Objective
Students will be able to define a ratio, write ratios in three forms (a:b, a to b, a/b), create equivalent ratios using ratio tables, and solve real-world proportion problems. Students will correctly solve at least 8 out of 10 ratio problems on an exit ticket.
Standards
- CCSS.MATH.CONTENT.6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
- CCSS.MATH.CONTENT.6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems.
Materials
- Colored counters (red and blue, 20 per student)
- Ratio table graphic organizer
- Whiteboard and marker per student
- Proportion word problem task cards (12 cards)
- Exit ticket worksheet
- Anchor chart: "Three Ways to Write a Ratio"
Warm-Up (5 minutes)
Display a bag with 3 red and 5 blue counters. Ask: "How can we describe the relationship between red and blue counters?" Accept all answers (there are more blue, 3 red and 5 blue, etc.). Then introduce the word "ratio" — a comparison of two quantities. Write the three ways: 3:5, 3 to 5, 3/5. Ask students to write the ratio of blue to red (5:3) on their whiteboards. Emphasize that order matters — the ratio of red to blue is different from blue to red.
Direct Instruction (12 minutes)
Explain ratios as comparisons that can describe part-to-part (red to blue), part-to-whole (red to total), or rates (miles per hour). Model each type:
- Part-to-part: In a class of 12 boys and 18 girls, the ratio of boys to girls is 12:18, which simplifies to 2:3.
- Part-to-whole: The ratio of boys to total students is 12:30, which simplifies to 2:5.
Introduce equivalent ratios using a ratio table. If the ratio of flour to sugar in a recipe is 3:1, build the table:
Show that each column is an equivalent ratio — we multiply both quantities by the same number. This is the same concept as equivalent fractions.
Then introduce solving proportions. If 3 apples cost $2, how much do 9 apples cost? Set up the ratio table: 3 apples = $2, so 9 apples = $6 (multiply both by 3). Also show cross-multiplication as a method: 3/2 = 9/x, so 3x = 18, x = 6.
Guided Practice (10 minutes)
Students solve 4 problems on their whiteboards:
- Write the ratio of vowels to consonants in the word "MATHEMATICS" in all three forms. (4:7, 4 to 7, 4/7)
- A bag has 6 red and 10 blue marbles. Write the ratio of red to total in simplest form. (6:16 = 3:8)
- Complete the ratio table: if 2 tickets cost $5, fill in costs for 4, 6, and 8 tickets. ($10, $15, $20)
- If a recipe uses 4 cups of flour for 2 dozen cookies, how many cups are needed for 5 dozen? (Set up 4/2 = x/5, cross multiply: 2x = 20, x = 10 cups)
Review each problem and address common errors: forgetting to simplify ratios, confusing part-to-part with part-to-whole, and setting up proportions with mismatched units.
Independent Practice (10 minutes)
Students work through 12 task cards placed around the room (gallery walk style). Each card has a real-world ratio or proportion problem:
- "A smoothie recipe calls for 2 cups of strawberries for every 3 cups of yogurt. If you use 9 cups of yogurt, how many cups of strawberries do you need?"
- "On a map, 1 inch represents 50 miles. Two cities are 3.5 inches apart on the map. How far apart are they in real life?"
- "A store sells 5 notebooks for $8. How much would 15 notebooks cost?"
Students record their work and answers on a recording sheet. They must show either a ratio table or cross-multiplication for each problem.
Assessment
- Formative: Check whiteboard responses during guided practice for correct setup and simplification.
- Summative: Exit ticket with 10 problems: 3 writing ratios, 3 completing ratio tables, 4 solving proportions. Target: 8 out of 10.
Differentiation
- Struggling learners: Use only whole-number, easy-to-multiply ratios (2:1, 3:1, 2:3). Provide a ratio table template with the multiplier filled in. Allow counters for building ratios physically. Start with part-to-part ratios only before introducing part-to-whole.
- ELL students: Create a visual vocabulary card set (ratio, proportion, equivalent, simplify, compare) with examples and pictures. Simplify word problem language. Provide sentence frames for explaining solutions: "The ratio of ___ to ___ is ___ because ___."
- Advanced learners: Introduce unit rates (price per item, speed in miles per hour). Include problems with decimals or fractions in the ratios. Challenge students to determine if two ratios are proportional and explain why or why not.
- Students with IEPs: Provide a step-by-step checklist for solving proportions. Allow calculator use for multiplication and division. Reduce task cards to 6 instead of 12. Offer extended time and a multiplication chart for reference.
Closure (3 minutes)
Quick class poll: "Give me a real-life example of a ratio." Take 3–4 responses (sports records, recipes, prices, map scales). Ask: "Why is it important that order matters in a ratio?" (Because 3 boys to 5 girls is different from 5 boys to 3 girls.) Preview tomorrow's lesson on unit rates and how ratios help us compare prices at the grocery store.