1 Introduction for teachers and parents In this book there are single-page exercises covering most of the mathematics work that children will meet in Year 6. On each page there are some examples/hints along with a set of questions for children to answer. There is space for answers to be written in the book. Each exercise ends with a challenging question for the more able children. At the end of each page, children are invited to answer a self-assessment question. Using this book The exercises in the Maths Practice Questions books are not intended to be used in the initial teaching of new mathematics topics. However, they can be used as: Homework activities - to consolidate work done in class. End of topic class activities - to give children the opportunity to check their understanding of a particular topic. Assessment tasks - allowing teachers to establish whether or not children are secure in their understanding of a topic. Year 6 Maths Practice Questions
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Transcript
1
Introduction for teachers and parents
In this book there are single-page exercises covering most of the mathematics work that children will meet in Year 6.
On each page there are some examples/hints along with a set of questions for children to answer. There is space for answers to be written in the book.
Each exercise ends with a challenging question for the more able children.
At the end of each page, children are invited to answer a self-assessment question.
Using this book
The exercises in the Maths Practice Questions books are not intended to be used in the initial teaching of new mathematics topics.
However, they can be used as:
Homework activities - to consolidate work done in class.
End of topic class activities - to give children the opportunity to check their understanding of a particular topic.
Assessment tasks - allowing teachers to establish whether or not children are secure in their understanding of a topic.
Year 6Maths Practice Questions
2
ContentsContents
Number and Place Value 3 - 7
Addition, Subtraction, Multiplication and Division 8 - 16
Fractions, Decimals and Percentages 17 - 27
Ratio and Proportion 28 - 31
Algebra 32 - 36
Measurement 37 - 42
Statistics 51 - 54
Appendix - Information for Parents 55 - 57
Notes 58 - 60
Geometry 43 - 50
Are you ready for
this?
3
Number and Place Value
A Place valueYou should know the values of all the digits in any number up to 10 million (10,000,000).
What is the value of these digits? Write your answers in words.1
6 in 1,640,225
5 in 1,249,560
8 in 8,745,645
Nine million, six hundred and forty-one thousand, eight hundred and twelve
Eight million, two hundred and ten
Can you write these numbers in digits?2
What is 10 million take away 1? Write your answer in digits.3
I’m confi dent I’m nearly thereI can read, write, order and compare numbers up to 10,000,000.
Can you put these numbers in order of size, smallest to largest?4
9,482,169 9,284,169 9,248,169 9,248,196 9,284,196
If you started with 3,445,841 and doubled it, what digit would be in the millions column?
What digit would be in the 1000s column?
5Brain
training!
ix d
e
t
9,641,812
8,000,210
9,999,999
9,248,169 9,248,196 9,284,169 9,284,196 9,482,169
6
1
4
I’m confi dent I’m nearly thereI can round numbers to the nearest
10, 100, 1000, 10,000, 100,000 or million.
Rounding numbersB
You should be able to round any whole number to the nearest 10, 100, 1000, 10,000, 100,000 or million.
E.g. 2,845,565 rounded to the nearest 1000 is 2,846,000
A city has a population of 3,684,266. Can you round this number to:1
the nearest 10?
the nearest 1000?
the nearest 100,000?
Which of these numbers would be rounded to 60,000 if they were rounded to the nearest 1000? Circle your answers.
2
60,584 59,842 60,499 59,444 60,010
John said that his home town had 30,000 people to the nearest 1000 living in it. What is the biggest number of people who could live there?
3
What is 3,999,999 rounded to the nearest 10?
What is 3,999,999 rounded to the nearest 100?
4
In a quiz Ali was asked to round this number so that it had one 6 in it: 4,445,525.
What answer would have been correct?
(You are only allowed to round to the nearest 10, 100, 1000, 10,000, 100,000 or million.)
5 Phew!
3,684,270
3,700,000
3,684,000
4,000,000
4,000,000
30,499
4,446,000
5
I’m confi dent I’m nearly thereI can do calculations containing negative numbers.
C Using negative numbers
Sometimes we need to think about numbers less than zero. This number line may help with some questions.
In the daytime, the temperature reached a maximum of 8 oC in Suzie’s garden. At night it fell to -4 oC. By how many degrees did the temperature fall?
1
oC
In a quiz Jon got 2 points for each correct answer, but he lost 2 points for each wrong answer.
In round 1 he got 2 questions correct and 4 wrong. How many points did he have?
2
Try these calculations:3
2 – 6 =
6 – 8 =
0 – 4 =
-1 + 4 =
-2 – 3 =
-4 + 4 =
The weather forecast said that it was 20 oC hotter in Madrid than in London. If it was -3 oC in London, what was the temperature in Madrid?
4
oC
Is 3661 rounded to the nearest 100 bigger or smaller than 3711 rounded to the nearest 10?5
(bigger or smaller)Help!
12
-4
-4
-2
-4
3
-5
0
17
r
6
I’m confi dent I’m nearly thereI can solve some number problems.
Solving problemsD
What is the temperature measured by this thermometer?
If the temperature fell by 18 oC what would it be?
1
-10 0 10 20 30 (oC)
oC
oC
Jason’s bank statement said that he had -£87.00 and that he must immediately bring his balance back to zero. For his birthday he received £90.
After paying back the bank, how much did he have left to spend?
2
£
Sara played a party game where she had to wear a blindfold. The others gave her these instructions:
“Take 6 steps back, three steps forward, two steps back, fi ve steps forward, three steps back.”
At the end she had to fi nd her way back to where she started. What did she need to do?
4
Are you serious?!
Would you rather win prize : £1501 rounded to the nearest £1000
or prize : £1890 rounded to the nearest £100?
3 A
B(A or B)
What were the prizes actually worth?
£1501 to the nearest £1000 =
£1890 to the nearest £100 =
£
£
13
-5
3.00
A
2000
1900
Take steps3 d
7
I’m confi dent I’m nearly thereI can answer word questions about number and place value.
Word problemsE
1 Blake was asked to fi nd the second largest number in this list: 1,236,544, 1,236,545, 1,236,455, 1,236,561 and 1,236,456.
What should his answer be?
3 Which is more, £84,865 rounded to the nearest 100 or £84,891 rounded to the nearest 10? Show how you worked this out.
4 In Alaska it is -21 oC. If the temperature increases by four degrees, what temperature will it be?
5 Rounded to the nearest 100, Zac has 62,500 stamps in his collection. What is the largest number of stamps he could have?
6 What is the answer to twenty-eight minus thirty-three?
7 What number do you need to add to minus two thousand eight hundred and twelve to get an answer of 50?
OK!
2 Yousef went to a football match where there were thirty-three thousand, eight hundred and sixty two people in the crowd.
What is this number rounded to the nearest hundred?
Answer (in digits):
1,236,456
33,900
£84,865
-17oC
62,549
-5
2862
3 3 8 6 2,
28 − 33 = -5
2 8 1 2
5 0
2 8 6 2
+
84865 o t 100
= 84900
84891 o t 10
= 84890
rounded to the nearest: 100
8
I’m confi dent I’m nearly thereI can do mental maths with bigger and bigger numbers.
Addition, Subtraction, Multiplication and Division
A Mental maths You will need to know your times tables and know how to simplify calculations.
E.g. 31 x 8 = 30 x 8 + 1 x 8 = 240 + 8 = 248
Try these calculations in your head:1
Now you are warmed up, try these:2
Hint: Start with 48 x 10
What about some division?3
x 220
+ 24 ÷ 8 – 10
Try this:4
If that was too easy, try this:
x 319
– 21 x 4 ÷ 12
Answer:
5 If 6372 ÷ 6 =
=
6000 ÷ 6 360 ÷ 6 12 ÷ 6+ +
1000 60 2+ + 1062=
What is 8256 ÷ 8?
Not now!
244 + 143 =
251 + 29 =
728 – 513 =
498 – 19 =
a)
c)
b)
d)
28 x 3 =
121 x 4 =
32 x 6 =
48 x 9 =
a)
c)
b)
d)
640 ÷ 8 =
132 ÷ 12 =
a)
c)
490 ÷ 7 =
1320 ÷ 12 =
b)
d)
387
280
215
479
192
432
84
484
80
11
70
110
40 64 8 -2
57 36 144 12
1032
+
+
+
+
8000 ÷ 8
1000
240 ÷ 8
30
16 ÷ 8
2
9
B Written addition and subtractionNow it’s time to try some bigger numbers!
I’m confi dent I’m nearly thereI can do written addition and subtraction.
Wow!
Let’s start with some addition.1
Now for a take-away!2
2 1 4
+ 1 3 5
6
2
4 4
+ 1 0 6
4
6
42 6
+ 1 3 2
8
4
3
5 9 9
+ 1 1 1
9
1
9 9
+
9
3
90 0
+ 9 9
2
9
4
3 2 4
– 1 1 3
6
4
1 4
– 6
0
4
42 2
– 1 1 1
0
6
4
6 8 2
– 2 9 1
4
4
0 0
– 5 4
0
3
40 0
– 0 0
0
2
7
1
Can you fi nd the missing numbers in this calculation?
3 4
– 6 4
9
2 9 91
3 4 9 8 4 5 9 21
5 5 1 011
7 1 1 01 1 1
5 0 0 11 1 1
0 0 0 21 1 1
1
2 1 1 2 3 1 0 4
1 1
4 0 7 6
0 13 1
3 9 1 0
5 1
5 9 9 8
6 9 9 1
3 4 5 7
3 9 9 1
8 32 9
8 94
9−
3
4
1
6
9 92 1
187
10
I’m confi dent I’m nearly thereI can multiply numbers with up to 4 digits
by 2-digit whole numbers.
Remember long multiplication?
1
2 4 2 1
x 1 4
6 8 494 2 1 023 8 9 43
C Written multiplication
Can you fi nd the missing numbers?4 1
x
4 5
4 5 63
2
9 4 00
3 9 63
I need a sleep!
Ready to try some?1
Now with 4-digit numbers.2
4 3
x 1
8
2
5 4
x 3
3
8
6 3
x 2
4
4
3 8
x
4
6
5 6
x 2
7
3
2 4
x 1
5
4
1 1 1
1
3 A bottle of water costs 56 p.
How much would it cost for 1250 bottles?Convert your answer from pence to £s.
£
8 7 1 6
3 8 04
2 5 651 1
3 1 6
6 8 02
2 1 65
5 1
1
1
2
1 1
42 4
2 9 06
6 3 40
33
1
2
4
11
1
4023582 82 0
4 5 02
4 3 07
9 1
1
1
4
1 1
02 1
3 4 01
0 4 16
7 2
3
3
4
1
1
111
700.00
2 3
2
3
1 2 5 0
5 6
7 1 53 0 0
22 5 0 06 1
0 0 0 07
x
11
1
11
I’m confi dent I’m nearly thereI can use short and long division to divide by 1 and 2-digit numbers.
D Written division22
5411
12
2 1
9
9277
3
321
6 r 1
or 216 1
12Short Division
2 2
1 3615
r 3Long Division
4 The town mayor wants to send out 3259 invitations to a special party in the town hall.
The invitations are in packs of 12.
How many packs will the mayor need?
How many invitations will be left over?
Tricky!
Try these using short division.1
Try these using long division. (They have remainders.)2
1 97 6 2 73 41 3 4 94 42 4
4 62 93 9 9 51 14 5
1 97 8
Use short division and write the remainders as fractions.3
2 58 7 4 69 7
272
5
82 112 6025 1 1 1 14
r 274 61 r 33 22
23
41 9
21 8
9
91 2
2 7
1120
28
1 1
21 3
5
8 2
3
8
32 1
82 23 1527
18
89
15 1
1 2 3 2 5 9
2 4
8 5
8 4
1 9
2 7 1 r 7
−
−
12
E Multiples, factors and prime numbers
2, 3, 5, 7, 11, 13, 17, 19, etc. are prime numbers. Their only factors are 1 and themselves.
1, 3 and 9 are factors of 9
The lowest common multiple of 3 and 4 is 12
2 is a common factor of 4 and 6
I’m confi dent I’m nearly thereI can fi nd multiples and factors and
I can recognise prime numbers.
Look at these pairs of numbers. Can you circle the common factors?1
(8, 12) 2 3 4 5 6
(36, 24) 2 3 4 5 6 7 8 9 10 11 12
Which of these numbers are prime numbers?3
3 16 23 27 31 39 43 77 83
Can you fi nd three common factors of 16 and 8 (not including 1)?2
Now we are going to fi nd some lowest common multiples for the following pairs of numbers. 4
The lowest common multiple of 6 and 9 is The lowest common multiple of 8 and 6 is
The lowest common multiple of 8 and 7 is
5 Multiply the lowest common multiple of 4 and 9 by the biggest common factor of 6 and 18.
What is your answer? x =
OK!
3, 6, 9, 12, etc. are multiples of 3
2 4 8
18
24
56
36 6 216
3 6x
1
6
2 63
13
I’m confi dent I’m nearly thereI know what order to do operations in.
F The order of operationsWhat is 2 + 3 x 6 – 1? Is the answer 29 or 19 or 25 or 17? Answer: 19
Why? Because we always do operations in this order: brackets, ÷ x, + –
Try these calculations:1
Now try these!2
4 + 6 ÷ 2 x 3 – 1 = 8 x 2 + 4 ÷ 2 + 6 =
Can you put the brackets into the calculations to make them correct?4
2 + 8 x 4 – 1 = 266 + 3 x 2 = 18
Can you add brackets to make this true?5
3 x 8 ÷ 2 + 4 = 4
Bring it on!
16 ÷ 4 + 2 =
12 + 8 ÷ 4 =
a)
c)
18 + 2 x 6 =
20 ÷ 2 + 3 =
b)
d)
To make things easier we can use brackets. Operations in brackets must be done fi rst.
Try these (the fi rst one is done for you).
3
88 x (3 – 2) =
12 ÷ (2 + 2) =
a)
c)
11 x (4 + 3) =
30 ÷ (11 – 5) =
b)
d)
8
6
14
30
13
12 24
3 5
77
( ) ( )
)( )(
14
G Estimating to check answersIt is always best to check your answers. One way you can do this is by estimating.
For example, Jon worked out that 6 x 199 = 1004. We can check this out by working out 6 x 200 which is 1200 and we see that Jon’s calculation must be wrong.
I’m confi dent I’m nearly thereI can make estimates to check my answers.
Mark these calculations by estimating. The fi rst is done for you.1
x8 99 = 792
x8 100 = 800
or x6 51 = 306
x6 =
or
Use your mental maths skills to estimate the answers to these calculations ( ≈ means “is approximately equal to”).
2
Estimate the answer to this calculation.4
89.9 + 2 x 5 – 50.006 =
Use a calculator for these. Sometimes a calculator gives answers to lots of decimal places. We have to decide how accurate the answer needs to be.
Example: 2 ÷ 3 = 0.66666666666. In most cases it would be good enough to write the answer as 0.67.
Try these:
3
800 cm ÷ 3 =
122 mm ÷ 5 =
4 litres ÷ 7 = Hint: Convert to ml before dividing.ml
Give me a break!
mm
cm
90.1 x 4 ≈
199.85 x 6 ≈
89.9 x 5 ≈
204.99 + 163.03 ≈
a)
c)
b)
d)
50 300
+999 399 = 1498
+ =
or +69 69 = 227
+ =
+ 69
+
or
70 70 70 210
360 450
1200 368
266.67
24.4
571.43
50
14001000 400
15
I’m confi dent I’m nearly thereI can solve problems involving addition, subtraction, multiplication and division.
Solving problemsH
1 Zak is 4 years older than Jody and Jody is 2 years younger than Mike. Zak is 10 years old.
How old is Mike? years old
2 In a game there are 3 sets of cards. Each set has 16 cards. The cards are mixed together and shared equally between 4 players.
How many cards does each player get? cards
3 Billy bought four boxes of cakes for his birthday party. Each box contained eight cakes. He carefully put fi ve cakes on each of six plates.
How many cakes were left over?
Sarah had £3 to spend on her lunch.4
drinks 40 p
sandwiches £1.60
fruit 50 p
biscuits 15 p
How many biscuits did she buy?
How much money did she have left?
She bought a drink, a sandwich and a piece of fruit.
She then bought as many biscuits as she could.
In a quiz, you get 3 points for a correct answer but you lose 2 points for a wrong answer. If you answer a ‘bonus’ question correctly your score is doubled, but if you get it wrong your score goes back to zero. Below is Jayne’s score card.
5
bonusbonusbonus
Question
Answers
1 2 3 4 5 6 7 8 9 10What is her score?
Help!
8
12
2
3
5 p
10
3 x 16 = 48
48 ÷ 4 = 12
4 x 8 = 32
5 x 6 = 30
6
5
5
1
0
2
+0
0
0
40 0.
.
.
.
£3 £2.50 = 50 p t
r u
3 x 15 p = 45 p
Z JM
(10) (6)(8)4
2
16
Word questionsI
I’m confi dent I’m nearly thereI can answer word questions using addition, subtraction,
multiplication and division.
1 What is three hundred and sixty-three minus one hundred and seventy-four?
2 How many would three more than 899,899 be?
3 8,420 spectators each spent £28 on a ticket for the match.
How much was spent on tickets in total? £
4 Susie said that 494 bananas could be shared equally between twenty-nine monkeys with no bananas spare. Do a calculation to fi nd out if this is correct.
Circle your answer: Correct or Incorrect
5 Use long division to work out the answer to 2473 divided by 12.
Write down any remainder as a fraction.
6 Counting up in ones, what is the next prime number you get after twenty-nine?
7 Which number is the lowest common multiple of twenty-seven and eighteen?
8 Write this in numbers so that it is correct: Eight times fi ve minus two equals twenty-four.
Let’sgo!
189
899,902
235,760
31
54
8 x (5 − 2) = 24
112206
6
7
1
− 4
8 9
1
15 3132
8 4 2 0
2 8
6 03 17 36
0 04861
6 07532
x
r 11 7
4 9 492
2 9
2 0 4
2 0 31
62 0
2 4 721 3
2 4
0 7 3
7 21
1 1
17
Fractions, Decimals and Percentages
I’m confi dent I’m nearly thereI can simplify fractions.
Simplifying fractionsA
We can simplify using what we know about equivalent fractions.
6
36
636
212
= 16
=
Can you simplify these fractions?1
816
= 936
=
1284
= 354
=
46
65
79
812
1015
1213
2030
Which of these fractions are equivalent to ? Circle your answers.22
3
Can you change all these fractions so that their denominator is 6?3
23
=6
318
=6
1012
=6
How many fractions can you fi nd that are equivalent to with a denominator less than 24?4 18
24
49168
= 147168
=
5 Try to simplify these fractions.
24
=6
a)
c)
b)
d)
OK!
1827
= =654
515
=3
=6
12
17
14
118
4 1 5
3 1 2 36 4
34
1520
1216
912
68
724
78
741
861
94
65
7
8==
18
I’m confi dent I’m nearly thereI can compare fractions by changing them so that
they have the same denominator.
B Comparing fractions
Is bigger or smaller than ? Answer: = so is smaller than .6
18
4
9
6
18
3
9
6
18
4
9
Now try doing the same with these, but you will need to choose the ‘common denominator.’2
23
35
715
smallest largest
Can you put these in order from largest to smallest?3
16
2 146
56
1 186
largest smallest
Now try putting these in order - no hints!4
34
812
1924
smallest largest
Stop it!
Can you put these fractions in size order from smallest to largest? Hint: Convert them all to twelfths.1
12
512
26
23
712
12 12 12 12 12
smallest largest
75
26
512
12
712
23
6 4 8
1015
915
715
715
23
35
812
1924
34
162
561
146
186
19
I’m confi dent I’m nearly thereI can add and subtract some fractions.
C Adding and subtracting fractions
We can use what we know about equivalent fractions.
For example:1
3 +2
9 =3
9 +2
9 =5
9
Try these:1
Now have a go with mixed numbers. The fi rst is done for you.3
13
1 +56
1 =43
+116
=86
+116
=196
=16
3
14
1 +18
=
+13
=
Now try these:2
16
4 − =12
2
4 What is the missing number in this calculation?
12
+ 18
=8
18 8
=
27
– 314
=14
314 14
=
16
+ 312
=
59
– 318
=
=
=
+
–
+
–
++ = = =
++ = = =36
2
2
1
Phew!
212
312
1018
318
512
718
54
156
178
43
108
156
178
86
278
236
38
56
3
3
26
4
4
5
1
25
6−
1
6
50
12=4
?
12−
50
12
24
12=
? = 26
20
I’m confi dent I’m nearly thereI can multiply two proper fractions together
and simplify my answer.
D Multiplying two fractions together
4 Which is bigger : a half of a quarter of a half or : a third of a third?
AB
Answer:
Tough!
To multiply fractions multiply the top numbers together then multiply the bottom numbers together.
23
x23
=491
4
1
4
1
2x = 1
8
2
3
2
3
1
2x = 2
6= 1
3
Use the diagrams to multiply fractions.2
14
x13
14
= 25
x23
25
=
One of the fractions could be a whole number such as = 2. Try these, the fi rst is done.1 2
1
13
x = x =2 13
21
23
a)
27
x = x =5c)
14
x = x =3b)
49
x = x =9 =d)
Now try these.3
23
x 34
= =
(simplify)
a)
29
x 37
= =
(simplify)
21c)
16
x 34
= =
(simplify)
b)
211
x 211
=d)
27
107
51
49
369
91
14
34
31
4
112
415
612
12
324
18
4121
663
2
B
1
2x
1
4
1
16=x
1
2
1
3x
1
3
1
9=
3 x 3
2 x 2
21
I’m confi dent I’m nearly thereI can divide proper fractions by whole numbers.
E Dividing fractions by whole numbers
Remember: Dividing by 2 is the same as multiplying by1
2
and dividing by 3 is the same as multiplying by .1
3
Let’s start with these.1
14
÷ 2 We know this is the same as:14
x 12
= 18
Now it’s your turn!
15
÷ 2 15
x ==
16
÷ 3 16
x ==
311
÷ 4 311
x ==
Now try these and remember to simplify your answers.2
911
÷ 3
1016
÷ 4
26
÷ 12
=
=
=
=
=
=
=
=
=
3 Luke ate of his birthday cake and shared the rest equally between himself and 6 friends.
How much cake did each of his friends get?
1
15
Try this...
x
x
x
12
13
14
110
118
344
911
1016
26
13
14
112
933
1064
272
311
532
136
215
÷14
15=7 x
14
15
1
7
= =14
105
2
15
22
F Changing fractions to decimals
is the same as 1 ÷ 2 = 0.51
2
I’m confi dent I’m nearly thereI can change fractions to decimals using division.
Tenths, hundredths and thousandths can easily be changed to decimals. Try these:1
110
= 0.1210
= 510
=
4100
= 14100
= 61000
=0.04 0.14 0.006
0.2 0.5
With some fractions we need to divide the top part (numerator) by the bottom part (denominator) to change them to decimals. Try these - the fi rst is done.
2
= 1 ÷ 41
4
1 04 0.0 2 5.
0 8.0 2 0.0 2 0.
3 04 0. = 3 ÷ 4 3
4
3 020 0. 9 020 0. = 3 ÷ 203
20 = 9 ÷ 209
20
Ready for some more? Try these if you dare!3
Woah!
0.625
0 7 5.2
0 4 5.10
0 1 5.10
0 3 7. 5
3 0 0. 086 4
0 6 2. 5
5 0 0. 082 4 = 3 ÷ 83
8 = 5 ÷ 85
8
23
I’m confi dent I’m nearly thereI can multiply decimals by whole numbers.
G Multiplying with decimals Remember: Keep your eye on the decimal point!
Let’s start by multiplying by 10, 100 and 1000.1
1.2 x 10 =
34.5 x 10 =
0.12 x 100 =
1.23 x 100 =
0.4 x 100 =
1.24 x 1000 =
Write down the value of the digit 2 in these numbers in words.2
24
1.264
4.812
8.020
Now let’s try some mental multiplication using decimals.3
0.6 x 7 =
0.03 x 8 =
0.002 x 4 =
0.5 x 9 =
0.08 x 6 =
0.006 x 12 =
Are you ready for some written multiplication? The fi rst is done for you.4
8 7
6 2 4.x 3
2.11
1 2 4.x 3
2 3 5.x 5
4 7 6.x 6
5 A boy bought 4 magazines at £1.34 each and 3 birthday cards at £1.65 each.
How much did he spend? £
No!
a)
c)
e)
b)
d)
f)
a)
c)
e)
b)
d)
f)
12
345
12
123
40
1240
4.2
0.24
0.008
4.5
0.48
0.072
o
o
o
o
10.31
3 7 1 2 1 1 1 7 2 5 2 84 53 6. . .
3x 4
6
41
5 1
6x 3
5
51
3 1 4 1 9 1
.
.
.
.
24
I’m confi dent I’m nearly thereI can do division calculations and write my answers as decimals.
H More written division - writing your answers as decimals
Look at the calculation 16 ÷ 5
Try this using short division.1
18 ÷ 5
1 85 0
91 ÷ 2
9 12 0
26 ÷ 4
2 64 0
Now have a go at these using long division. The answers will have 2 decimal places.2
66 ÷ 8 3.14 ÷ 278 ÷ 8
3 Jackie paid £2.88 for 6 cupcakes.
How much did each cake cost?
£
Ok let’s go!
1
1
0
0
0
5
3 2
6
5
1
1
.
.
.
or we can write the answer as
the decimal 3.2(3 x 5)
(0.2 x 5)
Remember that 1.0 is 10 tenths and 10 tenths ÷ 5 = 2 tenths
.
1
1
6
5
1
5
3r1 or 31
5
6 5.2
3 6.3
5 5.1
41
8 2
6 6 08
5
0
.
.9 7
7 8 08
5
0
.
.5 7
3 1 42
.
.1
4 8
2 8 864
.
.0
0.48
6 4 .
2 0.
1 6.
0 4 0.
0 4 0.
−
−
7 2 .
6 0.
5 6.
0 4 0.
0 4 0.
−
−
2 .
1 1.
1 0.
0 1 4.
0 1 4.
−
−
... .
25
I Rounding Remember: 4.6591 is 4.66 to two decimal places.
I’m confi dent I’m nearly thereI can round numbers to a sensible number of decimal places.
Try rounding these numbers to the nearest tenth (to one decimal place).1
11.94 6.88 3.06
Can you round these numbers to two decimal places?2
15.619 21.444
If you round these numbers to 1 decimal place, which one would not be 10.5? Circle your answer.3
10.48 10.54 10.501 10.45 10.555
5 Jake measured a bar of chocolate. It was 131 mm long. He wanted to break it into 5 equal pieces.
How long would each piece be to the nearest millimetre?
4 Sally wanted to share £4 between her 3 sisters. She worked out on her calculator that they would each get £1.333333.
What should she round the answer to so that she could give them their share?
6 A number with 3 decimal places is rounded down to 4.76. The original number didn’t have a 1, 2 or 3 in it.
What was the number?
Oh no!
11.9 6.9 3.10
15.62 21.44
£1.33
26 m
4.764
2 6
1 3 153
2
01.
.
26
I’m confi dent I’m nearly thereI can swap between fractions, decimals and percentages.
J Fractions, decimals and percentages
Remember: A fraction written in hundredths can easily be turned into a percentage or a decimal.
54
100= 0.54 = 54% 28
100= 0.28 = 28%
Write these fractions as percentages.1
20100
= %15100
= 80100
=% %
Now try these:2
410
= 610
= 910
=% % %
Use what you know about equivalent fractions to convert these fractions to decimals.3
14
= 25
= 1325
=
5 Zane had £120 and put 20% in the bank.
His sister Zara had £200 and put in the bank.
Who put the most money in the bank?
Show your working.
1
10 Tricky!
Can you fi ll in the spaces using your mental maths skills?4
Remember to simplify your fractions.
% DecimalFraction
40%
0.12
34
20 15 80
40 60 90
25 % 40 % 52 %
75 % 0.75
0.4
12 %
25325
Z
1 0 % of 1 2 0 = 1 2
2 0 % of 1 2 0 = 2 4
of 2 0 0 = 2 0110
=34 =75
100 0.75 = 75%
= =40100 0.440%
=40100
410 = 2
5=40%
=12100=0.12 12%
=12100
325
27
I’m confi dent I’m nearly thereI can answer word questions about fractions, decimals and percentages.
K Word questions
1 Which fraction is larger? Circle your answer.
Explain how you know.
53
or 169
2 Can you write down an equivalent fraction to that has a denominator of forty-fi ve?
5
9
3 Find a common denominator for these fractions and circle the largest.
23
= 34
= 56
=
4 Can you subtract one sixth from four ninths?
5 Work out the answer to three elevenths multiplied by four elevenths.
6 Sally and her brother have one seventh of a cake to share equally between them. What fraction of the cake will each of them get?
7 What is one fi fth as a decimal?
8 What is as a percentage?3
5
9 If you bought seven books at £3.89 each, how much would you spend?
£
Phew!
0.2
60 %
27.23
2545
812
912
1012
518
12121
114
=5
3
15
9
=5
9
25
45
−4
9
1
6= −
8
18
3
18
x3
11
4
11=
12
121
x1
7
1
2=
1
14
=1
5
20
100
=3
5
60
100 8x 7
3
93
7 6 26
.
.2 1
u t n of159
53
28
Ratio and Proportion
I’m confi dent I’m nearly thereI can scale up and down by multiplying and dividing.
A Scaling up and down
If you get a free drink with every six meals, how many meals would you need to buy to get two free drinks? Answer: 12
1 Ice creams cost 60 p each.
How much would 8 ice creams cost?
Answer:
3 In a bag of beads there were 3 red beads for every blue bead.
Altogether there were 80 blue beads. How many red beads were there? Answer:
2Serves 3 people
1 egg
50 g fl our
50 ml milk
Look at this recipe for Yorkshire puddings. How much fl our would you need to make puddings for 6 people?
g
Complete this: “for every egg you need fl our
and milk.”
g
ml
4 On a plane there were 6 airline workers and 240 passengers.
If each worker looked after the same number of passengers, how many passengers did each worker look after?
Answer:
Can you complete this chart showing the price of biscuits?5
Biscuits
Cost
1 2 3 4 5 6 7 8
£1.80
In a card game each player must have 7 red cards, 9 blue cards and 5 yellow cards. If there were 42 red cards in play, how many blue cards were in play?
6
Wow!
100
5050
£4.80
240
40
£0.30 £0.60 £0.90 £1.20 £1.50 £2.10 £2.40
54
x 6 0 = 4 8 08
x 8 0 = 2 4 03
0 ÷ 6 = 4 02 4
29
To fi nd of a number we divide it by 10.
I’m confi dent I’m nearly thereI can calculate percentages of whole numbers.
B Calculating percentages
What is 10% of 30? 10
100
1
1010% is = 1
10
So, 10% of 30 is: 30 ÷ 10 = 3
Try these:1
of10% 40 10% is 1
10of 40 is:1
1040 ÷ 10 =
of10% 90 10% is 1
10of 90 is:1
1090 ÷ 10 =
What is 10% of 80?
What is 5% of 80?
What is 25% of 80?
3
Hint: 10% + 10% + 5%
Here goes...
On a pie chart, Charlie wanted to show that 10% of children cycled to school. The angle around a point at the centre of a circle is 360o.
How big should angle A be?
4
On the same chart he wanted to show that 35% of the children travelled by car. What angle should he measure for this slice of pie chart?
o
10%
A
o
If 10% of 20 is 2, we know that 20% of 20 will be 4 (twice as much). Try these:2
of20% 50 =So
of30% 50 =
of10% 50: 10% is 1
10of 50 is:1
1050 ÷ 10 =Find
4
9
5
10
15
8
20
4
36
126
of 80 = 80 ÷ 10110
= 8
8 + 8 + 4 = 20
35% = 10% + 10%
+ 10% + 5%
= 36 + 36 + 36 + 18
= 126o
30
I’m confi dent I’m nearly thereI know what is meant by ‘scale factor.’
C Scale factor
S
The square has been enlarged by a scale factor of 2.
S
The sides of a triangle measure 3 cm, 4 cm and 5 cm. It is enlarged so that the shortest side is 36 cm. What is the scale factor?
How long will the longest side be on the enlarged triangle?
3
cm
On a map the distance from Sam’s home town to London was 30 mm.
The scale on the map was 1 mm for every 3 kilometres.
How far was it from Sam’s town to London?
4
km
2
B
Shape B has been enlarged and part of the enlarged shape is shown.
What scale factor has been used?
Can you enlarge shape A by a scale factor of 3?
Draw your answer on the empty grid.
(Be careful where you start!)
1
A
5 This picture of a tree is 9 mm high. If it is enlarged by a scale factor of 400, how high will the new picture be in metres?
Wow!
9 mm
4
12
60
90
3.6 m
31
I’m confi dent I’m nearly thereI can answer word questions about ratio and proportion.
D Word questions
1 To make fruit punch, Zoe mixed litre of orange juice with litre of mango juice and litre of pineapple juice. This made enough punch for three people.
How much mango juice would she need if she made punch for twenty-four people?
1
2
1
41
8
2 There are 30 members of a swimming club. 40% of them can swim 1500 m. How many members can swim 1500 m?
3 Zane was supposed to share a cake with his three sisters. One sister ate 30% and the other two ate 35% each.
How much was left for Zane?
4 Can you work out fi ve percent of one hundred and thirty?
5 On a pie chart Yousef wanted to show that 30% of people in his class had blue eyes.
What angle should he use for the ‘blue eye’ section?
6 A rectangle has sides of 6 cm and 8 cm. If it was enlarged by a scale factor of 28, what will the lengths of its sides be?
and
7 Amy’s dad takes two steps for every three of hers. When they walked to town, Amy counted her steps and she needed 582.
How many did her dad need?
Phew!
2 l
12
0 %
6.5
108o
168 m 224 m
388
x 8 = = 214
84
10% of 30 = 3
40% of 30 = 4 x 3 = 12
35 + 35 + 30 = 100
10% of 130 = 13
5% = 13 ÷ 2
10% = of 360 = 36110
110
of 360 = 108310
9 4x
8
2
3 8
1
9 4
5 8 231
12
2 8x 6
1 864
2 8x 8
2 426
32
Algebra
I’m confi dent I’m nearly thereI can use some formulae.
Using a formulaA
Example: The area of a rectangle is found using the formula: Area = length x width
5 cm
3 cm
Area = length x width = 5 cm x 3 cm = 15 cm2
Use the formula Area = length x width to answer these questions.
A rectangle has a length of 12 mm and a width of 6 mm. What is the area?
A rectangle has an area of 48 cm2 and a length of 12 cm. What is its width?
1 Area = length x width = =
x
mm2
Area = length x width 48 cm2 = 12 cm x The width is cm
2
The area of a triangle is found using the formula Area = base x height.1
2height
base
Can you calculate the areas of these triangles?
5 cm
6 cm
7 cm
14 cm
Adie made up a formula to work out the cost of his shopping.
Cost = A x 40 p + B x 50 p + P x 60 p
(A is the number of apples, B is the number of bananas and P is the number of pears).
He bought 6 apples, 6 bananas and 4 pears. Complete this to fi nd out how much he spent.
3
Cost =
=
+
+
+
+
x 40 p x 50 p x 60 p
= £
Brainstrain!
12 m 6 m
72
4 m
4
15 m2 49 m2
6 6 4
240 p 300 p 240 p 7.80
x 6 x 5 x 14 x 712
12
33
I’m confi dent I’m nearly thereI can describe number sequences and fi nd missing numbers.
Number sequencesB
What is the next number in this sequence? 1, 8, 15, 22, 29 ...?
Answer: 36 Why? Because the rule is ‘add 7’ to get the next number.
What is the rule to get the next term in this sequence?1
5 14 23 32
The rule for this sequence is ‘add 12’. Can you write the next three numbers?2
8 20 32
Can you spot the rule for this sequence and fi nd the next two numbers?3
34 29 24
Now try fi lling in the missing numbers in this sequence.4
9 30-12 2
This sequence involves subtracting the same number every time to get to the next term.5
3587
Can you fi nd the missing numbers?
Complete this: The rule for this sequence is ‘subtract ’
Impossible!
d
44 6856
19 14
-5 16 23
74 61 48
13
34
I’m confi dent I’m nearly thereI can solve missing number problems.
Missing number problemsC
John is three years older than Lisa. We can write a formula for this: J = L + 3 (J is John’s age and L is Lisa’s age)
If Lisa is ten (L = 10) then J = L + 3 = 10 + 3 = 13 John is 13
Try these:
Josh is six years older than Amy. If Josh is twelve, use this formula to work out Amy’s age.
1
J
12
=
=
+
+
6
6
A
So Amy is years old.
Michael is eleven years younger than Fran. If Michael is 8, how old is Fran?
M
8
=
=
–
–
11
11
F
So Fran is years old.
If4 –3 13 =x 44 What is the missing number?
2 The length of a box can be found using this formula.
4 x l = 64 cm
How long is the box? cmlength (l)
Harder!
The perimeter (P) of a rectangle can be found using this formula.
P = 2a + 2b
If a is 8 cm and b = 12 cm, what is the perimeter (P)?
3
P =
=
+
+
2 2 x
= cm
x
a
b
6
19
6
19
16
8
16
12
24 40
19
35
I’m confi dent I’m nearly thereI can fi nd pairs of numbers that work in equations
with two missing numbers.
Two missing numbersD
If A + B = 6, what values could the numbers A and B have?
0
6
6
1
5
6
2
4
6
3
3
6
4
2
6
5
1
6
6
0
6
A
B
A + B
A can be any whole number less than 7. B can be any whole number less than 9.
If A x B = 48 what are the values of A and B?
3
A = B =
Try this: If P x Q = 8 what values could the numbers P and Q have?1
P
Q
P x Q 8 8 8 8
If C + 2D is 12 can you list some possible whole number values for C and D? How many can you fi nd?
2
C
D
C + 2D 12 12 12 12 12 12 12
6
0
In a card game is worth 8 points.
Zac has these cards and a total of 28 points.
If is worth three times as many points as , how many
points is worth?
4 Help!
2 814
4 182
2
5
4
4
6
3
8
2
10
1
12
0
6 8
3
+ = 12
= 3 x
= 3
36
I’m confi dent I’m nearly thereI can answer word questions about algebra.
E Word questions
1 Lucy had a £1 coin and she spent 47 p. Which formula shows how much change (C) she got? Circle your answer.
C = £1 + 47 p C = 47 p – £1 C = £1 – 47 p
2 If you have £5 and you spend £2.64 can you write a formula to show how much change (C) you would get?
C =
3 What is the next number in this sequence?
Three, eleven, nineteen, twenty-seven,
Complete this: The rule for this sequence is .
4 Look at this equation: A + B = 29
If A is thirteen what is the value of B?
5 Ali is fi ve years older than Yogi. Using A for Ali’s age and Y for Yogi’s age, can you complete this formula?
= +
6 If P x Q = 12 what are the possible values of P and Q?
7 If A x B – 9 = 33 and if A is equal to six, what is the value of B?
Wow!
P
Q
P x Q 12 12 12 12 12 12
£5 − £2.64
35
d 8
16
A 5 Y
7
1
12
2
6
3
4
4
3
6
2
12
1
13 + B = 29
6 x B − 9 = 33
6 x B = 42
3, 11, 19, 27
+ 8
37
Measurement
A Units of measurement
Jake’s summer holiday lasted for 6 weeks and 3 days. How many days was this in total?
Answer: 6 weeks is 6 x 7 = 42 days. So 6 weeks and 3 days is 45 days.
I’m confi dent I’m nearly thereI can convert between different units.
Now try these.2
255 g is
6500 mm is
kg
m
2.4 m is
0.5 litres is
cm
ml
Five people shared a litre bottle of lemonade.
How much did each person get? Give your answer in millilitres.
3
ml
There are approximately 8 km in 5 miles. Can you complete this conversion chart?5
miles
kilometres
5
24
20
40
Try these to get started:1
There are
There are
There are
ml
cm
g
in a litre
in a metre and
in a kilogram
mm in a metre
6 Can you use short division to calculate approximately how many kilometres there are in 1 mile?
Phew...
4 Can you work out how many seconds are in 1 hour? (This will take 2 steps.)
seconds
Answer:
1000
100
1000
1000
0.255
6.5
240
500
200
3600
15
8 32
25
1.6 m
60 n 1
60 n 1 r
o 60 x 60 n 1 r
6.8 0
13
5
.
38
I’m confi dent I’m nearly thereI can use scales and graphs to help me do some conversions.
B Using scales and graphs to convert units
We can see from this scale that 2 cm = 20 mm.
0 10 20 30
0 1 2 3
mm
cm
Can you complete this to show how centimetres can be converted to metres?1
0
0 1 2
cm
m
Look at this diagram. It shows how inches can be converted into millimetres.
Find the values of , and in mm. (The fi rst one is done for you.)
3
0
0 1 2 30.25 0.5 0.75 1.25 1.5 2.251.75 2.5 2.75
6.35 12.7 19.0525.4 50.8 76.2
38.1BA C mm
inches
121110987654321
1 2 3 4 50
6
2Look at this graph. It shows how you can convert between miles and kilometres.
The dotted lines show that 5 miles = 8 km
Use a ruler and draw your own dotted lines to show how to do this conversion.
km
miles
1 7
5 4.+ 5
5.3
3
2
Ok!
6 .
B 1.75 inches 1 inch
mm
+
+
=
=
inch
mm = mm
C 2.5 inches +
+
=
=
inch
mm
2 inches
mm = mm
A 1.25 inches 1 inch
25.4 mm
+
+
0.25 inch
6.35 mm
=
= = 31.75 mm
BA C
1
100 200 300 400
43
2.5 miles = km4
0.75
0.5
19.0525.4 44.45
12.750.8 63.5
5 42
9 0
4 44
+
.
5
51
1 .
.
0 85
2 7
3 56
+
.
1
1 .
.
39
C Areas and perimeters
Question: Do all shapes that have the same perimeter have the same area? Soon you will know!
I’m confi dent I’m nearly thereI know that shapes can have the same areas
but different perimeters and vice versa.
Can you draw a shape with the same area as the shape below but a different perimeter?3
12 cm2Area =
Perimeter = cm
12 cm2Area =
Perimeter = cm
2
Look at these shapes. Which shapes have the same area?
Which shapes have the same perimeter?
and
and
6 cm
2 cm A
4 cm
3 cm B
2 cm
5 cm
2 cm1 cm
C
Work out the areas and perimeters of these two new shapes.1
rectangle
6 cm
2 cmsquare
4 cm
4 cm
cm2Area = cm Perimeter = cm2Area = cm Perimeter =
Complete this sentence. These shapes have the same but different .
Ouch!
16 14
16 16 12 16
A
B
B
C
40
I’m confi dent I’m nearly thereI can calculate the areas of triangles and parallelograms.
D Calculating areas of triangles and parallelograms
Area of triangle = base x height1
2
base
height
base
height
Area of parallelogram = base x height
Try calculating the areas of these triangles.1
10 cm
4 cm
12 cm
5 cm
Area x
x=
=
=cm2
height
4
base
x 10
1
2
1
2
Area x
x=
=
=cm2
base
x
1
2
1
2
height
6 cm
3 cm
8 cm
2 cm
Now try calculating the areas of these parallelograms.2
Area x
x=
=
=cm2
height
Area x
x=
=
=cm2
3 Some shapes are made up of triangles and parallelograms.
Can you fi nd the area of this shape?
Hint: Make a parallelogram and a triangle.
12 cm
4 cm
8 cm
Funtime!
Answer:
20
30512
183
base
6
162
base
8
t
40 m2
A a of a = x x t
= x 4 x 4
= 8 m2
A a of a m = x t
= 8 x 4 = 32 m2
T l a = 8 + 32 = 40 m2
1212
4 m 8 m
41
I’m confi dent I’m nearly thereI can calculate the volume of cuboids.
E Volumes Remember that the volume of a cuboid can be found using the formula:
Volume = length x width x height
1
3 cm
3 cm
2 cm
Volume =
=
=
x
x
x
x
cm3
length width height
Can you work out the volume of this cuboid?
3 A block of concrete measured 6 m by 3 m by 4 m.
What is its volume?
Think about units in your answer!
4 A sheet of steel measures 2 cm by 3 m by 3 m.
What is its volume? 3 m 3 m
2 cmAnswer:
Help!
2 Which of these cuboids A or B has the biggest volume?
Answer:
3 mm
5 mm
5 mm
4 mm
4 mm
4 mm
AB
3 2 3
18
A
72 m3
0.18 m3
V A =
5 m x 5 m x 3 m
= 75 m3
V B =
4 m x 4 m x 4 m
= 64 m3
2 m = 0.02 m
V =
0.02 m x 3 m x 3 m = 0.18 m3
V of =
6 m x 3 m x 4 m
= 72 m3
42
I’m confi dent I’m nearly thereI can answer word questions about measurement.
F Word questions
1 How would you write two thousand, six hundred and thirteen grams in kilograms?
kg
2 Eight people shared a kilogram of chocolate; how many grams of chocolate did each person get?
g
3 Zac needed some 14 cm lengths of string. His mum gave him a metre of string.
How many 14 cm pieces could he cut from it?
4 If one inch is 25.4 mm, how many millimetres are there in 12 inches?
mm
6 A rectangle has sides that measure 4 cm and 9 cm. A square has the same area as the rectangle.
How long would the square’s sides be?
5 A rectangle has sides that measure 18 mm and 9 mm.
What is the perimeter of the rectangle?
7 Can you write down the formula for the volume of a cuboid?
= x
8 The volume of a cuboid is 36 cm3. It is 3 cm long and 3 cm wide.
What is the height of the cuboid? cm OK!
= x x
2.613
125
7
54 m
6 m
304.8
h h t
4
2613 g
10 = 254.0 m
2 = 50.8 m
12 = 304.8 m
a of =
4 m x 9 m = 36 m2
36 = 3 x 3 x ?
36 = 9 x ?
r 27
1 0 041
2
1 08 02
0451
43
Geometry
A Drawing 2D shapes You will need a pencil, ruler, protractor and set square.
I’m confi dent I’m nearly thereI can draw some 2D shapes.
Draw this triangle to scale.
One line is drawn for you.
1
30 o
10 cm
Can you draw a rectangle with sides 45 mm and 85 mm in the space below?2
Try to draw a hexagon with these measurements.
One line is drawn for you.
3
120 o
3 cm
3 cm
10 cm
Not again!
44
3D shapesB
I’m confi dent I’m nearly thereI can recognise 3D shapes and make nets for them.
What 3D shape can you build using this net?
Answer: A cylinder
Can you spot which 3D shapes these nets are for?1
Jon wanted to make his own dice for a board game. 2
21
3
56
4 Can you write the numbers in their correct places on this net?
6
3Look at this square based pyramid.
Can you draw a net for it?
4 Try to draw a net for this 3D shape.
Hint: You can copy it on to a piece of paper, cut it out and try it if you want to!
Woah!
Hint: Opposite sides always add up to 7.
d r m
1 2 54
3
45
I’m confi dent I’m nearly thereI know some properties of shapes.
Properties of shapesC
The angles inside a quadrilateral
add up to 360o140o
40o
The angles inside a triangle add
up to 180o60o
30oRemember:
Which of these shapes have at least 2 pairs of parallel sides? Circle your answers.1
square pentagon hexagon rhombus trapezium
2 What quadrilateral is being described here?
It has 4 sides. Opposite sides are the same length and opposite angles are equal.
No!
This is a regular pentagon.
Can you write down the angle P and the length Q?
4
108o
PQ
o
Angle =P
cm
Length =Q
2 cm
3 Can you fi nd the missing angles?
50o
90oo
80o
120o
110o
=Ao=B
A
B
5
B
A70o
100o
15o
30o
Can you fi nd angles A and B?
o
Angle =A
o
Angle =B
m
40 50
2108
5095
70o + 15o + A = 180o
100o+ 30o + B = 180o
A of a
d p o 180o
A of a l
d p o 360o
46
I’m confi dent I’m nearly thereI can fi nd missing angles.
AnglesD
Can you fi nd angle B without measuring it?
2
oAngle =B
6 cm
6 cm
135oB
1 Without measuring it, work out the missing angle in this diagram.
A
130o
110o
90o
o
Angle =A
80o
140o140o
The angles meeting at
a point add up to 360o
Angles on a straight line add
up to 180o120o
60o
With two straight lines
vertically opposite
angles are equal
120o 120o
60o
60o
Can you write down angles C and D?3
152o
28o
C
D oAngle =C oAngle =D
4 Look at this regular hexagon. What are angles A and B?
o 60o
B
A
oAngle =A oAngle =B
What is the sum of all the interior angles in a hexagon?
Hint: B is an interior angle.
5 Look at this regular pentagon.
Can you work out angle A?
Now can you work out angle B?
Finally, what is angle C?
o
o
o
Wow!
AB
C
30
45
28 152
12060
720
72
54
108
360o = 110o + 130o + 90o + A
360o = 330o + A
120o x 6 = 720
A = 360o ÷ 5 = 72o
72o + B + B = 180o
B + B = 108o
A = 30o
B = 54o
47
E Properties of circles You will need to understand the words circumference, radius and diameter.
Draw arrows pointing to the parts of this circle.
1 circumference
diameter
radius
A circle has a diameter of 67 mm, what is the radius?2 mm
4 The diameter of this wheel with no tyre is 48 cm.
The radius of the wheel with its tyre is 30 cm.
cm
Here we go!
B
A3 In a maze Billy started at A and followed the path shown back to A.
If the radius of the maze is 6 m and the circumference is approximately 38 m, how far did Billy walk?
m
I’m confi dent I’m nearly thereI know what is meant by the radius, diameter and
circumference of a circle.
Can you work out the depth of the tyre?
depth of tyre
6
31
33.5
A o B g
= of 38 = 19 m
R h o e =48 m ÷ 2 = 24 m
D h of e =
30 m − 24 m = 6 m
12
B o A = 2 x
= 2 x 6 m = 12 m
48
I’m confi dent I’m nearly thereI can use coordinates in 4 quadrants.
1 Look at the grid to the left.
What are the coordinates of points C and D?
C has coordinates
D has coordinates
,
,
F Coordinates
3
2
1
-1
-2
-3
1 32-2-3 -1
y
x
B
A
D
C
A has coordinates (3 , 2)B has coordinates (3 , -2)
-3 2
-3 -2
Look at this shape. It is an isosceles triangle. What are the coordinates of point A?
A has coordinates:
3
,
6
5
4
3
2
1
-1
-2
-3
1 32-2-3 -1
y
x
Look at this parallelogram. What are the coordinates of corner P?
P has coordinates:
4
,
A
y
x
(-4 , 6) (8 , 6)
(6 , -5)P
Here we go...
2 -3 -6 -5
Plot these points on the grid and join them up to make an irregular hexagon.
(1 , 3) (2 , 0) (1 , -3)
(-1 , 3) (-2 , 0) (-1 , -3)
4
3
2
1
-1
-2
-3
-4
1 32 4-4 -2-3 -1
2y
x
49
I’m confi dent I’m nearly thereI can translate shapes and refl ect shapes in the axes of a grid.
G Translations and refl ections
Reminder: A translation is the same as sliding an object without rotating or refl ecting it.
1
1 32 4-4 -2-3 -1
y
x
5
4
3
2
1
-1
-2
-3
-4
-5
5-5
T
If triangle T is translated 2 units in the y direction and 1 unit in the x direction, what will its coordinates be?
, ,
,
If the triangle T in question 1 is translated -4 units in the x direction and -6 units in y direction,
what will its coordinates be?
2
, , ,
3
4
3
2
1
-1
-2
-3
-4
1 32 4-4 -2-3 -1
y
x
Refl ect this shape in the y-axis and draw the refl ection.
What are the coordinates of the vertices of the refl ected shape?
, ,
, ,
4
Describe in detail how you would get shape Z to the new position shown using refl ections and translations.
Look at shape Z on this grid.
1 32 4-4 -2-3 -1
y
x
5
4
3
2
1
-1
-2
-3
-4
-5
5-5
Is this a joke?Z
3 3 5 3
4 5
-2 -5 0 -5 -1 -3
2 0
2 4
3 2
1 2
R t n y-a
n 3 n x n
d -7 n y n.
50
I’m confi dent I’m nearly thereI can answer word questions about geometry.
H Word questions
1 Can you draw a rectangle with sides 15 mm and 85 mm in the space to the right?
2 Jack said that this is a net for a cuboid.
Can you draw it again and make it correct?
3 Two of the angles inside a triangle are 42o and 46o.
What is the third angle inside the triangle?
4 What shape is this? “It has four sides, two of them are parallel and the angles inside it are all different.”
5 Can you fi nish this sentence about quadrilaterals?
“The angles inside a quadrilateral add up to .”
6 A javelin is stuck in the ground at an angle of 62o. What would angle A be?62o A
7 Five logs were used to make a raft. The radius of each log was 27 cm.
How wide was the raft?width
OK!
92o
360o
118o
270 m
m
A of a
d p o 180o
A n a t
d p o 180o
D r of 1 g = 54 m
54 x 5 = 270 r
W h = 10 x
= 10 x 27 = 270
51
Statistics
A Pie charts
pet
nopet
90o
This pie chart shows that or 25% of the children in a class have no pets.
The angle you need to show will be = 90o
1
4
1
4
360
4
I’m confi dent I’m nearly thereI can take information from pie charts and I can make pie charts.
Hint: Think about a clock!
Let’sgo!
The eye colours in Class 6 are shown below:
Work out the angles for each sector and use a protractor to draw a pie chart.
3
Colour
Brown
Blue
Number of children
20
10
Angle
60 children were asked to name their favourite colour.
Their answers are shown in this table.
Can you fi nish the pie chart to show this information?
2 Favouritecolour
Blue
Red
Yellow
Orange
Pink
Number of children
15
5
10
20
10
The children in Class 6 were asked to choose their favourite sports. The pie chart shows the results.
What percentage of the children chose football?
What fraction of the children chose swimming?
What is the angle at the centre of the cricket and hockey sectors if they are both the same size?
1
football
swimmingcricket
hockey%
o
120o
240o
50
45
14
d
w
k
n
120o240o
52
I’m confi dent I’m nearly thereI can use line graphs.
B Line graphs
A line graph is a good way to show how something changes as time goes by.
1
140
120
100
80
60
40
20
1 2 3 4 5 6 7 8 9 100
XX
X XX X X
XX
XX
This graph shows how Sara’s height changed each year from when she was born.
How tall was she when she was 10?
Approximately how tall was she when she was 4?
By how much did Sara grow from when she was born to when she was 8?
cm
cm
cm
Age (years)
Hei
ght
(cm
)
3
Jack heated some water in a pan.
He used a thermometer to measure the temperature of the water every 15 seconds.
Can you plot a graph to show Jack’s results?
2100
90
80
70
60
50
40
30
20
10
15 30 45 60 75 90 105 1200
Time (seconds)
Tem
pera
ture
(o C
)
Time (secs)
Temp(oC)
0
20
15
25
30
40
45
55
60
70
75
80
90
90
105
95
120
100
Not again!
150
100
90
80
70
60
50
40
30
20
10
15 30 45 60 75 90 105 120 1350
XX
XX
X
XX
Time (seconds)
Tem
pera
ture
(o C
) Jill repeated Jack’s experiment in question 2, but she ran out of time and had to stop after 90 seconds.
Look at the line graph of her results.
Can you predict how long it would have taken for her water to boil at 100 oC?
secs
165
140
90
80
150
XX
X
X
X
X
XX
X
53
I’m confi dent I’m nearly thereI can calculate the mean of a set of numbers.
C Averages - the ‘mean’
The mean is a type of average. To work out the mean we add up all the numbers and then divide the total by how many numbers there are.
Four friends started collecting World Cup cards. Jon had 12, Lucy had 8, Ben had 6 and Sara had 14.
How many cards did they have between them?
Complete this to work out the mean number of cards.
1
mean =
=
÷
÷
total number of cards number of people
=
mean =
In a café Siân buys 2 cups of tea at £1.20 each, 1 cup of coffee at £1.40 and 2 cups of hot chocolate at £1.60 each.
How many drinks did she buy altogether?
How much did she spend altogether?
What was the average (mean) cost of each drink?
4
£
£
Yes!
Can you fi nd the mean of these groups of numbers?3
18 20 24 26 32 mean =
18 20 24 24 24 26 32 mean =
Can you fi nd the mean of this group of numbers?2
6 8 9 4 7 2
40
40 4 10
6
24
24
5
7.00
1.40
12 + 8 + 6 + 14 = 40
8 + 2 + 6 + 4 + 9 + 7 = 36
36 ÷ 6 = 6
120 ÷ 5 = 24
168 ÷ 7 = 24
£2.40 + £1.40 + £3.20
= £7.00
4
7 0
12
5
.
. 0
54
I’m confi dent I’m nearly thereI can answer word questions about statistics.
D Word questions
1 What fraction of children chose orange?
2 What fraction of children chose red?
3 What percentage of children chose blue?
4 If there were 28 children in the classhow many chose either blue or orange?
5 What percentage of children did not choose blue or orange?
6 What fraction of children chose yellow?
7 What percentage of children chose yellow?
orange
red
yellowblue
36o
54o
Look at this pie chart which shows the favourite colours of the children in Class 2.
Oh!
25 %
21
25 %
15 %
110
320
36o = of 360o110
= 25 %14
= 25 %14
of 28 = 2134
54360
27180
320= =
320
15100= = 15 %
12
55
Maths facts that children are expected to know by the end of Year 6
By the end of Year 6, children are expected to know the facts in this section. Parents can help by regularly asking questions to test their children's ability to recall these facts.
A little and often is often the best approach.
Appendix - Information for Parents
The times tables up to 12 x 12
Year 6 provides a good opportunity for children to consolidate their multiplication and associated division facts.
They should also try to increase their speed of recall.
A
0123456789
101112
xxxxxxxxxxxxx
1111111111111
=============
0123456789101112
1111111111111
xxxxxxxxxxxxx
0123456789101112
=============
0123456789101112
0123456789
101112
xxxxxxxxxxxxx
2222222222222
=============
024681012141618202224
2222222222222
xxxxxxxxxxxxx
0123456789101112
=============
024681012141618202224
0123456789
101112
xxxxxxxxxxxxx
3333333333333
=============
0369121518212427303336
3333333333333
xxxxxxxxxxxxx
0123456789101112
=============
0369121518212427303336
0123456789
101112
xxxxxxxxxxxxx
4444444444444
=============
04812162024283236404448
4444444444444
xxxxxxxxxxxxx
0123456789101112
=============
04812162024283236404448
56
0123456789
101112
xxxxxxxxxxxxx
5555555555555
=============
051015202530354045505560
5555555555555
xxxxxxxxxxxxx
0123456789101112
=============
051015202530354045505560
0123456789
101112
xxxxxxxxxxxxx
6666666666666
=============
061218243036424854606672
6666666666666
xxxxxxxxxxxxx
0123456789101112
=============
061218243036424854606672
0123456789
101112
xxxxxxxxxxxxx
7777777777777
=============
071421283542495663707784
7777777777777
xxxxxxxxxxxxx
0123456789101112
=============
071421283542495663707784
0123456789
101112
xxxxxxxxxxxxx
8888888888888
=============
081624324048566472808896
8888888888888
xxxxxxxxxxxxx
0123456789101112
=============
081624324048566472808896
0123456789
101112
xxxxxxxxxxxxx
9999999999999
=============
0918273645546372819099108
9999999999999
xxxxxxxxxxxxx
0123456789101112
=============
0918273645546372819099108
0123456789
101112
xxxxxxxxxxxxx
10101010101010101010101010
=============
0102030405060708090100110120
10101010101010101010101010
xxxxxxxxxxxxx
0123456789101112
=============
0102030405060708090100110120
0123456789
101112
xxxxxxxxxxxxx
11111111111111111111111111
=============
0112233445566778899110121132
11111111111111111111111111
xxxxxxxxxxxxx
0123456789101112
=============
0112233445566778899110121132
0123456789
101112
xxxxxxxxxxxxx
12121212121212121212121212
=============
01224364860728496108120132144
12121212121212121212121212
xxxxxxxxxxxxx
0123456789101112
=============
01224364860728496108120132144
57
Common factors
Children should be able to fi nd the factors of two numbers and say which factors the numbers have in common.
Example: 24 has factors 1 2 3 4 6 8 12 24
32 has factors 1 2 4 8 16 32
So the common factors of 24 and 32 are 1 2 4 and 8.
B
Decimals, fractions and percentages
By the end of Year 6 children should know these facts and be able to recall them instantly.
C
=
=
=
0.5
0.25
0.75
1
2
1
4
3
4
=
=
=
0.1
0.2
0.3
etc.
1
10
2
10
3
10
=
=
=
=
0.2
0.4
0.6
0.8
2
10
4
10
6
10
8
10
=
=
=
=
1
5
2
5
3
5
4
5
Try to use the following vocabulary:
How many tenths is 0.8?
How many hundredths is 0.12?
Write 0.75 as a fraction.
What is 75% as a fraction / as a decimal?
Change to a decimal.1
4
=
=
=
0.01
0.21
0.99
etc.
1
100
21
100
99
100
=
=
=
=
=
0.01
0.1
0.25
0.5
0.75
=
=
=
=
=
1%
10%
25%
50%
75%
1
100
10
100
25
100
50
100
75
100
=
=
=
1
4
1
2
3
4
Prime numbers and composite numbers
Children should know the prime numbers up to 50 which are:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
and know that a prime number has no factors apart from 1 and itself.
They should also know that numbers that are not prime numbers are composite numbers.
A composite number does have other factors apart from 1 and itself.