Year 9 · Unit 8 Ready for Year 10 · Lesson 5 of 6

Paths, borders and frames: area with letters

Write an area with brackets, expand it, and let the algebra say how the area behaves. A path round a pool is the outer rectangle take away the pool, and the x² terms cancel.

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 lesson 4: simplify (x + 2)(x + 3) − x².
    Answer5x + 6: x² + 5x + 6 take away x².
  2. A rectangle is 7 m by 4 m. Find its area and its perimeter.
    Answer28 m² and 22 m: 7 × 4 and 2 × (7 + 4).
  3. Solve 4x + 6 = 30.
    Answerx = 6: take 6 from both sides, then divide by 4.
Words used in this lesson
area
the space inside a flat shape, in square units. For example: a 3 m by 5 m rug has an area of 15 m².
perimeter
the distance all the way round a shape. For example: a 3 m by 5 m rug has a perimeter of 16 m.
border
a strip of the same width all round a shape. For example: the frame round a picture.
square metre
the area of a square 1 m by 1 m. For example: written m².
inner, outer
the shape inside, and the whole shape with its border. For example: a pool is inner; pool and path together are outer.

A path round a pool

A pool at a community centre in Durban is x m wide and (x + 3) m long. A path 2 m wide runs all the way round it. The centre may build a bigger pool next year, with the same 2 m path.

Predict first: each time x goes up by 1, does the path's area go up by the same amount, or by more each time, the way the pool's area does? Then explore the figure. (It opens with a 1 m path.)

4
1
26area of the path, m²
4x + 10the path's area in x
28area of the pool, m²

x = 4, path 1 m: outer 6 × 9 = 54, pool 4 × 7 = 28, path 26 square metres. In x, the path is 4x + 10.

Show me the steps
  1. x = 4, path 1 m: outer 6 × 9 = 54, pool 4 × 7 = 28, so the path is 26 square metres. In letters, (x + 2)(x + 5) − x(x + 3) = 4x + 10.
  2. x = 5, path 1 m: outer 7 × 10 = 70, pool 5 × 8 = 40, path 30. One more metre of x adds 4 square metres of path, as 4x + 10 says.
  3. x = 4, path 3 m: outer 10 × 13 = 130, pool 28, path 102. The path is 12x + 54, and 12 × 4 + 54 = 102.
  4. The pool's perimeter is 4x + 6 and the outer edge's is 4x + 6 + 8w: both grow by 4 for each metre of x.

Write the area with brackets, expand, then simplify. A rectangle (x + a) by (x + b) has area (x + a)(x + b) and perimeter 2(x + a) + 2(x + b). A path or border is the outer rectangle take away the inner one, so bracket the inner area before you subtract it.

Perimeter adds lengths, so it has no x². Area multiplies lengths, so it usually does, unless the x² terms cancel, as they do for a path of fixed width.

A common mistake: taking the inner area away without its bracket. −x(x + 3) is −x² − 3x, not −x² + 3x: the minus reaches both terms.

Worked examples

A rectangle is (x + 10) m by (x + 3) m. Write its area and its perimeter.

  1. Area: (x + 10)(x + 3) = x² + 13x + 30 square metres.
  2. Perimeter: 2(x + 10) + 2(x + 3) = 4x + 26 metres.
  3. Check at x = 2: 12 × 5 = 60 and 4 + 26 + 30 = 60; 2 × (12 + 5) = 34 and 8 + 26 = 34.

The hook's 2 m path.

  1. Outer rectangle: (x + 4) by (x + 7), area x² + 11x + 28.
  2. Path: x² + 11x + 28 − (x² + 3x) = 8x + 28 square metres.
  3. So each extra metre of x adds exactly 8 square metres of path, while the pool grows by more each time. Check at x = 5: 9 × 12 − 5 × 8 = 108 − 40 = 68 and 40 + 28 = 68.

Your go. The first step is given. A room is (x + 5) m by (x + 2) m. A square rug x m by x m lies on the floor. Find the area of floor not covered.

  1. Room minus rug: (x + 5)(x + 2) − x².
  2. Now expand, simplify and check with a value.
Check the answer

x² + 7x + 10 − x² = 7x + 10 square metres. Check at x = 4: 9 × 6 − 16 = 38 and 28 + 10 = 38.

Check before you go on

1. A rectangle is (x + 6) cm by (x + 2) cm. Which expression is its perimeter, in centimetres?

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. A square garden has sides of (x + 5) m. What is its area, in square metres, when x = 3?

3. A mat is (x + 6) m by (x − 2) m. Expand its area. What number is in front of x?

Level 2 · Building

4. A photo is x cm by (x + 2) cm. A border 1 cm wide goes all the way round it. What is the area of the border, in square centimetres, when x = 10?

5. A rectangle is (x + 7) cm by (x + 3) cm, and its perimeter is 40 cm. Find x.

Level 3 · Stretch

6. A square has sides of (x + 3) m. A rectangle is (x + 1) m by (x + 6) m. For which x do they have the same area?

7. A square garden is x m by x m. A path 2 m wide runs along one side and one end, making the garden and path together a square (x + 2) m by (x + 2) m. The path's area is 40 square metres. Find x.

Something to think about

With the 2 m path, is there a pool size where the path has exactly the same area as the pool? Try some values of x.

One way to think about it

The pool is x(x + 3) and the path is 8x + 28. At x = 8 the pool is 88 and the path 92; at x = 9 the pool is 108 and the path 100. So they are equal somewhere between 8 and 9. The equation x² + 3x = 8x + 28 keeps its x², so balancing alone cannot solve it: Year 10 shows how.

Exit ticket

8. A frame is x cm by (x + 5) cm. A border 3 cm wide goes all the way round it. Which expression is the area of the border?