Fraction Order

Overview

In this 20-minute lesson, students practise adding, subtracting, multiplying and dividing fractions (including proper, improper and mixed numbers). They will work through a hands-on card sort: matching fraction expressions to their simplified answers, then justifying matches using fraction concepts and BODMAS (order of operations).

Learning intentions

Students will:

• add and subtract fractions using common denominators and simplify results
• multiply fractions and simplify (including improper and mixed numbers)
• divide fractions using reciprocal reasoning and simplify results
• explain, using BODMAS, why operations are performed in the correct order

Success criteria

Students can:

• correctly match each fraction expression card to the simplified answer card
• show correct fraction setup (e.g., common denominator, converting mixed numbers to improper fractions)
• state the correct operation order using BODMAS when an expression contains brackets or multiple operations
• justify their chosen match with clear fraction reasoning (not guessing)

Curriculum links

• Mathematics Stage 3: Representing quantity fractions B — recognising and using fractions to represent quantity, including moving between mixed and improper forms when computing
• Mathematics Stage 3: Representing quantity fractions B — comparing and working with fractions using equivalence as part of adding/subtracting
• Mathematics Stage 3: Multiplicative relations B — constructing and completing number sentences involving multiplicative relations and applying BODMAS
• Mathematics Stage 3: Multiplicative relations B — selecting appropriate strategies to solve multiplication and division problems with fractions

Lesson structure (20 minutes)

1. 0–3 min · Warm-up prompt. Teacher writes: “What changes when we turn a mixed number into an improper fraction?” and a quick example: 2 1/3 =? Student answers using prior knowledge; teacher checks understanding of conversion.

2. 3–7 min · Mini direct teach: BODMAS + fraction rules. Teacher models two quick “think aloud” examples on the board:

• Example A (order of operations): (1 1/2) + (1/4) × (2/3)
• Example B (fractions basics): 3/4 ÷ 2/3 Teacher explicitly states: brackets first, then multiplication/division, then addition/subtraction; for division, the teacher explains “multiply by the reciprocal”.

3. 7–18 min · Hands-on card sort (matching). Teacher sets up one small set of cards (student-only). Each “expression” card must be matched to a “simplified result” card. Student sorts in three rounds, verbalising decisions:

• Round 1 (add/sub): expression cards with only addition/subtraction (may include mixed numbers).
• Round 2 (multiply): expression cards with multiplication (may require simplifying first or after).
• Round 3 (divide + BODMAS): expression cards that include division and at least one bracket or mixed operations, requiring BODMAS. For each match, the student picks up the answer card, places it beside the expression, and quickly checks simplification.

4. 18–20 min · Justification check (one match explanation). Student selects one card they matched and gives a 30–45 second justification:

• Which fraction rule(s) were used (common denominator, reciprocal, convert mixed numbers)?
• What BODMAS step determined the order?

Resources

• Fraction card sort set:
• Expression cards (proper/improper/mixed numbers; includes at least one with brackets and one with mixed operations)
• Answer cards (simplified fractions; some answers are improper fractions or mixed-number equivalents)
• Student recording sheet with two columns: “Expression” and “Simplified answer”
• Whiteboard or mini-whiteboard and marker
• Sentence starters (printed):
• “First I used BODMAS: ______.”
• “Because it’s division, I changed to multiplication by the reciprocal: ______.”
• “To add/subtract, I made the denominators the same (common denominator): ______.”

Assessment

• Teacher observation during the card sort: correct matching and whether the student checks simplification and conversion steps
• Targeted formative check: after each match, student explains one key reason (e.g., common denominator or reciprocal; or which BODMAS rule applied)
• Quick final evidence (18–20 min): one justification using both fraction concepts and BODMAS ordering

Differentiation

• Support:
• Sentence starters and a small “BODMAS reminder” strip on the desk: Brackets → Multiply/Divide → Add/Subtract
• Optional conversion card showing how to change mixed numbers to improper fractions
• Allow the student to use fraction bars/paper strips for quick common-denominator reasoning
• Extension (if the student finishes early within 20 minutes):
• Add one extra expression with nested brackets or a two-step operation (e.g., bracketed addition multiplied by a fraction), requiring careful BODMAS justification
• Language/SEN considerations:
• Keep explanations short and structured; focus on the key phrases “common denominator”, “reciprocal”, and “order of operations”
• If the student struggles with terminology, allow them to point to the relevant BODMAS step on the reminder strip while explaining

Example card content (for teacher to prepare)

Include 6–8 expression cards and 6–8 answer cards. Examples:

• (1 1/2) + (1/4)
• (2/3) + (5/6)
• (3 1/4) − (2/5)
• (2 1/3) × (3/5)
• (4/7) × (14/9)
• (3/4) ÷ (2/3)
• (1/2) + (1/6) × (3/2)
• (2 2/3) − (1/3) ÷ (1/2)

Ensure each answer card is simplified (e.g., 3/4 ÷ 2/3 → 9/8, which can be left as improper or converted to 1 1/8 consistently across the set).