Add first-degree monomials with whole-number coefficients

Ontario Grade 6 Mathematics

Algebra

Count equal variable groups by adding their whole-number coefficients

You already know how to add equal groups. If you have four groups of something and three more groups of the same thing, you have seven groups. Algebra uses a letter, called a variable, to stand for a number. A monomial is one term, such as

4x

. A first-degree monomial has a variable to the first power, such as

x

or

4x

. A whole number is 0, 1, 2, 3, and so on. In this lesson, you will add first-degree monomials whose coefficients are whole numbers.

What you will learn

Recognize a first-degree monomial and identify its coefficient and variable.
Add first-degree monomials that have the same variable part.
Explain why the variable part stays the same when these monomials are added.

From equal groups to a letter

Imagine that each bag holds the same unknown number of counters. Let

b

stand for the number of counters in one bag. Then

4b

means four groups of

b

counters. The letter does not tell us how many counters are in a bag. It names the amount in one group.

The number multiplying the variable is called the coefficient. In

4b

, the coefficient is

4

, and the variable is

b

. The coefficient tells how many equal groups there are. If the groups are joined, the number in each group stays the same.

If you add three more bags of the same size, you now have seven groups of

b

. The number in each group has not changed. So add the coefficients and keep the variable:

4b+3b=7b

.

Monomials with the same variable part are called like monomials. Their coefficients count equal-sized groups, so you can add those coefficients. In this lesson, we add monomials with whole-number coefficients.

4b+3b=7b

The coefficient counts the groups.
The variable part tells what kind of group is being counted.
For like monomials with whole-number coefficients, add the coefficients and keep the variable part.

See the groups, then write the sum

Think of each

x

as one tile of an unknown but fixed size. Then

5x

represents five

x

-tiles, and

2x

represents two more

x

-tiles. This is a model of equal groups, not a claim about the value of

x

.

All seven tiles stand for the same kind of group. The coefficients tell how many tiles there are. Adding tiles changes how many there are, not what each tile represents. So five groups of

x

plus two groups of

x

make seven groups of

x

.

When a coefficient is

1

, it may be left unwritten. So

x

means one group of

x

. A coefficient of

0

means no groups, so

0x

counts zero groups of

x

. These facts help us read terms correctly when a number is not shown.

5x+2x=7x

A model can show how many equal variable groups are counted.
The coefficient counts the groups.
$x$ means one $x$ -group.

A dependable way to add

First, identify the coefficient and variable in each term. Next, check that the variable parts are the same. If they match, add the whole-number coefficients and write the shared variable part once. The variable stays because the groups being counted are still groups of the same kind.

For three or more like monomials, include every coefficient in the addition. For example,

2y

,

5y

, and

4y

count eleven groups of

y

altogether. The coefficient of a term written as

y

is

1

, so remember to count it when it appears in a sum.

Check your answer by asking: Did I count every group? Did I keep the variable? Does the answer still describe the same kind of group? These questions help catch a missing coefficient or a missing variable.

Try these before looking at the quick check: find the sum of

3p+6p

, and then find the sum of

4q+q+2q

. For the second sum, remember that the unwritten coefficient of

q

is

1

.

2y+5y+4y=11y

Check that the variable parts match before adding.
When all terms have the same variable part, add every whole-number coefficient.
Keep the shared variable part in the answer.

Worked example

Add two like monomials

Add

4a+3a

.

Counting a-groups

The bars count four a-groups, three a-groups, and seven a-groups in total. They do not show the value of a.

Check the variable
Both terms use $a$ , so they count the same kind of group.
Add the coefficients
Four groups of $a$ and three more groups of $a$ make seven groups of $a$ . Keep $a$ in the answer.
$4+3=7$

Answer:

7a

Check: The answer counts seven groups of

a

, matching the four groups and three groups in the question.

Worked example

Add three like monomials

Add

2y+5y+4y

.

Three sets of y-groups

The bars count two, five, and four groups of y. Together they count eleven groups.

Check the variable
Each term has the variable $y$ , so every coefficient counts groups of the same kind.
Add all the coefficients
Two groups, five groups, and four groups make eleven groups. Keep the shared variable $y$ .
$2+5+4=11$

Answer:

11y

Check: The coefficients total eleven, and every term uses

y

.

Worked example

Include a zero coefficient

Add

0m+6m+2m

.

Counting m-groups

A zero coefficient counts no groups; six and two more make eight groups of m.

Read the zero term
A coefficient of zero means there are no groups in that term. The variable part is still $m$ .
$0m=0$
Add the coefficients
All three terms use $m$ . Add their coefficients, including zero, and keep $m$ .
$0+6+2=8$

Answer:

8m

Check: There are eight groups of

m

altogether. The zero term adds no groups.

Common mistakes and how to avoid them

Adding the coefficients but dropping the variable, such as writing

4a+3a=7

.

Correction: The total still counts groups of

a

. Write

7a

.

Multiplying the coefficients instead of adding them.

Correction: The terms count groups being joined, so add the coefficients.

Forgetting one coefficient in a sum with three terms.

Correction: Read each term and include every coefficient in the total.

Writing

q

as though it has no coefficient.

Correction: The unwritten coefficient is

1

. So

q

counts one group of

q

.

Lesson summary

A first-degree monomial has a variable to the first power.
The coefficient tells how many variable groups are counted.
To add like monomials with whole-number coefficients, add the coefficients and keep the shared variable.
A coefficient of zero means there are no groups in that term.

Check your understanding

Question 1

What is

5r+2r

?

$7r$
$7$
$10r$
$5r$

Show answer and explanation

7r

Both terms use

r

. Add the coefficients: five plus two is seven, so the sum is

7r

.

Question 2

What is

3c+4c

?

$12c$
$7c$
$7$
$3c$

Show answer and explanation

7c

Both terms count groups of

c

. Add three and four, then keep

c

.

Question 3

What is

2t+6t+3t

?

$11t$
$11$
$36t$
$8t$

Show answer and explanation

11t

All terms use

t

. The coefficients add to eleven, so the answer is

11t

.

Key terms

Variable: A letter that stands for a number.
Coefficient: The number that tells how many variable groups are counted.
Monomial: One algebraic term, such as $4b$ .
First-degree monomial: A monomial with a variable to the first power, such as $x$ or $4x$ .
Like monomials: Monomials with the same variable part, such as $3m$ and $8m$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation C2.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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C2.1 · Add first-degree monomials with whole-number coefficients

Ontario Grade 6 Mathematics

Count equal variable groups by adding their whole-number coefficients

What you will learn

From equal groups to a letter

See the groups, then write the sum

A dependable way to add

Add two like monomials

Add three like monomials

Include a zero coefficient

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review