Lesson 1: Variables

New Concept

A variable is a quantity that changes. In this lesson, you will learn to represent variables with letters, visualize their relationships using tables and graphs, and write equations to describe how one variable affects another.

What’s next

Get ready to apply this concept! You will start by interpreting graphs and tables on interactive cards, then move on to building your own equations.

Property

A variable is a numerical quantity that changes over time or in different situations. We can show the values of a variable in a table or a graph. By displaying the values of a variable in a table or a graph, we sometimes see trends or patterns in those values.

Examples

• A table tracking the height of a growing sunflower each week. The height is a variable because it increases over time.
• A graph showing the temperature at different times of the day. Temperature is a variable because it changes from morning to night.
• The number of students enrolled in a school each year is a variable, as the student population changes annually.

Explanation

A variable is simply a value that can change, like your height or the daily temperature. We use tables and graphs to organize these changing values, which helps us spot interesting trends, like how a plant grows taller over time.

Property

We often use a letter as a kind of short-hand to represent a variable.

Examples

• To track earnings, we can let $W$ represent the total wages earned and $h$ represent the hours worked.
• In a recipe, we might use $f$ for the amount of flour and $s$ for the amount of sugar needed.
• To study motion, we can use $v$ for velocity and $t$ for time, plotting their relationship on a graph.

Explanation

Think of letters as nicknames for variables to make writing math faster. It is much easier to write $d$ for distance than to spell out 'distance' repeatedly. On graphs, we assign one variable to the horizontal axis and the other to the vertical axis.

Property

Sometimes there is a simple mathematical relationship between the values of two variables. By studying the values in a table, we may be able to find a relationship between the values, and then write an equation relating the two variables.

Examples

• If movie tickets cost 12 dollars each, the total cost $C$ is related to the number of tickets $n$ by the formula $C = 12 \times n$.
• If you and a friend agree to split a restaurant bill equally, your share $S$ depends on the total bill $B$. The formula is $S = B \div 2$.
• A car travels at a constant speed of 60 miles per hour. The distance traveled $d$ is related to the time $t$ in hours by the formula $d = 60 \times t$.

Explanation

Often, two variables are linked by a consistent rule. If you know the value of one, you can calculate the other. For instance, the total cost of concert tickets is directly connected to how many tickets you buy. This rule can be written as an equation.

Property

A formula relating two variables is a type of equation, and an equation is just a statement that two quantities are equal.

Examples

• A table shows that the value of $y$ is always 8 more than the value of $x$. The equation that describes this relationship is $y = x + 8$.
• In a set of data, each $y$ value is always 5 less than its corresponding $x$ value. The formula for this is $y = x - 5$.
• A table shows that $y$ is always half of $x$. We can write this relationship as the equation $y = x \div 2$.

Explanation

A formula is like a math recipe with an equals sign, showing that two expressions are balanced. If you know the value of one variable, the formula tells you exactly what the other variable's value must be to keep things equal.