Many ways, one answer

The idea

Different ways of seeing the same pattern or region give algebraic expressions that look different but, when simplified correctly, become the same expression, so they are equal.

Four friends, four expressions

A pattern of circles has 3, 8, 15, 24, … circles at steps 1, 2, 3, 4. Four friends write the number at step k:

(k + 1)² − 1 k² + 2 × k k × (k + 1) + k k × (k + 2)

The four expressions look different. Are they?

• All four are different
• All four give the same number
• Only two of them agree

Four ways to count

Each way groups the same circles differently. Tap a way and watch the circles move.

Different looks, same expression

Different ways of seeing the same pattern or region give algebraic expressions that look different but, when simplified correctly, become the same expression, so they are equal.

(k + 1)² − 1 = k² + 2k + 1 − 1 = k² + 2k k² + 2 × k = k² + 2k k × (k + 1) + k = k² + k + k = k² + 2k k × (k + 2) = k² + 2k

So k² + 2k gives the number of circles at step k. Finding different ways to look at a problem is a creative process.

Notes

Different ways of seeing the same pattern give expressions that look different. When simplified correctly they become the same expression, so they are equal.

Check yourself

Use any way to find the number of circles at step 10.

Answer: 120

10 × 12 = 120. Check: (10 + 1)² − 1 = 121 − 1 = 120.

Which expression does not count the circles (3, 8, 15, 24, …)?

A square of side m + n has four m × n rectangles taken out. Tadang writes the area left as (m + n)² − 4mn. Find it for m = 3 and n = 7.

Answer: 16

(3 + 7)² − 4 × 3 × 7 = 100 − 84 = 16. Yusuf writes (n − m)² = 4² = 16. Same area.

Anusha writes the area as x² − xy. Aditya writes x(x − y). Are they equal?

• (k + 1)² − 1. It works. At k = 3 it gives 16 − 1 = 15.
• k² + 2 — correct. Yes! At k = 3 it gives 9 + 2 = 11, but step 3 has 15 circles. It should be k² + 2k.
• k × (k + 2). It works. At k = 3 it gives 3 × 5 = 15.
• No, one has a bracket. A bracket only groups the terms. x(x − y) = x² − xy by the distributive property.
• Only when y = 0. They are equal for every x and y, not only y = 0.
• Yes: x(x − y) = x² − xy — correct. Yes! The distributive property turns x(x − y) into x² − xy.