Solving Problems with Ratio — Year 6 Lesson Plan
National Curriculum: Mathematics KS2 (Year 6) — Pupils should be taught to solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts; to solve problems involving similar shapes where the scale factor is known or can be found; to solve problems involving unequal sharing and grouping using knowledge of fractions and multiples.
Overview
This lesson builds on pupils initial introduction to ratio earlier in Year 6 and develops their ability to apply ratio and proportion to solve a wider range of problems, including those involving scale, recipes, map reading and missing values. Pupils use the unitary method, scaling strategies and proportional reasoning to approach unfamiliar contexts. The lesson places a strong emphasis on mathematical reasoning and on communicating solutions clearly with justification.
Learning Objectives
- Solve problems involving the relative sizes of two quantities where missing values can be found using ratio
- Use the unitary method to solve proportion problems
- Apply ratio and proportion to real-world contexts including recipes, maps and scale diagrams
- Solve problems involving percentage increase and decrease using knowledge of ratio and proportion
Key Vocabulary
Suggested Lesson Structure
Display a simple recipe for eight biscuits: 200 g flour, 100 g butter, 50 g sugar. Ask pupils: How much flour would you need to make four biscuits? Sixteen biscuits? Twenty-four biscuits? Pupils discuss with a partner and record answers on mini whiteboards. Take feedback and establish the method: find the amount for one biscuit (divide by 8) then multiply. Name this the unitary method. Introduce the lesson: today we will use this method and others to solve more complex ratio and proportion problems.
Model four different problem types involving ratio and proportion. Type 1: Scaling a recipe up or down using the unitary method. Type 2: Sharing an amount in a given ratio (e.g. share 45 in the ratio 2:3 — find one part by dividing by the total number of parts, then multiply). Type 3: Using a scale on a map (e.g. a map has a scale of 1:25,000; a measured distance of 4 cm represents how many kilometres?). Type 4: A problem involving percentage increase expressed as a ratio (e.g. a price increases by 20%; find the new price as a ratio). For each type, model the solution step by step and emphasise showing all working.
Pupils work in pairs on a set of mixed ratio problems, one from each type modelled. They must first identify which type of problem it is before selecting a method. Encourage pupils to check their answers using a different method where possible (e.g. checking a sharing problem by adding the parts back together). Bring the class together to compare solutions for one problem — discuss if there were different correct methods and which was most efficient.
Pupils work independently on a graduated problem set. Problems include: sharing money and quantities in a ratio; solving a missing-value proportion problem; reading a scale diagram; finding the original amount when given a ratio and one part; and a multi-step problem combining ratio with another area (e.g. ratio and fractions, or ratio and percentage). Pupils must show all working and express answers in context (e.g. the shorter piece is 15 cm). Extension: Two quantities are in the ratio 5:3. The larger quantity is 40. Find the smaller quantity and express the ratio in another equivalent form.
Pose a two-part reasoning question: Paint is mixed in the ratio 3 parts red to 2 parts white. A pupil uses 12 litres of red paint. How much white paint is needed? Then: The same pupil wants to make 35 litres of this paint altogether. How much of each colour is needed? Ask pupils to solve both parts and then explain the difference in method between them. Discuss which approach is most efficient for each. Close by asking pupils to write one sentence explaining what the unitary method is and why it is useful.
Common Misconceptions
- Pupils often add or subtract values when scaling a ratio rather than multiplying or dividing; use the phrase always times or divide, never add or subtract when scaling and reinforce with counterexamples
- When sharing in a ratio, pupils sometimes divide by one of the ratio numbers instead of the total number of parts; use bar models to show that 2:3 means 5 equal parts in total, not 2 or 3
Prior Knowledge
Pupils should already be able to:
- Introduction to ratio notation and simplifying ratios from earlier in Year 6
- Ability to multiply and divide including with fractions and decimals
- Understanding of fractions, decimals and percentages and movement between them
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