Mathematics · Grade 5 · Fractions and Decimals: Different Names for the Same Parts · Term 2

Comparing and Ordering Fractions

Students will compare and order fractions with unlike denominators using strategies such as finding common denominators or comparing to benchmark fractions.

• Ontario Curriculum Expectations
• 5.NF.A.1

About This Topic

Visualizing fraction operations is about building a mental map of how parts of a whole interact. In Grade 5, students focus on adding and subtracting fractions with like denominators, while being introduced to the idea of unlike denominators through concrete models. They also explore improper fractions and mixed numbers, learning to see 5/4 as one whole and one quarter. This aligns with the Ontario curriculum's goal of developing a deep sense of quantity and magnitude.

When students can see that adding 1/5 and 2/5 simply means having three of those 1/5-sized pieces, the 'rule' about keeping the denominator the same becomes common sense rather than a mystery. This topic is particularly suited for active learning because it requires students to manipulate parts to form wholes. Students grasp this concept faster through structured discussion and peer explanation using tools like pattern blocks or fraction circles.

Key Questions

1. Compare two fractions with unlike denominators and explain which is greater.
2. Justify the use of a benchmark fraction to compare two given fractions.
3. Predict the order of a set of fractions when placed on a number line.

Learning Objectives

• Compare two fractions with unlike denominators, identifying the greater fraction using visual models or common denominators.
• Explain the strategy of using benchmark fractions (e.g., 0, 1/2, 1) to compare the relative size of two given fractions.
• Order a set of fractions with unlike denominators from least to greatest on a number line.
• Justify the placement of fractions on a number line by comparing them to benchmark fractions or finding common denominators.

Before You Start

Understanding Fractions as Parts of a Whole

Why: Students must first understand what a fraction represents before they can compare or order them.

Identifying Equivalent Fractions

Why: The ability to find equivalent fractions is a key strategy for comparing fractions with unlike denominators.

Representing Fractions on a Number Line

Why: Students need experience placing fractions on a number line to develop an intuitive sense of their magnitude and order.

Key Vocabulary

Unlike Denominators	Fractions that have different numbers in the bottom position, meaning the size of the pieces being considered are different.
Common Denominator	A shared multiple of the denominators of two or more fractions, allowing them to be compared or combined easily.
Benchmark Fraction	Familiar fractions like 0, 1/2, and 1 that are used as reference points to estimate or compare the value of other fractions.
Equivalent Fraction	Fractions that represent the same value or portion of a whole, even though they have different numerators and denominators.

Watch Out for These Misconceptions

Common Misconception

Adding both the numerators and the denominators (e.g., 1/4 + 1/4 = 2/8).

What to Teach Instead

Use fraction circles to show that two quarter-circles make a half-circle, not two eighths. Peer discussion where students explain that the denominator is the 'size' of the piece helps them realize the size doesn't change when you get more pieces.

Common Misconception

Thinking that a larger denominator means a larger fraction.

What to Teach Instead

Ask students if they would rather have 1/2 of a cake or 1/10 of a cake. Active modeling with paper folding (folding a paper twice vs. folding it four times) provides immediate visual proof that more folds result in smaller pieces.

Active Learning Ideas

Stations Rotation: Fraction Construction Site

Students move through stations using different materials (pattern blocks, Cuisenaire rods, and paper folding) to solve addition and subtraction problems. At each station, they must draw a 'blueprint' of their physical model to show how they reached the answer.

50 min·Small Groups

Inquiry Circle: The Recipe Swap

Groups are given a recipe that serves 4 people but need to adjust it for 6 or 2. This involves adding or subtracting fractional amounts of ingredients. They must use fraction circles to prove that their new measurements (e.g., 1 1/2 cups) are correct.

55 min·Small Groups

Think-Pair-Share: Beyond the Whole

The teacher shows a model of 7/4. Students work in pairs to find as many ways as possible to name that amount (e.g., 1 and 3/4, 4/4 + 3/4, 1.75). They share their 'names' with the class to build a collective understanding of improper fractions.

25 min·Pairs

Real-World Connections

• Bakers compare ingredient amounts when scaling recipes up or down, for example, deciding if 2/3 cup of sugar is more or less than 3/4 cup for a cake.
• Construction workers might compare measurements on blueprints, such as determining if a 5/8 inch pipe is larger or smaller than a 3/4 inch pipe for plumbing installations.
• In cooking, comparing fractions helps in portioning. For instance, a chef might need to know if 1/3 of a pizza is more than 2/5 of another pizza to distribute servings fairly.

Assessment Ideas

Exit Ticket

Provide students with two fractions, such as 3/4 and 5/6. Ask them to write one sentence explaining which fraction is larger and show one method they used to compare them (e.g., drawing, common denominator, benchmark).

Quick Check

Display a number line from 0 to 1. Write three fractions on the board (e.g., 1/3, 7/8, 1/2). Ask students to place these fractions on their own drawn number line and label each one, then write one sentence comparing two of the fractions.

Discussion Prompt

Pose the question: 'Imagine you have two identical chocolate bars. One is cut into 5 equal pieces and you eat 2 (2/5). The other is cut into 8 equal pieces and you eat 3 (3/8). Which bar did you eat more of?' Have students discuss their strategies for comparison.

Frequently Asked Questions

How do I teach improper fractions to Grade 5s?

Start with the 'full container' analogy. If a carton holds 4 eggs and you have 5 eggs, you have 5/4 of a carton, which is 1 full carton and 1/4 of another. Using concrete objects like egg cartons or pizza boxes helps students see that fractions greater than one are just a combination of wholes and parts.

Why is it important to use different models for fractions?

Some students find area models (circles/rectangles) intuitive, while others prefer linear models (number lines) or set models (groups of objects). By using a variety of hands-on tools, you ensure that every learner finds a representation that clicks. This variety is a key part of the Ontario curriculum's focus on multiple representations.

How can active learning help students understand fraction operations?

Active learning turns operations into 'doing.' Instead of just writing 3/8 + 2/8, students are physically combining three 1/8 pieces with two more. This physical action reinforces the concept that the denominator is a label for the unit. Collaborative problem-solving also allows students to catch each other's common errors, like adding denominators, in real time.

When should we move from models to symbols?

Only move to symbols once students can accurately predict the outcome using a mental model. If a student can explain why 1/2 + 1/4 is 3/4 using a picture, they are ready for the symbolic notation. In Grade 5, the emphasis should remain on the connection between the model and the numbers.