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
MathsUKwww.geekhero.co.uk
- 1.Solve the inequality 5 − x ≤ 2.
- 2.Kai writes the expression 8 − 3x + 5x² and says it has three terms. Ryan says it only has two terms because a number on its own is not a term. Work out the correct number of terms in the expression.
- 3.Point P has coordinates (5, 2). Point P is rotated 180° about the origin to point Q. Write down the coordinates of Q.
- 4.In the expression 5x² − 3x + 7, which of these is a single term of the expression?
- 5.Work out the value of m² + 3m when m = −2
- 6.A vertical line passes through the point (4, 7). Write down the equation of this line.
- 7.Solve 2(x + 3) = 10
- 8.Amelia's mother is three times as old as Amelia. Four years ago her mother was five times as old as Amelia was then. Work out Amelia's age now.
- 9.Which of these is a term, rather than a whole expression, an equation, or a formula?
- 10.Which of these numbers is a cube number?
- 11.The graph of y = 20/x is drawn for x > 0. As the value of x increases, what happens to the value of y?
- 12.Solve 2x + 7 = x + 13
- 13.Write down the coefficient of xy in the expression 4x²y − 7xy + 2.
- 14.A taxi firm charges a fixed fee of £3.50 plus £2.20 per mile. Work out the total cost of a journey of 6 miles.
- 15.A point has coordinates (−6, 9). Work out the sum of the x-coordinate and the y-coordinate.
Answer key
- (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.
- (b) 3 — Method: a term is any part of an expression separated from the rest by a + or − sign, including a number on its own. Working: 8 − 3x + 5x² splits at the + and − signs into 8, −3x and 5x² — that is three separate terms. Answer: 3. Ryan's answer of 2 comes from wrongly excluding the number 8, thinking a term must contain a letter. 5 comes from miscounting by treating the coefficients and powers as separate terms as well as the letters. 1 comes from treating the whole expression as a single term because it is written without brackets.
- (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).
- (c) 5x² — A term is one part of an expression, separated from the others by a + or − sign. The expression 5x² − 3x + 7 has three terms: 5x², −3x and 7. Distractor origins: 5x² − 3x combines two terms together, not a single term; −3x + 7 also combines two terms together; 5x² − 3x + 7 gives the whole expression, not a single term.
- (d) −2 — Method: substitute the value into both terms, working out the index and the multiplication before the addition. Working: m² = (−2) × (−2) = 4 and 3m = 3 × (−2) = −6, so the expression becomes 4 + (−6), which is −2. Answer: −2. The distractors: −10 comes from squaring −2 as −4, giving −4 + (−6); 10 comes from working out 3m as +6 and losing the minus sign, giving 4 + 6; −14 comes from working from left to right instead of multiplying first, giving (4 + 3) × (−2).
- (b) x = 4 — Method: every point on a vertical line has the same x-coordinate, so its equation sets x equal to a number, and that number is the x-coordinate of any point known to be on the line. Working: the line is vertical and passes through (4, 7), whose x-coordinate is 4, so every point on the line has x-coordinate 4 and the equation is x = 4. Answer: x = 4. The distractors: y = 7 is the horizontal line through the same point, obtained by fixing the y-coordinate instead of the x-coordinate; x = 7 uses the right form but takes the y-coordinate of the point rather than its x-coordinate; y = 4x treats the 4 as a gradient and gives a sloping line through the origin, which does not pass through (4, 7) at all.
- (d) x = 2 — Method: expand the bracket by multiplying both terms inside it by 2, then undo the addition and the multiplication in turn. Working: expanding gives 2x + 6 = 10; subtracting 6 from both sides gives 2x = 4; dividing both sides by 2 gives x = 2. Answer: x = 2. The distractors: x = 5 comes from dividing both sides by 2 first, reaching x + 3 = 5 and writing 5 without taking the 3 away; x = 8 comes from adding 6 to both sides instead of subtracting it, giving 2x = 16; x = 3.5 comes from expanding 2(x + 3) as 2x + 3, multiplying only the x by the 2, which leads to 2x = 7.
- (d) 8 — Method: write both present ages in terms of one letter, take four years off each of them, and form an equation from the comparison four years ago. Working: let Amelia be y, so her mother is 3y; four years ago they were y − 4 and 3y − 4, giving 3y − 4 = 5(y − 4); expanding gives 3y − 4 = 5y − 20; adding 20 and subtracting 3y from both sides gives 16 = 2y, and dividing by 2 gives y = 8. Checking: Amelia is 8 and her mother 24; four years ago they were 4 and 20, and 20 = 5 × 4. Answer: 8. The distractors: 4 comes from solving correctly but giving Amelia's age four years ago instead of her age now; 16 comes from stopping at 2y = 16 and giving 16 as her age; 2 comes from collecting the y terms by adding the 5y instead of subtracting it, giving 3y + 5y = 20 − 4 and so 8y = 16.
- (d) 5x² — 5x² is a single term: a product of a number and a power of a letter, with no + or − sign attached to anything else. A candidate who picks 5x² + 3x is choosing the expression that contains that term, rather than the term on its own. A candidate who picks 5x² = 20 is drawn in by the equals sign and misses that this makes it an equation. A candidate who picks A = 5x² thinks any statement with a letter and a number combined must be a formula.
- (b) 27 — 27 = 3³ (3 × 3 × 3), so it is a cube number. The other three, 9 = 3², 16 = 4² and 25 = 5², are all square numbers, not cube numbers — each is reached by multiplying a whole number by itself only twice, not three times.
- (b) y decreases towards zero but never reaches it — Method: as x gets larger, dividing 20 by a bigger number gives a smaller result, so y decreases; because 20/x can never be exactly zero for any positive x, the curve gets closer to zero without ever reaching it, so y decreases towards zero but never reaches it. Distractor origins: 'y increases towards a limit but never reaches it' has the relationship backwards, treating y as increasing when it is actually decreasing; 'y decreases at a steady rate and reaches zero' wrongly assumes the graph behaves like a straight line that eventually hits zero; 'y stays the same however large x becomes' wrongly assumes there is no change in y at all.
- (d) x = 6 — Method: with an unknown on both sides, first collect the x terms on one side by subtracting the smaller x term from both sides, then deal with the numbers. Working: subtracting x from both sides gives x + 7 = 13, and subtracting 7 from both sides gives x = 6. Answer: x = 6. The distractors: x = 20 comes from adding 7 to 13 instead of subtracting it once the x terms have been collected; x = 2 comes from collecting the x terms by adding them, giving 3x + 7 = 13 and then 3x = 6; x = −6 comes from subtracting 2x from both sides to get 7 = −x + 13, reaching −6 = −x and then copying the sign straight across instead of dividing by −1.
- (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.
- (d) £16.70 — The mileage charge is 2.20 × 6 = £13.20. Adding the fixed fee: £13.20 + £3.50 = £16.70. A candidate who forgets the fixed fee gives just the mileage charge, £13.20. A candidate who adds the fixed fee to the per-mile rate before multiplying by the number of miles, (3.50 + 2.20) × 6, gets £34.20. A candidate who rounds £2.20 down to £2 gets 2 × 6 + 3.50 = £15.50.
- (b) 3 — −6 + 9 = 3. A candidate who ignores the negative sign on the x-coordinate and adds the two positive values gets 6 + 9 = 15. A candidate who treats the y-coordinate as negative too gets −6 + (−9) = −15. A candidate who works out 9 − 6 correctly as 3 but then writes the answer with the wrong sign gets −3.
Build your own mix at the worksheet builder.