Year 8 · Unit 4 Expressions, equations and inequalities · Lesson 1 of 6

What does a number outside a bracket multiply?

A number outside a bracket multiplies everything inside it. An area made of two rectangles shows why, and a quick substitution shows when a tempting shortcut is wrong.

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. From Year 7, Unit 3, lesson 4, collecting like terms. Simplify 4a + 3 + 2a − 1.
    Answer6a + 2. 4a and 2a are like terms; 3 − 1 = 2.
  2. From Year 7, Unit 3, lesson 2, letters and expressions. What is 3n + 5 when n = 4?
    Answer17. 3 × 4 = 12, then 12 + 5.
  3. From Unit 3 (integers), lesson 4. Work out (−3) × (−7).
    Answer21. Two negative numbers multiply to a positive.
Words used in this lesson
expand
multiply out a bracket so it is gone. For example: 3(x + 2) = 3x + 6.
bracket
the ( ) that group terms together. For example: 3(x + 2).
term
one part of an expression, with its sign. For example: in 3x − 5, the terms are 3x and −5.
equivalent
has the same value for every x. For example: 2(x + 1) and 2x + 2.
substitute
put a number in place of a letter. For example: x = 4 in 3x + 1 gives 13.

One rectangle, or two?

A school garden bed is 3 m long. Across it there is a strip of beans, n metres wide, and then a strip of maize 5 metres wide. The gardener can find the area in two ways: the whole bed at once, 3 × (n + 5), or the beans and the maize separately, 3 × n + 3 × 5.

Predict first: if the bean strip is 2 m wide, work out both areas by hand. Will they match? Then move the sliders and test other widths.

3
4
5
27area of one rectangle, a × (n + b)
27area of two rectangles, a × n + a × b

3 × (4 + 5) = 27, and 3 × 4 + 3 × 5 = 12 + 15 = 27. Cutting the rectangle in two does not change its area.

Show me the steps
  1. a = 3, n = 4, b = 5: one rectangle gives 3 × (4 + 5) = 27; two rectangles give 3 × 4 + 3 × 5 = 12 + 15 = 27.
  2. Make the right part wider, a = 3, n = 4, b = 8: 3 × 12 = 36, and 12 + 24 = 36. Still equal.
  3. Make it taller, a = 5, n = 2, b = 6: 5 × 8 = 40, and 10 + 30 = 40.
  4. Every setting gives the same two areas, so a(n + b) = an + ab for any whole numbers.

A number outside a bracket multiplies every term inside it. a(n + b) = an + ab, and a(n − b) = an − ab.

Why: a(n + b) is a rectangle a tall and n + b wide. Cut it into a part n wide and a part b wide and you get two rectangles of area an and ab. The total area has not changed.

This is called expanding the bracket. The two forms look different but have the same value for every value of the letters, so they are equivalent.

The slip that looks right

Many people write 5(x + 2) = 5x + 2: they multiply the first term and forget the second. How far off is that, and does it depend on x? Set the sliders and see.

5
2
3
25the true value of a(x + c), which ax + ac also gives
17the slip, ax + c
8how far short the slip is

5 × (3 + 2) = 25 and 5 × 3 + 5 × 2 = 25, but the slip 5 × 3 + 2 = 17 is 8 short.

Show me the steps
  1. a = 5, c = 2, x = 3: 5 × (3 + 2) = 25 and 5 × 3 + 5 × 2 = 25, but the slip 5 × 3 + 2 = 17 is 8 short.
  2. Change x to 8 with a = 5 and c = 2: 5 × 10 = 50, and the slip 5 × 8 + 2 = 42 is still 8 short.
  3. Change c to 6 with a = 5: the slip is 24 short at every x.
  4. Set a = 1 and look at the gap.

To test an expansion, substitute a number. Put x = 1 into 3(x + 4) and 3x + 4: you get 15 and 7, so they are not equal. Putting in one value can show two expressions are different, but it cannot prove they are always equal: that needs the rectangle reasoning.

Worked examples

Expand 6(2y + 5).

  1. Multiply the first term: 6 × 2y = 12y.
  2. Multiply the second term: 6 × 5 = 30.
  3. Together: 12y + 30. Test with y = 1: 6 × 7 = 42 and 12 + 30 = 42.

Expand 5(3x − 4). 5 × 3x = 15x and 5 × (−4) = −20, so 15x − 20.

Expand −2(x − 6).

  1. −2 × x = −2x.
  2. −2 × (−6) = +12, because two negatives multiply to a positive.
  3. Together: −2x + 12.

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

  1. Expand: 2x + 10 + 3x.
  2. Collect like terms: 2x + 3x = 5x, so 5x + 10.

Your go. The first step is given. Expand and simplify 4(w + 2) + 3w.

  1. Expand the bracket: 4(w + 2) = 4w + 8.
  2. Finish by collecting like terms, then test with w = 1.
Check the answer

4w + 8 + 3w = 7w + 8. Test: 4 × 3 + 3 = 15 and 7 + 8 = 15.

Check before you go on

1. Which expression is equal to 4(x + 7)?

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 8(x + 3). It equals 8x + k. What is k?

3. Expand 3(2y − 5). It equals 6y − k. What is k?

Level 2 · Building

4. A bed is 4 m long. A strip of beans is x m wide and a strip of maize is 3 m wide, side by side. The area is 4(x + 3) = 4x + k square metres. What is k?

5. Expand −4(x − 3). It equals −4x + k. What is k?

6. Which expression is equal to 5(2a − 3) − a?

Level 3 · Stretch

7. Expand and simplify 3(x + 4) + 2(x − 1). It equals 5x + k. What is k?

8. Expand and simplify 2(3x + 5) − 4(x − 2). It equals 2x + k. What is k?

Something to think about

In the second figure, the slip ax + c is sometimes not wrong. Find every setting of a and c where it gives exactly the right value for every x. Why does that happen?

One way to think about it

Only a = 1 works with the sliders (c is at least 1 there). The gap is c × (a − 1), which is 0 only when a = 1 or c = 0. With a = 1, 1(x + c) is just x + c, so there is nothing for the multiplication to reach.

Exit ticket

9. Which pair of expressions has the same value for every x?