43483 NS4.1 Part 4 -...

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© State of New South Wales, Department of Education and Training (CLI) 2004

Mathematics Stage 4

NS4.1 Operations with whole numbers

Part 4 Mental strategies in calculating

Part 4 Mental strategies in calculating 1

Contents – Part 4

Introduction – Part 4..........................................................3

Indicators ...................................................................................3

Preliminary quiz – Part 4 ...................................................5

Tests of divisibility .............................................................9

One and two-digit division ...............................................17

Dividing by one digit numbers ................................................17

Dividing by two-digit numbers.................................................20

Order of operations .........................................................25

Simplifying calculations ...................................................29

Properties of operations..........................................................29

Aids to mental computation ....................................................32

Further mental strategies ................................................37

A spreadsheet activity .....................................................43

Suggested answers – Part 4 ...........................................51

Additional resources – Part 4 ..........................................55

Exercises – Part 4 ...........................................................57

Review quiz – Part 4 .......................................................69

Answers to exercises – Part 4.........................................75

2 NS4.1 Operations with whole numbers


Introduction – Part 4

This part explores some of the many ways simple calculations can be

done without using a calculator or traditional methods of calculations.

Some of these methods may have occurred to students, others may

appear new. There are a number of ‘short cuts’ that can be used to aid

computations. Many adults, especially those using calculations

frequently, have developed their own little ‘tricks’ to circumvent more

laborious number crunching. It is not cheating to arrive at the desired

result more quickly, rather it is showing an understanding of being able

to work mathematically.

These tricks are really strategies. This part also examines the

determining and applying various tests for divisibility and how these test

can be applied by you.

Indicators

By the end of Part 4, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• determining and apply tests of divisibility

• using an appropriate non-calculator method to divide two- and

three-digit numbers by a two-digit number

• applying a range of mental strategies to aid computation, such as;

– a practical understanding of associativity and commutativity;

– multiplying by larger numbers by first multiplying by smaller

ones;

– multiplying by an inconvenient number can be achieved as the

sum of multiplying by two smaller more convenient numbers;

– multiplying by repeated doubling;

– dividing by repeated halving;

– a practical understanding of the distributive law.


By the end of Part 4, you will have been given the opportunity to work

mathematically by:

• applying tests of divisibility mentally as an aid to calculation

• verifying the various tests of divisibility.


Preliminary quiz – Part 4

Before you start this part, use this preliminary quiz to revise some skills

you will need.

Activity – Preliminary quiz

Try these.

1 Write down all the factors of 12.

_______________________________________________________

2 Perform these divisions.

a 120 ÷ 8 =

b 450 ÷10 =

3 What is the remainder when 7 is divided into:

a 18?

b 52?

4 a Choose the number which does not divide exactly into 1340

i 2 ii 3 iii 5 iv 10

b Describe briefly how you worked this out.

___________________________________________________

___________________________________________________


5 Calculate these.

a 2 × 5 +1

b 3× 8 +2

c 5 × 9 +4

6 Do the following divisions, writing the remainders as fractions.

a 28 ÷ 5

b 39 ÷ 3

c425

d163

7

e 6 68)

f 8 95)

7 Tick those of the following that are true?

a 34 + 23 = 23 + 34

b 5 ×16 = 16 × 5

c 28 − 20 = 20 − 28

d 20 ÷ 4 = 4 ÷ 20

e 67 × 0 = 0

f 53 ×1 = 53


8 Do the following divisions and multiplications.

a 3 ×10 =

b 45 ×100 =

c 22.4 ×1000 =

d 1000 ÷10 =

e 2300 ÷100 =

f 3420 ÷1000 =

9 Eleven people are needed to form a cricket team.

There are 95 students in Year 7 at Ashes High School.

Mr Hatrick wants to make a number of cricket teams from Year 7.

What is the greatest number of teams he can make?

Will every student be in a team? Explain.

_______________________________________________________

_______________________________________________________

_______________________________________________________

These next two questions are written so you can describe methods for

tackling them. The answer is not as important as how you got there.

10 How could you work out 48 + 25? Describe two methods you could

use to mentally arrive at the answer.

_______________________________________________________

_______________________________________________________

_______________________________________________________


11 Five books cost $9.99 each. One way to find the answer is shown.

Describe another way to find the answer.

_____________________________________________

_____________________________________________

_____________________________________________

$9.99 ×

5

$49.95

Check your response by going to the suggested answers section.


Tests of divisibility

When you say that one number is divisible by another, you mean that it

goes into it evenly without leaving any remainder.

For example, 21 is divisible by 7 since 21 ÷ 7 = 3.

Similarly, 21 is also divisible by 3.

You know this because 7 × 3 = 21. Both 7 and 3 are factors of 21.

There is no remainderwhen 21 is divided by 7.

(Notice, when you use the phrase divides evenly here you are not talking

about an even number. You mean that there is no remainder left.)

But 24 is not divisible by 7. If you divide 7 into 24 you get a remainder.

Follow through the steps in this example. Do your own working in the

margin if you wish.

What numbers divide evenly into 24?

Solution

1, 2, 3, 4, 6, 8, 12, and 24.

All these numbers divide into 24 leaving no remainder.

Can you also see that these numbers are the factors of 24?

All factors of a number divide evenly into that number.


Activity – Tests of divisibility

Try this.

1 List the numbers that go evenly into 30.

_______________________________________________________


One rule you should already know about divisibility is that all even

numbers are divisible by 2. For this rule you only need to look at the last

digit of a number to see if it is 0, 2, 4, 6, or 8, because all even numbers

end in one of these digits.

You may also remember that any number which ends in a 0 or a 5 is

divisible by 5.

A summary of the divisibility rules you need to know is given in

the table. Of course there are many more but you are not required to

know them. You will be given opportunity to use some of them later in

this part.


Rule Example

1 All numbers are divisible by 1.

2 A number is divisible by 2 if the last digit inthe number is even. That is, the last digit is0, 2, 4, 6, or 8.

248378245 676But 715 is not divisible by 2

3 A number is divisible by 3 if the sum of itsdigits is divisible by 3.

681 is divisible by 3 since6 + 8 +1 = 15 which is divisible by 3.

7292 is not divisible by 3 since7 + 2 + 9 + 2 = 20 which is notdivisible by 3.

5 A number is divisible by 5 if it ends in 5 or 0. Both 240 and 33 925 are divisible 5

10 A number is divisible by 10 if it ends in 0. 1320

Testing if a number is divisible by 3 is not the same as testing if a

number is divisible by 2, 5 or 10. Why?

___________________________________________________________

___________________________________________________________

___________________________________________________________

Look at the cartoon below that describes using divisibility rules.

Check, using the method outlined in the cartoon to see if 192 is

divisible by 3.

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________


Oh, yeah. You must use adifferent rule. With 33, 3 + 3 = 6,and 6 is divisible by 3 so 33 isdivisible by 3.

For 2, 5 or 10 just check the lastdigit.

60 and 65 is divisible by 570 is divisible by 1028 is divisible by 2.

So 33 ends in a 3 so it’s divisibleby 3.

That doesn’t work for 43.

With 43 4 + 3 = 7 is notdivisible by 3 so 43 is notdivisible by 3.

A more complete table outlining some more divisibility rules is in the

Appendix for reference. There is no need to memorise it.

Use the rules of divisibility by 2, 3, 5 and 10 for this activity.



Try these.

2 a Show that 210 is divisible by 2.

___________________________________________________

b Show that 210 is divisible by 3.

___________________________________________________

c Show that 210 is divisible by 5.

___________________________________________________

d Show that 210 is divisible by 10.

___________________________________________________


Some divisibility tests are easier to use than others.

An interesting divisibility test is that of 11. A number is divisible by 11

if the sum of every second digit minus the sum of the remaining digits is

divisible by 11. The following example explains this.


margin if you wish.

Show that the numbers 5016, and 2 846 151 are divisible by 11.


Solution

Add every second digit. Then add the remaining digits as a

separate sum.

5 0 1 6

5 + 1 = 6

0 + 6 = 6

For 5016:

subtracting one sum from the other:

6 − 6 = 0 which is divisible by 11,

so 5016 is divisible by 11.

2 8 4 6 1 5 1

2 + 4 + 1 + 1 = 8

8 + 6 + 5 = 19

For 2 846 151:

subtracting one sum from the other:

19 − 8 = 11 which is divisible by 11,

so 2 846 151 is divisible by 11.

It doesn’t matter which sum you subtract from the other. 19 − 8 = 11 or

8 −19 = −11, both of which are divisible by 11.


Try these.

3 Which of these numbers is divisible by 11?

a 908 281

b 351 698


Dividing by 11 is not a rule you need to remember. However, you could

be given the information and asked to apply it, as was done above.

You have been practising using divisibility tests.

Go to the exercises section and complete Exercise 4.1 – Test of

divisibility.


You might like to investigate these ideas further.

Access related sites on more interesting tricks and activities like this by

visiting the LMP webpage below. Select Stage 4 and follow the links to

resources for this unit NS4.1 Operations with whole numbers, Part 4.

<http://www.lmpc.edu.au/mathematics>


One and two-digit division

You are already familiar with how to divide by one digit numbers.

Here you will refresh some of that learning.

Dividing by one digit numbers

You have already learned how to divide a number with two or more

digits by a single-digit number, and to write it in different ways.

For example, you can express dividing 25 by 4 as 25 divided by 4, divide

4 into 25, or how many 4s go into 25. These are different ways to say the

same thing.

Likewise in mathematics you can write this in different ways.

25 ÷ 4 4 25)254

The number that is being divided is called

the dividend. The number that is doing the

dividing is the divisor. The result is the

quotient.

quotient

divisor dividend

64 25

14

Four goes into 25 six times, with a remainder of 1. In the past you may

have written this as 6 r1. Now that remainder is just 1 part out of the

4 parts that make the whole, so you can also write it as the fraction 14

.


Here are 25 bananas. They make 6 groups of 4 bananas, with 1 banana

remaining. That remaining banana is 14

of a group of 4 bananas.

Another way of writing 25 ÷ 4 = 6 14 is 25 = 6 × 4 +1.

That is, 6 groups of 4 plus 1 more gives 25.

Depending on the question, either solution can be correct.


margin if you wish.

Change 74 days to

a weeks and days.

b weeks

Solution

a Each group of 7 days is one week. So divide 74 by 7.

The leftovers, or remainders, will be days.

Do your division as you learnt before.

74 days = 10 weeks 4 days.

10 r 47 74

b Those 4 remaining days make 47

of a week, so

74 days = 104

7 weeks

Even though you may have divided correctly, always check that you have

answered the question in the right way.


Activity – One and two-digit division

Try these.

1 Change 92 days to:

a weeks and days.

___________________________________________________

b weeks.

___________________________________________________

2 Without using a calculator, divide 452 by 5. Write your answer with

a remainder and also as a fraction

_______________________________________________________

_______________________________________________________


You have been practicing writing solutions with a remainder and also as

a fraction. Now you must learn to choose the appropriate answer,

depending on the question.

Go to the exercises section and complete Exercise 4.2 – One and two-digit

division.


Dividing by two-digit numbers

Dividing a number by a two-digit number is done the same way.


margin if you wish.

Change 135 minutes to

a hours and minutes.

b hours.

Solution

a Each group of 60 minutes makes 1 hour.

To change minutes to hours, divide the number of minutes

by 60. Any leftovers, or remainders, will be minutes.

You can see that 60 ×1 = 60, 60 × 2 = 120, 60 × 3 = 180.

So 60 goes into 135 twice with 15 left over.

You can write 135 minutes = 2 hours 15 minutes.

b The remaining 15 minutes is1560

=14

of an hour.

So 135 minutes is also 214

hours.

12

9

6

3

12

457

8

1011

15 minutes = hour14

This problem can easily be answered in this way. However, you can also

perform a long division, like the one below, to find the answer.

This method is not often recommended. Calculators can do difficult

divisions much faster.

120 ÷ 60 = 2 since 60 × 2 = 120.

You can see that 60 goes into 135 twice.

Write the 2 above the division bar, and write

120 below 135.

260 135

12015 remainder

Now subtract. This leaves us with 15 minutes remaining.


When answering this next activity, be sure to write the most appropriate

solution, by using either a remainder or a fraction.


Try this.

3 Change 215 seconds into minutes and seconds.

_______________________________________________________

_______________________________________________________


It is important to read the question carefully so you know the form of the

answer that is needed.

In the following activity it is best to write your answer as a mixed

numeral rather than using the letter ‘r’ for the remainder.


margin if you wish.

Divide 356 by 52.

Solution

Again you can use multiplication to help with this division.

You know there are two 50s in each 100, and you know there

are seven fifties in 350.

350 = 100 +100 +100 + 50

= 50 + 50 + 50 + 50 + 50 + 50 + 50

= 7 × 50

The numbers 52 and 356 are not far from 50 and 350. So you guess there

might also be seven 52s in 356. Check this by doing a multiplication.


Seven is obviously too large to fit into 356,

but six is just right.

527

364

526

312

Continue the division as before.

52 goes into 356 six times with a remainder

of 44. You can write this as

356 = 6 × 52 + 44.

652 356

31244 remainder

The answer is 64452

which is the same as 61113

.

You can use multiplication like this to help you with division by two-

digit numbers.

Remember, there can be simpler ways to find the answer to divisions.

The following is included to demonstrate long division. But if you can

find the answer in a simpler way, then do it.


margin if you wish.

Divide 973 by 45 expressing your answer as a mixed numeral.

Solution

Follow this division carefully.

Forty-five doesn’t go into the first digit (9) of 973, but it goes

into the first two digits (97) twice, because 2 × 45 = 90.

Write 2 above the 7 and 90 below the 97.

Subtract: 97 − 90 = 7.

Now bring down the 3 from the next

column to make 73. And repeat the

process.

245 973

907


Forty-five into 73 goes once. 45 × 1 = 45.

Write the 1 next to the 2, and the 45 below 73.

Subtract. 73 – 45 = 28.

Since this number is less than 45, you are left

with a remainder of 28, or 28 parts out of 45.

So 973 ÷ 45 = 212845

2145 973

90734528

2845

Answers written as a mixed numeral are always exact.


Try this.

4 Perform the division 355 ÷ 23 giving your answer as a mixed

numeral.


There are other methods that work for some numbers. For example, to

divide by 15 you can divide into groups of 3 first, then divide those into

groups of 5. So, you can divide by 15 by dividing by 3 then by 5 (or 5

then by 3).

You have been practicing division without using a calculator, and writing

your remainders as fractions.

Go to the exercises section and complete Exercise 4.3 – One and two-digit

division.


Order of operations

Suppose you had to calculate something like 6 + 5 × 2.

You could ask: “which way do I do this? There are two options!”

Option 1: you could add first. 6 + 5 × 2 = 11 × 2 = 22.

Option 2: you could multiply first. 6 + 5 × 2 = 6 + 10 = 16.

It seems as though the answer depends on which way you look at the

problem. But you can’t have this kind of uncertainty in mathematics.

To overcome any confusion, mathematicians have made some rules.

These rules tell you the correct order of operations with respect to

addition, subtraction, multiplication, and division.

• Expressions in grouping symbols are treated as one number and must

be calculated first. The grouping symbols are parentheses ( ),

brackets [ ], and braces { }. Sometimes these grouping symbols are

just called brackets.

• Multiplication and division must be completed before addition and

subtraction. Do these from left to right.

• Add and subtract, from left to right, whichever comes first.

A common way to remember the order of operations is the abbreviation

BODMAS.


Now to the problem at the top of the page. The rules tell you that

multiplication should be done before addition.

So the answer to 6 + 5 × 2 is 16, not 22. Check this out on the calculator.

All modern calculators have order of operations built into them.

When you have a bunch of operations of the same rank, you just work

from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but rather 5 × 4

because going from left to right you do the division (15 ÷ 3) first.

If you are not sure of this, test it using your calculator.

After looking at each example below, use your calculator to test if the

solution is correct.


margin if you wish.

Simplify the following:

a 5 + 42

b 12 – (10 – 8)2

c 45 + [3 × (4 + 2)]

d 16 − 3(8 − 3)2 ÷ 5

e 4 − 37 −12

f64 + 36

10


Solution

a

5 + 42 = 5 +16

= 21

42 means 4 × 4 and this needs to be done before addition.

b 12 − (10 − 8)2 =12 − 22

=12 − 4

= 8

You have to simplify inside the parentheses before you can

take the square.

c 45 + 3 × 4 + 2( )[ ] = 45 + 3 × 6[ ]= 45 +18

= 63

Don’t try to do brackets from left to right. Instead work

from the inside out.

d 16 − 3 8 − 3( )2÷ 5 = 16 − 3× 52 ÷ 5

= 16 − 3× 25 ÷ 5

= 16 −15

= 1

Simplify inside the parentheses first. The 3, outside the

parentheses, multiplies the result.

Now multiplication and division are on the same rank in

3 × 25 ÷ 5, so do this next.

e 4 − 37 −12 = 4 − 25

= 4 − 5

= −1

The square root sign groups together the 37 – 12.

It acts like brackets, so do this first before taking the square

root.

f64 + 36

10=

10010

= 10

The numbers in the numerator are grouped together by the

fraction line. Do these first before dividing.


Always remember that both the fraction line and square root sign act as

brackets.

The following activity reviews all the rules you have learnt.

Activity – Order of operations

Simplify these.

1 6(2 + 3) – 4

2 2 × 5 + 42

316 +1416 −14


Before doing any calculations, always think about in which order you

must do them. If you don’t, you may get the wrong answer.

Now practise these rules in the following exercises.

Go to the exercises section and complete Exercise 4.3 – Order of

operations.

Sometimes order does matter and sometimes it does not.

The next section gives a fuller explanation of why this is the case.


Simplifying calculations

Over the years you have been learning mathematics you found that

certain operations have interesting features. You know, the order in

which numbers are added does not make a difference to the answer.

Hence 3 + 5 is the same as 5 + 3. But the order in which numbers are

subtracted does make a difference to the answer.

For example 3 – 5 is not the same as 5 – 3.

Properties of operations

Here are some properties of operations. You do not need to remember

the names of the properties, just understand the properties themselves.

Commutative property

Addition and multiplication are commutative. This just means that

switching the order of two numbers being added or multiplied does not

change the result.

Example: 4 + 6 = 6 + 4

3× 2 = 2 × 3

Associative Property

Addition and multiplication are associative. This just means that the

order that numbers are grouped in addition and multiplication does not

affect the result.

Examples: 3 + 4 + 5( ) = 3 + 4( ) + 5

3× 4 × 5( ) = 3× 4( ) × 5


Distributive Property

The distributive property of multiplication over addition just means that

multiplication may be distributed over addition.

Example: 3× 4 + 5( ) = 3 × 4 + 3 × 5

This example shows that 3 lots of 4 and 5 added together is the same as

3 lots of 4 added to 3 lots of 5.

To make the distributive property clearer,

consider these 9 arrows. Four of them point to

the left, and five of them point to the right.

If you make three groups of them, there will be 27 arrows altogether.

This is 3 lots of the 4 and 5 added together. In other words, 3 × (4 + 5).

Now if you take these arrows and group them according to which way

they face, you get the following.

Can you now see that 3 × (4 + 5) = 3 × 4 + 3 × 5?


The distributive property of multiplication over subtraction just means

that multiplication may be distributed over subtraction.

Example: 2 × 10 − 6( ) = 2 ×10 − 2 × 6

This example shows that 2 lots of the difference between 10 and 6 is the

same as 2 lots of 10 minus 2 lots of 6.

Here are some other important properties.

• Adding 0 to a number leaves it unchanged.

Example: 21+ 0 = 21

The same is true for subtraction.

Example: 21− 0 = 21

• Multiplying any number by 0 gives 0.

Examples: 45× 0 = 0 (45 lots of zero)

0 × 2698 = 0 (no lots of 2698)

• Multiplying any number by 1 leaves the number unchanged.

One is called 1 the multiplicative identity.

Examples: 27 ×1 = 27 (27 lots of 1)

1× 43 772 = 43 772 (one lot of 43 772)

Using your knowledge of the properties above, complete the following

activity.


Activity – Simplifying calculations

Try these.

1 Write true (T) or false (F) for these statements.

a 7 ×15 = 15× 7 ______________________________________

b 12 + 78 = 78 +12 ___________________________________

c17854

=54178

_________________________________________

d 27 ÷1 = 1 _________________________________________

e 127 × 54 × 0 = 0 _____________________________________

f 14 45 −19( ) =14 × 45 −14 ×19 _________________________


Properties of operations can help you do calculations quickly.

This is explained in the next section.

Aids to mental computation

The properties of operations can assist us to do mental calculations

because they can help us to group numbers or split them in ways that

allow us to work with simpler numbers.

Choosing the best numbers to group together takes practice.



margin if you wish.

Simplify:

a 2 × 7 × 5

b 45 × 31 × 674 × 0

c 14 + 18 + 6

d 13 × 16

e 9 × 45

f 7 × 94 + 3 × 94

Solution

The solutions use the properties described above.

a 2 × 7 × 5 = 2 × 5× 7

= 10 × 7

= 70

Re-group the 2 and 5 together to give 10. It is easier to do 10 × 7.

b 45 × 31× 674 × 0 = 0

It doesn’t matter what the other multiplications give, when you

multiply by 0, the answer is 0.

c 14 +18 + 6 = 14 + 6 +18

= 20 +18

= 38

Notice that adding 14 and 6 together gives 20. It is now easier to

add on 18. You could do it mentally as 20 + 10 + 8

(since 10 + 8 is 18).


d

13 ×16 = 10 + 3( ) ×16

= 10 ×16 + 3 ×16

= 160 + 48

= 208

Use the distributive property.

Multiplying by 3 and by 10 is easier than multiplying by 13.

e

9 × 45 = 10 −1( ) × 45

= 10 × 45−1× 45

= 450 − 45

= 405

Use the distributive property.

Multiplying by 10 is easy. Then you just need to subtract one lot

of 45.

f 7 × 94 + 3× 94 = 7 + 3( ) × 94

= 10 × 94

= 940

Seven lots of 94 added to 3 lots of 94 is just 10 lots of 94.

This is the distributive property.

These are just some ways you can simplify mental calculations.

You can probably think of others.

In the following activity take time deciding which numbers to re-group,

before proceeding.



Try these.

2 Perform these calculations without a calculator, by re-grouping the

numbers. Show how you made these calculations easier.

a 23 + 45 + 7

b 12 × 15 – 2 × 15

c 25 × 18 × 4

d 9 × 36


The skill of re-grouping to simplify calculations can be practised in the

following exercise.

Go to the exercises section and complete Exercise 4.5 – Simplifying

calculations.


Further mental strategies

By now you should realise it is very easy to multiply by 10, 100, 1000,

and so on.

6 × 10 = 60 6 × 100 = 600 6 × 1000 = 6000

There is a range of mental strategies you can use to aid computations.

The following examples give just a few of these. You might be able to

think of others. Try them out to see that they work.


margin if you wish.

Simplify:

a 12 × 75

b 83 × 4

c 680 ÷ 20

d 88 + 54

e 224 – 85

f 67 × 50


Solution

a First method

12 × 75 = (2 +10) × 75

= 2 × 75 +10 × 75

= 150 + 750

= 900

One way to quickly find the answer is to use the

distributive property you learned earlier.

By breaking up 12 as 2 +10 allows you to easily do two

simpler multiplications.

Second method

12 × 75 = 2 × 6 × 75

= 2 × 450

= 900

Another method of multiplying a number by 12 is to first

multiply by 6, then double the result.

It is easier to multiply by single digit numbers than by

double digit numbers.

b 83 × 4 = 83× 2 × 2

= 166 × 2

= 332

When multiplying by 4 it is sometimes easier to think of

doubling, then doubling again.

Alternatively,

83 × 4 = 80 × 4 + 3× 4

= 320 +12

= 332

Or, just like in the previous example, the number can be

split into two parts.

Here 83 = 80 + 3.


c 680 ÷ 20 = 340 ÷10

= 34

To divide by 20, first halve the number and then divide

by 10.

d

88 + 54 = 88 +12 + 42

= 100 + 42

= 142

Alternatively

88 + 54 =100 −12 + 54

=100 + 54 −12

=100 + 42

=142

There are a number of short cuts you can try.

One is to bridge the gap between 88 and 100.

Break 54 into 12 + 42.

Then add 12 to 88.

The final answer is easily obtained.

Here are three ways Anna, Jenny and Lucas thought of

doing 88 + 54.

8890 100 142

+2 +10 +42

88 100 142

+12 +42


e

224 − 85 = 224 − 24 − 61

= 200 − 61

= 200 − 60 −1

= 140 −1

= 139

By taking 24 off 224 brings the number down to 200.

Since you needed to take off 85 altogether, you now only

need to take off 85 − 24 = 61 more.

When subtracting 61, you can simplify this further by

taking away 60, then taking away 1 more.

f 67 × 50 = 67 ×100 ÷ 2

= 6700 ÷ 2

= 3350

You can do this multiplication by finding 67 × 5 ×10 since

5 ×10 = 50.Another way is to multiply by 100 (which is double 50)

then halve the answer.

Some calculations lend themselves to certain strategies, others to several

strategies. Sometimes it may be easier not to use any strategy at all.

Always be on the look out to see if you can find some technique that may

help you to simplify a problem.

Activity – Further mental strategies

Try these.

Describe a strategy you can use to find

a 76 + 93

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


b 15 × 84

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Choosing a mental strategy to aid calculations can often involve being a

bit creative.

Practise using a variety of strategies in the following exercise.

Go to the exercises section and complete Exercise 4.5 – Further mental

strategies.


A spreadsheet activity

These three numbers 123, 45, and 67 890 use all 10 digits from 0 to 9.

Also,

212 = 21× 21

= 441

213 = 21× 21× 21

= 9261

Now the answers, 144 and 9261, use only the digits, 1, 1, 2, 4, 4, 6 and 9.

1 is used twice, 2 is used once, 4 is used twice, 6 is used once and 9 is

used once. The remaining digits (0, 3, 5, 7, and 8) are not used at all.

From this, the teacher posed the following question.

Is there a number whosesquare and cube uses all thedigits from 0 to 9 only once?

Obviously the answer is not 21 since there is a total of only seven digits

with some of them used more than once.

You can answer this question by using a calculator and pressing a lot of

buttons. But you will answer this question using a computer spreadsheet

like Microsoft Excel®.


Whilst using your computer, you will need to answer some questions.

As you are going along, use a piece of paper to cover the work ahead

because the solution is below each question. As you are reading the

instructions, slowly slide the piece of paper down the page. Once you

have completed the question, compare your answer to the solution

provided. If your answers are very different or if you do not understand a

given answer, contact your teacher for a more detailed explanation

Step 1 Open a computer spreadsheet document.

Step 2 Click on each cell and type in the headings as shown on the

spreadsheet copy below.

cellreference

menu

formula bar

active cell

Source: Screen shot Excel used by permission from Microsoft Corporation.

Can you see that what you are typing appears in the formula bar as well

as in the active cell? When you click into any cell you can go to the

formula bar to make corrections or changes. The cell reference tells you

where you are.

Step 3 In cell A2 type in 1. Press Return (or Enter).

You are now in cell A3.


Step 4 Type this instruction in cell A3: =A2+1.

Then press Return.

Notice that the number in cell A3 is 2.

Click on cell A3 again. Two is in cell A3 and =A2+1 is in the

formula bar above the spreadsheet.


The ‘=’ in the formula bar, tells the computer you have a formula and not

a value. In this case it is telling the computer to add 1 to the contents in

cell A2. The contents of cell A2 is 1, so the computer calculates cell A3

to be 1 + 1. The answer, 2, is put into the cell when you press the enter

(or return) button.

Look at the teacher’s question at the beginning of this section.

To answer the teacher’s question, you are going to use the spreadsheet as

a calculator to find the squares and cubes of all the integers from

1 to 100. You do not need to enter all the integers from 1 to 100

individually. There is a quicker way.


Step 5 Click on cell A3 and, while holding the mouse button down,

drag the cursor down the column until you reach cell 101.

Release the mouse button. These cells should now be

highlighted.


Step 6 While the cells are highlighted, click on Edit, then Fill – Down.


1 What do you notice? _____________________________________

_______________________________________________________


Solution

In column A, you now have all the numbers from 1 to 100 listed.

2 Why did you Fill – Down to cell A101 instead of to cell A100 for

the first 100 integers?

_______________________________________________________

_______________________________________________________

Solution

The first cell is labelled “number”, so 1 goes in cell 2. This means that 99

goes in cell 100 and 100 goes in cell 101.

Step 7 Click on cell B2 and type this instruction =A2^2. Then press

Return. (The caret, ^, is on the ‘6’ button on your keyboard.)

The caret is the computer’s way of raising to a power.

So =A2^2 tells the computer to take the value in A2 and square

it. The result of 12 is 1, so 1 is placed in the cell you are

currently in, that is, cell B2. Alternatively, you may wish to

enter: =A2*A2. (The asterisk * is on the ‘8’ button on your

keyboard.) The asterisk is the computer’s way of multiplying.

Step 8 Now you will need to Fill – Down in column B. You may

repeat steps 5 and 6 after clicking on cell B2. Alternatively, for

a quicker method, click on B2 to highlight the cell. You will

notice a thick border around it. If you place the cursor on the

bottom right corner of this border the cursor changes. Drag this

corner down the number of rows you want (to cell B101).

It has the same effect as Edit – Fill – Down.

3 Explain what the computer is doing in filling column B.

_______________________________________________________

Solution

The computer is squaring each number in column A, and putting the

result in column B.


4 What instructions would you give to find the values in column C?

_______________________________________________________

Solution

You would enter: =A2^3 into cell C2 and Fill – Down to cell C101.

Another way you could do it is to enter: =A2*A2*A2.

Step 9 When you have finished the third column, the values shown

below should appear.


Only the first 24 integers are shown in this copy of the spreadsheet.

You can see both the square and cube of 21, the example shown at the

beginning of this task. Now that you have the first 100 digits, you can

use it to answer the teacher’s question posed at the beginning. Just look

for a square and cube, that combined, use all the digits from 0 to 9 once

and only once. Name the number, using column A.


5 a What is the number?

___________________________________________________

___________________________________________________

b Write the square and cube of this number.

___________________________________________________

Solution

69 is the only number whose square and cube uses all the digits from 0 to

9. 692 = 4761 and 693 = 328 509 . Each digit is only used once.

Step 10 If you are able to print this list of squares and cubes, then do so.

At the top of your screen there may be a small icon that looks

like a printer. Press it to print a copy of your work.

Alternatively, you can click on File, then Print.

Step 11 You may save your work if you wish. Click on File and drag

down to Save As. You must decide where you wish to save

your work and you will need to name it. Name it – Squares and

Cubes. At the top of your screen there may be a small icon that

looks like a floppy disk. If you have already named your work,

just press this icon. Alternatively, click on File and drag down

to Save. The computer automatically saves it back to the same

place you chose when you clicked on Save As.

You will either need to have a printed copy of your spreadsheet or your

computer open at Squares and Cubes, to do the following activity.


Activity – A Spreadsheet activity

Try this.

1 Are there any squares and cubes you did not need to check? Why?

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Using the same spreadsheet, as above, answer the following exercise, and

attach a copy of your spreadsheet for your teacher.

Go to the exercises section and complete Exercise 4.7 – A Spreadsheet

Activity.

You have now finished the learning for this part.

Complete the review quiz for this part and return it to your teacher.

It contains problems based on work from throughout this entire part.


Suggested answers – Part 4

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.

Activity – Preliminary quiz

1 1, 2, 3, 4, 6, 12

2 a 15 b 45

3 a 4 b 3

4 a ii

b individual answers will vary

5 a 11 b 26 c 49

6 a 535

b 13 c 825

d 2327

e 1113

f 1178

7 a true b true c false d false

e true f true

8 a 30 b 4500 c 22 400 d 100

e 23 f 3.42

9 8 teams. Not everyone will be in a team as there are 7 students left

over.

10,11 Answers will vary



1 1, 2, 3, 5, 6, 10, 15, 30

2 a It is an even number

b The sum of the digits 2 + 1 + 0 = 3 which is divisible by 3.

c The number ends in either 5 or 0.

d The number ends in 0

3 a 908 281. Now 9 + 8 + 8 = 25 and 0 + 2 + 1 = 3.

Since 25 – 3 = 22 is divisible by 11, then so is the number.

b 351 698. Now 3 + 1 + 9 = 13 and 5 + 6 + 8 = 19.

Since 19 – 13 = 6 which is not divisible by 11, neither is the

number

Activity –One and two-digit division

1 a 13 weeks and 1 day. b 131

7 weeks.

2 90 r 2 and 902

5

3 60 × 3 =180 and 215 −180 = 35 ,

so the answer is 3 minutes 35 seconds

4 151023 (Calculation shown on the right)

23 355

15 1023

) 23 125 115 10

Activity – Order of operations

1

6 × 5 − 4 = 30 − 4

= 26

2 10 + 16 = 26

3302

= 15



1 a T b T c F d F

e T f T

2 a 23 + 7 + 45 = 30 + 45

= 75

b (12 − 2) ×15 = 10 ×15

= 150

c 25 × 4 ×18 = 100 ×18

= 1800

d (10 −1) × 36 = 10 × 36 −1× 36

= 36 − 36

= 360 − 30 − 6

= 324

Activity – Further mental strategies

1 (There are a variety of strategies you could try. Use whichever you

feel comfortable with, and which will get you to these answers.)

a 169 b 1260

Activity – A spreadsheet activity

1 In this activity you found the squares and cubes of all integers from

1 to 100. But you didn’t need to find all of these. After all, 52 = 25

and 53 = 125, give a total of only five digits and the question asked

for all ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to be used.

Also, there are some small numbers that give too few digits, such as

92 = 81 and 93 = 729 . Here there are only a total of 5 different

digits. This is not enough. You need to choose numbers that are

bigger.


Additional resources – Part 4

The following table gives the divisibility rules for the first twelve

counting numbers.

Rule Example

1 All numbers are divisible by 1.

2 A number is divisible by 2 if the lastdigit in the number is even. Thatis, the last digit is 0, 2, 4, 6, or 8.

248378245 676But 715 is not divisible by 2

3 A number is divisible by 3 if thesum of its digits is divisible by 3.

681 is divisible by 3 since 6+8+1 = 15 which isdivisible by 3.

7292 is not divisible by 3 since 7+2+9+2 = 20which is not divisible by 3.

4 A number is divisible by 4 if thenumber represented by the tworight-hand digits is divisible by 4

799 316 is divisible by 4 since 16 is divisible by4.

781 202 is not divisible by 4 since 02 is notdivisible by 4.

5 A number is divisible by 5 if it endsin 5 or 0.

Both 240 and 33 925 are divisible 5

6 A number is divisible 6 if it isdivisible by 2 and by 3.

462 is divisible by 6 since it is an even numberand 4+6+2 = 12, which is divisible by 3.

7 (the rule is too complicated)

8 A number is divisible by 8 if thenumber formed by the last threedigits is divisible by 8.

3104 is divisible by 8 since 104 is divisible by 8(104 ÷ 8 = 13).

9 A number is divisible by 9 if thesum of Its digits is divisible by 9.

4581 is divisible by 9 since 4 + 5 + 8 +1 = 18which is divisible by 9.

10 A number is divisible by 10 if itends in 0.

1320

11 A number is divisible by 11 if thesum of the digits in the evencolumns minus the sum of thedigits in the odd columns isdivisible by11.

(see examples earlier in these notes)

12 A number is divisible by 12 if it isdivisible by 3 and by 4.

8532 is divisible by 12 since 8 + 5 + 3 + 2 = 18which is divisible by 3, and 32 is divisible by 4.


You don’t need to know all these rules. Those dividing by 2, 3, 5, and 10

are the important ones. The others are given here for interest or

reference.

If you are curious about finding why these rules work, or for other rules

about dividing numbers, a quick search on the Internet using divisibility

rules will yield results.


Exercises – Part 4

Exercises 4.1 to 4.6 Name ___________________________

Teacher ___________________________

Exercise 4.1 – Test of divisibility

1 Circle the numbers not divisible by 2

a 46 b 195 c 6208 d 45 772

2 Explain why 26 475 is divisible by

a 3 __________________________________________________

b 5 __________________________________________________

3 Circle the following numbers that are divisible by 3?

(There may be more than one answer.)

a 261 b 705 c 101 001 d 962 422

4 A tennis ball manufacturer has just obtained an order to deliver

1000 tennis balls. He delivers them in cans containing 3 balls each.

a Can he deliver exactly 1000 balls? Explain.

___________________________________________________

___________________________________________________

b Suggest a solution so he can fulfill his order.

___________________________________________________

___________________________________________________


5 A number is divisible by 4 if the last two digits in the number is

divisible by 4. For example, 316 is divisible by 4 since 16 is

divisible by 4. Circle the following numbers that are divisible by 4?


a 712 b 822 c 14 780 d 316 832

6 Circle the following numbers that are divisible by 5?


a 852 b 14 315 c 14 001 d 60 190

7 A number is divisible by 6 if it is even and divisible by 3.

Circle the following numbers that are divisible by 6?


a 543 b 672 c 14 744 d 62 190

8 a If 783 is divisible by 3 explain why 873, 738, 387 and 378 are

all divisible by 3.

___________________________________________________

b A number is divisible by 9 if the sum of all its digits is divisible

by 9. Are all these numbers also divisible by 9? Why?

___________________________________________________

9 A number is divisible by 8 if the last three digits in the number is

divisible by 8. For example, 3016 is divisible by 8 since 016 is

divisible by 8. You will need this information for this question.

a Explain why 2032 is divisible by

i 2 ______________________________________________

ii 4 ______________________________________________

iii 8 ______________________________________________

b A number is divisible by 12 if it is divisible by 3 and 4.

Is 2032 also divisible by 12?

___________________________________________________


10 Show that 3 145 604 is divisible by 11.

(The rule is explained in the notes.)

_______________________________________________________

11 The number 11 creates interesting multiples: 11, 22, 33, and so on

are obvious.

a The numbers 121, 242, and 363 are also multiples of 11.

Write another 3-digit number that is a multiple of 11.

___________________________________________________

Here is an easy way to multiply 3-digit numbers by 11.

For example: 43 ×11.

Move the 4 to the hundreds place.

Leave the 3 in the units place.

Now add 4 + 3 = 7 and put it in the tens place.

So 43 ×11 = 473.

4 7 3

4 + 3( )

b Write down, without calculating, 26 × 11. _________________

(Now check with a calculator.)

c What about 28 × 11? What do you get? ___________________

Check it out. Can you explain it?

___________________________________________________

___________________________________________________

12 a On a calculator show that the following are all divisible by 4.

i 100 ii 500 ii 1900

b Are all whole hundreds divisible by 4? ____________________

Explain why. _________________________________________

___________________________________________________

c 1984 was a leap year and as such is divisible by 4.

If 1984 = 1900 + 84 , explain why you only need to look at the

last 2 digits of 1984 to test if it is divisible by 4. _____________

___________________________________________________


d What is the next leap year after the year 2050? ______________

Explain how you got your answer.________________________

___________________________________________________

e What event occurs during a leap year?_____________________


Exercise 4.2 – One and two-digit division

Do the following exercise without using a calculator.

1 A small hall has 92 chairs that must be arranged in rows of 9 chairs

in each row.

a How many complete rows are there?

b Find the number of chairs left over.

c How many rows do all of the 92 chairs make?

2. A gardener has 285 potted plants that are all the same size.

He needs to put them into containers for storage. Each container

holds 8 pots. He fills up all his containers, except for the last one.

How many more pots does he need so he can fill the last container?

3 Choose the correct answer:

At a football match, oranges were cut up in quarters for a team to eat

at half time. Someone ate a few of these quarters before they were

put in the bucket. If there were 29 pieces at half time, how many

oranges were the in the bucket.

a 29 b 71

4c 7 r 1.

4 Perform the following divisions, giving the remainder as a fraction.

a 57 ÷ 5 =

b 75 divided by 4 =

c divide 917 by 10 =

5 Example: 31 = 5 × 6 + 1. Complete the following.

a 66 = 7 × 9 + ____

b 88 = 20 × ____ + 8

c 572 = ____ × 23 + ____



a 57 ÷ 13 =

b 88 ÷ 15 =

c 345 ÷ 37 =

7 (Harder) Here are some further divisions if you, or your teacher,

feel you need the practice.) Carry out these divisions, giving your

answers as mixed numerals.

a 456 ÷ 32 =

b 209 ÷ 18 =

c 798 ÷ 25 =

d 700 ÷ 49 =

8 a Change 300 days to weeks and days.

___________________________________________________

b Change 64 months to years and months.

___________________________________________________

c Change 749 seconds to minutes and seconds.

___________________________________________________

9 A student is 845 weeks old. How old is he in years and weeks?

(Assume 1 year = 52 weeks.)


Exercise 4.3 – Order of operations

1 Use the order of operations rules to simplify the following.

Try these first without a calculator. You may wish to use the

calculator afterwards to check that you are correct.

a 12 + 5× 6 − 32 =

b 18 ÷ 3 + 3× 7 =

c 8 + 14 −10( )2=

d 56 − 4 × 8 − 3( )[ ] =

e 12 + 20 +16 =

f11 + 735 −17

=

2 a Is 23 + 3 × 4 the same as 23 + 3× 4( ) ? Why?

___________________________________________________

b Is 23 + 3 × 4 the same as 23 + 3( ) × 4 ? Why?

___________________________________________________


3 A student was asked to simplify 16 ÷ 2 × 8 − 3 × 4 − 2( )[ ] +1 and

correctly wrote the following. Circle what the student did at each

step and explain why.

a 16 ÷ 2 × 8 − 3 × 4 − 2( )[ ] +1 _____________________________

b = 16 ÷ 2 × 8 − 3× 2[ ] +1 ________________________________

c = 16 ÷ 2 × 8 − 6[ ] +1 ___________________________________

d = 16 ÷ 2 × 2 +1 _______________________________________

e = 16 +1=____________________________________________

f = 17 _______________________________________________

4 For 2 + 3× 7 − 5 place grouping symbols, where necessary, so the

answer is:

a 18

___________________________________________________

b 8

___________________________________________________

c 10

___________________________________________________

d 30

___________________________________________________


Exercise 4.4 – Simplifying calculations

1 Perform these calculations without a calculator. Show your method

you may then use your calculator to check that the method you used

was correct.

a 2 × 9 × 5 =

b 16 +12 + 4 =

c 36 × 0 × 9 × 346 × 367 =

d 39 + 54 +11 =

e 20 × 8 × 5 =

f 25 ×17 × 4 =

g 12 × 33 =

h 19 × 56 =

i 101× 63 =

j 98 × 26 =

k 7 × 67 + 3× 67 =


l 96 × 21 + 4 × 21 =

m 16 × 84 − 6 × 84 =


Exercise 4.5 – Further mental strategies

1 Perform these calculations without a calculator. Try and apply a

range of strategies. Then use a calculator to check your answers.

a 12 × 64 =

b 18 × 47 =

c 780 ÷ 4 =

d 960 ÷ 30 =

e 2280 ÷ 40 =

f 93 + 67 =

g 465 + 687 =

h 505 – 45 =

i 426 – 32 =

j 53 × 50 =

k 81 × 500 =


Exercise 4.6 – A Spreadsheet Activity

1 Consider the squares and cubes you did not need to check.

Write down how you could improve on this method to find the

answer to the teacher’s question more quickly.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

2 Perform your new, revised, activity, using the spreadsheet program

on your computer. Attach a copy of your spreadsheet, and forward it

to your teacher.

3 Comment on at least one other thing you learned about integers in

carrying out this activity.

_______________________________________________________

_______________________________________________________

You have now completed the exercises and tasks for this part.

Complete the review quiz section and return it to your teacher.


Review quiz – Part 4

Name ___________________________

Teacher ___________________________

You may refer to the divisibility rules given in the Appendix to answer

these questions. Except for the rules for 2, 3, 5, and 10, you do not need

to remember them.

1 How do you know that 45 890 is divisible by 2, 5, and 10?

_______________________________________________________

2 Is 45 890 evenly divisible by 3? How do you know?

_______________________________________________________

3 Circle the numbers that are divisible by 6?

a 6453 b 4590 c 19 342 d 30 468

4 Circle the numbers that are divisible by both 8 and 9?

a 3960 b 4698 c 6072 d 56 304


5 Which of the numbers 2, 3, 4, 5, 6, 8, 9, and 10, divide evenly into:

a 304?

___________________________________________________

b 740?

___________________________________________________

c 4884?

___________________________________________________

d 9999?

___________________________________________________

6 The number 323 01_ is known to be divisible by 11. A digit is

missing from that number. What is the digit?

_______________________________________________________


a 147 ÷ 23 =

b 858 ÷ 17 =

c 389 ÷ 46 =

d 920 ÷ 61 =

8 a Change 710 days to weeks and days.

___________________________________________________

b Change 840 minutes to hours and minutes.

___________________________________________________

c Change 94 months to years and months.

___________________________________________________


9 Use the order of operations rules to simplify the following.

a 82 + 9 × 4

b 76 − 55 ÷ 5

c 37 + 6 × 8 − 43

d 33 ÷ 3 + 3 × 7

e 69 − 7 × 12 − 6( )[ ]

f 100 − 64 + 36

g47 + 2333−13

h 134 + 52

10 Place grouping symbols in 14 + 8 −15× 4 so that the correct answer

is 28.


11 Perform these calculations without a calculator.

a 2 × 6 × 5 =

b 45 × 739 × 245 × 0 × 568 =

c 57 + 46 + 23 =

d 20 × 7 × 5 =

e 13 × 45 =

f 9 × 67 =

g 250 × 6 × 4 =

h 21 × 79 =

i 6 × 53 + 4 × 53 =


j 102 × 68 – 2 × 68 =

12 Describe two different ways you could calculate 348 + 125.

_______________________________________________________

_______________________________________________________

_______________________________________________________

13 Find two numbers less than 100 that have these properties.

When divided by 2, 4, or 5 they each leave a remainder of 1.

When the numbers are divided by 3 there is no remainder.


Answers to exercises – Part 4

This section provides answers to questions found in the exercises section.

Your answers should be similar to these. If your answers are very

different or if you do not understand an answer, contact your teacher.

Exercise 4.1 – Tests of divisibility

1 Only 195 (it is odd)

2 a 2 + 6 + 4 + 7 + 5 = 24 which is divisible by 3

b number ends in 5

3 A, B, C

4 a No since 3 doesn’t go evenly into 1000. One remains.

b He can deliver 1002 balls (334 cans) with the two extra balls

being ‘freebies’.

5 A, C, D

6 B, D

7 B, D

8 a The sum of the digits is always 18, which is divisible by 3

b Yes, since the sum of the digits (18) is divisible by 9

9 a i Even number

ii 32 (last two digits) is divisible by 4

iii 032 (last three digits) is divisible by 8

b 2032 is divisible by 4 as its last two digits, 32, is divisible by 4.

However, it is not divisible by 3 since 2 + 0 + 3 + 2 = 7 and 7 is

not divisible by 3. Hence, 2032 is not divisible by 12.

10 3 + 4 + 6 + 4 = 17, 1 + 5 + 0 = 6 and 17 – 6 = 11, which is divisible

by 11


11 a 484 (following this pattern)

b 286

c 308. Now 2 + 8 = 10, so placing this number between the 2 and

8, the 1 carries over to the hundreds position.

12 b Yes. Whole hundreds are divisible by 4, because if 100 is

divisible by 4, then any multiple of 100 is also divisible by 4.

For example 1900 is divisible by 4 because 1900 is 19 lots of

100, which is divisible by 4.

c 1900 is divisible by 4 because it is a multiple of 100, so you

don’t have to look at that part of the number. You only need to

look at the last two digits, 84, to test if it is divisible by 4.

d 2052 because 52, being the last two digits is divisible by 4. You

do not need to look at the first two digits because 2000 is a

multiple of 100 and thus is already divisible by 4.

e Olympic games.

Exercise 4.2 – One and two-digit division

1 a 10 b 3 c 102

9

2 3

3 B

4 a 1125 b 18 3

4 c 91 710

5 a 66 = 7 × 9 + 3 b 88 = 20 × 4 + 8

c 572 = 24 × 23 + 20

6 a 4 513 b 513

15 c 9 1237 d 7 7

26

7 a 14 14 b 1111

18 c 312325 d 14 2

7

8 a 42 weeks 6 days b 5 years 4 months

c 12 minutes 29 seconds

9 16 years 13 weeks


Exercise 4.3 – Order of operations

1 a 10 b 27 c 24 d 36

e 18 f 1

2 a Yes, the multiplication would have been done first anyway, so

the parentheses do not anything. They are redundant.

b No. The parentheses group the sum to be done first. The

answer now is 104, not 35.

3 a 4 − 2 = 2 The innermost grouping symbols are done first.

b 3× 2 = 6 The multiplication within the grouping symbols is

done next.

c 8 − 6 = 2 The subtraction within the grouping symbols is done

before working from left to right.

d16 ÷ 2 × 2 = 8 × 2

= 16 The division and multiplication are performed

from left to right.

d 16 + 1 = 17. Finally the addition is done.

f no answer required.

4 a 2 + 3 × 7 – 5 or 2 + (3 × 7) – 5. Parentheses make no difference.

b 2 + 3 × (7 – 5)

c (2 + 3) × (7 – 5)

d (2 + 3) × 7 – 5

Exercise 4.4 – Simplifying calculations

1 a 90 b 32 c 0 d 104

e 800 f 1700 g 396 h 1064

i 6363 j 2548 k 670 l 2100

m 840


Exercise 4.5 – Further mental strategies

1 a 768 b 846 c 195 d 32

e 57 f 160 g 1152 h 460

i 394 j 2650 k 40 500

Exercise 4.6 – A Spreadsheet Activity.

There are many ways to improve on this activity. The following is only

one way.

As there must be a total of 10 digits in the square and cube combined,

then the square needs to have, at most, 5 digits and the cube then has

5 digits. But there are no such numbers. More likely, the square will

have 4 digits with the cube having the remaining 6.

The first number for which this occurs is 47: 472 = 2209, 473 = 103 823.

The last is 99: 992 = 9801, 993 = 970 299. So the search can go on

between these two extremes.

Improving on an activity can teach you a lot about numbers.

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