A focused Grade 6 learning session for Two Quantities That Change Together using visual models, worked examples, strategy guidance, and a short mastery check.
Student goal: I can solve problems involving two quantities that change together.
Two Quantities That Change Together means choosing a model, naming what each number means, and explaining the strategy.
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 (\(C\)) | $\(18\) | $\(36\) | $\(54\) | $\(72\) |
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: \(C = 9p\)
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?
Answer: $\(2\) per hour
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 | \(0\) | \(25\) | \(50\) | \(75\) |
Answer: Pages read
Question: A worker earns \($15\) per hour. Which equation relates total earnings \(E\) to hours worked \(h\)?
Why it works: Earnings equal hourly rate multiplied by hours worked: \(E=15h\).
Answer: \(E=15h\)
Two Quantities That Change Together becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
Understand the model before choosing the operation.
Use scratch paper. The explanation matters more than simply seeing whether the choice is correct.
A worker earns \($15\) per hour. Which equation relates total earnings \(E\) to hours worked \(h\)?
Answer B: \(E=15h\)
Earnings equal hourly rate multiplied by hours worked: \(E=15h\).
A car travels at a constant speed of \(60\) miles per hour. Which variable is independent?
Answer B: Time driven
The independent variable is what you control or choose. Time is chosen first; distance depends on time.
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 \(p\)?
Answer D: \(C = 9p\)
Each pizza costs $\(9\). Check: when \(p=2\), \(C=9(2)=18\); when \(p=4\), \(C=9(4)=36\). The rule is \(C=9p\).
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?
Answer A: The cyclist's speed in miles per hour
In the equation \(d=20t\), \(20\) is the constant rate of change (the speed), and \(t\) is the time.
A table shows the cost of buying notebooks. Which equation best represents this relationship?
Answer A: \(\text{Cost} = 2 \times \text{Notebooks}\)
The cost is $\(2\) per notebook. Check: \(1\) notebook costs $\(2\); \(3\) notebooks cost \($6\). The equation is \(\text{Cost}=2n\).
A gym membership costs \($50\) per month. If \(m\) represents the number of months and \(T\) represents the total cost, which equation is correct?
Answer C: \(T = 50m\)
Total cost equals $\(50\) per month times the number of months: \(T=50m\).
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?
Answer A: $\(2\) per hour
The graph shows the line passes through \((2,4)\) and \((4,8)\). The slope is \(\frac{8-4}{4-2}=\frac{4}{2}=2\), so the rate is $\(2\) per hour.
The table represents the number of pages read in a book as a function of time. Which is the dependent variable?
Answer B: Pages read
The dependent variable depends on another variable. Pages read depends on time spent reading.
A store charges \($3\) per item. Complete the table. What are the missing costs?
Answer B: $\(15\) and $\(36\)
At $\(3\) per item: \(5\) items cost \(3 \times 5 = 15\) dollars; \(12\) items cost \(3 \times 12 = 36\) dollars.
The graph below shows a proportional relationship. Which equation matches this graph?
Answer A: \(y = 2x\)
The line passes through \((1,2)\) and \((3,6)\). The slope is \(\frac{6-2}{3-1}=\frac{4}{2}=2\), giving \(y=2x\).
A recipe uses \(2\) cups of flour for every \(3\) eggs. If you use \(6\) eggs, how many cups of flour do you need?
Answer D: \(4\) 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.
