Learning Goal: Represent relationships between two quantities using equations, tables, and graphs, identifying dependent and independent variables.
Vocabulary
independent variableThe input quantity that you choose or that changes on its own (often x), like number of hours.
dependent variableThe output quantity that depends on the input (often y), like total cost.
equationA rule relating two variables, like y = 8x, that lets you find one value from the other.
input/output tableA table that pairs each input value with its matching output value.
What You Need to Know
The independent variable is the input; the dependent variable depends on (changes because of) it.
Write the rule as dependent = (something) × independent, like cost c = 8h.
A table lists matching input/output pairs you get by using the equation.
On a graph, the independent variable goes on the x-axis and the dependent on the y-axis.
Find the rule by checking what you do to each input to get its output.
Example: if y = 5x, then when x = 3, y = 15 (substitute the input).
Worked Example I Do — watch how it works
A worker earns $8 per hour. Write an equation, identify the variables, and find earnings for 6 hours.
Step 1. Earnings depend on hours, so hours h is independent and earnings c is dependent: c = 8h.
Step 2. Substitute h = 6: c = 8 × 6.
Step 3. Multiply: c = 48, so 6 hours earns $48.
Answer: c = 8h; independent h, dependent c; $48 for 6 hours
Guided Practice We Do — try it together
1. Tickets cost $5 each. Write an equation relating total cost c to number of tickets t.
💡 Hint: Cost depends on how many tickets; multiply price by t.
2. In d = 60t (distance in miles, t in hours), which variable is independent?
💡 Hint: The input you choose is independent; distance depends on time.
3. For y = 3x, complete the pair when x = 4.
💡 Hint: Substitute 4 for x and multiply.
Independent Practice You Do — show your work
1. A book has 9 chapters per part. Write an equation for total chapters c with p parts.
2. For y = 7x, find y when x = 5.
3. In c = 2.5g (cost for g gallons), which variable is dependent?
4. A plant grows 2 cm per week. Write an equation for height h after w weeks.
5. For y = 4x, complete the table pair when x = 6.
MCAP-Style Practice
Directions: Answer each item the way you would on the MCAP. For selected-response items, fill in the circle (○) next to the correct answer. For constructed-response items, enter your answer and show your work. Every item below assesses standard 6.EE.C.9.
Item 1Selected Response6.EE.C.9
A printer makes 12 pages per minute. Which equation relates pages p to minutes m?
Select the correct answer.
Item 2Selected Response6.EE.C.9
In y = 6x, if x is the number of boxes and y is total items, which is the independent variable?
Select the correct answer.
Item 3Constructed Response6.EE.C.9
A pool fills at 10 gallons per minute. Write an equation for gallons g after t minutes, then find gallons after 7 minutes.
Enter your answer in the space provided. Show your work.
Enter your answer:
Teacher Answer Key (click to show)
Independent 1
c = 9p
Independent 2
y = 35
Independent 3
c (cost) is dependent
Independent 4
h = 2w
Independent 5
y = 24
Item 1 · 6.EE.C.9
B — Pages equal 12 times the number of minutes, so p = 12m.
Item 2 · 6.EE.C.9
B — x (boxes) is the input you select; y depends on x, so x is independent.
Item 3 · 6.EE.C.9
g = 10t; 70 gallons — Gallons equal 10 times minutes, so g = 10t; substituting t = 7 gives 70.
Neft Teacher · Grade 6 MCAP Mathematics Review · Standard 6.EE.C.9