Topics
1. Reasoning with Quantities and Expressions
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United States of America
Grades 9-12

1.03 Algebraic expressions

Lesson
Practice

Introduction

Mathematics is used to help us understand and explain our world. We can represent relationships and quantities using variables and expressions. In 6th grade, we were introduced to vocabulary that helps us communicate mathematically. Now, we will use the vocabulary for algebraic expressions to help us interpret the real-world meaning of different expressions.

Algebraic expressions

An expression is a mathematical statement that contains one or more numbers and variables joined together by operators and grouping symbols. An expression does not contain an equal sign or inequality symbol.

An algebraic expression is an expression that includes at least one variable.

Variable

A symbol, usually a letter, used to represent an unknown value.

Example:

Expression: 3x-y+2

Variables: x and y

Term

One part of an expression. Terms are separated by addition or subtraction.

Example:

Expression: 3x-y+2

Terms: 3x,-y and 2

Coefficient

The number or constant that multiplies a variable in an algebraic term.

Example:

Term: 3x+y and 1

Coefficient: 3

Constant term

A term that has a fixed value and as a result does not contain a variable.

Example:

Expression: 3x-y+2

Constant: 2

Like terms

Terms that have the same variables with the same exponents

Example:

Expression:

3x+5x^2 -2x -8

Terms:

3x, 5x^2,-2x, and -8

Like terms: 3x and -2x

Exponent

A number indicating how many times to use its base in a multiplication

Example:

2^3

The exponent is 3

Base

The number that is used in the repeated multiplication indicated by an exponent

Example:

2^3

The base is 2

Factor

A number or expression that another number or expression can be divided by with no remainder

Example:

2x\left(x+1\right)

The factors are 2, x, and (x+1)

Exploration

In order to write an expression that can be used to model the total cost of buying new school supplies, Mr. Okware defines the following variables:

Let x represent the cost of a folder, y represent the cost of a calculator, and z represent the cost of a pencil pack.

  1. What could the following expressions represent in this context?

    • x+y
    • x+y+z
    • 5y
    • 2x+10z
    • x+3y+4z
    • 4\left(4x + y + 2z\right)

  2. In this context, what do the coefficients describe?

  3. What expressions could we write that wouldn't make sense in this context?

Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Examples

Example 1

Bernie and Paula are painting the walls of their house. Bernie can paint x square meters an hour, while Paula can paint y square meters an hour. Bernie spends 4 hours and Paula spends 6 hours painting over the weekend. Together, they paint a total of 4x+6y square meters over the weekend.

Determine what 6y represents in the expression.

Worked Solution
Create a strategy

Read through the problem to see what 6 and y represent in context.

We can use this to help us determine what 6y represents.

Apply the idea

Paula paints for 6 hours and y is the number of square meters she can paint per hour.

The term 6y represents the number of square meters Paula painted over the weekend.

Reflect and check

To verify the units that are included in the term 6y, consider what will result from multiplying the units:

\text{h} \times \dfrac{\text{m}^2}{\text{h}}=\text{m}^2

The result of multiplying 6 hours and y meters squared per hour is 6y\text{ m}^2 which verifies that the term represents the number of square meters painted.

Example 2

A tennis tournament bracket begins with t teams. A team from each match is eliminated in each round until there is one winning team. The expression t\left(\dfrac{1}{2}\right)^x represents the number of teams left after x rounds.

Determine what the base of the exponent represents in the expression.

Worked Solution
Create a strategy

We can use this example of a bracket to help us visualize what is happening in the problem. Looking at the left side, this bracket begins with 16 teams. After the first round, there would only be 8 teams that move on. After the second round, only 4 teams would move on. This pattern will continue until there is one winning team.

A diagram with four levels. The first level is composed of 8 brackets. The second level has 4 brackets. The third lavel has 2 brackets. The fourth level has 1 bracket. Speak to your teacher for more information.
Apply the idea

In the given problem, there are t teams to begin with, and we multiply the number of teams by \dfrac{1}{2} after each round. To represent repeated multiplication, we use exponents.

The base of the exponent represents half the players being eliminated each round.

Reflect and check

Since we do not have an initial amount of teams, we do not know how many times we need to multiply by \dfrac{1}{2}. That is why the exponent is unknown.

Example 3

The perimeter of a rectangle can be expressed by 2\left(l+w\right). Explain what each of the factors represents.

Worked Solution
Create a strategy

First, we need to identify each of the factors. One factor is 2 and the other is \left(l+w\right).

We know that the perimeter of an object is the distance around the outside edges. A rectangle has 4 sides, but 2 are called lengths and 2 are called widths.

Apply the idea
\displaystyle \text{Perimeter}\displaystyle =\displaystyle l+w+l+w
\displaystyle =\displaystyle \left(l+w\right)+\left(l+w\right)
\displaystyle =\displaystyle 2\left(l+w\right)

This shows that the factor l+w represents adding the length and width, and the 2 means we added the length and width twice.

Reflect and check

Another way to represent the perimeter of a rectangle is 2l+2w. We can use the distributive property to see that this is an equivalent expression to the one given in the problem.

Example 4

Vincenzo runs a removalist company that charges \$ 37.50 per hour plus a one-off truck hire fee of \$ 150.00.

Write an expression that models how much he charges for a job that lasts a hours.

Worked Solution
Create a strategy

We need to look at the two values that affect the price of the job; the cost per hour of \$ 37.50 and the truck hire fee of \$ 150.00.

For each hour worked, Vincenzo charges an additional \$37.50. Let's consider a few cases:

Cost of working 1 hour: \$150+\$37.50

Cost of working 2 hours: \$150+\$37.50+\$37.50

Cost of working 3 hours: \$150+\$37.50+\$37.50+\$37.50

Notice that for each additional hour worked, we add an additional \$37.50. Multiplication is repeated addition, so we can multiply \$37.50 by the number of hours instead of adding repeatedly.

Apply the idea

Since the \$150 is a one-time fee, this value will remain constant. Next, we multiply \$37.50 by the number of hours which is a.

\text{Cost: } 37.5a+150

Reflect and check

For this problem, we would replace a with the number of hours Vincenzo works on a particular job, and the result would be the amount of money he makes on that job.

Idea summary

Expressions can be used to represent mathematical relationships. In an expression, sums often represent totals, coefficients and factors represent multiplication, and exponents represent repeated multiplication. When interpreting an expression in context, we can use the units to help understand the meaning.

Outcomes

A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.

A.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

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