year 8 maths

1B
Substitution and equivalence
13
PROBLEM-SOLVING AND REASONING
12, 15 12–16
12, 15–18
12 Give three expressions that are equivalent to 2x + 4y + 5 .
13 Copy and complete the following table.
x 3 0.25 −2
4x + 2 14 6
4 − 3x −5 −2
2 x − 4 8
14 a Evaluate x 2 for the following values of x . Recall that (−3) 2 = −3 × −3 .
i 3 ii −3 iii 7 iv −7
b Given that 25 2 = 625 , what is the value of (−25) 2 ?
c Explain why the square of a number is the square of its negative.
d Why is (−3) 2 = 9 but −3 2 = −9 ? What effect do the brackets have?
15 a Explain with an example why a ÷ (b × c) is not equivalent to (a ÷ b) × c .
b Does this contradict the associative law (see Key ideas)? Justify your answer.
c Is a ÷ (b ÷ c) equivalent to (a ÷ b) ÷ c ? Why or why not?
16 Are the following statements true for all values of p?
(Hint: try substituting p = 0, p = 1, p = 2 and p = 3 into each statement.)
a p + p = 2p b p × p = 2p c 6p − p = 6
d p × p = p 2 e p − p = 0 f p ÷ p = 0
17 a By substituting a range of numbers for a and b , determine whether (ab) 2 is equivalent to a 2 b 2 .
b Is (a + b) 2 equivalent to a 2 + b 2 ? Why or why not?
c Is √ab equivalent to √a × √b ? Why or why not?
d Is √a + b equivalent to √a + √b ? Why or why not?
e For pairs of expressions in a – d that are not equivalent, find a few values for a and b that make
them equal.
18 Sometimes when two expressions are equivalent you can explain why they are equivalent. For
example, x + y is equivalent to y + x because ‘the order in which you add numbers does not
matter’, or ‘because addition is commutative’. For each of the following pairs of expressions, try to
describe why they are equivalent.
a x × y and y × x b x + x and 2x c y − y and 0
d
1
2 × x and x ÷ 2 e a × 3a and 3a2 f k 2 and (−k) 2
ENRICHMENT
– –
19
Missing values
19 Find the missing values in the table below.
a 5 8 −20
b 2 1
a + b 10 10 7 −19
a + 2 b 17 0 11
a − b 1 13
a −
2b
−
29 20
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
Photocopying is restricted under law and this material must not be transferred to another party.