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© Boardworks Ltd 2006 1 of 51
N10 Written and calculator methods
KS3 Mathematics
© Boardworks Ltd 2006 2 of 51
A1
A1
A1A1
A1
A1
N10.1 Estimation and approximation
Contents
N10 Written and calculator methods
N10.2 Addition and subtraction
N10.3 Multiplication
N10.4 Division
N10.5 Using a calculator
N10.6 Checking results
© Boardworks Ltd 2006 3 of 51
Estimation four-in-a-line
© Boardworks Ltd 2006 4 of 51
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
© Boardworks Ltd 2006 5 of 51
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
© Boardworks Ltd 2006 6 of 51
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)
© Boardworks Ltd 2006 7 of 51
Estimating points on a scale
© Boardworks Ltd 2006 8 of 51
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
© Boardworks Ltd 2006 9 of 51
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
© Boardworks Ltd 2006 10 of 51
Contents
N10 Written and calculator methods
A1
A1
A1A1
A1
A1
N10.2 Addition and subtraction
N10.3 Multiplication
N10.4 Division
N10.5 Using a calculator
N10.6 Checking results
N10.1 Estimation and approximation
© Boardworks Ltd 2006 11 of 51
Adding and subtracting decimals
© Boardworks Ltd 2006 12 of 51
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
© Boardworks Ltd 2006 13 of 51
Sums and differences
© Boardworks Ltd 2006 14 of 51
Contents
N10 Written and calculator methods
N10.3 Multiplication
N10.4 Division
N10.5 Using a calculator
N10.6 Checking results
N10.2 Addition and subtraction
N10.1 Estimation and approximation
© Boardworks Ltd 2006 15 of 51
The grid method for multiplying whole numbers
© Boardworks Ltd 2006 16 of 51
The grid method for multiplying decimals
© Boardworks Ltd 2006 17 of 51
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?
© Boardworks Ltd 2006 18 of 51
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?
© Boardworks Ltd 2006 19 of 51
Drag and drop multiplication problem
© Boardworks Ltd 2006 20 of 51
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 =
© Boardworks Ltd 2006 21 of 51
Multiplying two-digit numbers
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 =
© Boardworks Ltd 2006 22 of 51
Multiplying two-digit numbers
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 =
© Boardworks Ltd 2006 23 of 51
Contents
N10 Written and calculator methods
A1
A1
A1A1
A1
A1
N10.4 Division
N10.5 Using a calculator
N10.6 Checking results
N10.2 Addition and subtraction
N10.3 Multiplication
N10.1 Estimation and approximation
© Boardworks Ltd 2006 24 of 51
Dividing decimals – Example 1
What is 259.2 ÷ 6?What is
259.2 ÷ 6?
Dividend Divisor
© Boardworks Ltd 2006 25 of 51
Using repeated subtraction
Start by finding an approximate answer:
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
© Boardworks Ltd 2006 26 of 51
Using short division
Start by finding an approximate answer:
259.2 ÷ 6 240 ÷ 6 = 40
2 5 9 . 260
2
41
3 .1
2
2.59 ÷ 6 = 43.2
© Boardworks Ltd 2006 27 of 51
Dividing decimals – Example 2
What is 714.06 ÷
9?
What is 714.06 ÷
9?
Dividend Divisor
© Boardworks Ltd 2006 28 of 51
Using repeated subtraction
Start by finding an approximate answer:
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
© Boardworks Ltd 2006 29 of 51
Using short division
Start by finding an approximate answer:
714.06 ÷ 9 720 ÷ 9 = 80
7 1 4 . 0 690
7
78
9 .3
33
4
714.06 ÷ 9 = 79.34
© Boardworks Ltd 2006 30 of 51
Drag and drop division problem
© Boardworks Ltd 2006 31 of 51
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?
© Boardworks Ltd 2006 32 of 51
Find the equivalent calculation
© Boardworks Ltd 2006 33 of 51
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.
© Boardworks Ltd 2006 34 of 51
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.
© Boardworks Ltd 2006 35 of 51
Contents
N10 Written and calculator methods
A1
A1
A1A1
A1
A1
N10.5 Using a calculator
N10.6 Checking results
N10.4 Division
N10.2 Addition and subtraction
N10.3 Multiplication
N10.1 Estimation and approximation
© Boardworks Ltd 2006 36 of 51
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?
© Boardworks Ltd 2006 37 of 51
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
© Boardworks Ltd 2006 38 of 51
Interpreting the calculator display
© Boardworks Ltd 2006 39 of 51
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?
© Boardworks Ltd 2006 40 of 51
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:
© Boardworks Ltd 2006 41 of 51
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.
© Boardworks Ltd 2006 42 of 51
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
© Boardworks Ltd 2006 43 of 51
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?
© Boardworks Ltd 2006 44 of 51
Contents
N10 Written and calculator methods
A1
A1
A1A1
A1
A1
N10.6 Checking results
N10.5 Using a calculator
N10.4 Division
N10.2 Addition and subtraction
N10.3 Multiplication
N10.1 Estimation and approximation
© Boardworks Ltd 2006 45 of 51
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.
© Boardworks Ltd 2006 46 of 51
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.
© Boardworks Ltd 2006 47 of 51
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.
© Boardworks Ltd 2006 48 of 51
Using inverse operations
We can use a calculator to check answers using inverse operations.
We can check the solution to
by calculating
128 × 7 ÷ 4
If the calculation is correct then the answer will be 224.
47
of 224 = 128
© Boardworks Ltd 2006 49 of 51
Using inverse operations
We can use a calculator to check answers using inverse operations.
We can check the solution to
6 ÷ 13 = 0.4615384 …
by calculating
13 × 0.4615384
If the calculation is correct then the answer will be 6.
© Boardworks Ltd 2006 50 of 51
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.
© Boardworks Ltd 2006 51 of 51
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.