Lesson 2.3: Variables

New Concept

A variable is a numerical quantity that changes. We will learn to spot patterns between variables, describe their relationships with math rules, and represent these rules using letters to form our first algebraic expressions.

What’s next

You'll start with practice cards to find patterns in tables and bar graphs. Then, you'll use interactive examples to write your first algebraic expressions.

Property

A variable is a numerical quantity that can take on different values at different times or in different situations. Quantities that do not change are called constants. When looking for a pattern between variables, check if a constant is added/subtracted, if they are multiplied/divided by a constant, or if their sum/product is constant.

Examples

• A student's final score is their test score plus 7 bonus points. If the test score is 85, the final score is $85 + 7 = 92$. The test score and final score are variables; the 7 bonus points are a constant.

• A phone plan costs 30 dollars per month. The total cost depends on the number of months. For 4 months, the cost is $30 \times 4 = 120$ dollars. The number of months and total cost are variables.

Property

We can use a bar graph to display the values of a variable. The height of the bar illustrates the value at each time or in each situation. A double bar graph can be used to compare the values of two different variables under the same conditions.

Examples

• A bar graph shows daily screen time. On Monday, the bar reaches 3 hours. On Wednesday, it reaches 2 hours. This means screen time was 1 hour longer on Monday than on Wednesday.

• A double bar graph compares the number of goals scored by Team A and Team B. In the first game, Team A's bar is at 4 and Team B's is at 2. Team A scored 2 more goals.

Property

We often use a single letter to represent a variable quantity. A letter used as a variable must always stand for a number. For example, a mathematical sentence like "Shawna's age = Jayson's age + 22" can be written with a variable as "Shawna's age = $a + 22$," where $a$ stands for Jayson's age.

Examples

• If a movie ticket costs 15 dollars, the total cost for $n$ people is $15 \times n$. For a group of 4 people, the cost is $15 \times 4 = 60$ dollars.

• A baker needs 3 cups of flour for each cake. To find the total flour needed for $c$ cakes, the expression is $3 \times c$. For 5 cakes, she needs $3 \times 5 = 15$ cups.

Property

To find a fraction of a number, you multiply by the fraction. For example, to find $\frac{2}{3}$ of a number, you can divide by 3 and then multiply by 2. To find a percentage of a number, you multiply the number by the decimal equivalent of the percent. For example, 5% of 80 is $0.05 \times 80 = 4$. To increase a number by a percentage, multiply by $(1 + \text{decimal})$. For example, an increase of 5% means multiplying by 1.05.

Examples

• A store offers a 20% discount on any item. For an item with price $p$, the discount is $0.20 \times p$. For a 50 dollar jacket, the discount is $0.20 \times 50 = 10$ dollars.

• A recipe requires an amount of sugar equal to $\frac{1}{4}$ the amount of flour. If you use $f$ cups of flour, you need $\frac{1}{4} \times f$ cups of sugar. For 6 cups of flour, you need $\frac{1}{4} \times 6 = 1.5$ cups of sugar.