13 In this question, assume all dimensions are in centimetres.
Jess and Pete have many rectangular tiles. Each tile has length \(a + b\) and width \(2b\).
Not to scale
(a) Jess joins three tiles together to make a larger rectangle, as shown.
Not to scale
(i) Write an expression for the perimeter of her rectangle. Give your answer in its simplest form. [2]
(ii) An expression for the area of her rectangle is \(6ab + 6b^2\).
Factorise this expression fully. [2]
(b) Pete joins some tiles together to make a different rectangle. The area of his rectangle is \(8ab + 8b^2\).
Draw a possible arrangement of tiles for Pete’s rectangle. Write down expressions for the length and for the width of the rectangle. [5]
Mark scheme (a)
Answer
Marks
Part marks and guidance
(i) \(6a + 10b\) or \(2(3a + 5b)\) final answer
2
M1 for \(6(a + b) + 2 \times 2b\) oe
If 0 scored SC1 for \(3a + 5b\) as final answer
M1 for EG \(a + b + a + b + a + b + a + b + a + b + a + b + 2b + 2b\) or \(2 \times (3a + 3b + 2b)\) etc
(ii) \(6b(a + b)\) final answer
2
B1 for \(6(ab + b^2)\) or \(b(6a + 6b)\) or \(3(2ab + 2b^2)\) or \(3b(2a + 2b)\) or \(2(3ab + 3b^2)\) or \(2b(3a + 3b)\)
Mark scheme (b)
Answer
Marks
Part marks and guidance
4 by 1 rectangle with \(4a + 4b\) and \(2b\) or 2 by 2 rectangle with \(2a + 2b\) and \(4b\) or 1 by 4 rectangle with \(a + b\) and \(8b\) stated or marked on rectangle
5
B4 for \(4a + 4b\) and \(2b\) or \(2a + 2b\) and \(4b\) or \(a + b\) and \(8b\) or B3 for rectangle drawn as (4 by 1) or (2 by 2) or (1 by 4) or B2 for one of \(2a + 2b\) or \(4a + 4b\) or \(4b\) or \(8b\) or
B1 for any rectangle of 3 or more tiles drawn with \(a+b\) or \(2b\) marked on individual tiles
Accept unsimplified throughout Once correct expression(s) seen, ignore incorrect simplification to answer line In answer space or intended as final length and width
Must clearly be answer
May be in attempt to factorise EG \(4b(2a + b)\) Accept unsimplified EG \(a+b + a+b\)
