82. The Area of a Product · Visualizing monomial multiplication

Try to break it

Notice that doubling one side doubles the area. Tripling a side triples the area.

How you build it

Construct rectangle ABCD with ruler and compass.

• Place point A as the bottom-left corner.
• Place point B to the right of A, level with it.
• Draw segment AB by clicking A then B.
• Erect a perpendicular to AB at A: click A, then click above it.
• Erect a perpendicular to AB at B: click B, then click above it.
• With centre A, draw an arc crossing the perpendicular at A. Click A, then click to set the radius.
• Mark point D where the arc meets the perpendicular at A.
• With centre B and the same radius, draw an arc crossing the perpendicular at B. Click B.
• Mark point C where the arc meets the perpendicular at B.
• Draw segment DC by clicking D then C to complete rectangle ABCD.

The proof, step by step

Prove that the rectangle area equals the product of its sides a and b.

• Identify the side lengths as 'a' and 'b'.
• The area formula for a rectangle is length × width.
• Substitute the side lengths: Area = a × b.

Worked example

A rectangular garden has length 'l' = 6x and width 'w' = 3y. What is the area of the garden in terms of x and y?

Multiply coefficients: 6 × 3 = 18. Multiply variables: x × y = xy. Area = 18xy.

• 9xy
• 18xy — correct
• 18x²y²
• 9x²y²