Fraction Problem Solving

Overview

In this lesson, children solve real-life problems using fractions by finding equivalent fractions, simplifying, comparing, and calculating fractions of amounts. They will explain their reasoning verbally before writing, in line with age-appropriate fraction expectations for Year 4.

Learning intentions

• Students will recognise and use equivalent fractions to solve problems.
• Students will simplify fractions in practical and written contexts.
• Students will compare fractions and justify which is larger/smaller.
• Students will find fractions of amounts (including non-unit fractions) in real-life scenarios.

Success criteria

• I can find an equivalent fraction and explain how I know.
• I can simplify a fraction and say what I divided by.
• I can compare fractions using models/benchmarks and explain my reasoning.
• I can find a fraction of an amount and check if my answer makes sense.

Curriculum links

• Number – fractions (including decimals): solve problems involving increasingly harder fractions to calculate quantities and fractions to divide quantities, including non-unit fractions where the answer is a whole number.
• Number – fractions (including decimals): recognise and show families of common equivalent fractions using diagrams.
• Number – fractions (including decimals): count up and down in hundredths; recognise tenths and hundredths (link used for checking understanding of “equal parts”).
• Number – fractions (including decimals): solve simple measure and money problems involving fractions and decimals to two decimal places (used in context, with fractions only in core tasks today).

Lesson structure (35 minutes)

1. 0–5 min · Hook & shared thinking. Teacher shows an image: “A recipe uses 3/4 of the flour. We only have 12/16 of the flour left. How much flour will I still need?” Teacher asks: “What information do we have, and what is the question asking?” Pupils think-pair-share; teacher listens for “need to compare” and “fractions of amounts”.

2. 5–10 min · Targeted whole-class questions (reasoning first). Display: “Is 6/8 equivalent to 3/4? Prove it.”

• Teacher question (Bloom: Understand/Apply): “What change would you make to 6/8 to get an equivalent fraction? What stays the same?”
• Teacher question (Bloom: Analyse): “If you simplify 6/8 to 3/4, which parts are grouped and which are removed? Why is that fair?”
• Teacher question (Bloom: Evaluate): “Would you rather compare 6/8 to 3/5 or to 2/3? Explain which benchmark helps you most.” Pupils answer using sentence stems (see Resources) and teacher prompts for justification.

3. 10–16 min · Modelling with ‘Prove it’ (copy example). Teacher models a problem with clear steps and verbal explanation before writing: Example to copy: “A runner drank 2/3 of a bottle. The bottle holds 9 litres. How many litres did the runner drink?”

• Teacher says aloud: “I know ‘2/3 of 9’ means I take 2 equal parts out of 3 equal parts of 9.”
• Teacher models: “First find 1/3 of 9: 9 ÷ 3 = 3 litres. Then 2/3 = 2 × 3 = 6 litres.”
• Teacher then simplifies/comparison cue: “I can check: 2/3 is less than 1 whole, so answer should be less than 9.” Sentence stems for verbal reasoning (timer 30–45 sec each):
• “I know this is 2/3 because…”
• “I found 1/3 by dividing the amount by 3 because…”
• “My answer makes sense because it is…” Timer: “Say your reasoning now” → pupils speak to partner → teacher selects 2 pupils to share before writing.

4. 16–24 min · Guided practice (layered questioning, active reasoning). Teacher groups questions on the board; pupils solve in books after verbal discussion. Problem set (real-life context): “A shop uses 5/6 of a roll of ribbon for prizes. The full roll is 18 m. How much ribbon is used?” Layered prompts:

• Level 1 (Remember/Understand): “What does the denominator 6 tell you about how to split the ribbon?”
• Level 2 (Apply): “Find 1/6 of 18. Don’t just write it—explain your method.”
• Level 3 (Apply/Analyse): “Now make 5/6. How many fifths of the ‘one-sixths’ do you take? Explain the ‘2 × …’ step if you use one.”
• Level 4 (Reason/Explain/Evaluate): “What fraction of the ribbon is left? Write it and justify using a diagram.” During work, teacher uses quick checks: “Turn to your partner: Where do you see each part in your working?” AfL: teacher circulates with a checklist: equivalent reasoning, correct use of denominator, clear explanation.

5. 24–31 min · Independent task (increasing complexity). Choose and complete all parts (supports mixed ability): Task: “You have 3/5 of a pizza left. The total pizza is 20 slices. You eat 7/10 of what you have. a) How many slices are 3/5 of 20? b) How many slices is 7/10 of that amount? c) Simplify and compare: is the remaining fraction closer to 1/2 or 2/3? Explain.” Complexity increase comes from: two fraction-of-amount calculations + comparison/justification + simplifying. AfL: after 4 minutes, teacher pauses: “Thumbs for confidence—who already knows the first step? Who needs help with ‘7/10 of that amount’?” Teacher gives a targeted hint to two groups.

6. 31–35 min · Plenary (White Rose-style check & misconceptions). Teacher displays two student-style answers:

• A: “1/3 of 9 is 6 because 9 − 3 = 6.”
• B: “2/3 of 9 is 3 because 2 + 3 = 5 parts.” Pupils decide: “Which is incorrect and why?” Teacher asks: “What misconception is shown?” Then quick whole-class verbal reasoning: “Timer 20 sec—say the correct method for 1/3 of 9.” Exit check (last 2 minutes): one question on board: “Simplify 12/16 and then compare it to 3/5— which is bigger? Explain in one sentence.”

Resources

• Fraction cards/paper strips (for 1/2, 1/3, 1/4, 1/5, 1/6, 2/3, 5/6, 7/10)
• Diagram sets (circles/rectangles divided into equal parts)
• Example worked copy (projected and also on one sheet)
• Timer for verbal reasoning (phone/stopwatch)
• Sentence stem strip for pupils
• Mini-whiteboards for misconception debate
• Checklist for teacher AfL during circulation

Assessment

• Use cold-call targeted reasoning during Q&A: “How do you know?” “What part of your diagram proves it?”
• During modelling and guided practice, listen for: correct link between denominator and equal parts, correct division/multiplication structure, simplification method.
• Exit ticket: simplify 12/16 and compare to 3/5 with one-sentence justification (not just the choice).

Differentiation

• Support: provide a fraction-of-amount scaffold table: “Amount ÷ denominator = 1 part; multiply by numerator = answer.” Include simplified fraction prompts and sentence starters.
• Support for EAL/SEN: allow oral rehearsal before writing; use more visuals (rectangles/circles) and vocabulary frames (“The denominator shows…”).
• Challenge: ask pupils to create their own real-life fraction problem using equivalent fractions (e.g., “Find 2/3 of 24” but presented in a different context) and include a check for reasonableness.
• Extension within task: in part (c), require simplification of the “remaining fraction” and justification using a common denominator.

Common misconceptions to address (built into questioning and plenary)

• Confusing numerator/denominator: pupils think the denominator is “how many to take” rather than “how many equal parts.”
• Treating “1/3 of 9” as subtraction instead of dividing by 3.
• Computing numerator and denominator operations incorrectly (adding parts or mixing multiplications).
• Skipping equivalent fractions/simplification when comparing: pupils compare fractions with different denominators without using a common representation.
• For “fraction of an amount”: finding a fraction of the original total instead of the fraction-of-a-fraction (part b of the task).