Which Deal Is Really Better? Ratio and Unit Rate Reasoning

• Project or write the opening “snack cart” choices and four comparison problems on the board.
• Post four signs around the room: Unit Rate, Equivalent Ratios, Not Enough Information, and Need to Check Units.
• Prepare a half-sheet exit ticket; otherwise, use usual notebooks and pencils.

Have students recall from memory: “A ratio compares ; a unit rate tells the amount for .” Take one precise response, then reveal that the tempting move is to compare 3 to 5 or $2.25 to $3.50 separately. Those do not answer which costs less per bar. Tell students that a fair comparison needs the same one-unit basis and the same units.

Model Snack Cart A with a ratio table: 3 bars/$2.25, then divide both quantities by 3 to get 1 bar/$0.75. Narrate: “I am not dividing only the price; I must preserve the relationship by making the number of bars equal to one. My unit rate is $0.75 per bar.” Model Cart B: $3.50 ÷ 5 = $0.70 per bar. Compare $0.70 per bar and $0.75 per bar; Cart B is better because each bar costs less.

Name the frequent error: choosing the package with more items or the lower total price. Explain that package sizes differ, so totals are not comparable. Use the familiar comparison: it is like judging two race times only after both racers have run the same distance; “per 1” gives both choices the same distance.

Show a second method briefly: Find an equivalent ratio for Cart A with 5 bars: 3 bars for $2.25 is 6 bars for $4.50, so 5 bars would cost $3.75 at that rate. Since $3.50 is less, B is better. Emphasize that either method works only when the comparison units match.

Read one situation at a time. Students move to the sign that names the best next move, then pair with someone nearby to justify their choice in a complete sentence. Call on pairs to explain; students may revise their location after hearing evidence.

1. “A faucet fills 18 liters in 3 minutes. Another fills 28 liters in 4 minutes.” Best move: Unit Rate. Students calculate 6 L/min and 7 L/min; the second fills faster.
2. “A recipe uses 2 cups of rice for 5 servings. How much rice for 15 servings?” Best move: Equivalent Ratios. Students explain that servings are multiplied by 3, so rice is also multiplied by 3: 6 cups.
3. “A runner travels 12 miles in 2 hours. A cyclist travels 30 miles in 2 hours. Who has the greater speed?” Best move: Unit Rate; compare 6 mi/h and 15 mi/h.
4. “One box has 12 red pencils and 8 blue pencils. Another has 15 red pencils and 10 blue pencils. Which box has more pencils?” Best move: Need to Check Units. Clarify that “more pencils” means totals, not a ratio comparison; both have 20.

Explicit check for understanding: Display, “Plan A earns $24 for 3 hours; Plan B earns $35 for 5 hours. Which pays more per hour?” Students hold up 1, 2, or 3 fingers for A, B, or same, then write one calculation. Scan before moving on. Address any student who chooses B because $35 is larger: $24 ÷ 3 = $8 per hour; $35 ÷ 5 = $7 per hour, so A pays more.

Students solve in notebooks, showing a ratio table or division equation, unit rate with units, comparison, and a written conclusion. Post the response frame: “ has a unit rate of per . has a unit rate of per . Therefore, is the better choice because .”

1. Movie A costs $18 for 3 tickets. Movie B costs $28 for 4 tickets. Which ticket price is lower?
2. A printer makes 45 pages in 5 minutes. Another makes 64 pages in 8 minutes. Which printer is faster?
3. A 6-pack of juice costs $4.50. A 10-pack costs $7.20. Which is the better buy? Explain why comparing only package price would mislead someone.

Circulate first to students who need an entry point: prompt them to circle “what is being measured,” underline the units, and complete a two-row ratio table before dividing. For students ready for more challenge, add: “A store offers 8 notebooks for $10.80, or a 25% discount on single notebooks priced at $1.90 each. Which option is cheaper per notebook, and what information must you calculate first?”