Comparing and Ordering Fractions

Active learning helps students grasp fraction comparison because it turns abstract numbers into tangible comparisons they can see and touch. When students work with real-world materials like price tags or grid shading, they build lasting mental models of value and equivalence.

• Year 7
• Mathematics
• 3 activities
• 20 min–60 min

Learning Objectives

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

Simulation Game

⏱ 60 min·Small Groups

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.

Prepare & details

Justify the need for a common denominator when comparing fractions.

Facilitation Tip: During The Pop-Up Shop, circulate with a notepad to jot down common conversion shortcuts students invent, then share these strategies with the class at the end.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Inquiry Circle

⏱ 45 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.

Prepare & details

Analyze different strategies for ordering a set of fractions.

Facilitation Tip: For the Best Value Challenge, assign clear roles to each group member to ensure accountability during the investigation phase.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 20 min·Pairs

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.

Prepare & details

Predict the relative size of two fractions without converting them to decimals.

Facilitation Tip: In The Percentage Change Mystery, give pairs exactly three minutes to discuss before sharing with the class to keep the think-pair-share focused and equitable.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers should start with concrete representations like 10x10 grids or play money before moving to symbolic notation. Avoid rushing to algorithms; instead, let students discover the need for common denominators or place value through guided discovery. Research shows that students who struggle benefit from visual-spatial representations before abstract symbols, while advanced students thrive when asked to justify their methods to peers.

What to Expect

Successful learning looks like students confidently converting between fractions, decimals, and percentages without prompting. They should explain their reasoning clearly, use multiple methods to verify answers, and apply these skills to unfamiliar problems like tax calculations or discount comparisons.

Watch Out for These Misconceptions

Common Misconception

During The Pop-Up Shop, watch for students who believe 0.5 is smaller than 0.15 because 5 is smaller than 15.

What to Teach Instead

Redirect them to the price tags. Have them compare $0.50 to $0.15 directly, then model writing the decimals with place value columns to make the comparison explicit.

Common Misconception

During the Best Value Challenge, watch for students who treat percentages as whole numbers rather than fractions of 100.

What to Teach Instead

Ask them to return to their 10x10 grid shading. Have them re-label each percentage as a fraction (e.g., 50% = 50/100) and simplify it to connect the ideas.

Assessment Ideas

Quick Check

After The Pop-Up Shop, 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, focusing on whether they identify the need for a common denominator.

Exit Ticket

After the Best Value Challenge, 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

During The Percentage Change Mystery, 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.

Extensions & Scaffolding

• Challenge students who finish early to create a new shopping scenario with a 15% discount and calculate the final price from memory.
• For students who struggle, provide fraction strips alongside the Best Value Challenge to make comparisons visual before calculations.
• Deeper exploration: Ask students to research and compare the GST rates in at least three different countries, then present their findings with conversions shown.

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.