C. Algebra · Algebra: Expressions, Equations and Coding

Multiplying Binomials with Tiles and Area Models

C1.4 C1.3 A1About 70 minutes

Learning goals

I can multiply two binomials using algebra tiles and an area model.
I can multiply every term of one binomial by every term of the other, including negative terms.
I can expand the square of a binomial and explain why (x + 5)² is not x² + 25.
I can use expanding to prove that two expressions are equivalent.

Words to know

binomial
A polynomial with two terms, such as x + 3 or 2x − 5.

product
The result of multiplying. The product of (x + 2) and (x + 3) is x² + 5x + 6.

area model (generic rectangle)
A rectangle divided into parts, one for each pair of terms, whose areas add up to the product. It doesn't need to be drawn to scale.

quadratic expression
A polynomial of degree 2, such as x² + 5x + 6.

perfect square
The square of a binomial, such as (x + 5)² = x² + 10x + 25.

Let's start at the backyard rink

A family's square backyard rink is x metres on each side. Next winter they want to make it 2 m longer in one direction and 3 m longer in the other. The new area is

(x + 2)(x + 3)

That is a product of two binomials. How much bigger is the new rink? To answer, we need to expand it.

Using algebra tiles

Make a rectangle with side lengths x + 2 and x + 3. Fill it with tiles. The tiles inside are the product.

A rectangle with sides (x + 2) and (x + 3) is filled by 1 x²-tile, 5 x-tiles and 6 unit tiles, so (x + 2)(x + 3) = x² + 5x + 6.

So the new rink's area is x² + 5x + 6 square metres. The old rink was x². The rink grows by 5x + 6 square metres. If x = 10, it grows by 5(10) + 6 = 56 m². (Check: 12 × 13 = 156 and 10 × 10 = 100; 156 − 100 = 56 ✓.)

Using an area model

Tiles get crowded with bigger numbers or negatives. An area model is a sketch of the same idea: write the terms of one binomial across the top and the other down the side. Multiply to fill each box, then add the boxes and collect like terms.

Every term of the first binomial gets multiplied by every term of the second. That's the distributive property used twice.

Area model for (2x + 1)(x + 3). The four boxes are 2x², 6x, x and 3. The two x-terms are like terms: 6x + x = 7x.

Worked example

Example 1: Multiply with a negative term

Expand (x + 4)(x − 1).

Area model boxes: x × x = x²; x × (−1) = −x; 4 × x = 4x; 4 × (−1) = −4.
Add the boxes: x² − x + 4x − 4.
Collect the x-terms: −x + 4x = 3x.
Check with x = 2: (6)(1) = 6, and 4 + 6 − 4 = 6 ✓.

x² + 3x − 4

Worked example

Example 2: Square a binomial

Expand (x + 5)², and show that it is not equal to x² + 25.

(x + 5)² means (x + 5)(x + 5).
Boxes: x², 5x, 5x and 25.
Add: x² + 10x + 25.
Counterexample for x² + 25: x = 1 gives (6)² = 36, but 1 + 25 = 26.

(x + 5)² = x² + 10x + 25

The square of (x + 5) has four parts, not two. Forgetting the two middle 5x pieces is the most common mistake with squares.

Worked example

Example 3: Difference of squares

Expand (x + 3)(x − 3). What do you notice?

Boxes: x², −3x, 3x, −9.
−3x + 3x = 0, so the x-terms cancel.

x² − 9. When the binomials differ only in the sign, the middle terms cancel. This pattern is called a difference of squares.

Worked example

Example 4: Finish the pool proof

In the patterns lesson, the border of an n-by-n pool was written as (n + 2)² − n² and also as 4n + 4. Prove they are equivalent.

Expand (n + 2)² = n² + 2n + 2n + 4 = n² + 4n + 4.
Subtract n²: n² + 4n + 4 − n² = 4n + 4. ✓

The expressions are equivalent for every n. The algebra proves what the table suggested.

Remember

Multiply each term of the first binomial by each term of the second: four products for two binomials.
Then collect like terms (usually the two middle ones).
(a + b)² is never a² + b² unless a or b is 0. Draw the square to remember the middle pieces.
In Grade 10 math you will work with quadratic expressions like these a lot, including factoring them back into binomials.

Hands-on activity

Mental math with binomials

You need: paper, a pencil, a calculator for checking.

Work out 12 × 13 with algebra: let x = 10. Then 12 × 13 = (x + 2)(x + 3) = x² + 5x + 6 = 100 + 50 + 6 = 156. Check with a calculator.
Do the same for 21 × 21: (x + 1)² with x = 20: 400 + 40 + 1 = 441.
Try 19 × 21: (x − 1)(x + 1) with x = 20: 400 − 1 = 399.
Make up three of your own and challenge someone at home to beat your mental math with a calculator.
If you have your algebra tiles, build (x + 1)(x + 4) and (x + 2)(x + 2). Both use 1 x²-tile. Which uses more unit tiles?

Model answer for step 5: (x + 1)(x + 4) = x² + 5x + 4 and (x + 2)(x + 2) = x² + 4x + 4. Both use 4 unit tiles; the first uses one more x-tile.

Just for fun

A mental-math trick that's secretly algebra: to square a number ending in 5, like 35, multiply the tens digit by the next number up (3 × 4 = 12) and put 25 on the end: 35² = 1225.

Why it works: (10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25.

Your turn

Expand (x + 1)(x + 4).

Expand (x + 3)(x − 2).

Expand (x − 5)(x − 2).

Expand (2x + 3)(x + 1).

Expand (x + 6)².

Expand (x + 7)(x − 7).

Which tiles would you need to build (x + 2)(x + 4)?

More practice: warm-up

Expand (x + 1)(x + 1).

Expand (x + 2)(x + 5).

List the four boxes of the area model for (x + 3)(x + 4).

Expand (x − 1)(x + 1).

Is (x + 3)² equal to x² + 9?

More practice: on your own

Expand (3x − 2)(x + 4) and check with x = 1.

Expand (2x − 5)².

A square rink is x m on each side. A new rink is 2 m longer and 3 m wider. How much area is added? Find it for x = 10.

Expand and simplify (x + 3)(x + 2) − x(x + 1).

Use (x + 1)² with x = 30 to find 31² in your head.

Challenge

Find whole numbers a and b so that (x + a)(x + b) = x² + 7x + 12. Explain how you found them.

Expand (x + 1)(x + 2)(x + 3). Check with x = 1.

Show that (n + 1)² − (n − 1)² = 4n for every n. Test it with n = 5.

Check your understanding

Quiz

Expand (x + 3)(x + 4).

Expand (x + 5)(x − 2).

Expand (x + 4)².

How many x-tiles are in the rectangle for (x + 1)(x + 5)?

Expand (x − 6)(x + 6).

Expand (2x + 1)(x + 2).

For parents and teachers

Big idea: Multiplying binomials extends the distributive property (C1.4): every term of one binomial multiplies every term of the other. It also lets students prove equivalences they suspected from tables or patterns (C1.3). These quadratic expressions are central to Grade 10 math.

Common difficulties:

Squaring each term: writing (x + 5)² as x² + 25. Draw the four-box square every time until the middle terms become automatic.
Sign errors with negative terms: (x − 5)(x − 2) has +10, not −10.
Forgetting to collect the two middle terms.

How to help: Try the mental-math activity together: 12 × 13 as (10 + 2)(10 + 3) is a fun way to see the area model at work. Ask your teen to check expansions with a small value like x = 1 or x = 2.