Introduction

Two Quantities That Change Together is an important Grade 6 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with two quantities that change together.

What Is Two Quantities That Change Together?

Two Quantities That Change Together means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Two Quantities That Change Together

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: The table below shows the relationship between the number of pizzas ordered and the total cost. What is the rule that relates cost \(C\) to the number of pizzas \(p\)?

Pizzas (\(p\))\(2\)\(4\)\(6\)\(8\)
Total Cost (\(C\))$\(18\)$\(36\)$\(54\)$\(72\)
  • A. \(C = 8p\)
  • B. \(C = 12p\)
  • C. \(C = 10p\)
  • D. \(C = 9p\)

Why it works: Each pizza costs $\(9\). Check: when \(p=2\), \(C=9(2)=18\); when \(p=4\), \(C=9(4)=36\). The rule is \(C=9p\).

Answer: \(C = 9p\)

Visual Model 2

Question: A table shows the cost of buying notebooks. Which equation best represents this relationship?

Notebooks\(1\)\(3\)\(5\)\(7\)
Cost ($)\(2\)\(6\)\(10\)\(14\)
  • A. \(\text{Cost} = 2 \times \text{Notebooks}\)
  • B. \(\text{Cost} = 3 \times \text{Notebooks}\)
  • C. \(\text{Cost} = \text{Notebooks} + 1\)
  • D. \(\text{Cost} = 2 \times \text{Notebooks} + 1\)

Why it works: The cost is $\(2\) per notebook. Check: \(1\) notebook costs $\(2\); \(3\) notebooks cost \($6\). The equation is \(\text{Cost}=2n\).

Answer: Cost = \(2 \times\) Notebooks

Worked Examples

Example 1

Question: The graph below shows the relationship between the number of hours worked and earnings. Based on the graph, how much does an employee earn per hour?

Example 1

  • A. $\(2\) per hour
  • B. $\(3\) per hour
  • C. $\(4\) per hour
  • D. $\(5\) per hour
  1. The graph shows the line passes through \((2,4)\) and \((4,8)\).
  2. The slope is \(\frac{8-4}{4-2}=\frac{4}{2}=2\), so the rate is $\(2\) per hour.

Answer: $\(2\) per hour

Example 2

Question: The table represents the number of pages read in a book as a function of time. Which is the dependent variable?

Time (hours)\(0\)\(1\)\(2\)\(3\)
Pages Read\(0\)\(25\)\(50\)\(75\)
  • A. Time in hours
  • B. Pages read
  • C. The reader
  • D. The book
  1. The dependent variable depends on another variable.
  2. Pages read depends on time spent reading.

Answer: Pages read

Example 3

Question: A store charges \($3\) per item. Complete the table. What are the missing costs?

Items\(2\)\(5\)\(8\)\(12\)
Cost$\(6\)?$\(24\)?
  • A. $\(12\) and $\(30\)
  • B. $\(15\) and $\(36\)
  • C. $\(18\) and $\(48\)
  • D. $\(24\) and $\(60\)
  1. At $\(3\) per item: \(5\) items cost \(3 \times 5 = 15\) dollars; \(12\) items cost \(3 \times 12 = 36\) dollars.

Answer: $\(15\) and $\(36\)

Real-World Word Problems

Problem 1

Question: A worker earns \($15\) per hour. Which equation relates total earnings \(E\) to hours worked \(h\)?

  • A. \(E=15+h\)
  • B. \(E=15h\)
  • C. \(E=\frac{15}{h}\)
  • D. \(E=h-15\)

Why it works: Earnings equal hourly rate multiplied by hours worked: \(E=15h\).

Answer: \(E=15h\)

Problem 2

Question: The distance \(d\) (in miles) traveled by a cyclist is related to time \(t\) (in hours) by the equation \(d = 20t\). What does the number \(20\) represent in this context?

  • A. The cyclist's speed in miles per hour
  • B. The distance traveled
  • C. The time in hours
  • D. The total hours of cycling

Why it works: In the equation \(d=20t\), \(20\) is the constant rate of change (the speed), and \(t\) is the time.

Answer: The cyclist's speed in miles per hour

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

A car travels at a constant speed of \(60\) miles per hour. Which variable is independent?

  • A. Distance traveled
  • B. Time driven
  • C. Speed of the car
  • D. Fuel consumption

Question 2

A gym membership costs \($50\) per month. If \(m\) represents the number of months and \(T\) represents the total cost, which equation is correct?

  • A. \(m = 50T\)
  • B. \(T = 50 + m\)
  • C. \(T = 50m\)
  • D. \(T = m - 50\)

Question 3

The graph below shows a proportional relationship. Which equation matches this graph?

Question 3

  • A. \(y = 2x\)
  • B. \(y = x + 2\)
  • C. \(y = 3x\)
  • D. \(y = x + 1\)

Question 4

A recipe uses \(2\) cups of flour for every \(3\) eggs. If you use \(6\) eggs, how many cups of flour do you need?

  • A. \(3\) cups
  • B. \(6\) cups
  • C. \(5\) cups
  • D. \(4\) cups

Question 5

A car rental charges a daily rate of \($40\) per day. Write an equation relating total cost \(C\) to number of days \(d\).

  • A. \(C = d + 40\)
  • B. \(C = 40d\)
  • C. \(d = 40C\)
  • D. \(C = \frac{40}{d}\)

Question 6

The table shows the relationship between side length and perimeter of a square. Which equation describes this relationship?

Side Length\(1\)\(2\)\(3\)\(4\)
Perimeter\(4\)\(8\)\(12\)\(16\)
  • A. \(P = 3s\)
  • B. \(P = 4s\)
  • C. \(P = s + 3\)
  • D. \(P = 2s + 2\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: Time driven

The independent variable is what you control or choose. Time is chosen first; distance depends on time.

Question 2

Answer: \(T = 50m\)

Total cost equals $\(50\) per month times the number of months: \(T=50m\).

Question 3

Answer: \(y = 2x\)

The line passes through \((1,2)\) and \((3,6)\). The slope is \(\frac{6-2}{3-1}=\frac{4}{2}=2\), giving \(y=2x\).

Question 4

Answer: \(4\) cups

If \(2\) cups of flour go with \(3\) eggs, then \(6\) eggs is twice as many, so you need \(2 \times 2 = 4\) cups of flour.

Question 5

Answer: \(C = 40d\)

Total cost equals the daily rate of $\(40\) times the number of days: \(C = 40d\).

Question 6

Answer: \(P = 4s\)

A square has \(4\) equal sides, so perimeter \(P = 4 \times s\) where \(s\) is the side length.

Timed practice quizzes

Related Quizzes

After studying Two Quantities That Change Together, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 6 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Two Quantities That Change Together becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.