Subtraction of Fractions and Mixed Numbers

Active learning builds confidence with fraction subtraction by giving students physical and visual ways to see why procedures work. Fraction strips and number lines make abstract rules concrete, especially for borrowing, which many students find confusing when taught only with symbols.

• Primary 5
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Calculate the difference between two fractions with unlike denominators by finding a common denominator and subtracting numerators.
2. Subtract mixed numbers with unlike denominators, applying the borrowing procedure when necessary.
3. Compare the steps required for subtracting mixed numbers versus adding mixed numbers.
4. Design a word problem that necessitates subtracting mixed numbers and requires interpretation of the fractional remainder.
5. Explain the concept of regrouping one whole into fractional parts when the numerator of the subtrahend is larger than the numerator of the minuend.

Problem-Based Learning

⏱ 35 min·Small Groups

Small Groups: Fraction Strip Borrow Challenge

Give each group colored fraction strips representing mixed numbers. Students build models for problems like 4 1/6 - 2 5/12, perform subtraction by aligning strips and borrowing wholes as needed. Groups record steps on mini-whiteboards and share one solution with the class.

Prepare & details

Differentiate between the steps for adding and subtracting mixed numbers.

Facilitation Tip: During Fraction Strip Borrow Challenge, circulate and ask groups to explain how their strips represent the borrowed whole before proceeding to subtraction.

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

⏱ 25 min·Pairs

Pairs: Subtraction Error Hunt

Provide cards with mixed number subtraction problems containing common errors, such as forgetting common denominators or improper borrowing. Pairs identify mistakes, correct them step-by-step, and create a correct version. Switch cards with another pair for verification.

Prepare & details

Explain the concept of 'borrowing' when subtracting mixed numbers with unlike denominators.

Facilitation Tip: For Subtraction Error Hunt, give pairs two completed problems with different errors so they must diagnose and articulate the mistake in their own words.

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 Mixed Number Race

Draw large number lines on the board. Divide class into teams; each team sends a member to plot mixed numbers, subtract by jumping intervals, and borrow visually if needed. Correct teams score points; debrief as a class.

Prepare & details

Design a word problem that requires subtracting mixed numbers and interpreting the result.

Facilitation Tip: In Number Line Mixed Number Race, require students to mark each step on the line so you can see their missteps in real time.

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

⏱ 20 min·Individual

Individual: Word Problem Creator

Students design original word problems requiring mixed number subtraction with borrowing, such as dividing fabric lengths. Solve their own problem, then swap with a partner for peer checking using fraction circles.

Prepare & details

Differentiate between the steps for adding and subtracting mixed numbers.

Facilitation Tip: When reviewing Word Problem Creator, check that students' scenarios require borrowing and that their solutions include clear fraction models.

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

Teach subtraction of fractions by starting with concrete models before moving to symbols. Use fraction strips and number lines to show why denominators must match and why borrowing is necessary. Research shows students retain procedures better when they first understand the underlying concepts through hands-on work and peer discussion. Avoid rushing to abstract steps before students have internalized the visual models and language of equivalent fractions.

What to Expect

Students will confidently find common denominators, recognize when borrowing is needed, and execute subtraction steps correctly. They will explain their reasoning using precise vocabulary and correct notation in both written and verbal forms.

Watch Out for These Misconceptions

Common Misconception

During Fraction Strip Borrow Challenge, watch for students who try to subtract denominators directly. Redirect them to place their strips side by side to see why common denominators are needed.

What to Teach Instead

Ask students to find strips that match the size of the fractions they're comparing before proceeding. Have them explain how the strips represent equivalent fractions, emphasizing that the value stays the same even though the pieces look different.

Common Misconception

During Subtraction Error Hunt, watch for students who claim borrowing isn't necessary when the fractional parts are close in value. Redirect them to model the problem with fraction strips to see the gap.

What to Teach Instead

Have pairs model the problem with strips and observe where the pieces don't align. Guide them to convert one whole into equivalent fractions to bridge the gap, then subtract.

Common Misconception

During Word Problem Creator, watch for students who forget to subtract 1 from the whole number after borrowing. Redirect them to check their final mixed number against the original minuend.

What to Teach Instead

Ask students to write the original and final mixed numbers side by side and circle the whole number part in each. Prompt them to explain why the whole number decreases by one after borrowing.

Assessment Ideas

Quick Check

After Number Line Mixed Number Race, present the problem 5 1/3 - 2 1/2. Ask students to show their work on mini whiteboards, focusing on how they found a common denominator and handled borrowing. Review responses to identify common errors.

Exit Ticket

After Word Problem Creator, provide students with two problems: 1) 7/8 - 1/4, and 2) 4 1/2 - 1 3/4. Ask them to write one sentence explaining the key difference in the procedure for solving each problem.

Discussion Prompt

During Fraction Strip Borrow Challenge, pose the question: 'Explain to your partner why you need to borrow from the whole number when subtracting 3 2/5 from 5 2/5. What does the 'borrowed' whole become?' Facilitate a brief class discussion to solidify understanding of the borrowing concept.

Extensions & Scaffolding

• Challenge: Create a set of three mixed number subtraction problems where the fractional parts require borrowing, and solve them using two different common denominators. Compare the results and explain why both methods are valid.
• Scaffolding: Provide fraction strips and pre-labeled number lines for students to use when solving 6 3/8 - 2 5/6. Ask them to label each step with the borrowed whole and new fraction.
• Deeper: Design a real-world scenario where a chef needs to adjust recipe quantities by subtracting mixed numbers, such as reducing a triple batch back to a single recipe.

Key Vocabulary

Common Denominator	A shared denominator for two or more fractions, which is necessary before adding or subtracting them.
Borrowing (in subtraction)	Regrouping one whole unit into equivalent fractional parts to enable subtraction when the top fraction is smaller than the bottom fraction.
Mixed Number	A number consisting of a whole number and a proper fraction, such as 3 1/2.
Improper Fraction	A fraction where the numerator is greater than or equal to the denominator, such as 5/4.