Adding and Subtracting Fractions

Active learning works for this topic because fractions with unlike denominators require students to move beyond procedural rules and engage with the concrete meaning of equivalence. Visual models and hands-on tasks help learners see why common denominators matter, building lasting understanding rather than temporary memorization.

• 6th-class
• Mastering Mathematical Reasoning
• 4 activities
• 25 min–40 min

Learning Objectives

1. Calculate the sum and difference of fractions with unlike denominators using both visual area models and abstract algorithms.
2. Explain the necessity of a common denominator for adding and subtracting fractions by referencing visual representations.
3. Compare the results of fraction addition and subtraction problems solved using visual models versus standard algorithms.
4. Apply the addition and subtraction of fractions with unlike denominators to solve multi-step word problems.
5. Analyze the relationship between equivalent fractions and the process of finding common denominators.

Problem-Based Learning

⏱ 25 min·Pairs

Pair Work: Fraction Strip Equivalence

Pairs cut and label fraction strips for denominators like 3 and 4. They align strips to find equivalent fractions with common denominators, then add or subtract by combining lengths. Partners record steps and check with drawings.

Prepare & details

Explain why a common denominator is essential for adding fractions but not for multiplying them.

Facilitation Tip: During Fraction Strip Equivalence, circulate to listen for pairs explaining how the strips show equal parts, reinforcing the idea that denominators must match before adding or subtracting.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 35 min·Small Groups

Small Groups: Area Model Pizza Sharing

Groups draw area models of pizzas divided into unlike fractions. They find common denominators to add or subtract slices for sharing problems, such as combining toppings. Each group presents one solution to the class.

Prepare & details

Analyze how visual area models can help us understand the process of fraction addition.

Facilitation Tip: During Area Model Pizza Sharing, ask groups to verbalize how the pieces combine to form a whole, which helps them see the connection between the model and the algorithm.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 40 min·Whole Class

Whole Class: Number Line Relay

Divide class into teams. Call out fraction addition or subtraction problems with unlike denominators. One student per team marks the first fraction on a large number line, next adds or subtracts to solve, passing a marker. First team correct wins.

Prepare & details

Apply addition and subtraction of fractions with different denominators to solve real-world problems.

Facilitation Tip: During Number Line Relay, stand at the starting line to observe how teams plot jumps, catching any missteps in unit alignment immediately.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 30 min·Individual

Individual: Recipe Adjustment Challenge

Students receive recipes with fractional ingredients using unlike denominators. They add or subtract to adjust for different servings, using visuals first then algorithms. Collect and share one adjusted recipe.

Prepare & details

Explain why a common denominator is essential for adding fractions but not for multiplying them.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

Start with visual models to establish why common denominators are essential, as research shows this builds stronger conceptual foundations than starting with abstract rules. Avoid rushing to the standard algorithm; instead, encourage students to verbalize their thinking while using models, which deepens understanding. Use peer discussions to normalize mistakes and corrections, helping students internalize the logic behind equivalence.

What to Expect

Successful learning looks like students confidently explaining why fractions must share the same unit size before combining them, using multiple representations to justify their steps. You will see students connecting visual models to abstract algorithms and applying their skills to real-world situations with little prompting.

Watch Out for These Misconceptions

Common Misconception

During Pair Work: Fraction Strip Equivalence, watch for students who add numerators and denominators separately or who ignore the need for equal-sized units.

What to Teach Instead

Ask pairs to physically align the strips and explain why the units must be the same size before combining. Have them sketch their strips on paper and label the parts to clarify the process.

Common Misconception

During Small Groups: Area Model Pizza Sharing, watch for students who assume the common denominator must always be the larger of the two denominators.

What to Teach Instead

Provide groups with fraction circles or rectangles to test different multiples, such as finding a common unit for thirds and halves. Ask them to compare using 6 versus 12 to see which is more efficient.

Common Misconception

During Whole Class: Number Line Relay, watch for students who subtract the whole number part first before dealing with the fractional parts.

What to Teach Instead

Direct teams to plot the numbers on the line and model borrowing from the whole, similar to whole number subtraction. Have them explain how the jumps on the line represent the correct process.

Assessment Ideas

Quick Check

After Pair Work: Fraction Strip Equivalence, present the problem: 'Sarah used 1/3 of a cup of sugar for cookies and 1/4 of a cup for a cake. How much sugar did she use in total?' Ask students to solve it first using an area model and then using the standard algorithm, writing both solutions on their whiteboards.

Exit Ticket

After Small Groups: Area Model Pizza Sharing, provide students with two fractions, e.g., 2/5 and 1/3. Ask them to write one sentence explaining why they need a common denominator to subtract these fractions, and then calculate the difference.

Discussion Prompt

During Whole Class: Number Line Relay, pose the question: 'When would it be more efficient to use a visual model to add fractions, and when is the abstract method better?' Facilitate a class discussion where students share examples and justify their reasoning using the relay as a reference point.

Extensions & Scaffolding

• Challenge: Ask students to create a real-world problem where adding fractions with unlike denominators would require borrowing a whole unit, then solve it using both a number line and the standard algorithm.
• Scaffolding: Provide fraction strips or pre-drawn number lines with labeled intervals for students who need extra support during the relay or strip activities.
• Deeper exploration: Introduce mixed numbers alongside fractions to explore how equivalence applies when whole units are involved, using the recipe challenge as a starting point.

Key Vocabulary

Unlike Denominators	Denominators in fractions that are different values, meaning the fractional parts are not of the same size.
Common Denominator	A shared denominator for two or more fractions, allowing them to be added or subtracted meaningfully because their parts are the same size.
Equivalent Fractions	Fractions that represent the same value or portion of a whole, even though they have different numerators and denominators.
Least Common Multiple (LCM)	The smallest positive number that is a multiple of two or more numbers, used to find the least common denominator.