Mathematics · Year 6 · Ratio and Proportion · Spring Term

Sharing in a Given Ratio

Students will solve problems involving the division of a quantity into two parts in a given ratio.

• National Curriculum Attainment Targets
• KS2: Mathematics - Ratio and Proportion

About This Topic

Proportional reasoning is the application of ratio and scaling to solve more complex problems involving grouping and sharing. In Year 6, this involves understanding direct proportion, where two quantities increase or decrease at the same rate. Students learn to use the 'unitary method' (finding the value of one unit) to solve problems, which is a powerful tool for comparing prices and adjusting quantities.

This topic is essential for developing a student's ability to reason mathematically about the world around them. It requires them to look at the relationship between variables and predict outcomes. Students grasp this concept faster through structured discussion and peer explanation, where they can debate whether two things are in proportion and test their predictions through collaborative investigations.

Key Questions

1. Explain how to determine the total number of parts when sharing in a ratio.
2. Analyze common errors when sharing quantities in a ratio and how to avoid them.
3. Construct a problem that requires sharing a total amount in a specific ratio.

Learning Objectives

• Calculate the value of one part when a total quantity is shared in a given ratio.
• Determine the amounts of each part when a total quantity is divided according to a specified ratio.
• Analyze and explain common errors made when sharing quantities in a ratio, such as confusing the order of the ratio or not accounting for all parts.
• Construct a word problem that requires sharing a given total amount into two parts according to a specific ratio.

Before You Start

Multiplication and Division

Why: Students need to be proficient with multiplication and division to calculate the value of one part and then scale it up for all parts.

Understanding Fractions

Why: Sharing in a ratio is closely linked to understanding fractions, as each part of the ratio can be represented as a fraction of the whole.

Key Vocabulary

ratio	A comparison of two or more quantities, showing their relative sizes. It is often written using a colon, for example, 2:3.
parts	The individual amounts that make up a whole when a quantity is divided according to a ratio. For a ratio of 2:3, there are 2 parts of one type and 3 parts of another, totaling 5 parts.
total parts	The sum of all the individual parts in a ratio. This represents the whole quantity being shared.
unitary method	A problem-solving strategy where you first find the value of one unit or part, and then use that to find the value of multiple units or parts.

Watch Out for These Misconceptions

Common Misconception

Assuming all relationships are directly proportional (e.g., if it takes 10 minutes to boil 1 egg, it takes 20 minutes to boil 2 eggs).

What to Teach Instead

This is a classic 'proportionality trap.' Use real-world examples and peer discussion to challenge these assumptions. Ask students to think about the logic of the situation, does the heat only work on one egg at a time?

Common Misconception

Struggling to find the 'unit rate' in a problem.

What to Teach Instead

Students often try to jump straight to the final answer. Teach the 'middle step' of finding what 1 unit is worth (e.g., if 5 items cost £10, find the cost of 1 item first). Collaborative investigations into 'Best Buys' make this step feel necessary and logical.

Active Learning Ideas

Inquiry Circle: The Best Buy Challenge

Provide students with different sized packages of the same product (e.g., 300g for £2.40 and 500g for £3.50). In small groups, they must find the price per 100g for each to determine which is the 'best buy' and present their findings.

40 min·Small Groups

Simulation Game: Currency Exchange

Set up a mock travel bureau where students must convert 'Travel Money' into different currencies using given exchange rates. They work in pairs to solve increasingly complex conversion problems for 'customers' (other students).

45 min·Pairs

Think-Pair-Share: Is it Proportional?

Present scenarios like 'the more you study, the higher your grade' or 'the more people painting a wall, the less time it takes.' Students discuss in pairs whether these are examples of direct proportion and why or why not.

20 min·Pairs

Real-World Connections

• Bakers use ratios to scale recipes. For example, to make a larger batch of cookies, they might need to increase the flour and sugar in a specific ratio, such as 3:2, to maintain the correct taste and texture.
• Financial advisors may help clients divide investments into different funds based on risk tolerance, using ratios to allocate percentages of a total portfolio. For instance, a client might have 60% in stocks and 40% in bonds, a ratio of 3:2.

Assessment Ideas

Quick Check

Present students with a scenario: 'Sarah and Tom share 20 sweets in the ratio 3:2. How many sweets does each person get?' Ask students to show their working, focusing on identifying the total number of parts and the value of one part.

Discussion Prompt

Pose the question: 'If two friends share £30 in the ratio 1:5, one friend gets £5 and the other gets £25. Is this correct? Explain why or why not, and what the correct answer should be.' Encourage students to articulate their reasoning about the total parts and the value of each part.

Exit Ticket

Give students a blank card. Ask them to write a word problem where a total amount (e.g., money, marbles, time) is shared in a ratio of 2:5. They must then solve their own problem, showing the steps clearly.

Frequently Asked Questions

How can active learning help students understand proportional reasoning?

Active learning strategies like the 'Best Buy Challenge' force students to use the unitary method in a meaningful way. Instead of just following a worksheet, they are acting as savvy consumers. This context makes the mathematical steps, dividing to find the unit rate and then multiplying to compare, feel like a natural problem-solving process rather than an abstract set of rules.

What is the 'unitary method'?

It is a technique where you first find the value of a single unit and then multiply that value to find the required amount. For example, if 3 pens cost 60p, you find the cost of 1 pen (20p) and then use that to find the cost of 7 pens (£1.40).

How does proportional reasoning relate to percentages?

Percentages are just a specific type of proportion where the 'whole' is always 100. Understanding that 20% is 20 for every 100 is the same as understanding a 1:5 ratio. Both rely on the same multiplicative thinking.

Why is it important to identify if variables are in direct proportion?

Because if they aren't, the standard rules of ratio and scaling won't work. Recognising direct proportion helps students avoid errors in science and geography, where they might otherwise assume a simple linear relationship that doesn't exist.