Expanding brackets with area models
An area model splits a multiplication into pieces you can do in your head. It works the same way for numbers and for algebra.
How it works. To find \(23 \times 14\), split \(23 = 20 + 3\) and \(14 = 10 + 4\). Draw a rectangle, find the area of each part, then add.
| \(\times\) | \(20\) | \(3\) |
|---|---|---|
| \(10\) | \(200\) | \(30\) |
| \(4\) | \(80\) | \(12\) |
\(23 \times 14 = 200 + 30 + 80 + 12 = 322\)
Questions
Part A: numbers
Use an area model to work out each product.
- \(23 \times 14\)
- \(36 \times 12\)
- \(47 \times 25\)
Part B: algebra
Use an area model to expand each expression. Collect like terms.
- \(3(x + 5)\)
- \(x(x + 4)\)
- \((x + 3)(x + 4)\)
- \((x + 5)(x + 2)\)
- \((x + 6)(x - 1)\)
- \((2x + 1)(x + 3)\)
- \((x + 4)^2\)
Worked solutions
1.
\(200 + 30 + 80 + 12\)
\(322\)
2.
Split \(36 = 30 + 6\) and \(12 = 10 + 2\): \(300 + 60 + 60 + 12\)
\(432\)
3.
Split \(47 = 40 + 7\) and \(25 = 20 + 5\): \(800 + 140 + 200 + 35\)
\(1175\)
4.
One row: \(3 \times x = 3x\) and \(3 \times 5 = 15\).
\(3x + 15\)
5.
One row: \(x \times x = x^2\) and \(x \times 4 = 4x\).
\(x^2 + 4x\)
6.
| \(\times\) | \(x\) | \(3\) |
|---|---|---|
| \(x\) | \(x^2\) | \(3x\) |
| \(4\) | \(4x\) | \(12\) |
\(x^2 + 3x + 4x + 12\)
\(x^2 + 7x + 12\)
7.
\(x^2 + 2x + 5x + 10\)
\(x^2 + 7x + 10\)
8.
Treat \(-1\) as a part with a negative "length": \(x^2 - x + 6x - 6\)
\(x^2 + 5x - 6\)
9.
\(2x \times x = 2x^2\), \(2x \times 3 = 6x\), \(1 \times x = x\), \(1 \times 3 = 3\)
\(2x^2 + 7x + 3\)
10.
\((x + 4)^2 = (x + 4)(x + 4) = x^2 + 4x + 4x + 16\)
\(x^2 + 8x + 16\)
Not \(x^2 + 16\). The area model shows why: there are two \(4x\) pieces.