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KS3 · Algebra

KS3.M-A1 Algebra: Expressions and Formulae

AQA AQA KS3 Mathematics · Calculator allowed · about 80 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The expression 5x + 3y - 2 is shown. Answer the questions below about this expression.

(a)What is the coefficient of x in the expression 5x + 3y - 2?(1)

A) 5
B) 3
C) x
D) 2

(b)How many terms are there in the expression 5x + 3y - 2?(1)

A) 2
B) 3
C) 4
D) 5

(c)Which of the following is an expression rather than an equation?(1)

A) 2x + 3 = 7
B) 2x + 3
C) 2x = 6
D) x = 3

Write an expression for each statement below. Use n, m or p as shown.

(a)A number, n, add 5.(1)

(b)Multiply a number, m, by 3, then subtract 2.(1)

(c)Divide a number, p, by 4, then add 7.(1)

x = 4 and y = 5.

(a)Find the value of 3x + 2y.(2)

(b)Find the value of x² - y.(2)

Simplify each expression by collecting like terms.

(a)3a + 5a(1)

(b)9b - 4b(1)

Simplify each expression fully. Take care with negative terms.

(a)5a - 8a(1)

(b)3x - 7 + 5x + 2(2)

Expand the brackets in each expression.

(a)3(x + 4)(1)

(b)5(2y - 3)(2)

(c)-2(3a + 5)(2)

Factorise each expression fully.

(a)6x + 9(2)

(b)10y - 15(2)

A plumber, Priya, uses the formula C = 25h + 40 to work out her total charge, where C is the cost in pounds and h is the number of hours she works.

(a)Work out the cost, C, when Priya works for 3 hours.(2)

(b)Priya charges a customer GBP 190 for a job. Work out how many hours, h, she worked.(2)

A rectangle has length (2x + 3) cm and width (x - 1) cm.

Diagram NOT accurately drawn

(a)Write an expression, in its simplest form, for the perimeter of the rectangle.(2)

(b)Given that x = 5, find the numerical perimeter of the rectangle in cm.(2)

Simplify each expression using the laws of indices.

(a)x² * x³(1)

(b)y⁷ / y²(1)

(c)(a²)³(1)

Expand and simplify each expression fully.

(a)3(x + 2) + 2(x - 1)(2)

(b)4(2y - 3) - 3(y - 5)(3)

Amy babysits and charges using the formula P = 6h + 2.50, where P is her payment in pounds and h is the number of hours she works.

(a)Find Amy's payment for 4 hours of babysitting.(2)

(b)One evening Amy is paid GBP 38.50. Work out how many hours she worked.(2)

(c)Tom says, 'If you double the number of hours, Amy's payment doubles too.' Use a calculation to show whether Tom is correct.(2)

An L-shaped garden is made from a rectangle of length 3x m and width 2x m, with a smaller square of side x m removed from one corner.

Diagram NOT accurately drawn

(a)Write an expression, in its simplest form, for the area of the L-shaped garden.(2)

(b)Given that x = 3, find the numerical area of the garden in m^2.(2)

Simplify each expression fully.

(a)(6x + 9)/3(2)

(b)(10y - 15)/5(1)

Factorise each expression fully.

(a)6x² + 9x(2)

(b)8x² - 12x(2)

Expand and simplify each expression fully.

(a)(x + 3)(x + 5)(3)

(b)(2x - 1)(x + 4)(3)

Rearrange each formula to make the stated letter the subject.

(a)Make x the subject of y = 3x + 2.(2)

(b)The formula for speed is v = u + at. Rearrange this formula to make a the subject.(2)

Show that (n + 1)² - (n - 1)² = 4n for all values of n.

(a)Show that (n + 1)² - (n - 1)² = 4n.(3)

A rectangular lawn has length (x + 6) m and width x m. A path of width 2 m is built all around the outside of the lawn.

Diagram NOT accurately drawn

(a)Write an expression, in its simplest form, for the area of the path.(4)

(b)Given that x = 8, find the area of the path in m^2.(2)

A pattern is made from matchstick squares joined in a row. Pattern 1 has 4 matchsticks, pattern 2 has 7 matchsticks, and pattern 3 has 10 matchsticks.

(a)Find an expression, in terms of n, for the number of matchsticks in pattern n.(2)

(b)Use your formula to find the number of matchsticks in pattern 12, and state whether a pattern in this sequence could ever have exactly 100 matchsticks, giving a reason.(3)

Mark scheme · KS3.M-A1 Algebra: Expressions and Formulae

Question 1

(a) B1 A cao
(a) Answer: A) 5
(b) B1 B cao
(b) Answer: B) 3
(c) B1 B cao
(c) Answer: B) 2x + 3

Question 2

(a) B1 n + 5 oe
(a) Answer: n + 5
(b) B1 3m - 2 oe
(b) Answer: 3m - 2
(c) B1 p/4 + 7 oe
(c) Answer: p/4 + 7

Question 3

(a) M1 3 x 4 (=12) and 2 x 5 (=10) seen, or 12+10 seen
(a) A1 22 cao
(a) Answer: 22
(b) M1 4² (=16) seen
(b) A1 11 cao
(b) Answer: 11

Question 4

(a) B1 8a cao
(a) Answer: 8a
(b) B1 5b cao
(b) Answer: 5b
(c) M1 x terms and y terms grouped correctly, e.g. 4x+2x and 3y+y seen
(c) A1 6x + 4y cao oe
(c) Answer: 6x + 4y

Question 5

(a) B1 -3a cao
(a) Answer: -3a
(b) M1 3x+5x (=8x) or -7+2 (=-5) seen
(b) A1 8x - 5 cao oe
(b) Answer: 8x - 5
(c) M1 -2y+5y (=3y) or 6-9 (=-3) seen
(c) A1 3y - 3 cao oe
(c) Answer: 3y - 3

Question 6

(a) B1 3x + 12 cao oe
(a) Answer: 3x + 12
(b) M1 one term correct, e.g. 10y or -15 seen
(b) A1 10y - 15 cao oe
(b) Answer: 10y - 15
(c) M1 one term correct with correct sign, e.g. -6a or -10 seen
(c) A1 -6a - 10 cao oe
(c) Answer: -6a - 10

Question 7

(a) M1 3 identified as the highest common factor, or partial factorisation e.g. 3(2x+3) attempted
(a) A1 3(2x + 3) cao oe
(a) Answer: 3(2x + 3)
(b) M1 5 identified as the highest common factor
(b) A1 5(2y - 3) cao oe
(b) Answer: 5(2y - 3)
(c) M1 4 identified as the highest common factor
(c) A1 4(2a + 3b) cao oe
(c) Answer: 4(2a + 3b)

Question 8

(a) M1 25 x 3 (=75) seen, or 75 + 40 seen
(a) A1 GBP 115 cao
(a) Answer: GBP 115
(b) M1 190 - 40 (=150) seen, or correct rearrangement 25h = 150
(b) A1 h = 6 cao
(b) Answer: h = 6 hours

Question 9

(a) M1 2[(2x+3)+(x-1)] or 2(2x+3)+2(x-1) seen
(a) A1 6x + 4 cao oe
(a) Answer: 6x + 4
(b) M1 length=13 and width=4, or 6(5)+4 substituted
(b) A1 34 cm cao
(b) Answer: 34 cm

Question 10

(a) B1 x⁵ cao
(a) Answer: x⁵
(b) B1 y⁵ cao
(b) Answer: y⁵
(c) B1 a⁶ cao
(c) Answer: a⁶

Question 11

(a) M1 both brackets expanded correctly: 3x+6 and 2x-2 seen
(a) A1 5x + 4 cao oe
(a) Answer: 5x + 4
(b) M1 4(2y-3) expanded correctly: 8y - 12
(b) M1 -3(y-5) expanded correctly with correct signs: -3y + 15
(b) A1 5y + 3 cao oe
(b) Answer: 5y + 3

Question 12

(a) M1 6 x 4 (=24) seen, or 24 + 2.50 seen
(a) A1 GBP 26.50 cao
(a) Answer: GBP 26.50
(b) M1 38.50 - 2.50 (=36) seen, or correct rearrangement 6h = 36
(b) A1 h = 6 cao
(b) Answer: h = 6 hours
(c) B1 correct comparison calculation, e.g. P(4)=26.50 and P(8)=50.50 found, or double of 26.50 (=53) found
(c) B1 correct conclusion that Tom is incorrect, with reference to the payment not doubling because of the fixed GBP 2.50 (ft from calculation)
(c) Answer: Tom is incorrect: for 4 hours P = GBP 26.50 but for 8 hours P = GBP 50.50, not GBP 53.00

Question 13

(a) M1 3x x 2x (=6x²) and x x x (=x²) both found, with subtraction set up
(a) A1 5x² cao oe
(a) Answer: 5x²
(b) M1 5 x 3² substituted, i.e. 5 x 9 seen
(b) A1 45 m² cao
(b) Answer: 45 m²

Question 14

(a) M1 6x/3 (=2x) and 9/3 (=3) both found, or 3(2x+3)/3 seen
(a) A1 2x + 3 cao oe
(a) Answer: 2x + 3
(b) B1 2y - 3 cao oe
(b) Answer: 2y - 3

Question 15

(a) M1 3x identified as the highest common factor
(a) A1 3x(2x + 3) cao oe
(a) Answer: 3x(2x + 3)
(b) M1 4x identified as the highest common factor
(b) A1 4x(2x - 3) cao oe
(b) Answer: 4x(2x - 3)

Question 16

(a) M1 at least two of the four products x², 5x, 3x, 15 correctly found
(a) M1 all four products correctly found (unsimplified), i.e. x² + 5x + 3x + 15
(a) A1 x² + 8x + 15 cao oe
(a) Answer: x² + 8x + 15
(b) M1 at least two of the four products 2x², 8x, -x, -4 correctly found
(b) M1 all four products correctly found (unsimplified), i.e. 2x² + 8x - x - 4
(b) A1 2x² + 7x - 4 cao oe
(b) Answer: 2x² + 7x - 4

Question 17

(a) M1 y - 2 = 3x, or equivalent correct first step
(a) A1 x = (y - 2)/3 cao oe
(a) Answer: x = (y - 2)/3
(b) M1 v - u = at, or equivalent correct first step
(b) A1 a = (v - u)/t cao oe
(b) Answer: a = (v - u)/t

Question 18

(a) M1 (n+1)² expanded correctly: n² + 2n + 1
(a) M1 (n-1)² expanded correctly: n² - 2n + 1
(a) A1 cso: fully correct simplification (n²+2n+1) - (n²-2n+1) = 4n shown with no errors, answer printed
(a) Answer: 4n (shown)

Question 19

(a) M1 outer length (x+10) and outer width (x+4) correctly found
(a) M1 outer area expanded correctly: x² + 14x + 40
(a) M1 lawn area x² + 6x found and subtraction set up: (x²+14x+40) - (x²+6x)
(a) A1 8x + 40 cao oe
(a) Answer: 8x + 40
(b) M1 8(8) + 40 substituted, i.e. 64 + 40 seen
(b) A1 104 m² cao
(b) Answer: 104 m²

Question 20

(a) M1 common difference of 3 identified, e.g. 3n seen
(a) A1 3n + 1 cao oe
(a) Answer: 3n + 1
(b) M1 3(12)+1 substituted correctly, i.e. 37 found
(b) M1 3n + 1 = 100 solved for n, i.e. n = 33 found (ft from part (a))
(b) A1 correct conclusion: yes, pattern 33 has exactly 100 matchsticks, since n=33 is a whole number (ft)
(b) Answer: Pattern 12 has 37 matchsticks; yes, pattern 33 has exactly 100 matchsticks

Algebra: Expressions and Formulae - Worksheets, Questions and Revision

KS3.M-A1 Algebra: Expressions and Formulae

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