Grade by Grade
C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Adding and Subtracting Monomials, and Adding Binomials, with Algebra Tiles

C2.1 A1About 45 minutes
Learning goals
  • I can use algebra tiles, including negative tiles, to model monomials and binomials.
  • I can add and subtract monomials with a degree of 1 that have integer coefficients.
  • I can add binomials with a degree of 1 that involve integers, using zero pairs.

Words to know

monomial
An expression with one term, like 4x, −3x or 7.
binomial
An expression with two unlike terms, like 3x − 2 or −x + 5.
degree of 1
The variable has an exponent of 1 (x, not x²).
coefficient
The number multiplied by the variable. In −3x the coefficient is −3.
like terms
Terms with the same variable part, like 3x and −5x, or 4 and −9.
zero pair
A positive tile and a matching negative tile. Together they are worth 0: x + (−x) = 0 and 1 + (−1) = 0.
simplify
Write an expression with as few terms as possible by combining like terms.

Remember from Grade 7

In Grade 7 you added and subtracted monomials like 3x + 5x = 8x and 7x − 3x = 4x, and you learned that unlike terms (like 5x and 6) stay separate. You also used zero pairs of integer counters: (+1) + (−1) = 0.

In Grade 8 you put those ideas together. The coefficients can be negative, and you'll add binomials like (3x − 2) + (−x + 5).

Let's start with a puzzle

A rectangle has a length of 3x − 2 and a width of −x + 5... wait, can a width be −x + 5? Yes, as long as x is a number that makes it positive (like x = 2, which gives a width of 3). Let's find length + width without knowing x. Algebra tiles will help.

x−x+1−1shaded = positive, unshaded with a minus sign = negative
Algebra tiles. A long tile is x and a small square is 1. Shaded tiles are positive. Unshaded tiles marked with a minus sign are negative.

How the tiles work

  • A long tile stands for x. A small square stands for 1.
  • Shaded tiles are positive. Unshaded tiles with a minus sign are negative.
  • A positive tile and a negative tile of the same shape make a zero pair. You can remove a zero pair (or add one) without changing the value.

To add: put all the tiles together, group like tiles, and remove zero pairs.

To subtract: take away the tiles you're subtracting. If you don't have them, add zero pairs first, then take them away.

3x − 2−x + 5Put them together, then remove zero pairsWhat is left: 2x + 3
(3x − 2) + (−x + 5): one x-tile pairs with the −x tile, and two unit tiles pair with the two −1 tiles. Crossed-out tiles are removed zero pairs. 2x + 3 is left.

So (3x − 2) + (−x + 5) = 2x + 3.

Without tiles: add the x-terms like integers, 3x + (−x) = 2x. Then add the constants, (−2) + 5 = 3.

Start with 2xAdd 3 zero pairs (the value is still 2x)Take away 3 negative x-tiles: 5x is left
2x − (−3x): there are no negative x-tiles to take away, so add 3 zero pairs first. Then take away the 3 negative x-tiles. 5x is left.

So 2x − (−3x) = 5x. Just like with integers, subtracting a negative is the same as adding its opposite: 2x − (−3x) = 2x + 3x = 5x.

Think integers

The coefficient rules are the integer rules you already know. Only the coefficients change; the variable stays the same.

  • −4x + 6x = 2x, because (−4) + 6 = 2.
  • −3x − 4x = −7x, because (−3) − 4 = −7.
  • 3x + 2 stays 3x + 2, because x-terms and constants are unlike terms.
Worked example

Example 1: Adding monomials with different signs

Simplify −4x + 6x.

  1. Tiles: 4 negative x-tiles and 6 positive x-tiles.
  2. Make 4 zero pairs and remove them.
  3. 2 positive x-tiles are left. (Or: (−4) + 6 = 2.)

2x

Worked example

Example 2: Subtracting a negative monomial

Simplify 5x − (−2x).

  1. Start with 5 positive x-tiles. There are no negative x-tiles to take away.
  2. Add 2 zero pairs (2 x-tiles and 2 negative x-tiles), then take away the 2 negative x-tiles.
  3. 7 positive x-tiles are left. (Or: add the opposite, 5x + 2x.)

7x

Worked example

Example 3: Adding two binomials

Simplify (2x + 4) + (3x − 7).

  1. x-terms: 2x + 3x = 5x.
  2. Constants: 4 + (−7) = −3. (With tiles: 4 zero pairs, 3 negative units left.)

5x − 3

Worked example

Example 4: All negative

Simplify (−x − 3) + (−2x + 1).

  1. x-terms: (−1x) + (−2x) = −3x.
  2. Constants: (−3) + 1 = −2.

−3x − 2

Worked example

Example 5: Perimeter

A triangle has sides 2x + 3, 2x − 1 and 3x + 4 (in cm). Write a simplified expression for the perimeter.

  1. Add all three sides: (2x + 3) + (2x − 1) + (3x + 4).
  2. x-terms: 2x + 2x + 3x = 7x.
  3. Constants: 3 + (−1) + 4 = 6.

Perimeter = 7x + 6 cm. Check with x = 4: the sides are 11, 7 and 16 cm, which add to 34 cm, and 7(4) + 6 = 34 ✓.

Hands-on activity

Make your own algebra tiles

You need: 2 or 3 sheets of thin cardboard or paper, a ruler, a pencil, and scissors (ask an adult).

  1. Cut about 10 long rectangles (about 2 cm by 7 cm) for x-tiles and 20 small squares (about 2 cm by 2 cm) for unit tiles.
  2. Shade one side of every tile and draw a minus sign on the other side. Shaded side up = positive. Minus side up = negative.
  3. Model each of these, remove zero pairs, and write the answer:
  • (2x + 3) + (x − 5)
  • (−3x + 4) + (2x − 4)
  • 4x − (−2x)
  1. Make up two binomials for a family member to add with your tiles. Check their answer.

Answers: 3x − 2, −x, 6x.

Keep your tiles. You'll use them again for solving equations.

Just for fun

Why did the x-tile and the −x tile stop hanging out?

Every time they got together, it added up to nothing.

Your turn

1

Simplify 3x + (−5x).

2

Simplify −2x + (−4x).

3

Simplify 6x − (−x).

4

Simplify −3x − 4x.

5

Simplify (4x + 2) + (x + 6).

6

Simplify (2x − 5) + (−3x + 1).

7

Simplify (−x + 7) + (−x − 7).

More practice: warm-up

1

Simplify 2x + 3x.

2

Simplify 8x − 5x.

3

Simplify x + (−x).

4

Which are like terms: 4x, 4, −2x?

5

Simplify (x + 1) + (x + 2).

More practice: on your own

1

Simplify −7x + 3x − (−2x).

2

Simplify (5x − 3) + (−2x − 6).

3

Simplify (−4x + 9) + (6x − 9).

4

A rectangle has a length of 3x + 2 and a width of x − 1. Write a simplified expression for its perimeter.

5

Sam wrote (2x + 3) + (x − 5) = 3x + 8. Find and fix the mistake.

Challenge

1

What binomial can you add to 3x − 4 to get −x + 2?

2

Show that (2x − 5) + (−3x + 1) and −x − 4 have the same value when x = −3.

3

A triangle has sides 2x − 1, x + 4 and 3x − 3 (cm). Its perimeter is 42 cm. Find x and the length of each side.

Check your understanding

Quiz

1.

Simplify −5x + 2x.

2.

Simplify 4x − (−3x).

3.

Simplify (x + 4) + (2x − 6).

4.

A zero pair is...

5.

Which expression cannot be written as one term?

6.

Simplify (−2x − 1) + (−x + 4).

For parents and teachers

Big idea: C2.1 asks Grade 8 students to add and subtract monomials with a degree of 1, and to add binomials with a degree of 1 that involve integers, using tools such as algebra tiles. Tiles make the zero-pair idea visible.

Common difficulties:

  • Combining unlike terms (3x + 2 = 5x). Tiles help: a long tile and a small square are different shapes.
  • Sign errors with constants ((2x + 3) + (x − 5) = 3x + 8). Treat the sign as part of the term: "+3 and −5".
  • Subtracting a negative. Show it with zero pairs, then connect to "add the opposite" from Grade 7 integers.
  • Changing the variable (3x + 5x = 8x²). Adding x-tiles never changes their shape.

How to help: Make the paper tiles together and let your child explain each step out loud. Ask "What zero pairs can you remove?" This lesson builds A1 skills: selecting tools and communicating.