Mathematics · Class 6 · Ratio and Proportion · Term 3

Equivalent Ratios

Discover how to find different ratios that represent the same comparison, similar to equivalent fractions.

TL;DR:Let's investigate how to keep things fair and balanced! We will explore how ratios help us make correct comparisons, just like when you share sweets equally with your friends.

• CBSE Learning Outcomes
• NCERT Class 6: Chapter 12 - Ratio and Proportion

About This Topic

This topic, Equivalent Ratios, is a crucial extension of the concepts of fractions and comparison, as outlined in the NCERT framework for Class 6. It builds directly upon students' understanding of equivalent fractions, transitioning them from part-to-whole comparisons to part-to-part or part-to-whole comparisons. Mastering this concept is fundamental for understanding proportionality, which is a cornerstone of later mathematical topics including percentages, direct and inverse variation, and trigonometry in higher classes. In the Indian context, this topic can be made highly relatable by using everyday examples such as mixing ingredients for a recipe, comparing prices of goods in a local kirana store, or understanding scales on a map of India. The focus should be on the multiplicative relationship between the terms of a ratio, ensuring students understand that a ratio represents a constant relationship, which remains unchanged when its terms are multiplied or divided by the same non-zero number.

Key Questions

1. Explain the method for finding an equivalent ratio.
2. Compare two ratios to determine if they are equivalent.
3. Justify why simplifying a ratio to its lowest terms is useful.

Learning Objectives

• Generate equivalent ratios for a given ratio by multiplying or dividing both terms by the same non-zero number.
• Simplify a given ratio to its simplest form by dividing by the Highest Common Factor (HCF).
• Compare two ratios to determine if they are equivalent.
• Solve simple word problems involving the application of equivalent ratios.
• Represent a ratio using concrete materials or diagrams.

Key Vocabulary

Ratio	A comparison of two quantities by division, showing how many times one value contains or is contained within the other.
Equivalent Ratios	Ratios that represent the same relationship or comparison. For example, 1:2 and 2:4 are equivalent.
Simplest Form	A ratio is in its simplest form when its terms have no common factor other than 1.
Terms of a Ratio	The two numbers that form the ratio. The first is called the antecedent and the second is called the consequent.

Watch Out for These Misconceptions

Common Misconception

Students try to find an equivalent ratio by adding or subtracting the same number from both terms (e.g., thinking 2:3 is equivalent to 2+2 : 3+2, which is 4:5).

What to Teach Instead

Explain that a ratio is a multiplicative comparison. Use a real example: if a recipe needs 2 cups of flour for 3 people, it will need 4 cups for 6 people (doubling both), not 4 cups for 5 people.

Common Misconception

Confusing the order of the quantities in a ratio. They might write the ratio of boys to girls the same as girls to boys.

What to Teach Instead

Emphasise that order matters greatly. The ratio of 'A to B' is written as A:B. Use a clear example: a ratio of 2 pens to 5 pencils is 2:5, while 5 pencils to 2 pens is 5:2, which are different comparisons.

Common Misconception

Believing that ratios with larger numbers are always 'bigger' (e.g., thinking 6:9 is greater than 2:3).

What to Teach Instead

Teach them to simplify ratios to their simplest form for comparison. Show that when 6:9 is simplified by dividing both terms by 3, it becomes 2:3, proving they are actually equivalent.

Active Learning Ideas

Collaborative Problem-Solving

Nimbu Pani Ratios

Provide students with a simple recipe for one glass of nimbu pani (e.g., 2 spoons sugar : 1 spoon lemon juice). Ask them to calculate the ingredients needed for 2, 4, or 10 glasses, thereby generating equivalent ratios.

20 min·Small Groups

Collaborative Problem-Solving

Ratio Match-Up Game

Create a set of cards with different ratios, some of which are equivalent (e.g., 2:3, 4:6, 8:12, 3:5). Students work in pairs to find and group all the equivalent ratio cards.

15 min·Pairs

Collaborative Problem-Solving

Visual Block Ratios

Give students two different colours of blocks or counters. Ask them to create a visual representation of a ratio, like 3 red blocks to 4 blue blocks (3:4), and then build a larger group that shows an equivalent ratio (e.g., 6 red to 8 blue).

15 min·Individual

Real-World Connections

• Scaling a recipe up or down when cooking for a different number of people.
• Understanding the scale on a map, where 1 cm on the map might represent 10 km in reality.
• Mixing paints, chemicals, or even drinks like squash in the correct proportions to get the desired result.
• Comparing prices at a shop to find the 'best buy', for example, comparing the cost of a 5-pack of biscuits to a 10-pack.
• In sports, comparing the win-loss record of two different cricket teams.

Assessment Ideas

Exit Ticket

Exit Slip: Give students a ratio like 4:6 and ask them to write one equivalent ratio and its simplest form before leaving the class.

Discussion Prompt

Think-Pair-Share: Pose a question like 'Are 3:5 and 9:15 equivalent? How do you know?'. Students think individually, discuss with a partner, and then share with the class.

Quick Check

A short quiz with a mix of problems: finding equivalent ratios, simplifying ratios, and solving one or two word problems.

Frequently Asked Questions

Why can we only multiply or divide to find an equivalent ratio? Why not add or subtract?

A ratio shows a relationship. If you have 1 teacher for every 20 students (1:20), for this relationship to stay the same, you must have 2 teachers for 40 students (2:40). If you just added 1 to both, you would have 2 teachers for 21 students, which is a different, much better, ratio for the students!

Is the ratio 3:4 the same as the fraction 3/4?

They are related but not always the same. A fraction 3/4 means 3 parts out of a total of 4 parts. A ratio of 3:4 could mean 3 boys for every 4 girls (a part-to-part comparison), where the total is 7 children. So, the context is very important.

How do I know if two ratios are equivalent?

There are two simple ways. First, you can simplify both ratios to their lowest terms. If the simplified forms are identical, the ratios are equivalent. Second, you can use the 'cross-multiplication' method: for ratios a:b and c:d, if a × d = b × c, they are equivalent.