Y7 – Autumn – Block 2 – Step 1 – Single function machines (number) Answers
Question Answer
1
2
a) 5b) 2c) 15d) 50
3
a)
b)
c)
d)
4 Tommy has doubled 8 instead of squaring it.
5
a)
b)
c)
d)
e)
f)
input output
39
952
133
856
56
56
56
22.5
22.5
22.5
22.5
22.5
22.5
72.5
–27.5
27.5
506.25
45
11.25
Y7 – Autumn – Block 2 – Step 1 – Single function machines (number) Answers (continued)
Question Answer
6
7
a) multiple possible answers, e.g. 1b) multiple possible answers, e.g. 2,000
c) 18,0007
8 Dora is correct because all the operations are equivalent.
9
a) possible answers: ÷ 8, × 18, × 0.125, + 56, squared
b) multiple possible answers, e.g.:
input 2, output 8, operations × 4, ÷ 14, cubed
19,198
13,818
10,201
19,999.8 ✔
Y7 – Autumn – Block 2 – Step 2 – Find the input given the output Answers
Question Answer
1
The inverse of addition is subtraction.The inverse of division is multiplication.The inverse of squaring is square root.
2
a) 10b) 30c) 2d) 200inverse operation
3 Annie has squared the output instead of using the inverse operation of square root.
4
a) 93b) 324c) 17d) 6e) 28.8f) 4.5
5
a)
b)
c)
d)
e)
f)
328
80
800
459
600
20
400
400
400
400
400
400
Y7 – Autumn – Block 2 – Step 2 – Find the input given the output Answers (continued)
Question Answer
6
a)
b)
c)
7
a) 29b) 1c) 5.5d) 100
8
a) 4b) 12
c) 14
d) – 12
9
multiple possible answers, e.g.:a)
b)
c)
d)
78.2
78.2
78.2
178.2
–21.8
21.8
20 × 5 28 + 72
400 ÷ 4 10,000
61 + 39 68 + 32
–1 + 101 –10 squared
Y7 – Autumn – Block 2 – Step 3 – Use letters to generalise number Answers
Question Answer
1
a) 5 + 5 + 5 = 3 × 5 = 5 × 3b) 7 + 7 + 7 + 7 = 4 × 7 = 7 × 4c)
d) 9 + 9 + 9 + 9 + 9 = 5 × 9 = 9 × 5e) c + c + c + c + c = 5 × c = c × 5f) d + d + d + d = 4 × d = d × 4
2 The other bar models represent:m + 4 (4 more than m)4 + m (m more than 4)
3multiple possible answers, e.g.:
a + a = 2aa means the same, so is equivalent, but is not the proper way of writing 2 lots of a
4a) The bar model shows 3 lots of 2b.b) 2b × 3 = 2b + 2b + 2b = 6b
5
a) 3gb) 3gc) !
3d) 3
!e) 7yf) jkg) k2h) 10di) 21aj) 20ck) 5a2l) 12r
4 4
✔
a a
Y7 – Autumn – Block 2 – Step 3 – Use letters to generalise number Answers (continued)
Question Answer
6
a) 2 × mb) m ÷ 2c) m squaredd) 2 ÷ me) g × hf) h ÷ g
7
a) "7
b) 3rc) 11pd) t2
e) #$
f) 3py
8In multiplication, the order of the terms does not matter. Multiplication is commutative.No, in division the order of the terms does matter. Division is not commutative.
Y7 – Autumn – Block 2 – Step 4 – Single function machines (algebra) Answers
Question Answer
1
a) 4gb) g + 4c) 4cd) c + 4e) 12nf) 3n + 4
2
a)b)c) ad) 2ae) 50cf) !
2
3
a) Worked out 2 + 3 instead of 2 × 3 6bb) Taken the 5 from 5y 5y – 5 c) Added 4 to the 3 in 3a 3a + 4d) Only 4a has been divided by 2 2a + 1
4
a) f – 7b) 4h – 4 c) r + 6d) 6r + 2
5
a)
b)
c)
20
20b
4b + 8
20b + 8
9
a + 6
2a + 6
2a + 11
5
3a
15𝒂2
3a + 5
Y7 – Autumn – Block 2 – Step 4 – Single function machines (algebra) Answers (continued)
Question Answer
6
a)
b)
c)
d)
7multiple possible answers, e.g.:2g + 1 → × 4 → 8g + 4
3a
9a
a
b – 2
2b
𝒃𝟐
d – 3
12c + 8
48c
3c
2d – 3
100d – 5
Y7 – Autumn – Block 2 – Step 5 – Find functions from expressions (single) Answers
Question Answer
1
a) + 5b) – 2c) × 4d) + 7e) – 5f) ÷ 3
2 Both functions involve 3 but the operations are different.
3
a)
The other function machine has added two terms that are not like terms.b)
The other function term has only multiplied the coefficients of b.
4
a) + gb) × fc) × yd) ÷ 5e) × rf) × 2bg) ÷ rh) – 5Part c) could also be ‘squared’.Part d) could also be ‘square root’.
5Ronh + 1 + 4h = 5h + 1, but 5 × (h + 1) = 5h + 5
× 3 + 3
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Y7 – Autumn – Block 2 – Step 5 – Find functions from expressions (single) Answers (continued)
Question Answer
6
a)
b)
7
a) × 3, + 8cb) ÷ 5, – 12yc) ÷ 20, – 28.5h
Other function machines with equivalent operations are possible, e.g. ÷ 13 instead of × 3
× 3
4
3c
6d
21f
3a + 3
÷ 4
4a2c54
14
28pb4
3y – 1
Y7 – Autumn – Block 2 – Step 6 – Substitute into single expressions Answers
Question Answer
1
a) 21b) 10c) 35d) 12e) 34f) 3.5
2a) Yes, in addition the order of the terms does not matter. Addition is commutative.b) No, in division the order of the terms does matter. Division is not commutative.
3 Rosie has worked out 12 – 7. The answer should be –5
4
a) p + 7 = 23p + 9.5 = 25.56 + p = 22p + p = 32
b) p – 3 = 13p – 20 = –420 – p = 4p – p = 0
c) 4p = 645p = 803.9p = 62.4p2 = 256
d) !2 = 8!
3.2 = 540! = 2.5
𝑝 = 4
5
a)
b) y + 5 and 5 + yc) If y = 1, y – 5 = –4 and 5 – y = 4d) y + 5 and 5 + y
Y7 – Autumn – Block 2 – Step 7 – 2-step function machines (number) Answers
Question Answer
1
a) 9b) 20c) 5d) 27
2The order of operations is different. If the input was 1, the output on the left would be 10, but the output on the right would be 28
3
a) 40b) 33c) 26d) 20e) 1f) 10g) 106.2h) 66.2i) –5j) –27
4
1 × 4 = 4, 4 + 8 = 12 ≠ 36No, she did not work backwards. She needs to do the inverse operations in the other order:36 – 8 = 2828 ÷ 4 = 7The input is 7
5
a) 7b) 7c) 38d) –1e) 6.2f) 10.75
6
a)
b) 39c)
d) 45.8e) Children can check that their partner is working out the inverse operations in the correct
order.
input
× 2 + 2
output
80
input
÷ 2 – 7
output
15.9
Y7 – Autumn – Block 2 – Step 8 – 2-step machines (algebra) Answers
Question Answer
1
a)
The other function machine has added 5 green blocks instead of 5 units after multiplying by 2
b) The function machines both show 2 × the input + 5 units, but the top function shows them as 5 separate units, whereas the bottom function machine shows them in a single bar.
c) different possible ways of representing this, e.g.:or
2
a) 6 has been added to the 4 of 4p. The answer should be 4p + 6b) Only g has been multiplied by 4, not the 6 as well. The answer should be 4g + 24c) Only 6 has been multiplied by 4, not the m as well. The answer should be 4m + 24
3
a) 4m + 1b) 4m + 4c) 6h – 2
4
bar model showing 3b + 6 = 3(b + 2)
5
a) 2.5d + 4b) 2e – 6.8c) !
2 + 1
6
a) 6g + 7
b) 2h + 34c) 7"
3 – 5
7
a) mb) 6rc) 9t
✔
5
5
Y7 – Autumn – Block 2 – Step 8 – 2-step machines (algebra) Answers (continued)
Question Answer
8
a)
b)
c) Function machines that use a combination of addition and subtraction or a combination of multiplication and division can be written as a single step.
✔
✔
x → × 5 → × 2 → 10x5x
x → × 10 → 10x
Y7 – Autumn – Block 2 – Step 9 – Find functions from expressions (2-step) Answers
Question Answer
1
a) + 3b) × 2c) – 5d) ÷ 2e) ÷ 4f) – 6
2
a) × 5 → + 2b) ÷ 5 → + 2c) ÷ 8 → + 6d) × 8 → + 6
3 Alex has put the operations the wrong way wrong. She should have put + 3, then × 7
4
a)
b)
c)
d)
Each pair of function machines uses the same operations but in the opposite order.
5
The function machines have the same operations but in the opposite order.
+ 2+ 2 × 6 × 6
× 8× 8 – 3 – 3
+ 4+ 4 ÷ 7 ÷ 7
÷ 6÷ 6 – 5 – 5
input output
y ÷ 3 + 5 !3 + 5
y
input
+ 5 ÷ 3
output
𝑦 + 53
Y7 – Autumn – Block 2 – Step 9 – Find functions from expressions (2-step) Answers (continued)
Question Answer
6
a)
b)
7a) × 2 → + 7b) ÷ 5 → + 4
8 2x is subtracted from 10, not subtract 10
× 5
+ 7
10b – 85c
3e4(2m + 7)
✔
Y7 – Autumn – Block 2 – Step 10 – Substitute into 2-step expressions Answers
Question Answer
1
a) 20b) 24c) 20d) 20
2
3
a) 3z + 7 = 252z – 5 = 710z + 6.3 = 66.3
b) 3(z + 7) = 392(z – 5) = 210(z + 5.3) = 113
c) 4 + z = 104 + 3z = 2220 – 2z = 8
d) ! " #5 = 1!3 + 2 = 4
3 + !2 = 6
4
a) She should have squared the 3 first.32 = 94 × 9 = 36
b) brackets: (4x)2c) 50
100100400
3
input
× 4 + 5
output
17
3
input
+ 5 × 4
output
32
Y7 – Autumn – Block 2 – Step 10 – Substitute into 2-step expressions Answers (continued)
Question Answer
5
a) 16.5b) 15.5c) –15.5d) 32
e) 0.03125 or 132
f) 8g) 44.5h) 4i) –4j) 256k) 768l) 2,304m) 4n) 16
6a) 5(m + 1) = 5m + 5b) 5m + 1 + 4 = 5m + 5
Y7 – Autumn – Block 2 – Step 11 – Generate sequences from a rule Answers
Question Answer
1
a)
b)
c)
2
a)
b)
c)
All the sequences go up by 5 each time, but the starting number is different.
3
a) –4, 2, 8, 14, 20b) 10, 13, 16, 19, 22c) –0.5, 0, 0.5, 1, 1.5d) 6.3, 10.3, 14.3, 18.3, 22.3e) 1.5, 1.75, 2, 2.25, 2.5f) 5.25, 5.5, 5.75, 6, 6.25Substitute n = 1 to 5 in the expressions. Some students may find the first term and then add the coefficient of n each time.
4a) 18, 16, 14, 12b) –3, –7, –11, –15
5
When the term in n is positive, the sequence is ascending.
6 The highest power of n is 1
4 8
7
13
5 9
12
206
10 155 2520
5 100 2015
12 177 2722
ascending descending ascending descending
✔ ✔ ✔ ✔
Y7 – Autumn – Block 2 – Step 11 – Generate sequences from a rule Answers (continued)
Question Answer
7
a) 3rdb) 190c) No, because all the terms are even.d) No
Solving the sequence rule for a value of 68 gives 4n = 68 + 10 = 78 and 78 is not a multiple of 4
8a) 770b) Any two terms that are 10 positions apart
Y7 – Autumn – Block 2 – Step 12 – Represent functions graphically Answers
Question Answer
1
a) line y = 3x + 1 plotted using a graphing programb) They have the same coordinates.c) The points for a sequences do not join up because we cannot have a decimal position of
a term.
2
a) They are both straight lines sloping upwards from left to right.b) They are both straight lines sloping downwards from left to right.c) They are both horizontal.
3The lines in part a), b) and c) are straight lines, so relate to linear sequences.The line in part d) is not a straight line, so relates to a non-linear sequence.
4
a) Both graphs go through (0, 0) and always slope upwards from left to right.y= 3xis a straight line and y= x3 is a curve.
b) y= 3xc) y= x3
5
6 The only power of x is 1
7
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