Mathematics · Grade 3 · Fractional Thinking · Term 2

Unit Fractions and Their Size

Students investigate how the size of a unit fraction changes as the denominator increases.

• Ontario Curriculum Expectations
• 3.NF.A.3.D

About This Topic

Comparing unit fractions is often counter-intuitive for Grade 3 students because the larger number (the denominator) actually represents a smaller piece. In the Ontario curriculum, students use concrete materials and number lines to visualize why 1/8 is smaller than 1/2. This understanding is crucial for developing a sense of 'fractional magnitude.'

By focusing on unit fractions (fractions with a numerator of 1), students can clearly see the relationship between the number of cuts made to a whole and the size of the resulting pieces. This unit encourages students to use benchmarks like 0, 1/2, and 1 to place fractions on a number line. This topic comes alive when students can physically model the patterns and compare different fractional lengths or areas side-by-side.

Key Questions

1. Explain why a larger denominator results in a smaller piece.
2. Design a number line to show the relationship between different unit fractions.
3. Justify how we determine which is larger when comparing two fractions with the same numerator.

Learning Objectives

• Compare the size of unit fractions with different denominators, explaining the relationship.
• Justify why a unit fraction with a larger denominator represents a smaller portion of a whole.
• Design a number line to accurately represent and order unit fractions.
• Analyze the relationship between the number of equal parts and the size of each part in a fraction.
• Create visual models to demonstrate the comparison of unit fractions.

Before You Start

Introduction to Fractions

Why: Students need a basic understanding of what a fraction represents (part of a whole) and how to identify the numerator and denominator.

Dividing a Whole into Equal Parts

Why: Students must be able to partition a whole into a specific number of equal parts to understand the concept of denominators.

Key Vocabulary

Unit Fraction	A fraction where the numerator is 1, representing one equal part of a whole.
Denominator	The bottom number in a fraction, which tells how many equal parts the whole is divided into.
Numerator	The top number in a fraction, which tells how many parts of the whole are being considered.
Whole	The entire object or set of objects being divided into equal parts.

Watch Out for These Misconceptions

Common Misconception

Students often think 1/10 is bigger than 1/2 because 10 is bigger than 2.

What to Teach Instead

Relate the denominator to the number of people sharing. Would you rather share a cake with 10 people or 2 people? Active role-playing of sharing scenarios helps students connect the denominator to the 'number of shares,' making the inverse relationship clear.

Common Misconception

Difficulty placing unit fractions on a number line between 0 and 1.

What to Teach Instead

Use a physical 'walking' number line on the floor. Have students take 4 equal steps to get from 0 to 1, then identify where '1 step' (1/4) landed. This physical movement helps them see fractions as numbers, not just shapes.

Active Learning Ideas

Simulation Game: The Pizza Party Dilemma

Students are told they can have one slice of pizza. They must choose between a pizza cut into 4 slices or 8 slices. They use paper circles to model the choice and must explain to their group why the 'smaller number' (4) gives them a 'bigger slice.'

30 min·Small Groups

Gallery Walk: Human Number Line

Each student is given a card with a unit fraction (1/2, 1/3, 1/4, etc.). They must work together to stand in order from smallest to largest. Once in line, they explain their position to 'visitors' who walk the line.

25 min·Whole Class

Think-Pair-Share: The Denominator Rule

Students look at 1/5 and 1/10. They think about which is larger and why. After sharing with a partner, they try to write a 'rule' for the class about what happens to the size of a piece when the denominator gets bigger.

20 min·Pairs

Real-World Connections

• When sharing a pizza, understanding unit fractions helps determine fair portions. If 8 friends share equally, each gets 1/8 of the pizza, which is smaller than if only 2 friends shared, where each would get 1/2.
• Bakers use fractions to measure ingredients precisely. A recipe calling for 1/4 cup of flour requires dividing the cup into four equal parts and using one, which is a smaller amount than 1/2 cup.

Assessment Ideas

Exit Ticket

Give students two identical rectangular strips of paper. Ask them to fold one strip into 4 equal parts and shade 1 part (1/4). Ask them to fold the second strip into 8 equal parts and shade 1 part (1/8). On the back, they should write one sentence explaining which shaded part is smaller and why.

Discussion Prompt

Present students with the fractions 1/3 and 1/6. Ask: 'Imagine you have two identical candy bars, one broken into 3 equal pieces and the other into 6 equal pieces. If you get to choose one piece from either bar, which piece would be bigger? Explain your thinking using the terms 'denominator' and 'whole'.

Quick Check

Draw a number line from 0 to 1 on the board. Ask students to come up and place the unit fractions 1/2, 1/4, and 1/8 on the number line. As they place each fraction, ask them to explain how they know its position relative to the others.

Frequently Asked Questions

Why do we only focus on unit fractions in Grade 3?

Focusing on unit fractions (1/n) allows students to master the core concept of the denominator's role without the added complexity of multiple parts. This builds a strong foundation for comparing more complex fractions in Grade 4.

What tools are best for comparing fractions?

Fraction strips, circles, and number lines are essential. Digital tools that allow students to 'cut' shapes into different numbers of pieces are also very effective for visualizing the change in size.

How can active learning help students compare unit fractions?

Active learning, such as the 'Human Number Line,' requires students to physically negotiate their understanding. To find their place, they must compare their fraction to others and justify their reasoning. This social negotiation surfaces misconceptions and forces students to use mathematical language to defend their position.

How can I include multicultural perspectives in fraction lessons?

Use examples of traditional breads from various cultures (naan, pita, bannock, tortillas). Discuss how these are shared among different numbers of family members, providing a culturally rich context for unit fractions and sharing.