e pl sa m g ew in Vi 6102RB Investigating Number patterns.indd 1
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e pl sa m g ew in Vi
Number patterns 3 – Investigating Number patterns
Published by R.I.C. Publications® 2012 Copyright© Paul Swan 2012 ISBN 9781922116079 RIC6102
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.
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CONTENTS
Addition Patterns.............................................................................................. 4–7 Patterns Within the Addition Table............................................................... 8–10 Place Value Patterns.................................................................................... 11–13
e
Noticing Nines............................................................................................... 14–15
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The 99 Times Table.............................................................................................16 Eleven Times.......................................................................................................17 Puzzling Patterns................................................................................................18
sa m
Function Machines....................................................................................... 19–21
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ew in
g
Patterns in Tables Charts............................................................................ 22–23
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Addition Patterns Add together.
+
0
0
0
1
2
4
5
6
7
8
9
7 4
1 2
3
2
3
5
==
2nd row 2de rij
==
3rd rij row 3de
==
4th row 4de rij
7 8 9
Refer to the table above.
sa m
6
2
1st 1ste row rij
pl
4
==
e
1
Add all the numbers in the 1st, 2nd, 3rd and 4th rows together.
b
Write down any patterns that you notice.
ew in
g
a
3
a
Predict the results.
Predict the results of adding the
b
Vi
numbers in the:
5th row
6th row
7th row
8th row
In the box, add the numbers in each row to check your predictions. 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 •
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Addition Patterns (continued) 4
Predict the results for adding:
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 =
+
+
+
g
+
+
h
+
pl
+
+
+
+
+
+
=
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
+
=
Vi
ew in
f
sa m
Create even more new rows!
g
5
e
a
i
+
+
+
+
+
+
+
+
+
=
j
+
+
+
+
+
+
+
+
+
=
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6102RB Investigating Number patterns.indd 6
10th diagonal
9th diagonal
8th diagonal
e
pl
You'll need these diagonals for the questions on the following page.
sa m 7th diagonal
4
0
1
5
.
2
Describe any patterns you notice.
of the page. The first answer (9) and the last (54), have already been done for you.
Add the numbers in the first 6 diagonals, beginning at the top right corner, and put the answers in the diagram at right top
Add diagonally.
g
ew in
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Complete this table of addition facts.
AC
b
a
2
1
More Addition Patterns 7
12345 678 8
C
6
9
%
–
3
x
÷
+ =
6 • Investigating Number patterns • © R.I.C. Publications® • www.ricpublications.com.au •
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More Addition Patterns (continued) 3 Complete the following diagonals. Estimate the answers of the:
• 8th diagonal
• 9th diagonal
• 10th diagonal
4
Go back further!
a
Tick the answer if you predicted correctly.
e
• 7th diagonal
Add the numbers along each diagonal. Write the answers in the boxes.
pl
b
sa m
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 :
Were you correct?
Yes
No
ew in
g
Total
5
Examine the diagonals; the first is listed below:
Vi
a
Now complete the diagonals from right to left, down from the top left corner.
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Patterns Within the Addition Table 1
Look at the following table of addition facts.
Draw a rectangle around a block of four numbers; e.g. a
Add all the numbers in
e
b
e.g. 5 + 6 + 6 + 7 =
sa m
pl
the box;
c
Divide the total by 4;
e.g.
d
Try the same thing with other blocks of four
ew in
What do you notice?
g
numbers.
e
÷ 4 =
2
Look for a relationship that will allow you to work out the total of the four
Vi
a
Use the space below.
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 •
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Patterns Within the Addition Table (cont.) 3
Even more patterns.
b
sa m
pl
e
a
What other rules or patterns have you discovered? Write them down.
g
Vi
ew in
nge e l l a h C
Work out the total of the 100 numbers in the addition table.
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More Patterns in the Addition Table 1
Look at the following addition table.
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.
pl
e
x = b Step 2: Multiply the numbers in the top right and bottom left corners of the cross.
x = Step 3: Subtract the smaller answer from the larger.
sa m
c
–
2
Refer to the table above.
=
Draw some crosses of your own on the addition table.
b
Multiply the numbers in opposite corners of the cross and then subtract the smaller from the larger number.
g
a
c
=
x
=
x
=
x
=

=

=
ew in
x
Make it bigger.
Vi
3
What did you notice?
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. a
b
x
=
x
=
x
=
x
=

=

=
What did you notice about the answers?
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Place Value Patterns – 1 1 Try the following multiplications.
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 a
e
b
pl
26 x 1 =
See a pattern?
3
Describe the pattern you can see.
In what ways are the answers the
g
same?
ew in
In what ways are the answers
4
b
Describe what happens when you multiply by a number less than one.
Describe a method for doing calculations, like those above, in your head.
Vi
different?
Describe what happens when you one.
c
a
How is that?
multiply by a number greater than
b
sa m
a
A calculator will help.
5
Try these without a calculator first.
Then check your answers with a calculator.
a
39 x 10 =
b
39 x 100 =
c
39 x 0.01 =
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Place Value Patterns – 2
12345 678 AC
1
c
43 ÷ 100 =
d
43 ÷ 1 000 =
e
43 ÷ 10 000 =
= +
A calculator will help.
2 Patterns. Describe any patterns you notice.
f
43 ÷ 0.1 =
g
43 ÷ 0.01 =
h
43 ÷ 0.001 =
i
43 ÷ 0.0001 =
j
43 ÷ 0.00001 =
g
ew in
b
3
e
43 ÷ 10 =
x –
pl
b
2 .
∏
9 6
sa m
43 ÷ 1 =
5
Try the following divisions.
a
a
%
8
4
0
1
C
7
Try these without a calculator first.
Vi
3
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 =
Check your answers with a calculator.
12 • Investigating Number patterns • © R.I.C. Publications® • www.ricpublications.com.au •
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Place Value Patterns – 3 1
Try the following multiplications.
a
173 x 27 =
b
A calculator will help.
17.3 x 27 =
173 x 2.7 =
f
1.73 x 27 =
c
173 x 0.27 =
g
17.3 x 2.7 =
d
173 x 0.027 =
h
1.73 x 2.7 =
e
e
a
Write about any patterns you notice.
sa m
b
In what ways are the answers the same?
In what ways are the answers different?
g
c
ew in
d
pl
2 Determine.
3
237 x 16 = 3792. Use this information to help work out the answers to:
23.7 x 16 =
Vi
a
How does the position of the decimal point change the answer?
e
237 x 160 =
b
2.37 x 16 =
f
2370 x 16 =
c
23.7 x 1.6 =
g
2370 x 1.6 =
d
237 x 0.16 =
h
0.237 x 160 =
4
Explain how patterns can help to work out answers to questions like those above.
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Noticing Nines – 1 1
You'll be familiar with this?
a
Complete the nine times table. What happens when you add the digits? e.g. 2 x 9 = 18 1 + 8 = 9
1 x 9 = 2 x 9 =
9 =
9
9
1 + 8 = =
e
3 x 9 =
=
pl
4 x 9 =
=
5 x 9 =
sa m
6 x 9 = 7 x 9 = 8 x 9 = 9 x 9 =
= = = =
What do you notice about the answers?
ew in
What do you notice about the tens?
g
10 x 9 =
=
What do you notice about the ones?
b
Vi
Record your observations.
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 •
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Noticing Nines – 2
12345 678 AC
1
5 2 .
∏
9
x
6
–
3
= +
List the answers to the nine times table.
odd 1 x 9 9 = (
)
Write down whether the answer is odd or even.
6 x 9 =
(
)
7 x 9 =
(
)
(
)
3 x 9 =
(
)
8 x 9 =
(
)
4 x 9 =
(
)
9 x 9 =
(
)
5 x 9 =
(
)
10 x 9 =
(
)
2
7
3
6
2 a
When the digits in the answer are split, more patterns may be found.
pl
8
What happens when you add the digits along the diagonal?
sa m
1
5
4
6
3
7
2
8
1
9
0
b
8
Try adding the digits along the other diagonal; i.e.
2
Write about what you notice.
3
What happens when the numbers along each of the diagonals are subtracted? (Always take the smaller number away from the bigger.)
g
5
ew in
4
e
2 x 9 =
Complete the following multiplications.
a
736 x 9 =
6624
b
437 x 9 =
=
=
Vi
4
6 + 6 + 2 + 4 = 18 =
c
615 x 9 =
=
=
d
336 x 9 =
=
=
e
167 x 99 =
=
=
5
%
8
4
0
1
C
7
9
Add the digits in the answer and keep adding until a single digit is found.
What do you notice when the digits are added?
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12345678
The 99 Times Table 2 a
AC
C
÷
%
x
7
8
9
4
5
6
–
1
2
3
=
0
.
+
You have discovered patterns in the table of 9. You can also see patterns in the table of 99.
What do you notice about the units, tens and hundreds?
Complete the 99 times table.
b
2 x 99 = 198
3 x 99 = 297
4 x 99 =
sa m
99
1 x 99 =
What happens when you add the digits in the answer?
pl
1
e
5 x 99 =
3 Try to write a rule to determine whether a number
6 x 99 =
is divisible by nine without leaving a remainder. Share and discuss your ideas with a friend.
7 x 99 =
g
8 x 99 = 9 x 99 =
ew in
10 x 99 = 11 x 99 = 12 x 99 = 13 x 99 = 14 x 99 =
Vi
15 x 99 = 16 x 99 = 17 x 99 = 18 x 99 = 19 x 99 = 20 x 99 = 21 x 99 =
16 • Investigating Number patterns • © R.I.C. Publications® • www.ricpublications.com.au •
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12345678
Eleven Times
AC
1 Notice what happens when the table is extended further.
b
12 x 11 =
c
13 x 11 =
d
14 x 11 =
c
17 x 11 =
3
=
0
.
+
Yes 3
No
Try some more to check.
a
18 x 11 =
b
19 x 11 =
c
What happens when you reach
sa m
16 x 11 =
2
2 Do you think the pattern will continue?
15 x 11 =
b
–
1
Do you know the pattern in the eleven times table? 11, 22, 33, 44, 55, 66, 77, 88, 99
Predict the following. a
x
9 6
e
11 x 11 =
÷
%
8 5
pl
a
C
7 4
20 x 11?
Check your answers.
Try these multiplications. 51 x 11 =
b
27 x 11 =
ew in
a
g
4
31 x 11 =
d
34 x 11 =
e
43 x 11 =
f
72 x 11 =
Vi
c
To multiply any twodigit 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.
5 What do you think you would need to do to multiply 68 by 11?
6
Now try these multiplications.
a
59 x 11 =
c
77 x 11 =
b
82 x 11 =
d
93 x 11 =
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Puzzling Patterns
12345 678 AC
1
a
5 2 .
÷
9
x
6
–
3
= +
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. can You a e us tor ula calc re. he
99 x 12 =
d
4 x 2 178 =
4 x 21 978 =
99 x 23 =
99 x 34 =
4 x 219 978 =
99 x 45 =
4 x 2 199 978 =
99 x 56 =
4 x 21 999 978 =
99 x 67 =
99 x 78 =
pl
sa m
1 x 9 + 2 =
12 x 9 + 3 =
123 x 9 + 4 =
1 234 x 9 + 5 =
e
1 x 8 + 1 =
12 x 8 + 2 =
123 x 8 + 3 =
1 234 x 8 + 4 =
B ou ut w tt o yo he rk ur re se st lf!
ew in
g
12 345 x 8 + 5 =
e
b
%
8
4
0
1
C
7
= 987 654
12 345 x 9 + 6 =
= 1 111 111 1 234 567 x 9 + 8 =
12 345 678 x 9 + 9 =
Vi
1 234 567 x 8 + 7 =
9 x 9 + 7 =
98 x 9 + 6 =
f
9 x 1 089 =
987 x 9 + 5 =
9 x 10 989 =
9 876 x 9 + 4 =
9 x 109 989 =
9 x 1 099 989 =
c
98 765 x 9 + 3 =
= 8 888 888
9 876 543 x 9 + 1 =
18 • Investigating Number patterns • © R.I.C. Publications® • www.ricpublications.com.au •
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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.
5, 6, 7, 8, 9,
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
h
2
15, 16,
adds 10
 5
10, 11, 12, 13, 14
x 2
10, 12, 14, 16, 18
Explain how you worked out the answers to questions (g) and (h).
Vi
x 4
ew in
g
+ 10
g
a
pl
Insert the missing numbers for each function machine and explain what the machine is doing.
sa m
1
e
The + 5 machine adds five.
3
Try this function machine.
+ 8
14, 15, 16, 17, 18
Write a ‘function machine’ question and give it to a friend to answer. (Remember to write the answers on a separate sheet of paper.)
nge Challe
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Function Machines – 2 Insert the missing numbers for each machine and give a brief explanation of what each machine does. 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
Take a closer look at the numbers coming in and going out of the machines. Write about any patterns you notice.
Investigate what occurs when two function machines are placed together. 1, 2, 3, 4, 5
b
10, 11, 12, 13
c
What did you notice?
3, 4, 5, 6, 7
+ 3
+ 5
+ 5
Could you replace two machines with one? Try these challenging questions to see if you are right.
6, 7,
Yes
No
Vi
4
+ 2
ew in
a
g
3
20, 40, 80, 100
sa m
2
pl
a
e
1
a
What happens when two subtraction machines are joined? Write questions of your own to find the answer.
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 •
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Function Machines – Finding the Program
adds 3
pl
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.
e
This machine is multiplied by six
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
d
10, 20, 30, 40
e
10, 11, 12, 13
f
3, 4, 5, 6, 7
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
sa m
a
6, 12, 18, 24, 30 5, 10, 15, 20
g
5, 6, 7, 8, 9
Vi
ew in
30, 40, 50, 60, 70
n Challe
ge!
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
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Patterns in Tables Charts
a
Complete the tables chart.
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
4
4
8
5
5
10
6
6
12
7
7
14
8
8
16
9
9
18
pl
e
x
sa m
1
Staring at the top lefthand corner of your tables chart‚ draw larger and larger
1
1
2
4
3
2
4
6
8
9
12
2
2
4
6
3
6
4
3
6
9
4
8
12 16
Predict the totals for the next 3 squares. Check your predictions.
1
2
3
4
2
4
6
8
10 12
3
6
9
12 15 18
4
8
12 16 20 24
5
10 15 20 25 30 35
1
2
3
4 8
5
6
5
6
7
10 12 14
1
2
3
4
5
2
4
6
2
4
6
8
10
3
6
9
3
6
9
12 15
4
8
4
8
12 16 20
5
10 15 20 25 30
6
12 18 24 30 36 42
10 15 20 25
6
12 18 24 30 36
7
14 21 28 35 42 49
5 c
3
Add the numbers in each square.
Vi
b
2
2
ew in
1
1
g
squares.
12 15 18 21
12 16 20 24 28
Now try an 8 x 8 and 9 x 9 square. What set of numbers is formed?
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Patterns in Tables Charts (continued) A square number is formed when a number is multiplied by itself; for example, 5 x 5 = 25. Twentyfive is a square number. A raised 2 is used to show when a number is to be squared; e.g. 52 = 5 x 5 or 25.
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
pl
3
sa m
2
Draw a diagram to show 32.
ew in
g
2
1
e
Twentyfive is called a square number because if you tried to draw a rectangle with the dimensions 5 x 5 it would form a square.
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
4
1
3
Try these.
a
18
b
25
c
39
d
67
e
50
f
69
g
100
h
111
Vi
9
9 + 9 + 4 + 1 = 23
Make some up for someone to try.
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