Draw brackets as rectangles. Expand them, tidy them up and factorise them with an area model. Test every answer with a real number, solve puzzles and beat the clock.
Start here. A rectangle is easy to measure when you split it into pieces. To work out 3 × 14, split 14 into 10 + 4. Now you have two pieces: 3 × 10 = 30 and 3 × 4 = 12. Together they make 42. That is called the distributive law: 3 × (10 + 4) = 3 × 10 + 3 × 4.
Brackets with x. Sometimes a length is a mystery. We call it x. A rectangle with 3 rows, each row an x-block plus 4 little squares, is written 3(x + 4). There are 3 x-blocks and 3 × 4 = 12 little squares, so 3(x + 4) = 3x + 12. This is called expanding the bracket. Multiply the outside number by every part inside.
Minus inside a bracket. For 3(x − 2) start with 3 rows of x, then cut away a strip that is 3 tall and 2 wide. That strip is 3 × 2 = 6 squares, so 3(x − 2) = 3x − 6. In the pictures the cut-away strip is hatched.
Tidying up. After expanding you can add the plain numbers together: 2(x + 3) + 4 = 2x + 6 + 4 = 2x + 10.
Factorising. Factorising is expanding backwards. Look at 6x + 9. Both parts can be divided by 3, so 3 is a common factor. Write 3 outside a bracket and divide each part by 3: 6x + 9 = 3(2x + 3). Take out the biggest common factor you can find.
Two brackets. (x + 2)(x + 3) is a rectangle with four pieces: an x by x square (x²), two pieces 2x and 3x, and a little 2 × 3 = 6 rectangle. Add the two middle pieces, 2x + 3x = 5x, and you get x² + 5x + 6. With a minus, such as (x − 2)(x + 3), a hatched strip is cut away. It holds the −2x and −6 pieces. Maths calls that negative area.
Always test. Put a real number in place of x. If 3(x + 4) = 3x + 12 is right, both sides give the same answer for every x. Try x = 2: 3 × 6 = 18 and 3 × 2 + 12 = 18. 3x + 4 would give 10, so it is wrong.
Common mix-ups. 3(x + 4) is NOT 3x + 4: the 3 must multiply the 4 as well. (x + 2)(x + 3) is NOT x² + 6: do not forget the two middle pieces. A minus sign stays with its number.
Try this above. In Play, press the + and − buttons and watch the pieces change. Switch to Minus to see a strip cut away. Then change x is and watch both sides stay equal.
To expand a bracket means to multiply out the bracket so that no bracket is left. For example 3(x + 4) expands to 3x + 12.
To factorise means to write an expression with a bracket by taking out a common factor. For example 6x + 9 factorises to 3(2x + 3). It is the opposite of expanding.
The 3 means three rows. Every row has an x-block and four little squares, so there are 3 x-blocks and 12 little squares. Only multiplying the first part leaves out 8 squares. Test it with x = 2: 3(2 + 4) = 18, but 3 × 2 + 4 = 10.
When a bracket has a minus, a strip is cut away from the picture. The area of that strip is taken off the total, so it counts as a negative piece. In the game it is drawn hatched.
Just starting: split a rectangle in two and expand a(x + c). I know the basics: expand a(x − b), tidy up and factorise. Challenge me: expand double brackets with the four-piece grid and factorise simple quadratics.
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