Area models make expanding brackets obvious
If you can multiply \(23 \times 14\) with a rectangle, you can expand \((x + 3)(x + 4)\). It is the same picture.
Numbers first
Split each number into parts: \(23 = 20 + 3\) and \(14 = 10 + 4\). Draw a rectangle with those side lengths and work out the area of each piece.
\[ 23 \times 14 = 200 + 30 + 80 + 12 = 322 \]Then algebra
Now split \((x + 3)\) and \((x + 4)\) the same way. The four pieces are \(x^2\), \(3x\), \(4x\) and \(12\).
\[ (x + 3)(x + 4) = x^2 + 3x + 4x + 12 = x^2 + 7x + 12 \]The classic mistake: \((x + 4)^2 = x^2 + 16\). Draw the area model and you will see two \(4x\) pieces that this answer leaves out. The correct expansion is \(x^2 + 8x + 16\).
The free worksheet below has ten questions, starting with numbers and finishing with brackets.
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Years 7–10
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Expanding brackets with area models
Year 9 · Algebra · Worksheet
Ten questions that move from multiplying numbers to expanding algebraic brackets.
Open: Expanding brackets with area models