Universal Studios Rate of Change Adventure

Materials Needed

• Pencil and paper
• Calculator
• Practice worksheet
• Graph paper or a digital graphing tool
• Ruler, optional
• Timer, optional

Learning Objectives

By the end of this 15-minute lesson, the learner will be able to:

• Calculate the average rate of change over a specified interval using a table or graph.
• Interpret the rate of change using appropriate units and a real-world meaning.
• Identify the slope and initial value of a linear function from a table or graph.
• Use the equation average rate of change = change in output ÷ change in input.

Success Criteria

I can:

• Choose two points or two rows from the specified interval.
• Use the formula (y₂ − y₁) ÷ (x₂ − x₁).
• State what the rate means in the context of the problem.
• Identify the initial value as the output when the input is 0.

Introduction: Hook and Purpose — 1 minute

Hook: Imagine planning a day at Universal Studios. The temperature rises from 72°F at 10:00 a.m. to 84°F at 2:00 p.m. How quickly did the temperature increase on average?

Today, you will calculate and explain rates of change using Universal Studios situations involving ticket costs, ride wait times, distance, and attendance.

Key vocabulary:

• Rate of change: How much one quantity changes compared with another quantity.
• Average rate of change: The overall change across an interval.
• Slope: The rate of change of a line.
• Initial value: The output value when the input is 0.

Body

I Do: Teacher or Parent Models — 4 minutes

Example 1: Input-Output Table

A Universal Studios parking lot charges a flat fee plus an hourly fee.

Step 1: Select two points. Use (0, 30) and (3, 45).

Step 2: Find the change in output. $45 − $30 = $15.

Step 3: Find the change in input. 3 − 0 = 3 hours.

Step 4: Divide.

Average rate of change = $15 ÷ 3 = $5 per hour.

Interpretation: The parking cost increases by an average of $5 for every additional hour. The initial value is $30, the cost when 0 hours have passed.

Example 2: Graph or Two Points

A line representing distance traveled through a Universal Studios tour passes through the points (1, 0.8) and (4, 3.2), where x is time in hours and y is distance in miles.

Average rate of change:

(3.2 − 0.8) ÷ (4 − 1) = 2.4 ÷ 3 = 0.8 miles per hour.

Interpretation: The tour group traveled an average of 0.8 miles per hour between hours 1 and 4.

Initial value reminder: If the line crosses the y-axis at (0, 2), then the initial value is 2 miles.

Quick check: If the average rate of change is negative, what does that mean? Expected response: The output decreases as the input increases.

You Do: Independent Practice — 5 minutes

Complete the 10-question worksheet below. Show your formula, substitution, answer, and interpretation when requested. You may use a calculator.

We Do: Check and Discuss — 3 minutes

Review the answers together. For each selected question, explain:

1. Which two input-output values were used?
2. What was the change in output?
3. What was the change in input?
4. What units belong in the answer?
5. What does the answer mean in the Universal Studios context?

Think-pair-share option: Choose one answer that represents an increasing rate and one that represents a decreasing rate. Explain how the sign of the rate affects the situation.

10-Question Practice Worksheet

Universal Studios Rate of Change

Directions: Use the formula below. Include units when possible.

Average rate of change = (change in output) ÷ (change in input)

1. A Universal Studios ticket costs $125 for one day and $175 for a two-day ticket. What is the average increase in cost per additional day?
2. A parking fee table is shown below. Find the average rate of change from 1 hour to 4 hours.

   Hours, x	Cost, y
   0	$30
   1	$35
   2	$40
   3	$45
   4	$50

3. A ride wait time decreases from 75 minutes at 11:00 a.m. to 45 minutes at 2:00 p.m. Find the average rate of change in minutes per hour. Interpret your answer.
4. A Universal Studios souvenir shop records sales:

   Customers, x	Sales, y
   10	$180
   20	$360
   30	$540

   Find the average sales per customer from 10 to 30 customers.
5. A park shuttle travels 6 miles in 20 minutes and 15 miles in 50 minutes. Find the average rate of change in miles per minute.
6. A line representing daily park attendance passes through (2, 18,000) and (5, 30,000). Find its slope and interpret it.
7. The cost of a Universal Studios food plan is modeled by C(d) = 42d + 25, where d is the number of days. Identify the rate of change and the initial value. Explain what each means.
8. A graph of ride wait time contains the points (1, 20) and (5, 60), where x is the number of hours after opening and y is wait time in minutes. Find the average rate of change and interpret it.
9. A family’s remaining budget is represented by the table:

   Hours in Park, x	Remaining Budget, y
   0	$300
   2	$250
   4	$200

   Find the average rate of change from 0 to 4 hours. What does the negative sign mean?
10. Create a real-world Universal Studios example with two points that has an average rate of change of 4. Include labels, units, and one sentence explaining your example.

Exit Ticket — 2 minutes

1. A Universal Studios express pass costs $80 at 10:00 a.m. and $110 at 1:00 p.m. Find the average rate of change in cost per hour.
2. If a linear function has the point (0, 25), what is its initial value? Explain what the point means.
3. In one sentence, explain why units are important when interpreting a rate of change.

Assessment

Formative Assessment

• Listen for correct use of “change in output divided by change in input.”
• Ask the learner to explain the meaning of a positive and negative rate.
• Check that the learner includes units and uses the specified interval.
• Use the quick check and think-pair-share explanations to identify misconceptions.

Summative Assessment

Use the worksheet and exit ticket. The learner demonstrates mastery by correctly solving at least 8 of 10 worksheet questions and at least 2 of 3 exit-ticket questions, including a correct interpretation with units.

Answer Key

1. $50 per day.
2. ($50 − $35) ÷ (4 − 1) = $15 ÷ 3 = $5 per hour.
3. (45 − 75) ÷ 3 = −30 ÷ 3 = −10 minutes per hour. The wait time decreases by 10 minutes per hour on average.
4. ($540 − $180) ÷ (30 − 10) = $360 ÷ 20 = $18 per customer.
5. (15 − 6) ÷ (50 − 20) = 9 ÷ 30 = 0.3 miles per minute.
6. (30,000 − 18,000) ÷ (5 − 2) = 12,000 ÷ 3 = 4,000 visitors per day.
7. Rate of change: $42 per day. Initial value: $25. The plan increases by $42 per day and starts with a $25 fee.
8. (60 − 20) ÷ (5 − 1) = 40 ÷ 4 = 10 minutes per hour.
9. ($200 − $300) ÷ (4 − 0) = −100 ÷ 4 = −$25 per hour. The family spends an average of $25 per hour.
10. Answers will vary. Example: Points (1, $20) and (3, $28) represent a snack stand earning money. The rate is ($28 − $20) ÷ (3 − 1) = $4 per customer or time unit, depending on the labels.

Exit Ticket Answers

1. ($110 − $80) ÷ (3 − 0) = $10 per hour.
2. 25. The function’s output is 25 when the input is 0.
3. Units explain what the rate means, such as dollars per hour, minutes per ride, or visitors per day.

Differentiation and Adaptations

• Support: Provide the formula with blanks: (____ − ____) ÷ (____ − ____). Allow the learner to highlight input values and circle output values.
• Support: Begin with intervals that include x = 0, then move to intervals such as 1 to 4.
• Visual support: Plot the table values on graph paper and draw a line through them.
• Verbal support: Have the learner explain the answer using the sentence frame: “For every 1 ___, the ___ changes by ___ ___.”
• Extension: Write an equation for one linear situation in the form y = mx + b and explain the meanings of m and b.
• Choice: The learner may complete calculations by hand, use a graphing tool, or create an original Universal Studios pricing or ride-wait-time scenario.

Closure and Recap

Ask the learner to complete this verbal recap:

“To find average rate of change, I subtract the ______ values and divide by the change in the ______ values. The rate tells me ______. In a linear function, the slope is ______ and the initial value is ______.”

Expected responses: output, input, how quickly one quantity changes compared with another, the rate of change, the output when the input is 0.