Printable · GCSE Foundation · ages 14-16
Algebraic notation worksheet — GCSE Foundation
Fifteen questions on "algebraic notation" — DfE statement A1. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Algebraic notation worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- 1.A packet contains s sweets. The sweets are shared equally between f friends. Write an expression for the number of sweets each friend receives.
- 2.Which expression means the same as a × a × a?
- 3.Write down the expression that means the same as w ÷ 6.
- 4.A box holds n pencils. A shop has 6 full boxes and 4 loose pencils. All of these pencils are shared equally between 2 classes. Write down the expression for the number of pencils each class receives.
- 5.Which expression means 'triple c, then add double d'?
- 6.A rectangular field has length (3a + 2) metres and width (a − 1) metres. Write a simplified expression for the perimeter of the field.
- 7.p = 4. Work out the value of 2p³.
- 8.Write down the expression that means the same as 3 × y × y.
- 9.Which expression means 'add 3 to n, then multiply the result by 4'?
- 10.a = 3 and b = 5. Work out the value of a²b.
- 11.a = 2 and b = 5. Work out the value of ab².
- 12.A taxi charges a £3 fixed fee plus £2 for each mile travelled. Write an expression, in pounds, for the total cost of a journey of n miles.
- 13.Simplify k + k + k + k + k.
- 14.A ribbon of length L metres is cut into 5 equal pieces, and then 3 metres is removed from one piece. Write down an expression, in metres, for the length of that piece after 3 metres is removed.
- 15.Write down the expression that means the same as 'subtract 5 from n, then divide the result by 2'.
Answer key
- (a) s/f — Sharing s sweets equally between f friends means dividing the total by the number of friends, written as a fraction: s/f. Writing f/s divides the wrong way round, sharing the number of friends between the sweets instead of the sweets between the friends. Writing s − f mistakes sharing for taking away, subtracting the number of friends from the number of sweets. Writing sf multiplies the two quantities together, which would make the total larger rather than splitting it into smaller equal parts. The number of sweets each friend receives is s/f.
- (c) a³ — a × a × a means a multiplied by itself three times, which is written using powers as a³ — the small 3 shows how many times a is multiplied by itself. Writing 3a instead uses the 3 as a coefficient, as if the expression meant a + a + a (three lots of a added together) rather than three a's multiplied together. Writing a² only accounts for two of the three a's being multiplied together, missing one factor. Writing 3 + a treats the repeated multiplication as an addition of 3 and a, which has no connection to the original expression. The expression that means a × a × a is a³.
- (c) w/6 — Method: a ÷ b is written as a fraction a/b, with the number being divided (w) on top. Working: w ÷ 6 = w/6. Answer: w/6. 6/w comes from writing the numbers the wrong way round, putting the 6 on top instead of w. 6w comes from reading the ÷ sign as ×, multiplying instead of dividing. w − 6 comes from reading ÷ as −, subtracting instead of dividing.
- (c) (6n + 4)/2 — Six full boxes hold 6 lots of n pencils, which is 6n, and the 4 loose pencils are added on, so the shop has 6n + 4 pencils altogether. Sharing them equally between 2 classes divides that whole total by 2, and brackets are what show that the division applies to all of it: (6n + 4)/2. Without the brackets, 6n + 4/2 halves only the loose pencils; 6(n + 4)/2 adds the loose pencils to every box before the division; 2(6n + 4) doubles the total instead of halving it.
- (c) 3c + 2d — Method: 'triple c' is 3c, 'double d' is 2d, and 'add' joins the two separate terms with a plus sign. Working: 3c + 2d. Answer: 3c + 2d. 2c + 3d comes from swapping which letter gets tripled and which gets doubled. 6cd comes from multiplying the two terms together instead of adding them, and also multiplying the coefficients (3 × 2 = 6). 5(c + d) comes from adding the coefficients (3 + 2 = 5) and applying that single number to both letters together, as if c and d always came as a pair.
- (c) 8a + 2 — The perimeter of a rectangle is twice the length plus twice the width: P = 2(3a + 2) + 2(a − 1) = (6a + 4) + (2a − 2) = 8a + 2. Answering 5a adds the length and width once each but doubles only one of them, missing that a rectangle has two of each side. Answering 8a + 6 distributes the 2 into (a − 1) correctly as far as 2a, but then adds 2 instead of subtracting it, as though the bracket had been (a + 1). Answering 12a + 8 uses the length for all four sides instead of using the length twice and the width twice, as if the field were a square with side (3a + 2). The perimeter of the field is (8a + 2) metres.
- (d) 128 — In 2p³ the index belongs to p only, so cube p first and multiply by the coefficient afterwards. Cubing gives 4 × 4 × 4 = 64, and then 2 × 64 = 128. Cubing the coefficient as well would mean working out (2 × 4)³, which is 512. Reading the index as an instruction to multiply by 3 gives 2 × 4 × 3 = 24, and ignoring the coefficient altogether leaves 64.
- (a) 3y² — There is one 3 in the product and the letter y appears twice. A number that multiplies a letter is written in front of it as the coefficient, so the 3 goes at the front. The two ys multiply each other, so the letter carries an index of 2. The shortest correct way of writing the product is therefore 3y². Note that (3y)² would mean 3 × y × 3 × y, which squares the coefficient as well, while 3 + y² turns a multiplication into an addition and y³ treats the coefficient as a third factor of y.
- (c) 4(n + 3) — 'Add 3 to n' must happen before 'multiply the result by 4', so the addition needs brackets to show it happens first: 4(n + 3). Writing 4n + 3 multiplies n by 4 immediately and only adds the 3 afterwards, which reverses the order the words describe. Writing n + 3 × 4 only multiplies the 3 by 4, and adds n as a separate, unmultiplied term — it treats 'the result' as just the 3, not the whole of n + 3. Writing 3(n + 4) keeps the correct structure but swaps which number is added and which is multiplied. The expression for 'add 3 to n, then multiply the result by 4' is 4(n + 3).
- (a) 45 — a²b means a × a × b, so the index applies to a only and b is multiplied on afterwards. Substituting the values gives 3 × 3 = 9, then 9 × 5 = 45. Reading a²b as (ab)² gives (3 × 5)², which is 15² = 225 and squares b as well. Substituting the two values the wrong way round works out 5 × 5 × 3 = 75, and reading the letters written side by side as an addition gives 9 + 5 = 14.
- (a) 50 — Method: in ab², only the b is squared, so square b first, then multiply by a. Working: b² = 5² = 25, then a × b² = 2 × 25 = 50. Answer: 50. 100 comes from squaring the product ab instead of just b: (2 × 5)² = 100. 20 comes from squaring a instead of b: a² × b = 4 × 5 = 20. 10 comes from ignoring the square altogether and working out a × b = 2 × 5 = 10.
- (b) 2n + 3 — Method: multiply the cost per mile by the number of miles to get an expression, then add the fixed fee as a separate term. Working: n miles at £2 each is 2n; add the £3 fixed fee: 2n + 3. Answer: 2n + 3. 3n + 2 comes from swapping the fee and the rate round, treating £3 as the rate per mile and £2 as the fixed fee. 5n comes from adding the fee and the rate together first (3 + 2 = 5) and multiplying the result by n, instead of keeping the fixed fee as its own term. 2n − 3 comes from subtracting the fixed fee instead of adding it.
- (a) 5k — Method: adding a letter to itself repeatedly means counting how many of that letter you have, written as a coefficient in front of the letter. Working: k + k + k + k + k is five lots of k, written 5k. Answer: 5k. k⁵ comes from treating repeated addition as repeated multiplication (raising to a power) instead of counting copies. 5 + k comes from adding the count of terms (5) to a single k instead of multiplying. k/5 comes from dividing instead of counting how many k's there are.
- (c) L/5 − 3 — Each of the 5 equal pieces is L/5 metres long, and removing 3 metres from one piece gives L/5 − 3. Subtracting the 3 metres before dividing by 5, (L − 3)/5, divides the removed length between all 5 pieces instead of taking it from just one. Dividing only the 3 by 5 instead of dividing L by 5, L − 3/5, divides the wrong number. Writing 5/L − 3 inverts the fraction, swapping which number is the numerator.
- (a) (n − 5)/2 — Method: do the subtraction first, keep it together as a single bracket, then divide that whole bracket by 2. Working: 'subtract 5 from n' is (n − 5); 'divide the result by 2' means the whole bracket goes over 2, giving (n − 5)/2. Answer: (n − 5)/2. n/2 − 5 comes from dividing n by 2 first and only then subtracting 5, the wrong order. 2(n − 5) comes from multiplying by 2 instead of dividing. (5 − n)/2 comes from subtracting n from 5 instead of subtracting 5 from n, the wrong way round.
Build your own mix at the worksheet builder.