C. Algebra · Patterns, Expressions, Equations, Inequalities, Coding and Modelling

Variables and Adding Monomials

C2.1 A1About 40 minutes

Learning goals

I can explain what a term, a coefficient and a monomial are.
I can tell which terms are like terms.
I can add monomials with a degree of 1, using tools like algebra tiles.
I can check a sum by substituting a number for the variable.

Words to know

variable
A letter or symbol that stands for a number we don't know, or a number that can change, like x or n.

term
A number, a variable, or a number times a variable, like 7, x or 4x. Terms are separated by + or − signs.

coefficient
The number multiplied by the variable. In 4x, the coefficient is 4. In x, the coefficient is 1.

monomial
An expression with only one term, like 5, n or 9n. (Mono means one.)

degree of 1
The variable appears just once in the term, like 3x. It is not multiplied by itself.

like terms
Terms with the same variable, like 3x and 5x. Like terms can be added together into one term.

algebra tiles
Tiles used to model expressions: a long tile for x and a small square tile for 1.

Remember from Grade 5

In Grade 5 you learned that 3 × n can be written 3n, and you translated among words, expressions and pictures. You showed that different-looking expressions can be equivalent (always equal).

In Grade 6 you will learn the names of the parts of an expression, and you will add expressions like 3x and 2x.

Let's start with mittens

Ms. Cardinal's class is collecting mittens for a winter clothing drive. Every box holds the same number of pairs. Nobody has counted a box yet, so we call that number x.

Monday: 3 boxes, so 3x pairs.
Tuesday: 2 boxes, so 2x pairs.

Altogether: 3x + 2x. That's 5 boxes, so 5x pairs. We just added two monomials!

Algebra tiles. Three long x-tiles plus two long x-tiles equals five long x-tiles: 3x + 2x = 5x.

The parts of an expression

Look at 4x + 3.

It has two terms: 4x and 3.
x is the variable.
4 is the coefficient of x (the number multiplied by x).
3 is a number term with no variable (sometimes called a constant, because it doesn't change).

A monomial is an expression with just one term, like 4x, 3, x or 12n. So 4x + 3 is not a monomial: it is the sum of two monomials.

In Grade 6, all our monomials have a degree of 1. That means the variable appears just once, like 4x. (Later grades meet terms like x × x.)

Adding like terms

Like terms have the same variable. To add them, add the coefficients and keep the variable.

3x + 2x = 5x (3 x-tiles and 2 x-tiles make 5 x-tiles)
4n + n = 5n (remember, n means 1n)

Unlike terms can't be joined into one term. 3x + 4 has x-tiles and 1-tiles, which are different kinds. It stays 3x + 4.

Three long x-tiles and four small 1-tiles. They are different kinds of tiles, so 3x + 4 cannot be written as one term.

Sum	Like terms?	Result
6a + 2a	yes (both a)	8a
x + x + x	yes (each is 1x)	3x
5m + 4	no (m-term and number)	5m + 4
2x + 3y	no (different variables)	2x + 3y
7 + 9	yes (both numbers)	16

Add like terms by adding coefficients. Leave unlike terms separate.

Worked example

Example 1: Add and check

Add 4n + 3n. Check by substituting n = 2.

Like terms: both have n. Add the coefficients: 4 + 3 = 7.
So 4n + 3n = 7n.
Check with n = 2: 4n + 3n = 8 + 6 = 14. And 7n = 7 × 2 = 14 ✔.

4n + 3n = 7n.

Worked example

Example 2: Three terms

Add a + 5a + 2a.

a has a coefficient of 1 (it means 1a).
1 + 5 + 2 = 8.

a + 5a + 2a = 8a.

Worked example

Example 3: Mixed terms

Add 3x + 2y + 4x.

Group the like terms: 3x and 4x are like terms. 2y is different.
3x + 4x = 7x.
2y has nothing to join, so it stays.

3x + 2y + 4x = 7x + 2y. With tiles, you'd have 7 x-tiles and 2 y-tiles in separate groups.

Worked example

Example 4: Perimeter

A rectangle has a length of 3b and a width of b. Write its perimeter as one monomial. Find it when b = 5 cm.

Perimeter = length + width + length + width = 3b + b + 3b + b.
Add the coefficients: 3 + 1 + 3 + 1 = 8. So the perimeter is 8b.
When b = 5 cm: 8 × 5 = 40 cm. Check: the sides are 15 cm and 5 cm, and 15 + 5 + 15 + 5 = 40 ✔.

Perimeter = 8b, which is 40 cm when b = 5 cm.

Worked example

Example 5: A common mistake

Malik says 5m + 4 = 9m. Is he right?

5m and 4 are unlike terms: one has a variable and one is just a number.
Test with m = 10: 5m + 4 = 50 + 4 = 54. But 9m = 90.
54 is not 90, so the two expressions are not equal.

No. 5m + 4 can't be simplified. It stays 5m + 4.

Adding monomials

Only like terms combine: same variable.
Add the coefficients; the variable stays the same.
A variable on its own has a coefficient of 1.
Check your answer by substituting a small number for the variable. If both sides match, you're on track.

Just for fun

Joke: Why won't 3x and 4 ever join together into one term?

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

Hands-on activity

Make your own algebra tiles

You need: paper or thin cardboard, scissors, a pencil, and a ruler. (Or use craft sticks or pencils for x-tiles and buttons for 1-tiles.)

Cut about 10 long strips (about 2 cm by 6 cm) and write x on each. Cut about 10 small squares (2 cm by 2 cm) and write 1 on each. If you want a second variable, cut a few strips of a different length and write y.
Model each sum with tiles, then push the like tiles together and write the result:

2x + 4x
3x + 2 + x
x + 2y + 3x + y

Mystery check: your grown-up secretly picks a number for x (for example, x = 3). Replace each x-tile with that many 1-tiles. Count. Does the total match your answer?

What to look for: 6x; 4x + 2; 4x + 3y. With x = 3, 4x + 2 = 14 tiles.

Grown-up note: help with or supervise scissors. Keep buttons away from babies and toddlers.

Your turn

Add 2x + 6x.

Add 5n + n.

Add 4a + 3a + 2a.

Can 7y + 2 be written as one term? Explain.

Add 3p + 5q + 2p.

A triangle has three equal sides, each s long. Write its perimeter as a monomial.

Check 6x + 2x = 8x by substituting x = 3.

More practice: warm-up

What is the coefficient in 9k?

Add x + x.

Add 10m + 5m.

Which two are like terms: 4x, 4y, 7x?

How many terms are in 3n + 8?

More practice: on your own

Add 12w + 8w + w.

Add 2x + 3 + 5x + 4.

A rectangle is 4c long and 2c wide. Write its perimeter as a monomial, then find it when c = 3 m.

Raj says 3x + 4y = 7xy. Is he right? Explain.

Liam buys 4 packs of hockey cards, then 3 more packs. Each pack has c cards. Write the total as a sum of monomials, then add. How many cards if c = 8?

Challenge

Find the missing monomial: 3x + ☐ + 2x = 10x.

A pentagon has sides 2a, 2a, 3a, 3a and a. Write its perimeter as a monomial. If the perimeter is 55 cm, how long is a?

Write three different sums of monomials that each equal 12n.

Check your understanding

Quiz

What is 5x + 4x?

Which pair are like terms?

What is the coefficient in 8p?

Add: 2y + 7 + 3y

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

Which is not a monomial?

For parents and teachers

Big idea: Grade 6 students add monomials with a degree of 1 that involve whole numbers, using tools (C2.1). A monomial is a single term like 4x or 7. Like terms (same variable) are added by adding the coefficients: 3x + 2x = 5x. Unlike terms, like 5m + 4, stay separate.

Common difficulties:

Combining unlike terms (5m + 4 = 9m). Tiles help: x-tiles and 1-tiles are different shapes, so they can't make one pile.
Forgetting that x means 1x (x + 4x = 4x instead of 5x).
Adding the variable too (3x + 2x = 5xx). Only the coefficients are added.

How to help: Make paper algebra tiles together, or use craft sticks for x and buttons for 1. Have your child check sums by substituting a number. Social-emotional learning (A1): algebra can feel strange at first. Remind your child that x is just a number we haven't counted yet, like the mittens in an unopened box.