Proportion Ratios Mastery

Overview

Today you will strengthen your understanding of part:part and part:whole ratios, link ratio to fractions, and use ratios to solve direct and inverse proportion problems. You will also practise interpreting and using graphs/algebraic representations, and connect proportional reasoning to percentage change (original value problems).

Learning intentions

• WALT express quantities as part:part and part:whole ratios and simplify them.
• WALT interpret ratios as multiplicative relationships, connecting them to fractions.
• WALT solve direct proportion and inverse proportion problems using tables, equations, and graphs.
• WALT solve original value percentage problems using proportion reasoning.

Success criteria

• I can form and simplify a part:part and part:whole ratio from a given context.
• I can convert a ratio to a fraction (and vice versa) and explain what each part represents.
• I can set up and solve direct proportion and inverse proportion problems correctly.
• I can use my method to explain (or check) an answer, including using a graph/algebra step.

Curriculum links

• Ratio, proportion and rates of change: part:part and part:whole ratios; express division as ratios
• Ratio, proportion and rates of change: multiplicative relationship expressed as ratio or fraction
• Ratio, proportion and rates of change: solve direct and inverse proportion (graphical and algebraic representations)
• Ratio, proportion and rates of change: relate ratio language to fractions and linear functions; solve percentage change including original value problems

Lesson structure (60 minutes)

1. 0–8 min · Do Now (retrieval practice). Teacher shows 4 short questions; students answer on mini-whiteboards: a) Simplify 12:18 b) Is 3/5 a ratio? Write it as a ratio c) If 2x:5 = 14:35, find x (ratio equivalence) d) Original value: “20% off makes 48”. Find the original. Students: quick answers, show working briefly; teacher circulates and notes misconceptions (unsimplified ratios, wrong “original value” setup).

2. 8–20 min · Teach (worked example + model answer). Teacher explains ratio-to-fraction and proportion setup. Worked example 1 (ratio → fraction): In a class, 6 are boys and 9 are girls. Write the part:whole ratio and corresponding fraction for boys. Teacher step: total = 6+9=15; boys part:whole = 6:15, fraction = 6/15 = 2/5. Quick check question: “If the ratio is 2:7 (part:whole), what fraction is that?” Model example 2 (direct vs inverse):

• Direct: 3 pens cost £4.50. Cost of 5 pens?
• Inverse: If 6 workers take 10 hours to paint, how long for 10 workers (same work)? Teacher models:
• Direct: constant unit rate, multiply by 5/3.
• Inverse: workers × time = constant. Set up 6×10 = 10×t → t=6. Teacher adds a link: direct proportion graphs are straight lines through the origin; inverse proportion graphs curve.

3. 20–30 min · Guided practice (teacher prompts). Students complete Questions A–C in pairs, teacher leads first one. Question A (part:part + part:whole): “Tom has 8 red beads and 12 blue beads.”

• Write red:blue (part:part)
• Write red:(red+blue) (part:whole)
• Convert red:(red+blue) into a fraction, then into simplest form. Teacher prompt: “What is the total? Keep the order the same.” Question B (direct proportion table): “y is directly proportional to x. When x=4, y=15. Find y when x=7.” Teacher prompt: “What is the multiplier from 4 to 7?” Question C (inverse proportion equation): “x and y are inversely proportional. When x=5, y=12. Find y when x=8.” Teacher prompt: “Write the constant product.”

4. 30–45 min · Independent practice (differentiated worksheets). Students choose the correct tier based on teacher grouping (high-set still gets a scaffold card they can use). Tier 1 (Support) – “Strategy card” allowed:

• Task 1: Simplify and interpret ratios in context (1 part:part + 1 part:whole).
• Task 2: One direct proportion problem using a table.
• Task 3: One inverse proportion problem using product method. Tier 2 (Core, high-set):
• Task 1: Two ratio questions including ratio to fraction conversion.
• Task 2: Direct proportion with an algebraic equation setup (y = kx).
• Task 3: Inverse proportion solved algebraically and checked by substituting back. Calculator guidance: calculators only for arithmetic; not for reasoning steps. Dyslexia-friendly presentation: worksheet uses short lines, generous spacing, key words highlighted (RATIO, DIRECT, INVERSE, ORIGINAL), and a step box for “Set up → Solve → Check”.

5. 45–55 min · Plenary (graph + percent link + exit ticket). Teacher draws two small sketches: one straight line through origin, one curved decreasing.

• Students hold up mini-whiteboard: “Which sketch is direct proportion and why?”
• Next: original value percentage reasoning: “£80 after a 25% increase. What was the original?” Teacher prompt: “Is it ‘increase from’ or ‘increase to’?” Exit ticket (5 questions total, 2 marks each for accuracy and method):

1. Write part:part ratio and simplify.
2. Convert part:whole ratio to simplest fraction.
3. Direct proportion: solve for missing value.
4. Inverse proportion: solve and show the product constant.
5. Original value percent problem: find original.

Resources

• Printed ratio/proportion practice sheets (Tiers 1–2) with clear spacing
• Step-by-step strategy cards (Direct: unit rate/multiplier; Inverse: constant product)
• Mini-whiteboards + pens + erasers
• Graph sketch cards (straight-through-origin vs inverse curve)
• Timer for Do Now and independent work
• Dyslexia-friendly slide/board display (short lines, key words bolded, colour highlighting)
• Optional: audio playback of question text (teacher reads instructions; students can listen via classroom device)

Assessment

• Formative checks: mini-whiteboard answers in Do Now and plenary
• Teacher questioning during guided practice (students explain “what is the total?” and “what stays constant?”)
• Exit ticket marking against success criteria (especially: ratio simplification, correct proportional model, correct original value setup)

Differentiation

• Support (Tier 1):
• Ratio sentence stems: “Part:whole means : ( + ____).”
• Strategy cards for direct/inverse (include “constant” reminders).
• Worked example reprinted on the sheet margin.
• Misconception checks: reminders not to add ratios; not to use addition when multiplying/dividing is required.
• High-set core (Tier 2):
• Require algebra setup: define k and show substitution/check.
• Encourage full explanation: “I multiply/divide because…” not just answers.
• EAL/SEN:
• Vocabulary emphasis in colour: DIRECT, INVERSE, ORIGINAL, FRACTION, SIMPLIFY.
• Pair students strategically so one can model aloud; allow “talk first then write” for equations.
• Dyslexia-friendly reading options:
• Use fewer words per line on slides; highlight key words; provide an audio option for instructions; offer a “reading buddy” for the first guided question.

Extension (for advanced learners)

• Direct vs inverse comparison: “A and B vary. A is directly proportional to time; B is inversely proportional to time. At time 2, A=12 and B=10. Find A and B at time 5 and interpret which increases faster.”
• Algebraic set-up + graph interpretation: students sketch the graph type, then write the equation form (direct: y=kx; inverse: xy=k) and solve for the missing variable.
• Challenge mixed problem (percent + inverse): “A discount changes a price, then the updated price is used in an inverse proportion scenario. Find the final price.” (Students must show both setups and one check by substitution.)

Quick success-criteria checklist (students use in exit ticket review)

• Simplified ratio correct (part:part and/or part:whole)
• Ratio ↔ fraction conversion correct
• Direct proportion uses correct constant (multiplier/unit rate)
• Inverse proportion uses constant product and correct equation
• Percentage original value set up correctly