Fast and Furious: Rearranging Formulas for Rate Problems

Materials Needed

• Pencil and paper or a whiteboard
• Calculator
• Timer or stopwatch
• Optional: toy car, ball, or another object that can move
• Optional: colored pens or highlighters

Learning Target

I can rearrange formulas and use these formulas to solve rate problems.

Standard

MA.912.AR.1.2: Rearrange equations or formulas to isolate a quantity of interest.

Success Criteria

By the end of the lesson, I can:

• Identify the quantity I need to find.
• Rearrange a rate formula to isolate that quantity.
• Substitute values and include correct units.
• Explain whether my answer makes sense in the situation.

Introduction: Bell Work and Hook — “Ready, Set, Go!” (2 minutes)

Bell Work: Reinforce Prior Knowledge

Answer these questions without using a formula sheet:

1. A car travels 120 miles in 3 hours. What operation would you use to find its speed?
2. Complete the formula: rate = ______ ÷ ______
3. If d = rt, what do you think you would do to find t?

Check: The answers are:

• 120 ÷ 3 = 40 miles per hour
• distance ÷ time
• Divide both sides by r, giving t = d ÷ r.

Hook: Imagine you are planning a fast road trip. You know the distance and how fast you can travel, but you need to know how long the trip will take. How can one formula answer all three questions: distance, rate, and time?

Lesson Overview

Today we will use the formula:

d = rt

• d = distance
• r = rate or speed
• t = time

We will rearrange the formula so that the quantity we need is alone on one side.

Body: I Do — Teacher/Parent Modeling (4 minutes)

Example 1: Find Distance

A race car travels at 80 miles per hour for 2.5 hours. How far does it travel?

1. Identify the quantity of interest: d.
2. The formula already has d isolated: d = rt.
3. Substitute the values: d = 80(2.5).
4. Solve: d = 200 miles.

Think aloud: “The units are miles per hour multiplied by hours. The hours cancel, leaving miles, which is the correct unit for distance.”

Rearranging the Formula

To isolate r:

d = rt
Divide both sides by t:
r = d/t

To isolate t:

d = rt
Divide both sides by r:
t = d/r

Memory strategy: Cover the letter you want to find in the rate triangle:

d is above r and t.

• Cover d: multiply r × t.
• Cover r: divide d ÷ t.
• Cover t: divide d ÷ r.

Body: We Do — Guided Practice (4 minutes)

Work through each problem together. For every problem, say:

“What am I finding? Which rearranged formula do I need? Do the units make sense?”

Problem 1: Find Rate

A cyclist travels 36 miles in 3 hours. What is the cyclist’s average rate?

1. Quantity of interest: r
2. Rearranged formula: r = d/t
3. Substitute: r = 36/3
4. Answer: 12 miles per hour

Problem 2: Find Time

A train travels 210 miles at a rate of 70 miles per hour. How long does the trip take?

1. Quantity of interest: t
2. Rearranged formula: t = d/r
3. Substitute: t = 210/70
4. Answer: 3 hours

Quick Check: Why is the answer to Problem 2 in hours? Discuss your answer aloud or write one sentence.

Body: You Do — “Fast and Furious” Challenge (3 minutes)

Solve the following problems independently. Show the formula, substitution, answer, and units.

1. Speed Challenge: A go-kart travels 150 meters in 10 seconds. What is its rate in meters per second?
2. Time Challenge: A remote-control car travels 96 feet at 12 feet per second. How long does it take?
3. Distance Challenge: You ride your bike at 15 miles per hour for 1.5 hours. How far do you travel?

Answer Key:

• 1. r = d/t = 150/10 = 15 meters per second
• 2. t = d/r = 96/12 = 8 seconds
• 3. d = rt = 15(1.5) = 22.5 miles

Application Choice: Create Your Own Race (Optional extension or use if additional time is available)

Choose one option:

• Real-world option: Time a toy car, ball, or person moving a measured distance. Calculate the rate.
• Design option: Create a fictional race involving a car, spaceship, animal, or athlete. Give two of the three values—distance, rate, or time—and ask someone else to find the missing value.
• Digital option: Use a calculator or spreadsheet to test how changing the rate or time affects distance.

Assessment

Formative Assessment

• Check bell work responses.
• Ask the learner to explain why t = d/r.
• Check whether the learner identifies the quantity of interest before calculating.
• Review units after each guided practice problem.

Summative Exit Ticket (2 minutes)

A delivery driver travels 180 miles at an average rate of 60 miles per hour.

1. Rearrange d = rt to isolate t.
2. Use the rearranged formula to find the travel time.
3. Explain how you know your answer is reasonable.

Expected response:

t = d/r = 180/60 = 3 hours. The answer is reasonable because traveling 60 miles each hour for 3 hours gives 180 miles.

Differentiation and Support

• For additional support: Provide the formula triangle, highlight the quantity being found, and allow the learner to use a calculator.
• For language support: Use sentence frames such as, “I am finding ____, so I use ____.”
• For hands-on learners: Measure an actual distance and time a moving object.
• For advanced learners: Solve problems involving decimal values, unit conversions, or average speed over multiple parts of a trip.
• For independent learners: Ask the learner to create and solve a three-part rate problem, then check the solution by substituting it back into d = rt.

Conclusion and Recap (Final minute)

Complete this recap aloud or in writing:

• To find distance, I use d = rt.
• To find rate, I rearrange the formula to get r = d/t.
• To find time, I rearrange the formula to get t = d/r.
• I always identify the quantity of interest, substitute carefully, label my units, and check whether my answer makes sense.

Final reflection: Which part of rearranging formulas feels easiest now, and which part would you like to practice again?