Non-Unit Fractions (e.g., 2/3, 3/4)

Active learning works for non-unit fractions because visual and hands-on experiences help students move beyond abstract symbols to concrete understanding. Students need to see, touch, and discuss fractions to grasp that the denominator defines the size of each share, not the numerator. This physical engagement turns confusing rules into clear, memorable insights.

• 3rd Year
• Mathematical Foundations and Real World Reasoning
• 3 activities
• 15 min–25 min

Learning Objectives

1. Represent non-unit fractions, such as 2/3 or 3/4, as a sum of unit fractions.
2. Compare the value of a non-unit fraction to a unit fraction with the same denominator.
3. Construct visual models, like fraction bars or circles, to represent given non-unit fractions.
4. Explain the relationship between a non-unit fraction and its corresponding unit fraction using precise mathematical language.

Formal Debate

⏱ 20 min·Whole Class

Formal Debate: Which Would You Rather?

Present scenarios like 'Would you rather have 1/2 of a giant chocolate bar or 1/10 of the same bar?' Students must choose a side, use fraction tiles to prove their choice, and then debate their reasoning with someone who chose the opposite (if anyone did!).

Prepare & details

Explain how 3/4 is related to 1/4.

Facilitation Tip: During the Structured Debate, assign clear roles like 'presenter,' 'questioner,' and 'illustrator' to keep all students engaged.

Setup: Two teams facing each other, audience seating for the rest

Materials: Debate proposition card, Research brief for each side, Judging rubric for audience, Timer

• Analyze
• Evaluate
• Create
• Self-Management
• Decision-Making

Gallery Walk

⏱ 25 min·Small Groups

Gallery Walk: The Fraction Wall Build

In small groups, students build a large fraction wall on the floor using colored tape or long strips of paper. Once finished, they move around the wall in pairs, identifying which 'bricks' are longer or shorter and recording their findings using < and > symbols.

Prepare & details

Construct a model to show 2/3 of a pizza.

Facilitation Tip: For the Gallery Walk, have students label their fraction walls with sticky notes to explain their choices and invite peer feedback.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: Fraction Number Line

Give pairs a set of unit fraction cards (1/2, 1/3, 1/4, 1/10). They must work together to place them on a 0-to-1 number line. They then explain to another pair why 1/10 is closer to zero than 1/2, despite 10 being a 'bigger' number.

Prepare & details

Differentiate between a unit fraction and a non-unit fraction.

Facilitation Tip: During the Think-Pair-Share, provide blank number lines on paper for students to fold and mark as they discuss their reasoning.

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

Experienced teachers approach non-unit fractions by prioritizing visual models over symbolic procedures, as research shows this deepens understanding. Avoid rushing to rules like 'bigger denominator means smaller fraction' without context, as this can reinforce misconceptions. Instead, use repeated opportunities for students to compare fractions using fraction circles, bars, and number lines to build intuitive sense-making.

What to Expect

Successful learning looks like students confidently comparing fractions by reasoning about denominators and using terms like numerator and denominator correctly. They should explain their thinking using visual models and articulate why 3/4 is larger than 1/2 without relying on memorized rules. Students should also demonstrate flexibility in representing fractions in multiple ways.

Watch Out for These Misconceptions

Common Misconception

During the Structured Debate, watch for students who assume 1/2 and 1/4 are equal because they have the same numerator.

What to Teach Instead

Use the pizza sharing analogy during the debate. Ask, 'If you share a pizza with 2 friends, do you get more or less than if you share it with 4 friends?' Have students model this with fraction circles to see the size difference.

Common Misconception

During the Think-Pair-Share: Fraction Number Line, watch for students who order fractions by numerator size instead of denominator size.

What to Teach Instead

Have students fold a string number line in half to mark 1/2, then fold it again to mark 1/4. Ask them to place 1/3 by estimating its position relative to these folds to see that 1/3 is not between 1/2 and 1/4.

Assessment Ideas

Quick Check

After the Gallery Walk, give students pre-drawn fraction bars and ask them to shade 3/5 of the bar. Then, ask them to write one sentence explaining how their shaded bar relates to 1/5.

Discussion Prompt

During the Structured Debate, pose the question: 'How is 5/8 different from 1/8, and how are they related?' Encourage students to use fraction manipulatives or drawings to support their explanations and use the terms numerator and denominator.

Exit Ticket

After the Think-Pair-Share: Fraction Number Line, give each student a card with a non-unit fraction (e.g., 2/4, 3/3, 4/6). Ask them to draw a model representing this fraction and then write the fraction as a sum of unit fractions.

Extensions & Scaffolding

• Challenge students to create a new fraction wall with six different fractions and justify its order in a written reflection.
• Scaffolding for students who struggle: Provide pre-divided circles with cut lines to help them physically see the shares before comparing.
• Deeper exploration: Ask students to find real-world examples of non-unit fractions (e.g., 3/4 cup, 2/3 of a mile) and bring them to class to discuss how the fraction relates to the whole.

Key Vocabulary

Unit Fraction	A fraction where the numerator is 1, representing one equal part of a whole. Examples include 1/2, 1/4, or 1/8.
Non-Unit Fraction	A fraction where the numerator is greater than 1, representing multiple equal parts of a whole. Examples include 2/3, 3/4, or 5/6.
Numerator	The top number in a fraction, indicating how many equal parts of the whole are being considered.
Denominator	The bottom number in a fraction, indicating the total number of equal parts the whole is divided into.