11 =
Predict the following.
a 15 x 11 =
b 16 x 11 =
c 17 x 11 =
Check your answers.
4 Try these multiplications.
a 51 x 11 =
b 27 x 11 =
c 31 x 11 =
d 34 x 11 =
e 43 x 11 =
f 72 x 11 =
2 Do you think the pattern will
continue?
Yes
3 Try some more to check.
a 18 x 11 =
b 19 x 11 =
5 What do you think you would need to
do to multiply 68 by 11?
c What happens when you reach
20 x 11?
No
To multiply any two-digit
number by 11, take the two
digits of the original number
to form the first and last digits
of the number; e.g. 42 x 11
gives 4_2. The middle digit is found by adding
these two digits; e.g. 4 + 2 = 6 answer = 462.
Viewing sample
6 Now try these multiplications.
a 59 x 11 =
b 82 x 11 =
c 77 x 11 =
d 93 x 11 =
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 17

0 . +
Puzzling Patterns
1 2 3 4 5 6 7 8
AC C % ÷
9 x
4 5 6 –
1 2 3 =
1 Complete the first three questions in each sequence using a calculator.
Use any patterns you discover to complete the rest of the questions
without a calculator. Remember to check your answers.
a 99 x 12 =
99 x 23 =
99 x 34 =
99 x 45 =
99 x 56 =
99 x 67 =
99 x 78 =
b 1 x 8 + 1 =
12 x 8 + 2 =
123 x 8 + 3 =
1 234 x 8 + 4 =
12 345 x 8 + 5 =
1 234 567 x 8 + 7 =
c 9 x 1 089 =
9 x 10 989 =
9 x 109 989 =
= 987 654
You can
use a
calculator
here.
But work
out the rest
yourself!
d 4 x 2 178 =
4 x 21 978 =
4 x 219 978 =
4 x 2 199 978 =
4 x 21 999 978 =
e 1 x 9 + 2 =
12 x 9 + 3 =
123 x 9 + 4 =
1 234 x 9 + 5 =
12 345 x 9 + 6 =
1 234 567 x 9 + 8 =
12 345 678 x 9 + 9 =
f 9 x 9 + 7 =
98 x 9 + 6 =
987 x 9 + 5 =
9 876 x 9 + 4 =
98 765 x 9 + 3 =
= 1 111 111
Viewing sample
9 x 1 099 989 =
= 8 888 888
9 876 543 x 9 + 1 =
18 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Function Machines – 1
When numbers pass through a function machine, they change according to the way
the machine has been programmed. For example, the function machine below has been
programmed to add five (+5). Note what happens when numbers pass through the machine.
1 Insert the missing numbers for each function machine
and explain what the machine is doing.
a 5, 6, 7, 8, 9, + 10
b 100, 101, 102, 103, + 1
c 10, 11, 12, 13, 14 - 1
d 20, 30, 40, 50, 60 ÷ 2
e 20, 30, 40, 50, 60 ÷ 10
f 3, 6, 9, 12, 15 x 4
g - 5 10, 11, 12, 13, 14
h x 2 10, 12, 14, 16, 18
2 Explain how you worked out the answers to questions (g) and (h).
3 Try this function machine.
The + 5 machine
adds five.
15, 16, adds 10
Viewing sample
+ 8 14, 15, 16, 17, 18
Challenge
Write a ‘function machine’ question and give it to a friend to answer.
(Remember to write the answers on a separate sheet of paper.)
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 19

Function Machines – 2
1 Insert the missing numbers for each machine and give a
brief explanation of what each machine does.
a 10, 12, 14, 16, 18 ÷ 2
b 10, 12, 14, 16, 18 x 1 ⁄2
c 10, 20, 30, 40, 50 x 10
d 100, 200, 300, 400 ÷ 10
e x 2 20, 40, 80, 100
2 Take a closer look at the numbers coming in and going out of the machines.
Write about any patterns you notice.
3 Investigate what occurs when two function machines are placed together.
a 1, 2, 3, 4, 5 + 2 3, 4, 5, 6, 7 + 3
b 10, 11, 12, 13 + 5 + 5
c What did you notice?
4 Could you replace two machines with one?
Try these challenging questions to see if you are right.
6, 7,
Viewing sample
a What happens when two subtraction machines are joined?
Write questions of your own to find the answer.
Yes
No
b What happens when an addition and a subtraction machine are joined?
Write questions of your own to find the answer
20 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Function Machines – Finding the Program
Try to work out what each of the following function machines
do and write the program on the side of the machine. Explain in
words what the function machine is doing.
a 1, 2, 3, 4, 5 4, 5, 6, 7, 8
b 1, 2, 3, 4, 5 11, 12, 13, 14, 15
c 2, 4, 6, 8, 10 6, 12, 18, 24, 30
d 10, 20, 30, 40 5, 10, 15, 20
e 10, 11, 12, 13 5, 6, 7, 8, 9
f 3, 4, 5, 6, 7 30, 40, 50, 60, 70
g 3, 4, 5, 6, 7 0.3, 0.4, 0.5, 0.6, 0.7
h 6, 7, 8, 9 7.5, 8.5, 9.5, 10.5
i 10, 12, 14, 16 7.5, 9.5, 11.5, 13.5
j 3, 6, 9, 12, 15 1, 2, 3, 4, 5
Challenge!
This
machine
is multiplied
by six
adds 3
Viewing sample
Write function machine questions with
missing functions. Give them to a friend to solve.
Remember to write the missing functions on a
separate sheet of paper.
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 21

Patterns in Tables Charts
1 Complete the tables chart.
x
1 2 3 4 5 6 7 8 9
a Staring at the top left-hand corner of your tables chart‚ draw larger and larger
squares.
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
1 2 3 4
1 2 3 2 4 6 8
1 2 2 4 6 3 6 9 12
1 2 4 3 6 9 4 8 12 16
Add the numbers in each square.
2 3 4 5 6 7 8 9
4
6
8
10
12
14
16
18
6 8 10 12 14 16 18
b Predict the totals for the next 3 squares. Check your predictions.
1 2 3 4 5 6 7
1 2 3 4 5 6 2 4 6 8 10 12 14
1 2 3 4 5 2 4 6 8 10 12 3 6 9 12 15 18 21
2 4 6 8 10 3 6 9 12 15 18 4 8 12 16 20 24 28
3 6 9 12 15 4 8 12 16 20 24 5 10 15 20 25 30 35
4 8 12 16 20 5 10 15 20 25 30 6 12 18 24 30 36 42
5 10 15 20 25 6 12 18 24 30 36 7 14 21 28 35 42 49
Viewing sample
c Now try an 8 x 8 and 9 x 9 square. What set of numbers is formed?
22 • Investigating Number patterns • © R.I.C. Publications ® • www.ricpublications.com.au •

Patterns in Tables Charts (continued)
A square number is formed when a number is multiplied by itself; for example,
5 x 5 = 25. Twenty-five is a square number. A raised 2 is used to show when a number is to
be squared; e.g. 5 2 = 5 x 5 or 25.
Twenty-five is called a square number because if you tried to draw a rectangle with the
dimensions 5 x 5 it would form a square.
2 Draw a diagram to show 3 2 .
Square numbers have many interesting properties. Some numbers may be written as the sum
of four square numbers. For example‚ 23 may be written as the sum of four square numbers.
9 9
4 1
9 + 9 + 4 + 1 = 23
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
3 Try these.
a 18 b 25 c 39 d 67
Viewing sample
e 50 f 69 g 100 h 111
Make some up for someone to try.
• www.ricpublications.com.au• © R.I.C. Publications ® • Investigating Number patterns • 23

Viewing sample