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Key words
Understanding decimals
Know the value of each digit in a decimal number with up to
3 decimal places
Read a number written in words, and then write it as a decimal
Add and subtract 0.1 and 0.01 to or from a decimal number
digit
decimal point
units
tenths
hundredths
thousandths
decimal place
The decimal point separates the whole numbers from the parts of whole numbers.
8.475
These are units
These are tenths
The tenths digit is 4. There are 4 tenths or 10.
These are hundredths
The hundredths digit is 7. There are 7 hundredths or 100.
These are thousandths
The thousandths digit is 5. There are 5 thousandths or 1000.
4
7
5
Decimal numbers can have different numbers of decimal places :
16. 83 has 2 decimal places
105. 9 has 1 decimal place
2. 485 has 3 decimal places
8.47
8.4
8.5
8.475
8.47
8.48
On a number line, 8.47 is between 8.4 and 8.5 and 8.475 is between 8.470 and 8.480.
Example 1
Write the value of 0.1 more than 3.62.
3.62 0.1 3.72
Example 2
Add one tenth to six tenths.
What needs to be added or subtracted to change 5.64 to 5.75?
5.75 is one tenth and one hundredth
more than 5.64, so we need to add
0.1 and 0.01, i.e. 0.11.
0.11, since 5.64 0.11 5.75
Example 3
7
The number is: 24 1000 100 10
7
3
6
24 0.007 0.03 0.6
24 0.637 24.637
. 0
14
0
3
6
Write this as a decimal number: 24 1000 100 10.
7
Maths Connect 1R
. 0
3
. 6
. 6
3
7
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Exercise 2.1 .............................................................................................
Write the value of the digit underlined:
a) 6.387
b) 143.704
c) 7.83
d) 249.3
e) 1.408
Write these as decimal numbers:
a)
b)
c)
d)
e)
fifteen, three tenths and four hundredths
two thousandths, nine hundredths and five tenths
fourteen, seven tenths and six thousandths
twenty seven and thirty five thousandths
eleven and seven hundredths
Watch out for numbers which
contain the digit zero!
35 thousandths is 3 hundredths
and 5 thousandths.
Write the answers as units, tenths, hundredths and thousandths in words:
a) 4.17 0.01
b) 18.35 0.01
Write these as decimal numbers:
a) 8 0.4 0.06 0.007
c) 0.008 0.5 60
1
7
e) 10 9 100
c) 9.478 0.01
b) 7 0.09 10 0.3
4
5
3
d) 13 100 10 1000
6
7
f) 1000 4 100
Convert fractions
to decimals first.
i) Write the value of each position shown on the number lines:
17
(a)
(c)
(d)
(b)
(e)
(f )
4.1
18
(g)
4.2
(i)
(h)
Write the value of:
ii) 0.01 more than (c)
iv) 0.01 less than (f)
Calculate the following:
a) 4.7 0.1
d) 8.462 0.01
(j)
iii)
v)
1
less than (d)
10
1
more than (g)
100
b) 3.62 0.1
e) 2.1 0.01
c) 5.73 0.01
e) 5.926 0.1
What needs to added or subtracted to change 13.42 to:
a) 13.32
b) 14.52
c) 13.419
d) 13.531
How many numbers with 2 decimal places between 4.5
and 5.5 contain the digit 7?
How many numbers with 2 decimal places between and
7.2 and 8.2 contain the digit 4?
e) 13.31 ?
The digit 7 can be in the tenths
place or in the hundredths place.
Investigation
You need a counter and digit cards. Use the digits 0, 7, 4, 2 and the counter to
represent the decimal point.
Investigate how many decimal numbers you can make
with up to 3 decimal places.
Each digit can only be used once in each number.
Here are three: 7.04, 2.407, 74.2
Be systematic: start with
numbers with 1 decimal
place, then numbers with 2
decimal places, and so on.
Understanding
decimals 15
Number 1: Proportion
15
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Key words
Multiplying and dividing by 10, 100,
1000
digit
decimal point
decimal place
Know how to multiply whole numbers and decimals by 10, 100 or 1000
Know how to divide whole numbers and decimals by 10, 100 or 1000
When multiplying by 10, 100 or 1000, the digits move one, two, three places to the
left on a place value grid.
When dividing by 10, 100 or 1000, the digits move one, two, three places to the right
on a place value grid.
Th H
T
U
4
4
4
7
7
8
.
.
.
Example 1
7
8
h
7
8
th
(4.78 ⫻ 10)
8
(4.78 ⫻ 100)
.
4
t
4
7
8
(4.78 ⫼ 10)
(4.78 ⫻ 1000)
0
Convert 0.45 litres into millilitres.
To convert into m, multiply by 1000.
1 1000 m
0.45 1000 450 m
Example 2
Convert 367 metres into kilometres.
1000 m 1 km
To convert m into km, divide by 1000.
367 m 1000 0.367 km
Exercise 2.2 ..........................................................................................
Complete the following calculations:
a) 35.8 10
e) 5.01 1000
b) 4.76 100
f) 736 1000
c) 236 100
g) 15.32 100
Complete these calculations. Each must have an answer of 14.6.
a)
c)
e)
g)
16
10
1000
1000
20
Maths Connect 1R
b)
d)
f)
h)
100
10
100
200
d) 4.86 10
h) 0.5 100
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Find
a)
b)
c)
d)
e)
in each of the following:
427.3 10 100
15.8 10 10
7.34 1000 10
68.2 100 10
5.9 5900 100
Convert:
a) 650 cm into m
c) 350 m into km
1
e) 2 km into m
b) 85 cm into mm
d) 75 mm into cm
f) 255 m into litres
1 m 100 cm
1 cm 10 mm
1 km 1000 m
Ten thousand people go to a music concert in Hyde Park.
If the tickets cost £16.95 each, how much money is raised?
The proceeds of the concert are split between 100 charities.
How much does each charity receive?
Gary walks to and from work each day, five days a week. The distance from home to
work is 2.9 km. How far does he walk to and from work in a year, if he has 4 weeks’
holiday a year?
Ten keen gardeners invest in a lawnmower between them which costs £365. Each
gardener uses it for 100 hours in a year. What is the cost per hour of the lawnmower for
each gardener?
Investigations
Use the digits 2 and 4 and a decimal point, together with
any of 1, ÷1, 10, ÷10, 100, ÷100, 1000, ÷1000.
Investigate how many true statements like this you can
make:
Be systematic, e.g. try all
statements which contain
two signs, then two signs, then one of each sign.
24 100 2.4 10
Write down any 3-digit number, and divide it by 100.
Divide the same number by 99.
Compare the two answers. What do you notice?
Investigate for different 3-digit numbers.
Multiplying and dividing by 10, 100, 1000 17
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Key words
Comparing and ordering decimals
decimal
digit
unit
Know how to compare two decimals and decide which is the larger, and
which the smaller
Know how to place a set of decimal numbers in order, smallest to largest
Know how to order a set of measurements which are in different units
To compare decimals , look at the whole number parts first. If they are the same,
compare the tenths digits . If the tenths digits are the same, you will need to
compare the hundredths digits.
Positions on a number line show the order clearly.
To compare measurements, you may need to change them to a common unit first.
1 m 100 cm
Example 1
1 cm 10 mm
1 km 1000 m
Put these decimal numbers in order with the smallest first:
3.25, 3.3, 3.245
3.245 3.25
3.3
3.2
3.3
To compare these decimals:
a) start with the units digit (all the same),
b) then compare the tenths digit (3.3 is the largest,
the others have the same digit),
c) then compare the hundredths digit (5 is larger
than 4, so 3.25 is larger than 3.245).
The order is 3.245, 3.25, 3.3
Example 2
Put these measurements in order with the smallest first:
45 cm, 0.4 m, 360 mm,
0.4 m
0.36 m
1
m,
2
0.4 m
Convert all the measures
to a common unit, for
example to metres.
0.5 m
0.45 m
0m
1m
1
45 cm 0.45 m, 0.4 m 0.4 m, 360 mm 0.36 m, 2 m 0.5 m
The order is: 0.36 m, 0.4 m, 0.45 m, 0.5 m or 360 mm, 0.4 m, 45 cm, 21 m
Example 3
Write this pair of numbers with either or between them: 2.35 and 2.53.
2.35 2.53
18
Maths Connect 1R
2.35 is less than 2.53
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Exercise 2.3 .............................................................................................
Write these numbers in order, smallest to largest:
a) 3.2, 3.16, 3.24, 3.12, 3.5
c) 0.04, 0.6, 0.12, 0.4, 0.62
b) 17.7, 17.63, 17.36, 17.07, 17.6
d) 9.94, 10.04, 10.1, 19.9, 10.01
Write each pair, with either or between them:
a)
c)
e)
g)
i)
5.2 and 5.17
6.05 and 6.1
0.015 and 0.02
3.029 and 3.03
0.1671 100 and 1670 100
b)
d)
f)
h)
j)
means ‘less than’
4.32 and 4.23
means ‘more than’
9.308 and 9.31
7.15 and 7.146
45.6 10 and 0.465 10
0.09 10 and 8900 1000
Write the number that is exactly halfway between these:
a) 14 and 15
c) 3.09 and 3.1
e) 2 and 5
b) 11.7 and 11.8
d) 7.35 and 7.364
f) 3 and 6
3.26 is halfway between two numbers. What is the larger of these two numbers if the
smaller is:
a) 3.2
d) 3.255
b) 3.25
e) 3.205?
c) 3.258
Put the following sets of measurements in order, largest first:
a)
b)
c)
d)
1
0.43 m, 45 cm, 420 mm, 2 m
0.015 km, 1.4 m, 16 m, 145 cm
1
650 mm, 0.7 m, 75 cm, 2 m
0.07 km, 75 m, 0.065 km, 80 m
14.7 x 14.9
What possible values can x have if it has:
a) 1 decimal place
b) 2 decimal places
c) 3 decimal places?
For parts b) and c) give five answers only.
Remember to change to a
common unit first.
means ‘less than or equal to’.
x could be any value from 14.7
up to 14.9.
Investigation
Use the four digit cards 3, 5, 2, 9, together with a decimal
point card. Make as many decimal numbers as you can
between 3 and 4. The numbers can have 1, 2 or 3 decimal
places. Finally, put them in order, smallest first.
Be systematic. For example:
start with numbers with 1
decimal place, then with 2
decimal places, and so on.
Comparing and ordering decimals 19
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Order of operations
Know the order of the operations , , , in a calculation
Understand the commutative law
Key words
operation
addition
subtraction
multiplication
division
commutative
When rearranging a written calculation, it is important to keep the sign or operation
with the number.
1 2 7 3 can be rearranged as 1 7 3 2.
1 6 4 2 can be rearranged as 1 2 4 6.
When an expression includes addition and subtraction and multiplication and
division , work from left to right. Do the divisions and multiplications first, then the
additions and subtractions.
6 4 7 13
6 28 13
Addition and multiplication are commutative .
It does not matter which order you do them in: 2 3 3 2 and 2 3 3 2
Subtraction and division are not commutative.
The order in which you do them is important: 4 2 2 4 and 4 2 2 4
Example
Calculate 8 7 12 3
8 7 12 3 56 4
52
Do the division and
multiplication first, then
do the subtraction.
Exercise 2.4 .............................................................................................
Work out the answer to these calculations. Rewrite each calculation in a different
order, and check that the answers are the same.
a) 8 7 3
b) 4 5 9 6
c) 47 21 33 19
d) 17 5 6 1
e) 6 9 3 4
f) 18 5 4 1
g) 5.8 4.9 2.7
h) 4.7 3.2 1.6
1
1
1
3
1
1
i) 12 2 22
j) 64 2 4 42
Work out the answer to these calculations. Rewrite each calculation in a different
order, and check that the answers are the same.
a) 5 2 3
b) 6 4 2
c) 7 3 2
d) 8 5 3
e) 16 2 4
f) 48 3 4
20
Maths Connect 1R
Remember to work
from left to right.
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g) 160 5 10
i) 12 2 3 4
k) 48 6 4 2
h) 144 4 12
j) 40 5 8 4
l) 2 120 6 5
These are the answers to the calculations in the following questions:
18 19 40 27
9
4 19 6
25
7
8 20 15 44
5
except that they include an extra answer by mistake. Complete each calculation, then find
the extra answer.
a) 6 4 3
b) 3 5 10
c) 16 5 2
d) 4 2 1
e) 2 8 3
f) 12 3 2
g) 11 6 1
h) 4 2 7
i) 5 4 1
j) 24 2 10
k) 37 35 5
l) 29 27 3
m) 8 4 0
n) 10 6 5
Write ‘True’ or ‘False’ for each of these. First guess what the answer will be. Then do the
calculations and compare the answers with your guesses.:
Remember
a) 12 7 6 12 6 7
b) 7 12 6 12 6 7
to work from
c) 9 8 5 2 9 8 5 2
d) 12 6 30 12 30 6
left to right.
e) 11 7 3 11 3 7
f) 35 5 2 35 2 5
g) 24 2 4 24 4 2
h) 16 4 3 2 16 3 4 2
Complete these calculations:
a)
c)
e)
g)
i)
k)
5423
6253
12 6 10 5
24 2 4 4
7546
60 10 40 5
b)
d)
f)
h)
j)
l)
9234
8 4 15 3
7 2 18 3
18 1 3 4
3 8 40 5
9947
Insert the missing operation signs to make the following correct:
a) 5 4 1 5
1
4
c) 19 7 12 5 19
12
e) 18 3 6 18
6
3
5
7
b) 8 6 4 2 8
2
4
d) 12 3 2 12
2
3
f) 120 6 2 10 120
10
6
6
2
Investigation
Calculate 3 7 4 8.
Investigate different ways of writing this calculation without changing the value of
the answer, e.g. 8 4 7 3.
Choose your own calculation with mixed operations, and investigate different ways
of writing the same calculation.
Order of operations 21
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Key words
Brackets and powers
Understand the meaning of a bracket
Know the order of operations in a calculation involving , , , ,
brackets and powers
Know how to use the bracket keys and ‘x2’ key of a calculator
ab
Understand a horizontal line in expressions like: as meaning division
c
squared
powers
bracket
operation
32 is read as ‘three squared ’ or ‘three to the power of 2’.
It is shorthand for 3 multiplied by itself, i.e. 3 3 32.
A bracket tells you that the contents must be worked out first, before any other
operations . Examples of calculations involving brackets are: (3 5) 2 or (4 1)2 6.
35
A horizontal line acts like a bracket. For example: means (3 5) 2.
2
The order for operations is:
brackets
powers
division and multiplication
addition and subtraction
Example 1
Calculate (3 5)2 8 4
(3 5)2 8 4
82 8 4
64 8 4
64 32
Start with brackets.
Powers next.
Multiplication next.
32
Example 2
8
Calculate 4 53
8
4 53
8
4 2
44
8
22
Maths Connect 1R
Start with hidden brackets.
Division next.
Finally addition.
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Exercise 2.5 .............................................................................................
Complete these calculations involving brackets:
a) 3 4 (5 3)
b) 87 (5 4) 7
c) 100 5 (6 7)
d) (51 39) 9 17
e) (3.6 2.1) (11 9)
f) (4.9 3.7) (13 9)
Check each calculation using the bracket keys on a calculator.
Copy these statements, putting in brackets to make them true.
a)
c)
e)
g)
i)
3 5 2 16
12 3 2 2
16 4 3 4
86342
27 5 9 5 88
b)
d)
f)
h)
j)
4 5 7 48
14 10 3 2
12 3 0 12
28 4 3 14 58
100 3 6 4 78
Calculate the value of:
For 9 (6 2) 4, press:
950
17 38
b) a) 9 [( 6 2 )]
35 15
5
42
18 21
c) 3
d) 4 62
13
Check each calculation using the bracket keys and x2 key on a calculator.
Calculate the value of:
a) 42 10
b) 8 92
e) 62 9
f) 82 52 62
i) 28 (9 7)2
4 12
m) 2
(1 3)
q) (7 3)2 25
j) (2 5) 22
c) 22 3
82
g) 16
k) (4 32) 1
n) 9 (7 6)2
o) 72 (5 2)2
r) (32 22)2
s) (5 2 42)2
d) 72 42
h) (8 3)2
4
Calculate the
contents of the
bracket first.
l) 2 (2 52) 5
(42 1)
p) 6 32
Use 0 – 9 digit cards.
Deal two cards to make a 2-digit target number such as 47.
Deal four more cards to use as ‘ammunition’ such as 1, 3, 8 and 5.
Using different operations, brackets and powers, see how close you can get to the target,
and score points for the difference. You are trying to get a low score.
For example
but
(8 5) 1 3 44
(1 8) 5 3 48
scores 3 points because 47 44 3
scores 1 point because 48 47 1.
Shuffle the cards and play again.
Play ten rounds. What is your total score?
Can you get a score under 20?
Investigation
Choose the digit 2 and any other three digits, for example 3, 4 and 7.
You may use any of the four operations: , , , .
You can make square numbers using the digit 2.
Using each operation and sign only once each time, investigate how many different
answers you can make, for example:
(2 4) 3 7 11
7 (4 3) 2 9
7 (3 4)2 151
Brackets and powers 23
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Key words
Adding
column method
estimate
carrying
Add whole numbers using a standard written method
Add decimal numbers using a standard written method
Estimate answers and use a calculator to check answers
Use known facts to work out answers to calculations
When adding using a column method :
● write an estimate of the answer by rounding each value
● write the numbers in columns underneath each other, making sure each digit is in
its correct column
● start adding from the right, carrying into the next column on the left if necessary
● compare the answer with the estimate as a check.
Example 1
Add 5.86, 3.8 and 15.66
Choose numbers that are
close to the ones given that
are easy to add up.
Estimate: 6 4 16 26
5.86
3.8
Make sure the decimal
points line up to check that
the place value of the digits
in each column is right.
15.66
25.32
12 1
25.32 is close to 26.
Compare the answer with the estimate as a check.
Example 2
39.46 7.75 47.21. Use this to find:
a) 39.46 17.75
b) 3.946 0.775
c) 394.6 77.5
d) 47.21 7.75
a) 39.46 17.75 57.21
The second number in the
addition is 10 more than
the original, so add 10 to
the answer.
b) 3.946 0.775 4.721
c) 394.6 77.5 472.1
d) 47.21 7.75 39.46
If a b c, then c b a.
24
Maths Connect 1R
Each number in the
1
addition is 10 of the
original, so, divide the
answer by 10.
Each number in the
addition is 10 times the
original, so multiply the
answer by 10.
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Exercise 2.6 .............................................................................................
Complete these additions:
a) 5691 387
b) 173 4158 934
d) 52.17 8.3
e) 0.86 17.59
Use a calculator to check each answer.
c) 2671 3428 1795
f) 3.85 15.6 27.34
Remember to estimate first.
52.17 8.3 60.47
Use this calculation to write the answers to these, without performing any written
calculations:
a) 52.17 18.3
b) 52.17 108.3
c) 5.217 0.83
d) 52.175 8.3
e) 60.47 8.3
f) 60.47 52.17
If a 4.583, b 16.79, c 143.8, d 25.37, e 9.648, calculate these totals:
i) a b
v) a c e
ii) c d
vi) a b d e
iii) e a
vii) b c d e
iv) b c d
viii) a b c d e
Complete these addition pyramids.
The number in each brick is found by adding the two directly below it:
a)
b)
1.35
2.17
4.36
7.09
8.37
6.9
3.59
17
Jason has £50. He buys a book that costs £14.45, together with a bag
costing £23.76. How many files could he buy at £2.95 each with the
change? How much money will he have left?
Use each of the digits 1 to 8 to make two decimal numbers of the form
.
Find pairs of numbers that will have these totals:
a)
b)
ⵧⵧ.ⵧⵧ
ⵧⵧ.ⵧⵧ
⫹ⵧ ⵧ . ⵧ ⵧ
1 0 8 . 1 8
⫹ⵧ ⵧ . ⵧ ⵧ
1 2 3 . 5 7
Use 1–9 digit cards to create a 5-digit number and a 4-digit number. The total is 41 814.
What are the numbers? Explain your method.
Invent a problem to match these calculations:
a) 1436 2351 378
b) 15.35 9.72 8.9
Investigation
Use a set of 1–9 digit cards, and three counters.
Shuffle the cards, and deal them out to create three numbers with 2 decimal places,
using the counters as decimal points.
4 . 3 9
7 . 6 1
2 . 8 5
Add them together.
Find the total of the digits in the answer.
estimate 15
Continue adding the digits until you
4.39
make a 1-digit number.
7.61
See the example on the right.
2.85
Total of digits: 1 4 8 5 18
Repeat the activity several times.
14.85
189
What do you notice?
1 1
Adding 25