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.Write down the expression that means the same as m × m × m × n.
- 2.Write down 0.2x with its coefficient written as a fraction in its simplest form.
- 3.a = 2 and b = 5. Work out the value of a²b.
- 4.A rectangular field has length (3a + 2) metres and width (a − 1) metres. Write a simplified expression for the perimeter of the field.
- 5.Simplify k + k + k + k + k.
- 6.Write down the expression that means the same as m² × m × 3.
- 7.Which expression means the same as 3 × a × b?
- 8.p = 4. Work out the value of 2p³.
- 9.n = 3. Work out the value of (2n)².
- 10.Write down the expression that means the same as w ÷ 6.
- 11.Which expression means 'triple c, then add double d'?
- 12.Write down the expression that means the same as (x + 7) ÷ 4.
- 13.Write down the coefficient of xy in the expression 4x²y − 7xy + 2.
- 14.Write down the expression that means the same as p + p + p + q.
- 15.Which expression means 'y squared, multiplied by 3'?
Answer key
- (d) m³n — m × m × m × n means three m's multiplied together, then multiplied by n, which is written as m³n. Putting the power of 3 on n instead of m, mn³, puts the power on the wrong letter. Writing 3mn treats the repeated multiplication as if it were repeated addition, as if it meant 3 × m × n. Writing m³ + n adds the n instead of multiplying it in.
- (a) x/5 — Read the decimal as a fraction first: 0.2 is 2 tenths, so the coefficient is 2/10. Dividing the numerator and the denominator by 2 cancels this to 1/5, and a coefficient of 1/5 in front of a letter is written x/5. Reading the digit 2 as 'a half' gives x/2; reading 0.2 as 1/20 gives x/20; dividing only the denominator by 2 leaves 2/5, which is the coefficient 0.4 and twice as large as it should be.
- (b) 20 — a²b means a² multiplied by b: a² = 2² = 4, and 4 × 5 = 20. Reading the expression as (ab)² instead of a²b gives (2 × 5)² = 100. Squaring b instead of a, 2 × 5² = 50, squares the wrong letter. Adding a² and b instead of multiplying them, 2² + 5 = 9, uses the wrong operation.
- (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.
- (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.
- (d) 3m³ — Method: multiply the powers of m by adding their indices, then bring the number coefficient to the front. Working: m² × m has indices 2 and 1; add them to get 3, giving m³, then × 3 gives 3m³. Answer: 3m³. 3m² comes from multiplying the indices instead of adding them: 2 × 1 = 2, giving m², then × 3 = 3m². m³ comes from correctly combining the m's but dropping the coefficient 3. m⁶ comes from multiplying the index by the coefficient instead of writing the coefficient in front: taking the 2 in m² and the 3 to give m raised to the power 2 × 3, which is m⁶, with the lone m left out.
- (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.
- (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.
- (d) 36 — (2n)² means the whole of 2n is squared, so with n = 3: (2n)² = (2 × 3)² = 6² = 36. Answering 18 instead works out 2n² — squaring only the n and then multiplying by 2 — which is a different expression because the brackets around 2n are missing. Answering 12 squares only the coefficient, treating (2n)² as 2² × n = 4 × 3 = 12, and forgets to square the n as well. Answering 9 ignores the coefficient of 2 altogether and works out n² on its own. The value of (2n)² when n = 3 is 36.
- (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) 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) (x + 7)/4 — Method: dividing an expression by a number is written as a fraction, with the whole expression on top. Working: (x + 7) ÷ 4 = (x + 7)/4. Answer: (x + 7)/4. 4/(x + 7) comes from writing the numbers the wrong way round, putting 4 on top. 4(x + 7) comes from reading ÷ as ×, multiplying instead of dividing. (x + 7) − 4 comes from reading ÷ as −, subtracting instead of dividing.
- (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.
- (a) 3p + q — Repeated addition of the same letter is written as a multiple of that letter, so p + p + p is 3 lots of p, which is 3p. The letter q is added once only, so it stays as a separate term and the result is 3p + q. Writing 3pq multiplies the q by 3 and by p as well; p³ + q records repeated multiplication rather than repeated addition; 3(p + q) multiplies both letters by 3.
- (a) 3y² — 'y squared, multiplied by 3' means the square is applied to y only, and the result is then multiplied by 3, written as 3y². Writing y³ mistakes the multiplication by 3 for an extra factor of y, adding to the power instead of using a coefficient. Writing (3y)² squares the whole of 3y, including the 3, which gives 9y² rather than 3y² — the square should apply to y alone. Writing 3 + y² adds the 3 instead of multiplying by it. The expression for 'y squared, multiplied by 3' is 3y².
Build your own mix at the worksheet builder.