Lesson 2 of 4

Expanding brackets.

About 25 minutes · 5 practice questions

Why a number before brackets multiplies every term inside: a product is the area of a rectangle, and a rectangle whose side is split is cut into parts. And what a minus before brackets does.

Step 1 of 5 · Cutting a rectangle

Step 1 of 5

Cutting a rectangle

A product of two numbers is the area of a rectangle: $7 \times 13$ ⁠ is a rectangle $13$ ⁠ wide and $7$ ⁠ high, with $91$ ⁠ squares in it. Cut the side $13$ ⁠ into $10 + 3$ ⁠, and the rectangle is cut into two smaller rectangles, each easier to work out. Predict the result, then watch the cut.

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Step 2 of 5

A number before brackets

The same idea works when a side has a letter in it. $3(x + 2)$ ⁠ is a rectangle $3$ ⁠ high and $x + 2$ ⁠ wide, because a multiplication hides between a number and brackets. The top side is split into $x$ ⁠ and $2$ ⁠, so the rectangle is cut into two parts, each with its area written on it. Predict what they will say.

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Step 3 of 5

Checking with a number

Expanding brackets means writing the same expression without brackets, and this is the distributive law: $a(b + c) = ab + ac$ ⁠. The value does not change, only the form. A good way to check yourself: put a number in place of x in both forms, and see that they give the same number. Answer, then choose Split into parts under Look inside: the area for the current x is written under the board.

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Step 4 of 5

A minus before brackets

A minus before brackets is a multiplication by $-1$ ⁠. Like any number before brackets, it multiplies every term inside. On the board a part with a minus is drawn pale, with an outline: its area is taken away. Its size on the board is its true size, and the sign is written on it. Predict, then see the two parts.

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Step 5 of 5

A minus inside the brackets

When there is a subtraction inside the brackets, the number outside still multiplies each term together with the sign in front of it: in $4(2x - 3)$ ⁠ the $4$ ⁠ meets $2x$ ⁠ and also $-3$ ⁠. Answer, then choose Split into parts under Look inside and check each part. Move x: the areas change, and the pattern stays.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

The rectangle is $7 \times 13$ ⁠, and the side $13$ ⁠ is cut into $10 + 3$ ⁠. Predict: what is $7 \times 13$ ⁠?

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Practice · Predict, then look

The left side is $3$ ⁠, and the top side $x + 2$ ⁠. Predict: what is $3(x + 2)$ ⁠ without brackets?

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Practice · Calculate

Now $x = 5$ ⁠. Work out $3(x + 2)$ ⁠, and check that $3x + 6$ ⁠ gives the same number.

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Practice · Predict, then look

A minus in front of brackets is a multiplication by $-1$ ⁠. The board’s sides are $-1$ ⁠ and $x - 4$ ⁠. Predict: what is $-(x - 4)$ ⁠ without brackets?

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Practice · Calculate

Expand the brackets in $4(2x - 3)$ ⁠. What is the term with no x?