Why settle the sign before calculating?

For directed numbers, decide the sign first and then work with the magnitudes: same signs add and keep the sign, opposite signs take the sign of the larger magnitude, and multiplication or division gives positive for like signs and negative for unlike signs.

What order should be used when substituting?

Put brackets round every substituted value, then work through brackets, indices, multiplication and division, and finally addition and subtraction. Keep −3² (which is −9) apart from (−3)² (which is 9).

S1 Maths: Directed Numbers and Algebra (Challenge, 10 Questions)

2026-11-06

Tutor Prep Guide · S1 Mathematics

Ten challenge items on multi-step work and working backwards: index and bracket order with negative bases, evaluating a two-letter expression, the nth term of an arithmetic sequence, simplifying three brackets, recovering a repeated addition from its result, finding a product from a sum and a difference, and a temperature problem with directed numbers.

Quick answer Answers in order: 2, 29, 23, 8x – 17, 3, 6, 12, -21, 6 hours, 3a + 4b. Brackets and indices come first, then multiplication and division, then addition and subtraction.

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 45 minutes
Difficulty mix	Challenge 10
Chapter	Chapter: S1 Maths: Directed Numbers and Algebra
Level	S1

02 Questions (10 items)

ChallengeWork out (-2) to the fourth power divided by (-4), minus (-3) times 2.
ChallengeFind the value of (a + b) squared times c plus ab when a = -1, b = -2 and c = 3.
ChallengeA sequence runs -4, -1, 2, 5, and each term is 3 more than the one before. Find the tenth term.
ChallengeSimplify 3(2x – 3) – 2(4 – x).
ChallengeStarting from -5, the same number is added four times and the result is 7. Find the number.
ChallengeThree times a number minus 7 equals the number plus 5. Find the number.
ChallengeWork out minus 2 cubed times (-3) squared, divided by (-6).
ChallengeIf x + y = -4 and x – y = 10, find the value of xy.
ChallengeThe temperature is 8 °C at noon and falls to -7 °C by midnight, dropping 2.5 °C each hour on average. How many hours pass?
ChallengeSimplify (5a – 2b) – (3a – 7b) + (a – b).

03 Answers and one-line notes

#	Answer	One-line note
1	2	(-2) to the fourth power is 16, so 16 / (-4) = -4, and (-3) x 2 = -6, giving -4 – (-6) = 2.
2	29	a + b is -3, whose square is 9; 9 x 3 = 27 and ab = 2, so the value is 29.
3	23	The tenth term is the first term plus nine steps of 3, so -4 + 27 = 23.
4	8x – 17	Expanding gives 6x – 9 – 8 + 2x, which collects to 8x – 17.
5	3	The total increase is 7 – (-5) = 12, so each of the four steps adds 12 / 4 = 3.
6	6	Writing the number as n gives 3n – 7 = n + 5, so 2n = 12 and n = 6.
7	12	Minus 2 cubed is -8 and (-3) squared is 9, so -72 divided by -6 gives 12.
8	-21	Adding the two equations gives 2x = 6, so x = 3 and y = -7, which makes xy equal to -21.
9	6 hours	The total fall is 8 – (-7) = 15 °C, and dividing by 2.5 °C per hour gives 6 hours.
10	3a + 4b	Removing the brackets gives 5a – 2b – 3a + 7b + a – b, which collects to 3a + 4b.

04 What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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