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.
Algebra worksheet — GCSE Foundation
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- 1.Work out the equation of the straight line through the points (−3, 4) and (1, −8).
- 2.A rectangle has length (2x + 5) cm and width (x − 2) cm. Work out an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.
- 3.Solve the inequality 5 − x ≤ 2.
- 4.The point C is 6 units to the right of the origin and 0 units up. Write down the coordinates of C.
- 5.A plant is 4 cm tall in week 1. Each week after that, its height increases by 5 cm. Work out the first week in which the plant is taller than 30 cm.
- 6.Work out the value of 3x² − 4x when x = −2.
- 7.Solve 3x − 1 = 8
- 8.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x
- 9.The diagram shows a straight line passing through the origin, drawn on a numbered grid. Which of these points lies on the line?
- 10.The formula for the energy stored in a stretched spring is E = (1/2)kx², where E is in joules, k is the spring constant in N/m and x is the extension in m. A spring with a spring constant of 40 N/m is stretched so that its extension is 0.3 m. Work out the energy stored in the spring.
- 11.A quadratic curve has a minimum turning point at (3, −4). Which of these statements about the curve must be true?
- 12.A box contains 6 chocolates. Write down an expression for the total number of chocolates in c boxes.
- 13.Which expression means the same as 3 × a × b?
- 14.On Monday, a runner covers 15 km in 2.5 hours. On Tuesday, she covers 12 km in 1.5 hours. Using the formula speed = distance ÷ time, work out on which day she ran faster, and by how much.
- 15.Write down the coefficient of xy in the expression 4x²y − 7xy + 2.
Answer key
- (a) y = −3x − 5 — Gradient = (−8 − 4) ÷ (1 − (−3)) = −12 ÷ 4 = −3. Using the point (1, −8): −8 = −3(1) + c, so c = −5, giving y = −3x − 5. A candidate who drops the negative sign on the gradient, using m = 3 instead, would then solve −8 = 3(1) + c to get c = −11, writing y = 3x − 11. A candidate who makes a sign error isolating c, writing c = 5 instead of −5, would write y = −3x + 5. A candidate who mixes up both mistakes — keeping the correct gradient but the wrong, positive value of c from the flipped-gradient calculation — would write y = −3x + 11.
- (b) 6x + 6 — Perimeter = 2[(2x + 5) + (x − 2)] = 2(3x + 3) = 6x + 6. A candidate who adds the length and width but forgets to double for the perimeter gets 3x + 3. A candidate who makes a sign error and adds 2 instead of subtracting it before doubling gets 2[(2x + 5) + (x + 2)] = 6x + 14. A candidate who multiplies the length and width instead of adding them, confusing the perimeter formula with the area formula, and then doubles that product, gets 2(2x + 5)(x − 2) = 4x² + 2x − 20.
- (c) x ≥ 3 — Method: collect the number terms first; the x term is negative, so the final step multiplies both sides by −1, and that is the one step that turns the inequality sign round. Working: subtracting 5 from both sides of 5 − x ≤ 2 gives −x ≤ −3; multiplying both sides by −1 turns −x into x and −3 into 3, and because the multiplier is negative the ≤ becomes ≥, so x ≥ 3. Answer: x ≥ 3. The distractors: x ≤ 3 comes from multiplying by −1 without turning the sign round, the commonest slip on this type; x ≤ −3 comes from reading −x ≤ −3 as though the minus sign could simply be rubbed off the left-hand side; x ≥ −3 comes from turning the sign round correctly but leaving the right-hand side at −3 instead of multiplying it by −1 as well.
- (a) (6, 0) — Method: a pair of coordinates records movement from the origin, the across movement written first and the up or down movement second. Working: C is 6 units to the right of the origin, so the across number is 6; it is 0 units up, so the up number is 0; written in order that gives (6, 0), a point on the x-axis. Answer: (6, 0). The distractors: (0, 6) comes from writing the two movements the wrong way round; (0, 0) comes from reading '0 units up' as meaning the point never left the origin at all, which ignores the movement across; (7, 0) comes from counting the origin itself as the first unit while counting 6 units to the right.
- (b) Week 7 — The height after n weeks is 4 + 5(n − 1) = 5n − 1. The plant is taller than 30 cm when its height, 5n − 1, is more than 30, written 5n − 1 > 30; add 1 to both sides (5n > 31) and divide by 5, which gives n > 6.2, so the first whole week is week 7, where the height is 5 × 7 − 1 = 34 cm. Rounding 6.2 down to week 6 without checking gives a height of only 5 × 6 − 1 = 29 cm, which is not yet taller than 30 cm. Counting one week too many gives week 8. Working out 30 ÷ 5 = 6 and then subtracting 1 from the week number instead of from the height gives week 5.
- (b) 20 — 3x² − 4x = 3(−2)² − 4(−2) = 3(4) − (−8) = 12 + 8 = 20. A candidate who makes a sign error on −4x, treating −4 × −2 as −8 instead of +8, gets 12 − 8 = 4. A candidate who squares −2 but keeps the result negative, using 3 × (−4) = −12 for the first term, but correctly works out −4x = −4 × (−2) = 8, gets −12 + 8 = −4. A candidate who squares the whole term 3x, working out (3 × −2)² − 4 × (−2), gets (−6)² + 8 = 36 + 8 = 44.
- (d) x = 3 — Method: undo the operations on the left in reverse order — deal with the subtraction of 1 first, then with the multiplication by 3. Working: adding 1 to both sides gives 3x = 9, and dividing both sides by 3 gives x = 3. Answer: x = 3. The distractors: x = 9 comes from stopping at 3x = 9 and writing 9 as the value of x; x = 6 comes from subtracting 3 from 9 instead of dividing by 3; x = 27 comes from multiplying 9 by 3 instead of dividing by 3.
- (a) −10 — Method: multiply each bracket out, treating the second bracket as being multiplied by −3 because it is subtracted, then collect like terms. Working: 4(2x − 1) = 8x − 4 and −3(x + 2) = −3x − 6, so the expression becomes 8x − 4 − 3x − 6 − 5x; the x terms give 8x − 3x − 5x = 0, so no term in x survives, and the numbers give −4 − 6 = −10. Answer: −10. The distractors: 2 comes from expanding −3(x + 2) as −3x + 6, leaving the numbers −4 + 6; 5x − 10 comes from forgetting the final −5x, so the x terms give 8x − 3x = 5x; −7 comes from multiplying the 4 over only the first term of its bracket, giving 8x − 1 and so the numbers −1 − 6.
- (c) (2, 6) — Method: substitute the x-coordinate of each point into the rule for the line (y = 3 × x) and compare it with the point's y-coordinate. Working: the line passes through the origin and rises 3 squares for every 1 square across, so at x = 2 the line's y-value is 3 × 2 = 6, giving the point (2, 6). Answer: (2, 6). Distractor refutation: (2, 3) comes from counting only 3 squares up in total between the origin and x = 2, instead of 3 squares up for every 1 square across, halving the true rise. (3, 2) comes from swapping the x-coordinate and the y-coordinate round. (2, 5) comes from a miscounted gridline, landing one square below the line.
- (a) 1.8 — E = (1/2)kx² = 0.5 × 40 × 0.3² = 0.5 × 40 × 0.09 = 1.8 joules. 3.6 comes from forgetting the (1/2) at the front, 40 × 0.09. 72 comes from squaring kx together instead of squaring only x, 0.5 × (40 × 0.3)² = 0.5 × 144 = 72. 6 comes from using x instead of x², 0.5 × 40 × 0.3.
- (b) It crosses the x-axis, since the minimum is below it. — A minimum turning point at (3, −4) means the lowest value the curve reaches is y = −4, which is below the x-axis (y = 0); since the curve opens upward from there, it must rise up through y = 0 on both sides, crossing the x-axis twice. Saying it does not cross confuses 'the minimum is negative' with 'the whole curve stays negative' — a minimum below the axis guarantees the curve rises above it elsewhere. Saying it touches the x-axis once at (3, −4) mistakes the turning point itself for a root — the turning point is not on the x-axis at all here, since its y-coordinate is −4, not 0. Saying it is impossible to tell ignores that the two facts given — that the turning point is a minimum, and that its y-coordinate is negative — are together enough to decide the number of crossings without knowing the equation.
- (a) 6c — Each box has 6 chocolates, so c boxes have 6 × c = 6c chocolates. A candidate who adds the number of boxes to the number per box instead of multiplying gets 6 + c. A candidate who subtracts instead of multiplying gets c − 6. A candidate who divides instead of multiplying gets c ÷ 6.
- (a) 3ab — 3 × a × b means 3, a and b are all multiplied together, and in algebraic notation this is written with no multiplication signs: 3ab. Writing 3 + a + b turns every multiplication into an addition, giving a completely different expression. Writing a³b misreads the 3 as a power on a rather than as a coefficient in front of both letters. Writing 3a + b multiplies the 3 by a correctly but then adds b instead of also multiplying it in. The expression that means 3 × a × b is 3ab.
- (a) Tuesday, by 2 km/h — Monday's speed is 15 ÷ 2.5 = 6 km/h and Tuesday's speed is 12 ÷ 1.5 = 8 km/h, so Tuesday was faster, by 8 − 6 = 2 km/h. A candidate who works out the correct speeds but mislabels which day is faster gets Monday, by 2 km/h. A candidate who divides 15 ÷ 2.5 incorrectly as 5 instead of 6 gets a difference of 8 − 5 = 3 km/h, still crediting Tuesday. A candidate who forgets to find Monday's speed and gives Tuesday's speed itself as the difference states Tuesday, by 8 km/h.
- (a) −7 — The coefficient of xy is the number multiplying that exact term, including its sign: in 4x²y − 7xy + 2, the term in xy is −7xy, so the coefficient is −7. Answering 7 correctly picks out the number but drops the minus sign that belongs to it. Answering 4 picks the coefficient of the x²y term instead of the xy term — x²y and xy are different terms because their powers of x differ. Answering −5 combines −7 with the constant term +2, as though the coefficient of xy included the number with no letter attached to it. The coefficient of xy is −7.
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