© Boardworks Ltd 2006 1 of 51 N10 Written and calculator methods KS3 Mathematics.

© Boardworks Ltd 2006 1 of 51

N10 Written and calculator methods

KS3 Mathematics


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N10.1 Estimation and approximation

Contents


N10.2 Addition and subtraction

N10.3 Multiplication

N10.4 Division

N10.5 Using a calculator

N10.6 Checking results


Estimation four-in-a-line


Martin uses his calculator to work out 39 × 72.

The display shows an answer of 1053.

How do you know this answer must be wrong?

“is approximately equal to”

39 × 72 40 × 70 = 2800

The product of 39 and 72 must therefore end in an 8.

9 × 2 = 18.9 × 2 = 18.

Estimation

Also, if we multiply together the last digits of 39 and 72 we have


3.5 × 17.5 can be approximated to:

4 × 20 = 80

3 × 18 = 54

4 × 17 = 68

or between 3 × 17 = 51 and 4 × 18 = 72

How could we estimate the answer to 3.5 × 17.5?

Estimation


4948 ÷ 58 can be approximated to:

5000 ÷ 60 = ?

5000 ÷ 50 = 100

4950 ÷ 50 = 99

or 4800 ÷ 60 = 80

How could we estimate the answer to 4948 ÷ 58?

Estimation

(60 does not divide into 5000 exactly)


Estimating points on a scale


Using points on a scale to estimate answers

Jessica is trying to estimate which number multiplied by itself will give the answer 32.

She knows that 5 × 5 = 25 and that 6 × 6 = 36.

The number must therefore be between 5 and 6.

She draws the following scales to help her find an approximate answer.

25 26 27 28 29 30 31 32 33 34 35 36

5 65.64


Use Jessica’s method to estimate which number multiplied by itself will give an answer of 40.

We know that 6 × 6 = 36 and that 7 × 7 = 49.

Draw a scale from 36 to 49.

Underneath, draw a scale from 6 to 7.

36 37 38 39 40 41 42 43 44 45 46 47 48 49

6 76.31

Using points on a scale to estimate answers


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N10.4 Division





Adding and subtracting decimals


Jack is doing some DIY.He buys a 3m length of wood.Jack needs to cut off two pieces of wood –one of length 0.7m and one of length 1.92m.

a) What is the total length of wood which Jack needs to cut off?b) What is the length of the piece of wood which is left over?

0.7

1.92

0

+

a)

1

262.

b)

– 2.62

3.002

191

83.0

Jack needs to cut off 2.62m

altogether.

The left-over wood will

measure 0.38m (or 38cm).

Adding and subtracting decimals


Sums and differences


Contents



N10.4 Division






The grid method for multiplying whole numbers


The grid method for multiplying decimals


Using the standard column method

Start by finding an approximate answer:

2.28 × 7 2 × 7 = 14

2.28 × 7 is equivalent to 228 × 7 ÷ 100

228× 7

65

91

15

Answer

2.28 × 7 = 1596 ÷ 100 = 15.96

What is 2.28 × 7?


Using the standard column method

Again, start by finding an approximate answer:

392.7 × 0.8 400 × 1 = 400

392.7 × 0.8 is equivalent to 3927 × 8 ÷ 100

3927× 8

65

12

47

31

Answer

392.7 × 0.8 = 31416 ÷ 10 ÷ 10

= 314.16

What is 392.7 × 0.8?


Drag and drop multiplication problem


Multiplying two-digit numbers

Calculate 57.4 × 24.

Estimate: 60 × 25 = 1500

Equivalent calculation: 57.4 × 10 × 24 ÷ 10

= 574 × 24 ÷ 10

574

× 24

11480

2296

13776

Answer: 13776 ÷ 10 = 1377.6

4 × 574 =

20 × 574 =



Calculate 23.2 × 1.8.

Estimate: 23 × 2 = 46

Equivalent calculation: 23.2 × 10 × 1.8 × 10 ÷ 100

= 232 × 18 ÷ 100

232

× 18

2320

1856

4176

Answer: 4176 ÷ 100 = 41.76

8 × 232 =

10 × 232 =



Calculate 394 × 0.47.

Estimate: 400 × 0.5 = 200

Equivalent calculation: 394 × 0.47 × 100 ÷ 100

= 394 × 47 ÷ 100

394

× 47

15760

2758

18518

Answer: 18518 ÷ 100 = 185.18

7 × 394 =

40 × 394 =


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N10.4 Division







Dividing decimals – Example 1

What is 259.2 ÷ 6?What is

259.2 ÷ 6?

Dividend Divisor


Using repeated subtraction


259.2 ÷ 6 240 ÷ 6 = 40

259.266 × 40– 240.0

19.26 × 3– 18.0

1.26 × 0.2– 1.2

0

Answer: 43.2


Using short division


259.2 ÷ 6 240 ÷ 6 = 40

2 5 9 . 260

2

41

3 .1

2

2.59 ÷ 6 = 43.2


Dividing decimals – Example 2

What is 714.06 ÷

9?

What is 714.06 ÷

9?

Dividend Divisor


Using repeated subtraction


714.06 ÷ 9 720 ÷ 9 = 80

714.0699 × 70– 630.00

84.069 × 9– 81.00

3.069 × 0.3– 2.70

0.369 × 0.04– 0.36

0Answer: 79.34


Using short division


714.06 ÷ 9 720 ÷ 9 = 80

7 1 4 . 0 690

7

78

9 .3

33

4

714.06 ÷ 9 = 79.34


Drag and drop division problem


Writing an equivalent calculation

This will be easier to solve if we write an equivalent calculation with a whole number divisor.

We can write 36.8 ÷ 0.4 as 36.8

0.4=

×10

368

×10

4

36.8 ÷ 0.4 is equivalent to 368 ÷ 4 = 92

What is 36.8 ÷ 0.4?


Find the equivalent calculation


Dividing by two-digit numbers

Calculate 75.4 ÷ 3.1.

Estimate: 75 ÷ 3 = 25

Equivalent calculation: 75.4 ÷ 3.1 = 754 ÷ 31

Answer: 75.4 ÷ 3.1 = 24.32 R 0.08

75431– 620 20 × 31

134– 124 4 × 31

10.0– 9.3 0.3 × 31

0.70– 0.62

0.080.02 × 31

= 24.3 to 1 d.p.


Dividing by two-digit numbers

Calculate 8.12 ÷ 0.46.

Estimate: 8 ÷ 0.5 = 16

Equivalent calculation: 8.12 ÷ 0.46 = 812 ÷ 46

Answer: 8.12 ÷ 0.43 = 17.65 R 0.1

81246– 460 10 × 46

352– 322 7 × 46

30.0– 27.6 0.6 × 46

2.40– 2.30

0.100.05 × 46

= 17.7 to 1 d.p.


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N10.4 Division





Solving complex calculations mentally

What is ?3.2 + 6.8

7.4 – 2.4

3.2 + 6.8

7.4 – 2.4=

10

5= 2

We could also write this calculation as: (3.2 + 6.8) ÷ (7.4 – 2.4).

How could we work this out using a calculator?


Using bracket keys on the calculator

What is ?3.7 + 2.1

3.7 – 2.1

We start by estimating the answer:

3.7 + 2.1

3.7 – 2.1 3

6

2=

Using brackets we key in:

(3.7 + 2.1) ÷ (3.7 – 2.1) = 3.625


Interpreting the calculator display


Finding whole number remainders

Sometimes, when we divide, we need the remainder to be expressed as a whole number.

For example, 236 eggs are packed into boxes of 12.

Using a calculator: 236 ÷ 12 = 19.66666667This is 19.6 recurring or 19.6

.Number of boxes filled = 19

.Number of eggs left over = 0.6 × 12 = 8

How many boxes are filled?

How many eggs are left over?


Finding whole number remainders

Find the remainder if this answer was obtained by:

a) Dividing 384 by 60 0.4 × 60 = 24

b) Dividing 160 by 25 0.4 × 25 = 10

c) Dividing by 2464 by 385 0.4 × 385 = 154

My calculator display shows the following:


Working with units of time

What is 248 days in weeks and days?

Using a calculator we key in:

2 4 8 ÷ 7 =

Which gives us an answer of 35.42857143 weeks.

We have 35 whole weeks.

To find the number of days left over we key in:

– 3 5 = × 7 =

This give us the answer 3.

248 days = 35 weeks and 3 days.


Converting units of time to decimals

When using a calculator to work with with units of time it can be helpful to enter these as decimals.For example:

7 minutes and 15 seconds = 7 1560

minutes

= 7 14

minutes

= 7.25 minutes

4 days and 18 hours = 4 1824

days

= 4 34

days

= 4.75 days


Find the correct answer

Four people used their calculators to work out .9 + 30

15 – 7

Tracy gets the answer 4.

Fiona gets the answer 4.875.

Andrew gets the answer –4.4.

Sam gets the answer 12.75.

Who is correct?

What did the others do wrong?


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N10.4 Division





Making sure answers are sensible

When we complete a calculation, whether using a calculator, a mental method or a written method we should always check that the answer is sensible.

Use checks for divisibility when you multiply by 2, 3, 4, 5, 6, 8 and 9. For example, if you multiply a number by 9 the sum of the digits should be a multiple of 9.

Make sure that the sum of two odd numbers is an even number.

When you multiply two large numbers together check the last digit. For example, 329 × 842 must end in an 8 because 9 × 2 = 18.


Using rounding and approximation

We can check that answers to calculations are of the right order of magnitude by rounding the numbers in the calculation to find an approximate answer.

Sam calculates that 387.4 × 0.45 is 174.33. Could this be correct?

387.4 × 0.45 is approximately equal to 390 × 0.5 =195

This approximate answer is a little larger than the calculated answer but since both numbers were rounded up, there is a good chance that the answer is correct.


Using inverse operations

We can use a calculator to check answers using inverse operations.

We can check the solution to

34.2 × 45.9 = 1569.78

by calculating

1569.78 ÷ 34.2

If the calculation is correct then the answer will be 45.9.





by calculating

128 × 7 ÷ 4

If the calculation is correct then the answer will be 224.

47

of 224 = 128





6 ÷ 13 = 0.4615384 …

by calculating

13 × 0.4615384

If the calculation is correct then the answer will be 6.


Using an equivalent calculation

Another way to check answers to calculations is to use an equivalent calculation.

For example, we can check

698 × 11 = 7678

with

(700 – 2) × 11 = 7700 – 22

or

698 × (10 + 1) = 6980 + 698.


Using an equivalent calculation

Write two calculations that are equivalent to 22 × 98.

We can either write 22 as (20 + 2) or we can write 98 as (100 – 2).

This give us two equivalent calculations:

(20 + 2) × 98 = 1960 + 196

and

22 × (100 – 2) = 2200 – 44

The answer to all three equivalent calculations is 2156.

© Boardworks Ltd 2006 1 of 51 N10 Written and calculator methods KS3 Mathematics.

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