L2-01 · Name quantities with variables
A variable is a letter that stands for a quantity that can change, like a number of tickets or a cost in dollars. An equation states a rule that links variables. Some rules are set by people, like a price list. Others only describe a pattern in data, and a pattern is not the same as a law of cause.
Level 2 · Grades 7–9 · about 45 minutes
Big idea. A variable is a letter that stands for a quantity that can change, like a number of tickets or a cost in dollars. An equation states a rule that links variables. Some rules are set by people, like a price list. Others only describe a pattern in data, and a pattern is not the same as a law of cause.
Before this lesson. L1-01, L0-06
Grown-ups. The guide for adults is at the end of this page.
What you will learn¶
I can name a quantity with a variable and give its unit.
I can write a rule like y = 3x + 2 and use it to fill in a table and draw a graph.
I can explain what the coefficient and the constant mean in a real situation.
Story: The ticket rule¶
Maya is selling tickets for her school’s science night. The made-up prices are 3 dollars per ticket plus a 2 dollar booking fee for each order. The fee is the same whether you buy 1 ticket or 10. Leo asks, “So how much do 4 tickets cost?” Maya works it out: 3 times 4 is 12, plus 2 is 14 dollars.
Maya does not want to do this sum for every customer. She names the quantities. Let x be the number of tickets in an order. Let y be the total cost in dollars. Now the rule fits on one line: y = 3x + 2. The 3 is the coefficient, the price of each ticket. The 2 is the constant, the fee you pay once.
She fills in a table for 0 to 4 tickets. Each row becomes a point (x, y): (0, 2), (1, 5), (2, 8), (3, 11), (4, 14). On a graph the points sit on a straight line. Each step of +1 across goes +3 up, because each extra ticket adds 3 dollars. The line meets the y axis at 2, the fee. You can check any point: for x = 3, the rule gives 3 times 3 plus 2, which is 11.
Leo has a rule of his own. He wrote down made-up numbers from his basketball team: hours of practice per week and points per game. A line through the dots is close to points = 2 times hours + 4. “So if I practice 30 hours a week, I will score 64 points?” he jokes.
Maya laughs. “The ticket rule is true because we set the prices. Your rule only describes the players you wrote down.” Maybe players who already score well enjoy practice more. Maybe taller players do both. And nobody on the team practiced anywhere near 30 hours, so the rule says nothing about that. An equation can describe a pattern without being a law of cause.
A table for Maya’s made-up ticket prices turns into points, and the points line up on y = 3x + 2.
What the animation shows. Using Maya’s made-up ticket prices, a table fills in one row at a time: x is tickets and y is dollars. Each row becomes a point on the graph: (0, 2), (1, 5), (2, 8), (3, 11), (4, 14). A straight line is drawn through all five points and labeled y = 3x + 2. A small step marked +1 across, +3 up shows that each extra ticket adds 3 dollars, the coefficient. Then the point where the line starts, at 2 dollars for 0 tickets, is circled. That is the booking fee, the constant.
Hands-on: Three price plans on one grid¶
About 20 minutes.
You need:
Graph paper
A ruler
Pencils in three colors
A calculator (optional)
Steps:
Here are three made-up monthly plans, close to real ones. Phone data: 15 dollars a month plus a 10 dollar start fee. Bike share: 10 dollars a month plus a 25 dollar sign-up fee. Streaming: 8 dollars a month, no fee.
Let x be the number of months and y be the total cost in dollars. Write a rule for each plan, like y = 15x + 10.
Make a table for each plan for 0 to 6 months.
Draw axes on graph paper. Put months on the across axis (0 to 6) and dollars on the up axis (0 to 100).
Plot each plan’s points in its own color and join them with a ruler.
For each line, mark where it meets the y axis. Check that it matches the fee in the rule.
Find the month where the phone line and the bike line cross. Check it by putting that x into both rules.
Write one sentence: for a short time, which plan of those two is cheaper, and which is cheaper for a long time?
Practice together¶
An imagined bowling alley charges 5 dollars per game plus 4 dollars for shoes. Name the variables with units, write the rule, and find the cost of 3 games.
Check your answer
Let x be the number of games and y be the total cost in dollars. The rule is y = 5x + 4. For 3 games, y = 5 times 3 plus 4, which is 19 dollars. The coefficient 5 is the price per game. The constant 4 is the shoe fee, paid once.
Try it yourself¶
1. A taxi in an imagined city charges 2 dollars per kilometer plus 3 dollars to start. Write the rule. What does a 6 kilometer ride cost? Where does the line meet the y axis?
Check your answer
Let x be kilometers and y be dollars. Then y = 2x + 3. A 6 kilometer ride costs 2 times 6 plus 3, which is 15 dollars. The line meets the y axis at 3, the starting charge.
2. Kai’s made-up data show that on hot days his family’s fan runs longer. A line through the dots is roughly hours = 0.5 times temperature minus 8, with temperature in degrees Celsius. Is this rule a price list, like Maya’s, or a pattern? What else should you ask?
Check your answer
It is a pattern found in data, not a rule someone set. It may hold only for the temperatures Kai saw. You should ask what else changes on hot days, and whether the rule still works far outside the data.
Talk about it¶
Why does Maya’s rule work for any number of tickets, while Leo’s rule may not work for 30 hours of practice?
Why is it important to say the unit of a variable? What could go wrong if one person counts in dollars and another in cents?
Can you think of a rule in your life that is set by people, and a pattern that only shows up in data?
Check yourself¶
1. In y = 3x + 2 for Maya’s tickets, what does the 2 stand for?
A. The price of each ticket
B. The number of tickets
C. The booking fee, paid once
Show the answer
C. The booking fee, paid once
The constant 2 is added once, whatever x is. It is the fee. The coefficient 3 is the price of each ticket, and x is the number of tickets.
2. On the graph of y = 3x + 2, what happens to y each time x goes up by 1?
A. y goes up by 3
B. y goes up by 2
C. y doubles
Show the answer
A. y goes up by 3
Each step of +1 across goes +3 up. One more ticket adds 3 dollars. The 2 only sets where the line starts.
3. Leo’s made-up team data fit points = 2 times hours + 4. Which statement is fair?
A. The rule must be wrong because it is not about prices.
B. The rule describes a pattern in his data, not a proven cause.
C. Practicing 30 hours a week will give 64 points.
Show the answer
B. The rule describes a pattern in his data, not a proven cause.
A rule fitted to data describes the players he recorded. Other things could explain the pattern, and 30 hours is far outside the data.
Words to know¶
- variable
- A letter, like x or y, that stands for a quantity that can change. Always say what it counts and in what unit.
- coefficient
- The number that multiplies a variable. In y = 3x + 2, the coefficient is 3.
- constant
- A number in a rule that stays the same, like the 2 in y = 3x + 2.
For grown-ups: about this lesson
Why it matters. Writing an expression with a variable that stands for a quantity in a real situation is the Common Core expectation in 6.EE.6, and this lesson extends it to two-variable linear rules NGA Center and CCSSO, 2010. OpenStax treats y = a + bx with a as the y-intercept and b as the slope, which here are the fee and the price per ticket Barbara Illowsky and Susan Dean, 2023. The second half of the story separates a rule fixed by definition (a price list) from a line fitted to observations, which describes a pattern and supports no causal claim by itself.
Watch for these mix-ups.
The fee is part of every ticket. (The constant is added once per order; only the coefficient multiplies x.)
A variable is just an unknown number to solve for. (Here it names a quantity that changes, with a unit.)
Any equation that fits data is a law. (A fitted line describes the recorded data; a cause needs more evidence, and the line may fail outside the data range.)
The line starts at the origin. (It starts at the constant: 2 dollars for 0 tickets.)
To make it easier. Use only the ticket rule. Fill in the table for 0 to 4 tickets and plot the five points before drawing the line.
To make it harder. Find the exact crossing of the phone and bike plans by solving 15x + 10 = 10x + 25. Then ask whether the streaming line ever crosses the phone line for positive x, and explain why from the rules.
Limits. All prices and the basketball data are made up. Real price plans often have extra conditions, such as limits or discounts, that a single straight-line rule leaves out.
On the computer. Grown-ups can see where this idea leads later in Lab: Confounding and intervention, written for older learners.
Sources for this guide. NGA Center and CCSSO, 2010Barbara Illowsky and Susan Dean, 2023
Next: L2-02 · Functions map inputs to outputs
- NGA Center and CCSSO. (2010). Common Core State Standards for Mathematics. Common Core State Standards Initiative. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
- Barbara Illowsky and Susan Dean. (2023). 12.1 Linear Equations. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/12-1-linear-equations