Printable · GCSE Foundation · ages 14-16
Algebra worksheet — GCSE Foundation
Fifteen questions across the algebra statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Answer key: Algebra worksheet — GCSE Foundation
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- (c) −8 — (−2)³ = (−2) × (−2) × (−2) = −8, since multiplying three negative numbers gives a negative result. A candidate who forgets the sign of a negative number when cubing it might treat (−2)³ as if it were 2³ = 8. A candidate who multiplies −2 by 3 instead of cubing it might get −2 × 3 = −6. A candidate who combines both mistakes — multiplying by 3 and dropping the sign — might get 2 × 3 = 6.
- (a) −2 and 2 — Method: the graph crosses the x-axis where y = 0, so read the two crossing points straight off the curve. Working: the curve meets the horizontal axis two squares to the left of the origin and two squares to the right, at x = −2 and x = 2. Answer: −2 and 2. Distractor refutation: −4 and 4 comes from reading the value where the curve crosses the y-axis, −4, and using that value and its positive partner as the x-axis crossings instead. −2 and 3 comes from misreading the right-hand crossing point one square out along the grid, taking it where the curve is already above the axis. 0 and 2 comes from confusing the lowest point of the curve, which sits on the y-axis at x = 0, with one of the crossing points.
- (c) x + 18 — Expand −3(2x − 6) by multiplying both terms by −3: −3 × 2x = −6x and −3 × (−6) = 18, giving 7x − 6x + 18 = x + 18. Writing x − 18 comes from not flipping the sign of the −6 inside the bracket, so −3 × (−6) is treated as −18 instead of +18. Writing x + 6 comes from forgetting to multiply the −6 by 3, only carrying its sign. Writing 13x − 18 comes from treating the whole bracket as being added rather than subtracted, so 3(2x − 6) = 6x − 18 is added to 7x.
- (c) 28 — Method: substitute values of n into 5n − 2 and check which of the options matches. Working: for n = 6, 5 × 6 − 2 = 30 − 2 = 28, so 28 is a term of the sequence, the 6th term. Answer: 28. 27 comes from using the rule 5n − 3 instead of 5n − 2. 30 comes from using 5n on its own, forgetting to subtract 2 at all. 20 comes from applying the subtraction before the multiplication, working out 5 × (n − 2) instead of 5n − 2.
- (a) 14 — Substituting x = 3: y = 5(3) − 1 = 15 − 1 = 14. A candidate who stops after the multiplication and forgets to subtract 1 gets 15. A candidate who adds 1 instead of subtracting gets 16. A candidate who incorrectly treats the expression as 5 × (3 − 1) gets 10.
- (a) 216 — The cube numbers are formed by cubing 1, 2, 3, 4, 5, ..., so the next one is 6³ = 216. A candidate who squares 6 instead of cubing it gets 6² = 36. A candidate who assumes the sequence doubles each time gets 125 × 2 = 250. A candidate who uses 5 × 6² instead of 6 × 6 × 6 gets 180.
- (a) 27 — Method: let the smaller number be n, so the larger number is n + 9. Form the equation n + (n + 9) = 45. Working: simplify: 2n + 9 = 45, so 2n = 36, giving n = 18. The larger number is 18 + 9 = 27. Answer: 27. 18 is the smaller number, found correctly, but given as the answer to the question asking for the larger number. 22.5 comes from ignoring the "9 more" part and simply halving the total, 45 ÷ 2. 36 comes from a sign error when isolating the constant, treating the equation as 2n = 45 + 9 instead of 2n = 45 − 9.
- (b) A straight line. — y = 3x + 2 is a linear function, since the highest power of x is 1, so its graph is a straight line with gradient 3 and y-intercept 2. Saying it is a U-shaped curve confuses a linear graph with a quadratic graph, which has an x² term. Saying it decreases then increases describes a curve with a turning point, which a straight line does not have. Saying it gets closer to an axis but never reaches it describes a reciprocal graph, y = k/x, not a linear one.
- (c) −8 — The nth term is the first term plus (n − 1) lots of the common difference: 40 + 8 × (−6) = 40 − 48 = −8. A candidate who uses 9 lots of the common difference instead of 8 gets 40 + 9 × (−6) = −14. A candidate who treats the common difference as +6 instead of −6 gets 40 + 8 × 6 = 88. A candidate who uses only 7 lots of the common difference gets 40 + 7 × (−6) = −2.
- (b) c = k/6 — k = 6c means c has been multiplied by 6, so to make c the subject, divide both sides by 6: c = k/6. Writing c = 6k multiplies by 6 again instead of undoing the multiplication. Writing c = k − 6 mistakes multiplying for adding, and subtracts 6 rather than dividing. Writing c = 6 − k reverses the order as well as the operation. The correct rearrangement is c = k/6.
- (d) 2x = 10 — Method: collect the x-terms on one side and the constants on the other, keeping the equals sign balanced. Working: 5x − 3x = 8 + 2, giving 2x = 10. Answer: 2x = 10. 2x = 6 comes from a sign error moving the constant, subtracting instead of adding: 8 − 2 = 6. 8x = 6 comes from adding the x-terms instead of subtracting them (5x + 3x = 8x), alongside the constant sign error. 2x = −10 comes from a sign error moving the constant the other way: −2 − 8 = −10.
- (b) (−5, −2) — A rotation of 180° about the origin reverses the sign of both coordinates, so Q = (−5, −2). A candidate who reverses the sign of only the y-coordinate, as if reflecting in the x-axis, gets (5, −2). A candidate who reverses the sign of only the x-coordinate, as if reflecting in the y-axis, gets (−5, 2). A candidate who swaps the coordinates instead of reversing their signs gets (2, 5).
- (d) x = 4 or x = −9 — Since (x − 4)(x + 9) = 0, either x − 4 = 0, giving x = 4, or x + 9 = 0, giving x = −9. A candidate who takes the sign shown in each bracket rather than its opposite gets x = −4 or x = 9. A candidate who makes both solutions negative gets x = −4 or x = −9. A candidate who drops both signs and makes both solutions positive gets x = 4 or x = 9.
- (b) 6 — Method: for any point that lies on y = k/x, the value of k is found by multiplying the x-coordinate and the y-coordinate together, since k = x × y. Working: k = 2 × 3 = 6. Answer: k = 6. Distractor refutation: 1.5 comes from dividing the y-coordinate by the x-coordinate instead of multiplying them. 5 comes from adding the two coordinates instead of multiplying them. 9 comes from misreading the point's x-coordinate as 3 instead of 2, then multiplying 3 × 3.
- (b) 3, 6, 9, 12 — Method: substitute the positions n = 1, 2, 3 and 4 into the rule in turn, because a position-to-term rule gives each term from its own position number. Working: 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9 and 3 × 4 = 12. Answer: 3, 6, 9, 12. The distractors: 3, 9, 27, 81 comes from reading 3n as 3 multiplied by itself n times and so multiplying by 3 at every step; 0, 3, 6, 9 comes from starting the count at n = 0, which shifts every term one place; 4, 5, 6, 7 comes from reading 3n as n + 3 and adding 3 to each position number instead of multiplying by 3.
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