Year 9 · Unit 8 Ready for Year 10 · Lesson 1 of 6

Every term times every term: the area grid

Two brackets multiplied together make a rectangle with four parts. Draw the grid, find the area of each part, and collect what is alike.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Expand 4(x + 7).
    Answer4x + 28: the 4 multiplies both the x and the 7.
  2. Simplify 6x + 2 + x + 9.
    Answer7x + 11: 6x and x are like terms, and 2 + 9 = 11.
  3. Work out x × x and x + x when x = 6.
    Answer36 and 12. x × x is x²; x + x is 2x. They are different things.
Words used in this lesson
expand
multiply out the brackets. For example: (x + 1)(x + 5) becomes x² + 6x + 5.
term
one part of an expression, joined to the rest by + or −. For example: x², 6x and 5 are the terms of x² + 6x + 5.
like terms
terms with the same letter part, which can be collected. For example: 4x and 5x make 9x.
grid
a table with one row and one column for each term. For example: a 2 by 2 grid for two brackets of two terms.
double brackets
two brackets multiplied together. For example: (x + 1)(x + 5).

A bigger courtyard

A square courtyard in Marrakesh is x metres along each side. The owners extend it 3 m one way and 2 m the other, so the new courtyard is (x + 3) m by (x + 2) m. Amira says the new area is x² + 6 square metres. Jonas says it is x² + 5x + 6.

Predict first: who is right, or is neither? Then set a = 3 and b = 2 on the grid and look at what each of them has counted.

1
2
3
x² + 3x + 2the four cells, collected
20the whole rectangle at this x
9how far short x² + ab is

(x + 1)(x + 2) = x² + x + 2x + 2 = x² + 3x + 2. At x = 3: 4 × 5 = 20, and 9 + 3 + 6 + 2 = 20.

Show me the steps
  1. a = 1, b = 2, x = 3: the cells are x², x, 2x and 2, so (x + 1)(x + 2) = x² + 3x + 2. At x = 3: 4 × 5 = 20, and 9 + 3 + 6 + 2 = 20.
  2. a = 4, b = 5, x = 3: the cells are x², 4x, 5x and 20, so (x + 4)(x + 5) = x² + 9x + 20. At x = 3: 7 × 8 = 56, and 9 + 12 + 15 + 20 = 56.
  3. Same grid: x² + 20 is only 9 + 20 = 29 at x = 3, which is 27 short. The missing 27 is the two strips, 12 + 15.
  4. Whatever a and b are, the two strips are ax and bx, so they collect to (a + b)x, and the corner is a × b.

Every term in one bracket multiplies every term in the other. A grid keeps track: one column for each term of the first bracket, one row for each term of the second, and one cell for each product. Then collect the like terms.

For the courtyard the cells are x², 3x, 2x and 6, so (x + 3)(x + 2) = x² + 5x + 6 square metres. Jonas is right. Amira's x² + 6 counts the big square and the corner but leaves out the two strips, 3x and 2x.

Why four parts?

The grid is the single-bracket rule from Year 8 used twice. Split the first bracket into its two terms: (x + 3)(x + 2) = x(x + 2) + 3(x + 2). The first part gives x² + 2x (the top row of cells) and the second gives 3x + 6 (the bottom row). Two brackets of two terms each always give 2 × 2 = 4 products before anything is collected.

A common mistake: writing x × x as 2x. It is x², the area of an x by x square. Test with x = 5: 5 × 5 = 25, but 2 × 5 = 10.

Worked examples

Expand (x + 7)(x + 3).

  1. Grid cells: x × x = x², 7 × x = 7x, x × 3 = 3x, 7 × 3 = 21.
  2. Collect: 7x + 3x = 10x.
  3. x² + 10x + 21. Check at x = 2: 9 × 5 = 45 and 4 + 20 + 21 = 45.

Expand (3x + 2)(x + 5).

  1. Grid cells: 3x × x = 3x², 3x × 5 = 15x, 2 × x = 2x, 2 × 5 = 10.
  2. Collect: 15x + 2x = 17x.
  3. 3x² + 17x + 10. Check at x = 1: 5 × 6 = 30 and 3 + 17 + 10 = 30.

Your go. The first step is given. Expand (x + 8)(x + 3).

  1. Top row of the grid: x × x = x² and 8 × x = 8x.
  2. Now fill the bottom row yourself, collect, and check with a value of x.
Check the answer

Bottom row: 3x and 24. Collect 8x + 3x = 11x: x² + 11x + 24. Check at x = 2: 10 × 5 = 50 and 4 + 22 + 24 = 50.

Check before you go on

1. Which is (x + 6)(x + 2) written without brackets?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Expand (x + 4)(x + 6) and collect like terms. What number is in front of x?

3. A rectangle is (x + 5) cm by (x + 8) cm. Find its area, in square centimetres, when x = 2.

Level 2 · Building

4. (x + 2)(x + k) = x² + 9x + 14. What is k?

5. Which is (2x + 3)(x + 4) written without brackets?

Level 3 · Stretch

6. (x + a)(x + 6) = x² + 13x + c, where a and c are whole numbers. What is c?

7. A square photo is x cm by x cm. It is enlarged by adding 3 cm to its width and 5 cm to its height. By how many square centimetres does its area grow when x = 10?

Something to think about

Could (x + a)(x + b) ever be exactly x² + ab for every value of x? What would a and b have to be? (Whole numbers can be negative.)

One way to think about it

The grid always adds the strips ax and bx, so the expansion is x² + (a + b)x + ab. That matches x² + ab for every x only if a + b = 0, so b is the negative of a, as in (x + 3)(x − 3). Lesson 2 puts negative numbers in the grid and lesson 3 looks at this pair closely.

Exit ticket

8. Ravi's grid for (x + 5)(x + 9) has the cells x², 5x, 9x and 14. Which is true?