Mathematics · Grade 6 · The Number System and Rational Quantities · Term 1

Dividing Fractions by Fractions: Algorithm Practice

Practicing the standard algorithm for dividing fractions and mixed numbers.

• Ontario Curriculum Expectations
• 6.NS.A.1

About This Topic

The standard algorithm for dividing fractions by fractions, often called 'keep, change, flip,' requires students to keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal before multiplying. In Grade 6, students practice this with proper fractions, improper fractions, and mixed numbers, while justifying each step and comparing it to visual models. This builds directly on multiplication of fractions and supports the Ontario curriculum's focus on rational numbers.

Fluency with the algorithm helps students tackle multi-step problems efficiently, such as determining how many batches of cookies a recipe yields or dividing lengths in construction plans. By evaluating when models suffice versus when the algorithm saves time, students develop strategic competence. Visual aids like fraction circles initially illustrate the 'why,' transitioning to procedural reliability.

Active learning benefits this topic greatly, as partners can share fraction strips to model divisions before applying the algorithm, groups solve contextual problems with physical items like yarn for measurements, and class discussions unpack sample errors. These methods make procedures intuitive, foster justification skills, and connect math to practical scenarios students encounter.

Key Questions

1. Justify the steps in the 'invert and multiply' algorithm for fraction division.
2. Evaluate the efficiency of using the algorithm versus drawing models for division.
3. Construct solutions to real-world problems involving division of fractions.

Learning Objectives

• Calculate the quotient of two fractions using the invert and multiply algorithm.
• Explain the mathematical reasoning behind the invert and multiply algorithm for fraction division.
• Compare the efficiency of using the invert and multiply algorithm versus visual models for solving fraction division problems.
• Construct word problems that require the division of fractions or mixed numbers to solve.
• Solve real-world problems involving the division of fractions and mixed numbers by applying the standard algorithm.

Before You Start

Multiplying Fractions and Mixed Numbers

Why: Students must be proficient in multiplying fractions and mixed numbers to apply the invert and multiply algorithm.

Understanding Fractions and Mixed Numbers

Why: Students need a solid grasp of what fractions and mixed numbers represent to understand the process of division and reciprocals.

Key Vocabulary

Reciprocal	Two numbers are reciprocals if their product is 1. For example, the reciprocal of 3/4 is 4/3.
Invert and Multiply Algorithm	The procedure for dividing fractions: keep the first fraction, change the division sign to multiplication, and multiply by the reciprocal of the second fraction.
Mixed Number	A number consisting of a whole number and a proper fraction, such as 2 1/2.
Quotient	The result obtained by dividing one quantity by another.

Watch Out for These Misconceptions

Common Misconception

Invert both fractions or the first one instead of the divisor.

What to Teach Instead

Students often flip the wrong fraction, leading to incorrect products. Modeling with area diagrams or number lines in pairs shows the divisor's reciprocal multiplies correctly. Group discussions of swapped examples clarify the 'flip' rule quickly.

Common Misconception

No need to convert mixed numbers to improper fractions first.

What to Teach Instead

Leaving mixed numbers causes computation errors during multiplication. Hands-on conversion practice with fraction strips helps students see the whole-part structure. Partner checks during algorithm steps build accuracy and confidence.

Common Misconception

The result of dividing fractions is always smaller than the parts.

What to Teach Instead

Dividing a large fraction by a small one yields larger quotients, confusing some. Real-world tasks like dividing fabric lengths reveal this pattern. Collaborative problem-solving exposes and corrects the assumption through examples.

Active Learning Ideas

Pairs: Invert and Multiply Race

Pairs receive cards with fraction division problems. One partner models with fraction bars, the other applies the algorithm and justifies steps. Switch roles after three problems, then compare results and discuss efficiencies. Collect cards for whole-class share.

25 min·Pairs

Small Groups: Recipe Division Challenge

Provide recipes with fractional ingredients. Groups divide quantities to scale for different servings using the algorithm, convert mixed numbers as needed, and verify with drawings. Present solutions and vote on the most efficient method.

35 min·Small Groups

Whole Class: Error Analysis Carousel

Post sample problems with intentional algorithm errors around the room. Students rotate in pairs, identify mistakes, correct them, and explain using reciprocal properties. Debrief as a class to reinforce justifications.

40 min·Pairs

Individual: Mixed Number Marathon

Students complete a timed set of 10 mixed number divisions, self-checking with simplified answers provided. Follow with partner swaps to peer-review justifications for efficiency over models.

20 min·Individual

Real-World Connections

• Bakers use fraction division to determine how many batches of cookies can be made from a given amount of dough, if each batch requires a specific fractional amount of dough.
• Construction workers might divide lengths of material, such as pipes or lumber, into smaller fractional sections, requiring division of fractions to calculate the number of pieces they can get.

Assessment Ideas

Exit Ticket

Provide students with the problem: 'A recipe calls for 3/4 cup of flour per batch. If you have 6 cups of flour, how many batches can you make?' Ask students to solve using the invert and multiply algorithm and write one sentence explaining why they changed the division to multiplication.

Quick Check

Present students with two division problems: one with proper fractions (e.g., 1/2 ÷ 1/4) and one with mixed numbers (e.g., 3 1/2 ÷ 1/2). Ask them to solve both using the algorithm and circle the problem they found easier to solve and why.

Discussion Prompt

Pose the question: 'Imagine you need to cut a 5-foot ribbon into pieces that are each 1/3 of a foot long. How many pieces can you cut? Explain how you would solve this problem, first by drawing a model, and then by using the invert and multiply algorithm. Which method is more efficient for this problem and why?'

Frequently Asked Questions

How do you justify the invert and multiply algorithm for fraction division?

Relate it to the definition of division as multiplying by the reciprocal: dividing by 3/4 equals multiplying by 4/3. Use visual models like partitioning rectangles to show why it works, then practice with number lines. Students justify by rewriting problems, e.g., 'How many 1/4s in 3/2?' becomes 3/2 x 4/1, building conceptual links before procedural fluency.

What are common mistakes when dividing fractions by fractions?

Errors include flipping the wrong fraction, skipping mixed number conversions, or forgetting to simplify. Address with scaffolded practice: start with unit fractions, add visuals, then full algorithm. Peer review sheets where students circle errors and explain fixes reinforce accuracy and self-monitoring skills.

What real-world problems use dividing fractions?

Examples include scaling recipes (divide ingredient fractions by serving size), sharing materials (how many 1/3 meter ropes from 5/2 meters), or rates (divide distances by time fractions). Pose problems like 'Divide 3/4 kg of flour by 1/8 kg per loaf' to show practical efficiency of the algorithm over drawings for larger numbers.

How can active learning help students master fraction division algorithms?

Active methods like fraction strip manipulations let students physically model divisions before algorithms, revealing the reciprocal's role. Group recipe challenges apply procedures contextually, while error carousels promote peer teaching. These build justification skills, compare model versus algorithm efficiency, and make abstract steps tangible, boosting retention and confidence in rational number operations.