Mathematics · Year 7 · Proportional Reasoning · Term 2

Comparing and Ordering Fractions

Students will compare and order fractions with different denominators.

• ACARA Content Descriptions
• AC9M7N04

About This Topic

Decimals and percentages are the primary tools for financial literacy and data comparison in the modern world. In Year 7, students learn to convert between fractions, decimals, and percentages and apply these skills to solve problems involving discounts, taxes, and interest (AC9M7N06, AC9M7N07). This topic bridges the gap between classroom math and the 'real world' of shopping, banking, and statistics. It is particularly relevant in the Australian context when discussing the Goods and Services Tax (GST) or comparing interest rates on savings accounts.

Because decimals and percentages are so prevalent in daily life, students often have informal knowledge that can be used. This topic particularly benefits from hands-on, student-centered approaches where students can simulate shopping trips or manage a classroom budget. Students grasp these concepts faster through structured discussion and peer explanation, especially when comparing different ways to calculate a 15% discount or a 10% GST increase.

Key Questions

1. Justify the need for a common denominator when comparing fractions.
2. Analyze different strategies for ordering a set of fractions.
3. Predict the relative size of two fractions without converting them to decimals.

Learning Objectives

• Compare fractions with unlike denominators by finding common denominators.
• Order a set of fractions with unlike denominators from least to greatest and vice versa.
• Justify the necessity of a common denominator for accurate fraction comparison.
• Analyze and explain at least two different strategies for ordering fractions.
• Predict the relative size of two fractions without converting them to decimals.

Before You Start

Equivalent Fractions

Why: Students must be able to generate equivalent fractions to find common denominators.

Introduction to Fractions

Why: Students need a foundational understanding of what a fraction represents (part of a whole) and the roles of the numerator and denominator.

Multiples and Factors

Why: Understanding multiples is essential for finding common denominators and the least common multiple.

Key Vocabulary

Common Denominator	A shared denominator for two or more fractions, which is a multiple of all the original denominators. It allows for direct comparison of fraction sizes.
Least Common Multiple (LCM)	The smallest positive number that is a multiple of two or more numbers. It is used to find the least common denominator.
Equivalent Fractions	Fractions that represent the same value or proportion, even though they have different numerators and denominators. For example, 1/2 and 2/4 are equivalent.
Improper Fraction	A fraction where the numerator is greater than or equal to the denominator, representing a value of one or more.
Mixed Number	A number consisting of a whole number and a proper fraction, such as 1 3/4.

Watch Out for These Misconceptions

Common Misconception

Thinking that 0.5 is smaller than 0.15 because 5 is smaller than 15.

What to Teach Instead

Use a place value chart or 'decimal money' (dollars and cents). Comparing $0.50 to $0.15 makes the value clear. Collaborative sorting activities where students order decimals from smallest to largest help surface this error.

Common Misconception

Believing that a percentage is a whole number rather than a fraction of 100.

What to Teach Instead

Use 10x10 grids to shade in percentages. Seeing that 5% is only 5 squares out of 100, while 50% is half the grid, reinforces the 'per cent' (per hundred) concept. Peer teaching using these grids is very effective.

Active Learning Ideas

Simulation Game: The Pop-Up Shop

Students create a small 'store' with items and prices. They must calculate 10%, 20%, and 25% discounts for a 'sale' and then add 10% GST to the final price. Classmates 'shop' at different stores to check the accuracy of the calculations.

60 min·Small Groups

Inquiry Circle: Best Value Challenge

Groups are given flyers from different Australian supermarkets. They must compare the prices of similar items that are on sale (e.g., one is 20% off, another is 'buy one get one half price') to determine which is the best value per unit.

45 min·Small Groups

Think-Pair-Share: The Percentage Change Mystery

Students are asked: 'If a price increases by 50% and then decreases by 50%, do you get back to the original price?' They solve it individually, discuss their surprising results with a partner, and then explain the logic to the class.

20 min·Pairs

Real-World Connections

• Bakers often need to compare recipe ingredient amounts given in fractions with different measurements, such as comparing 1/2 cup of flour to 2/3 cup of sugar to ensure the correct proportions for a cake.
• Construction workers use fractions to measure and cut materials like wood or pipes. They might need to compare lengths like 3/4 inch and 5/8 inch to ensure precise fits.
• When dividing a pizza or a cake among friends, you might compare how much each person ate using fractions like 1/4 or 1/3, needing to understand which fraction represents a larger portion.

Assessment Ideas

Quick Check

Present students with two fractions, such as 2/3 and 3/4. Ask them to write down the steps they would take to determine which fraction is larger, requiring them to identify the need for a common denominator.

Exit Ticket

Give students a set of three fractions (e.g., 1/2, 3/5, 2/3). Ask them to order these fractions from least to greatest and provide a brief written justification for their ordering, highlighting their method.

Discussion Prompt

Pose the question: 'Imagine you have two recipes, one calling for 1/3 cup of butter and another for 2/5 cup of butter. How can you tell which recipe needs more butter without using a calculator?' Facilitate a class discussion comparing strategies.

Frequently Asked Questions

How can active learning help students understand decimals and percentages?

Active learning grounds decimals and percentages in familiar contexts like money and shopping. By participating in simulations or 'best buy' investigations, students see the immediate utility of these concepts. This practical application helps them understand that percentages are relative values, making the math feel less like a chore and more like a life skill.

What is the easiest way to calculate 10% of a number?

To find 10%, you simply move the decimal point one place to the left (or divide by 10). This is a 'benchmark' percentage that makes it easy to find others, like 5% (half of 10%) or 20% (double 10%).

Why do we have GST in Australia?

The Goods and Services Tax (GST) is a 10% tax on most goods and services sold in Australia. It is used by the government to fund public services like schools, hospitals, and roads.

How do you convert a decimal to a percentage?

Multiply the decimal by 100 and add the '%' symbol. For example, 0.75 becomes 75%. An easy way to remember this is that 'percent' means 'out of 100.'