Grade by Grade
C. Algebra · Patterns, Integers, Equations, Inequalities, Coding and Modelling

Adding and Subtracting Monomials

C2.1 A1About 45 minutes
Learning goals
  • I can explain what a monomial with a degree of 1 is.
  • I can tell like terms from unlike terms.
  • I can add and subtract monomials using algebra tiles or drawings.
  • I can simplify an expression by combining like terms.

Words to know

monomial
An expression with just one term: a number (like 7), a variable (like x) or a number times a variable (like 4x).
degree of 1
The variable is not multiplied by itself. 4x has a degree of 1; x × x (written x²) does not.
coefficient
The number multiplied by the variable. In 4x, the coefficient is 4. In x, the coefficient is 1.
constant
A term with no variable, like 7.
like terms
Terms with the same variable, like 3x and 5x, or two constants, like 2 and 9.
simplify
Rewrite an expression with fewer terms by combining like terms: 3x + 5x becomes 8x.
algebra tiles
Tiles that model algebra. A long tile is x and a small square is 1.

Remember from Grade 6

In Grade 6 you learned that a monomial is one term, like 4x or 7, and you added monomials with a degree of 1 using tiles: 3x + 2x = 5x. You also learned that 5m + 4 can't be combined into one term.

In Grade 7 you will add and subtract monomials and simplify longer expressions.

Let's start with a hockey bag

Ava has 3 bags of pucks and 2 loose pucks. Her brother has 2 bags and 4 loose pucks. Every bag holds the same number of pucks, but nobody knows how many.

Together they have 5 bags and 6 loose pucks: 5x + 6, where x is the number of pucks in a bag. You can't add the bags and the loose pucks into one number, because you don't know x. Bags go with bags, loose pucks go with loose pucks. That's the main idea of today's lesson!

Like terms

Like terms have the same variable: 3x and 5x are like terms. 4 and 9 are like terms too (both constants). But 3x and 3 are unlike terms, and so are 3x and 3y.

  • To add like terms, add the coefficients: 3x + 5x = 8x.
  • To subtract like terms, subtract the coefficients: 7x − 3x = 4x.
  • The variable stays the same. 3x + 5x is 8x, not 8x².
  • Unlike terms stay separate. 5x + 6 cannot be written as one term.
3x + 2+2x + 4=5x+ 6
Algebra tiles. Three x-tiles and 2 unit tiles plus two x-tiles and 4 unit tiles make five x-tiles and 6 unit tiles: 5x + 6.
7x − 3x: remove 3 x-tiles4 x-tiles are left: 4x
Seven x-tiles. Three are crossed out (taken away). Four are left: 7x − 3x = 4x.

No tiles? Make them!

Cut long strips and small squares from cardboard or paper. Long strip = x. Small square = 1. Or just draw: a tall rectangle for x and a small square for 1.

Worked example

Example 1: Add with tiles

Simplify (3x + 2) + (2x + 4).

  1. Model 3x + 2: three x-tiles and two unit tiles. Model 2x + 4: two x-tiles and four unit tiles.
  2. Put the x-tiles together: 3x + 2x = 5x.
  3. Put the unit tiles together: 2 + 4 = 6.

5x + 6

Worked example

Example 2: Subtract like terms

Simplify 7x − 3x and 8y − y.

  1. 7x − 3x: start with 7 x-tiles and take away 3. 4x are left.
  2. 8y − y: remember y means 1y. 8y − 1y = 7y.

7x − 3x = 4x and 8y − y = 7y

Worked example

Example 3: Simplify a longer expression

Simplify 4x + 5 + 3x − 2.

  1. Find the like terms. x-terms: 4x and 3x. Constants: +5 and −2.
  2. Combine the x-terms: 4x + 3x = 7x.
  3. Combine the constants: 5 − 2 = 3. (With tiles: start with 5 unit tiles and take 2 away.)

7x + 3

Worked example

Example 4: Two different variables

Simplify 6a + 2b − a + 3b.

  1. a-terms: 6a − a = 6a − 1a = 5a.
  2. b-terms: 2b + 3b = 5b.
  3. 5a and 5b are unlike terms, so they stay separate.

5a + 5b (not 10ab!)

Worked example

Example 5: Perimeter

A rectangle is 3x long and 2x wide. Write its perimeter in simplest form. Find the perimeter when x = 4 cm.

  1. Perimeter = 3x + 2x + 3x + 2x.
  2. Add the coefficients: 3 + 2 + 3 + 2 = 10. Perimeter = 10x.
  3. When x = 4 cm: 10 × 4 = 40 cm. Check: the sides are 12 cm and 8 cm, and 12 + 8 + 12 + 8 = 40 ✓.

Perimeter = 10x, which is 40 cm when x = 4 cm.

Common mistakes

  • 5x + 3 is not 8x (unlike terms).
  • 4x − x is 3x, not 4 (x means 1x).
  • 2a + 3b is not 5ab.
  • Check by choosing a number for x: if x = 2, then 4x + 5 + 3x − 2 = 8 + 5 + 6 − 2 = 17, and 7x + 3 = 14 + 3 = 17 ✓.
Just for fun

Why didn't 5x and 6 combine into one term?

They just weren't each other's type. They're unlike terms!

Hands-on activity

Make your own algebra tiles

You need: a cereal box or thin cardboard, a ruler, a pencil and scissors (ask a grown-up to help with cutting).

  1. Cut 10 long strips (about 1 cm by 4 cm). Each strip is x.
  2. Cut 15 small squares (1 cm by 1 cm). Each square is 1.
  3. Model 2x + 5. Then add 3x + 1. What do you have? (5x + 6)
  4. Model 6x + 4. Take away 2x + 3. What's left? (4x + 1)
  5. Ask a grown-up to say an expression, like "4x plus 7 minus x minus 2". Build it, simplify it and say the answer. (3x + 5)
  6. Check one answer by deciding that each strip is worth 3: count the total value before and after simplifying.

Your turn

1

Simplify 6x + 2x.

2

Simplify 9n − 4n.

3

Simplify 5m + 3 + 2m.

4

Simplify 8x + 7 − 3x − 2.

5

Can 4x + 3y be simplified? Explain.

6

Simplify 10k − k.

7

You have 4 x-tiles and 5 unit tiles. You take away 1 x-tile and 2 unit tiles. Write what's left.

More practice: warm-up

1

Simplify x + x + x.

2

Simplify 2a + 5a.

3

Simplify 7t − 2t.

4

Are 3p and 3 like terms?

5

Simplify 4 + 6 + 2x.

More practice: on your own

1

Simplify 3a + 4b + 2a + b.

2

Simplify 12y + 9 − 5y − 9.

3

A triangle has sides 2x, 3x and 4x. Write its perimeter. Find the perimeter when x = 5 cm.

4

Ben says 6x − 2 = 4x. Is he right? Explain.

5

Start with 8n + 6. Take away 3n + 1. What is left?

Challenge

1

A rectangle has a perimeter of 16x. Its length is 5x. What is its width?

2

Fill in the blank: 4x + ___ + 3 = 9x + 7.

3

Aiden has 3x + 5 hockey cards and Bea has 2x + 12, where x is the number of cards in a pack. Write their total in simplest form. Who has more cards if x = 10? If x = 5?

Check your understanding

Quiz

1.

Simplify 5x + 3x.

2.

Simplify 9y − y.

3.

Which pair are like terms?

4.

Simplify 6m + 4 − 2m + 1.

5.

A square has sides of length x. What is its perimeter?

6.

What is the coefficient in 12k?

For parents and teachers

Big idea: Grade 7 students add and subtract monomials with a degree of 1 that involve whole numbers, using tools such as algebra tiles (C2.1). Grade 6 only added; subtracting and simplifying longer expressions are the new steps. Only like terms combine: x-tiles with x-tiles, unit tiles with unit tiles.

Common difficulties:

  • Combining unlike terms, like 5x + 3 = 8x or 2a + 3b = 5ab.
  • Forgetting the invisible 1: 4x − x = 3x, not 4.
  • Losing the subtraction sign on a constant: in 4x + 5 + 3x − 2, the 2 is subtracted.

How to help: Make paper tiles together. When your child simplifies, ask them to check by picking a number for x and evaluating both versions (they should match). This lesson builds A1 skills: using tools to model ideas, checking work and explaining reasoning.