Materials & Prep
- Post four station cards around the room and place one recording sheet at each pair of desks.
- Prepare a board or slide with the opening comparison and the worked example.
- Students need pencils; calculators are not needed.
Opening
0–7 minDisplay“Two snack stands sell water bottles. Stand A: 6 bottles for $9. Stand B: 8 bottles for $11. Which stand is the better buy? Stand by A or B.” Students move to the side of the room labeled A or B, then commit to a reason with a partner.
Ask“Is the stand with more bottles automatically the better deal? Is the lower total price automatically the better deal?” Take two quick explanations without confirming the answer. Recall: “What does a ratio tell us?” Listen for “a comparison of two quantities.”
ConfrontReveal that Stand A costs $1.50 per bottle and Stand B costs $1.375 per bottle, so B is the better buy even though its total cost is higher. Students revise their position if needed. State that today they will make fair comparisons by finding the amount for one unit.
Direct instruction
7–15 minDefineA unit rate is a rate written with a denominator of 1, such as dollars per 1 bottle or miles per 1 hour. It lets us compare situations with different-sized groups fairly.
Worked exampleCompare two bike rental companies. Company A charges $18 for 3 hours. Company B charges $28 for 4 hours.
Think aloud“I cannot compare $18 and $28 alone because the rental times differ. I need the cost for the same amount of time, one hour. For A, I divide both parts of the ratio by 3: $18 ÷ 3 = $6 and 3 hours ÷ 3 = 1 hour. A costs $6 per hour. For B, $28 ÷ 4 = $7 and 4 ÷ 4 = 1, so B costs $7 per hour. Since $6 per hour is less, A is the better deal.” Record $18/3 h = $6/1 h and $28/4 h = $7/1 h.
MisconceptionStudents may divide only the cost or choose the smaller total price. Emphasize that a rate compares two quantities, so both quantities must be scaled by the same factor. It is like cutting equal-size pieces from two pizzas: comparing one slice from each is fair only when each slice represents the same fraction of its pizza.
Check for understandingDisplay: “Gym A charges $24 for 3 visits. Gym B charges $30 for 5 visits. Which costs less per visit?” Students show 1 finger for A or 2 fingers for B, then explain the division. Expected: A is $8 per visit; B is $6 per visit; B costs less.
Guided practice
15–25 minStudentsForm pairs and rotate through two of the four posted comparison cards, spending about four minutes at each. At each card, students record the unit rate for both choices, circle the better comparison, and write one sentence beginning, “ is better because it costs/travels/produces per , compared with per .”
Post
- Trail mix A: 12 ounces for $6. Trail mix B: 15 ounces for $9.
- Phone plan A: 4 GB for $20. Phone plan B: 6 GB for $27.
- Printer A: 45 pages in 3 minutes. Printer B: 56 pages in 4 minutes.
- Juice A: 3 bottles for $7.50. Juice B: 5 bottles for $11.50.
CirculateAsk, “What is your one unit? What does your unit rate mean in this situation?” Require pairs to point to both quantities in their division before calculating.
ScaffoldGive students who need it a rate table with columns labeled “total amount,” “number of units,” and “amount for 1 unit.” Model the first row: $6 ÷ 12 oz = $0.50 per oz. Partners may say the comparison before writing it.
ExtensionChallenge ready pairs to compare Trail mix A and B using an equivalent ratio for 30 ounces rather than unit price. Ask why this method reaches the same decision.
Independent work
25–39 minHave studentsReturn to seats and complete the focused problem set individually. They may stand at a posted “check point” after Problem 2 to compare only their unit labels with a partner, then return to finish.
- A bus travels 126 miles in 3 hours. A train travels 220 miles in 4 hours. Which is faster? Show each unit rate and justify.
- Store A sells 7 notebooks for $10.50. Store B sells 9 notebooks for $12.60. Which is the better buy? Explain what your unit rate means.
- A recipe uses 4 cups of flour for 24 cookies. Another uses 5 cups of flour for 35 cookies. Which recipe uses less flour per cookie? Show the unit rates in cups per cookie.
- A student says, “The train is faster because 220 is greater than 126.” Write a response that uses unit rates to show whether the claim is correct.
Look forCorrect units and reasoning, not only correct quotients. Expected answers: bus 42 mi/h, train 55 mi/h, train; A $1.50/notebook, B $1.40/notebook, B; first 1/6 cup/cookie, second 1/7 cup/cookie, second; the train is faster because 55 mi/h is greater than 42 mi/h.
SupportOffer fraction strips or a blank ratio table for Problem 3. Students can compare 1/6 and 1/7 by imagining one cup of flour split among 6 or 7 equal cookies: more equal shares means less flour in each cookie.
Closing
39–45 minExit ticket“Two streaming services offer downloads. Service A: 18 downloads in 3 minutes. Service B: 28 downloads in 4 minutes. Which downloads faster? Find both unit rates and write one sentence that justifies your choice.”
Students submit before leaving. Expected: A is 6 downloads/minute; B is 7 downloads/minute; B is faster. Sort responses quickly into secure, unit-label error, and reasoning error to form the next lesson’s opening group.