Printable · GCSE Foundation · ages 14-16
Expressions, equations, formulae, identities and inequalities worksheet — GCSE Foundation
Fifteen questions on "expressions, equations, formulae, identities and inequalities" — DfE statement A3. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Expressions, equations, formulae, identities and inequalities worksheet — GCSE Foundation
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- 1.Which of these is an expression?
- 2.Rosie is asked whether 4(x + 3) = 4x + 12 is an equation or an identity. Which statement is correct? Give a reason for your answer.
- 3.In the expression 5x² − 3x + 7, which of these is a single term of the expression?
- 4.Which of these is a term, rather than a whole expression, an equation, or a formula?
- 5.A lift has a weight limit: it must carry no more than 630 kg. Let w be the total weight, in kg, carried by the lift. Which inequality describes this?
- 6.The perimeter, P, of a rectangle with length l and width w is calculated using P = 2l + 2w. What kind of mathematical statement is P = 2l + 2w?
- 7.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.
- 8.Meera writes the statement 3(x + 4) = 3x + 12. Which of these correctly describes what she has written, with a reason?
- 9.The formula for converting a temperature from degrees Celsius, C, to degrees Fahrenheit, F, is F = 1.8C + 32. Work out F when C = 20, and identify what kind of mathematical statement F = 1.8C + 32 is.
- 10.A plumber charges a £40 call-out fee plus £25 for each hour worked. The total charge, £C, for a job lasting h hours is given by C = 40 + 25h. Work out the charge for a job lasting 3 hours, and name what kind of statement C = 40 + 25h is.
- 11.A gardener charges a fixed fee of £15 plus £8 for each hour worked. The total charge, £T, for a job lasting h hours is given by T = 15 + 8h. Work out the charge for a job lasting 4 hours, and identify what kind of statement T = 15 + 8h is.
- 12.Which of these is a formula, rather than an expression, an equation, or an identity?
- 13.Which of these is a factor of 15x + 20?
- 14.Look at the statement 2x + 5 = 17. Considering whether it is an expression, an equation, a formula, or an identity, which classification is correct?
- 15.Ben writes n + 4 ≤ 9. Which statement describes what Ben has written? Give a reason for your answer.
Answer key
- (c) 5y − 2 — An expression is a collection of terms with no equals sign and no inequality sign in it: nothing is being claimed about two sides. Here 5y − 2 on its own fits that description. The line with an equals sign and one unknown letter to find is an equation; the line joined by a comparison sign is an inequality; the line that links two different letters so that one can be worked out from the other is a formula.
- (a) An identity, because it is true for every value of x — Expanding the brackets multiplies both terms inside by 4, giving 4x + 12, which is exactly the right-hand side. The two sides are therefore equal whatever x is, and a statement true for every value of the letter is an identity. An equation is true only for particular values, and trying to solve this one leads to 0 = 0, which places no restriction on x at all. Being able to expand brackets is not what makes a statement an identity, since any equation with brackets can be expanded. Nor is it a formula: a formula links two different quantities, and only one letter appears here.
- (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) 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.
- (d) w ≤ 630 — 'No more than 630 kg' means the weight can be exactly 630 kg or anything less, so the correct inequality is w ≤ 630, using 'less than or equal to' to include the limit itself. Writing w < 630 excludes 630 kg itself, as though the limit could not be reached exactly. Writing w ≥ 630 reverses the direction, describing a minimum weight rather than a maximum. Writing w > 630 both reverses the direction and excludes the boundary value. The inequality describing the lift's weight limit is w ≤ 630.
- (c) a formula — A statement that shows how to calculate one quantity from others, using an equals sign, is a formula. P = 2l + 2w tells us how to work out the perimeter, P, from the length and width, so it is a formula. Distractor origins: 'an equation' confuses a formula linking several letters with an equation solved for one unknown value; 'an identity' confuses a formula with a statement that is true for every value of a single variable; 'an expression' forgets that an expression has no equals sign at all.
- (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.
- (a) An identity, true for every value of x — Expanding the bracket on the left gives 3x + 12, which matches the right-hand side exactly, so the statement is true for every value of x — this makes it an identity. A candidate who reasons that any statement with an equals sign must be an equation picks that option, missing that an equation is only true for particular value(s) of x, not all of them. A candidate who confuses an identity with a formula, because both relate two expressions, picks the formula option — but a formula connects two different quantities, such as area and side length, not two equivalent forms of the same expression. A candidate who assumes it can be solved for a single value of x, as with a normal equation, picks that option, not realising there is no single solution here.
- (d) 68, and it is a formula — Substituting C = 20 gives F = 1.8 × 20 + 32 = 36 + 32 = 68. The statement F = 1.8C + 32 is a formula because it expresses a general relationship between F and C that holds for any value of C, not just C = 20. A candidate who gets the correct value but calls it an equation is confusing "has an equals sign" with "is an equation" — an equation is only true for particular value(s) of the unknown, whereas this relationship is used to calculate rather than to be solved. A candidate who forgets to multiply C by 1.8 and just adds 32 to 20 gets 52. A candidate who multiplies correctly but forgets to add 32 gets 36.
- (b) £115, and C = 40 + 25h is a formula — Substitute h = 3 into the rule, multiplying before adding. The hours cost 25 × 3 = 75, and adding the call-out fee gives 40 + 75 = 115, so the charge is £115. The rule itself links two different quantities, C and h, and lets one be worked out from the other, so it is a formula; an expression would have no equals sign in it. Adding the fee before multiplying gives 65 × 3 = 195, which charges the call-out fee three times over, and stopping at 40 + 25 leaves £65, the charge for a single hour.
- (b) £47, and T=15+8h is a formula (relates T and h). — The charge is £15 fixed plus £8 for each hour: T=15+8h, so for h=4, T=15+8×4=15+32=47. Because T=15+8h relates two different quantities, the total charge T and the number of hours h, it is a formula, not an equation — an equation is solved for one particular value of an unknown, but this relationship holds for every value of h a job might last. A candidate who forgets to include the fixed £15 fee would compute only 8×4=32. A candidate who adds the three numbers in the formula together instead of multiplying the hourly rate by the number of hours would compute 15+8+4=27. A candidate who correctly finds the charge but mistakes the formula for an equation, treating it as something to be solved for one specific value of h rather than a general relationship between T and h, would pick the correct charge with the wrong classification.
- (d) V = lwh — V = lwh is a formula: it relates one quantity, the volume V, to others, the length, width and height, in a way that is true generally. A candidate who picks lwh has chosen the expression, not a full statement relating two quantities. A candidate who picks lwh = 60 has chosen an equation, since it is only true for particular values of l, w and h that multiply to 60. A candidate who picks 2(l + w) ≡ 2l + 2w has chosen an identity, mistaking the identity symbol ≡ for a sign that makes a statement a formula.
- (a) 5 — A factor of the whole expression must divide every term exactly. 15x + 20 = 5(3x + 4), so 5 is a factor. Distractor origins: 3 divides 15x exactly but does not divide 20 exactly; 4 divides 20 exactly but does not divide 15x exactly; 10 also divides 20 exactly but does not divide 15x exactly, so it is a factor of only one term, not of the whole expression.
- (d) An equation, true for one value of x — 2x + 5 = 17 is only true when x = 6, so it is an equation. A candidate who ignores the equals sign and focuses only on the x terms and numbers present picks the expression option. A candidate who sees an equals sign and assumes it must relate two different quantities, as a formula does, picks that option, missing that there is only one quantity, x, involved here. A candidate who wrongly checks whether both sides look similar in structure rather than testing specific values picks the identity option, missing that the two sides are not equal for every value of x.
- (d) An inequality, because ≤ compares the two sides — The symbol ≤ means 'is less than or equal to', so the statement compares the sizes of the two sides instead of saying they are equal: that makes it an inequality. Solving it gives n ≤ 5, a whole range of values rather than the single value an equation would give. An identity has to be true for every value of the letter, and this fails at n = 6, so it is not one. A formula works one quantity out from another, and there is only one letter here.
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