Fraction Problems

Overview

Today’s lesson helps pupils solve real-life problems using fractions and decimals to two decimal places, choosing the correct operation and explaining their reasoning. The learning builds from knowing what fractions represent and focuses on calculating quantities and interpreting answers in context.

Learning intentions

• Solve measure and money problems involving fractions and decimals to two decimal places.
• Use diagrams and fraction/decimal equivalence to work out quantities in real-life contexts.
• Explain and justify the steps taken using mathematical language and reasoning.

Success criteria

• I can represent the problem using a bar model or array diagram.
• I can calculate the required fraction/quantity and give the answer to two decimal places when needed.
• I can explain my method clearly using fraction reasoning words (e.g., “out of”, “equal parts”, “so each part is…”).
• I can check if my answer makes sense in context (reasonable magnitude and units).

Curriculum links

• Number – fractions (including decimals): solve simple measure and money problems involving fractions and decimals to two decimal places.
• Number – fractions (including decimals): solve problems involving increasingly harder fractions to calculate quantities.
• Add/subtract fractions with the same denominator using diagrams (for “remainder” steps when needed).
• Count up and down in hundredths / recognise hundredths when dividing (used when converting tenths to hundredths).

Lesson structure (40 minutes)

1. 0–5 min · Hook and active reasoning

• Show an image prompt: “A 1 litre bottle is shared: 3/5 of it is used for a plant, and you pay £2.40 for the rest. How much is the full bottle worth? How much was used?”
• Teacher asks whole-class questions (no hands up at first, then choose pupils):
• “What does ‘3/5 of it’ mean in this situation? What would you expect the amount to be compared with 1 litre?”
• “If someone wrote ‘3 ÷ 5 = 0.6’, what part of the litre would that describe? Explain why.”
• Layered: “What representation could help you see 3/5? What would each equal part be?”
• AfL: note pupils who confuse “of” with “divide by”, and those who can’t link fractions to parts of a quantity.

2. 5–15 min · Modelling (prove it) with timer + stems

• Board model copy-style (see “Copy example” section below). Teacher models one problem carefully.
• Example problem for modelling: “A ribbon is 2.40 m long. You use 3/5 of it. How much ribbon is used?”
• Teacher says: “I will think aloud first, then you will speak, then we write.”
• Steps with targeted questions:
• “Prove it verbally: how do you know 3/5 means three equal parts out of five?”
• “Active reasoning Q1 (Bloom: understand): What is the ‘whole’ here? What is the ‘part’?”
• “Active reasoning Q2 (Bloom: apply): How do we calculate 3/5 of 2.40? What operation do we choose?”
• “Active reasoning Q3 (Bloom: analyse): If 1/5 is 0.48, how can we use that to get 3/5?”
• Timer routine: 20 seconds Think → 30 seconds Talk with a partner → teacher picks two pupils to justify before writing.
• Sentence stems for verbal reasoning:
• “I know ___ because ___.”
• “To find ___, I calculate ___.”
• “Each part is ___, so ___ parts is ___.”
• “This makes sense because ___ (units / size).”
• AfL checks during modelling:
• Listen for correct language linking 1/5 to three-fifths.
• Check everyone understands why the decimal places stay consistent (answer in metres to 2 d.p. where required).

3. 15–26 min · Guided practice task (layered questions)

• Pupils work in pairs on a set of 3 questions increasing complexity. They must show a diagram and a written explanation.
• Q1 (lower): “You buy a £10.00 gift card. You use 2/5 of it. How much is left? (Give the answer to two decimal places.)”
• Q2 (middle): “A recipe uses 3/10 of a 500 g packet. How many grams are used?”
• Q3 (higher): “A taxi costs £6.50 for the first 3/8 of the journey, then the rest costs £12.00. What is the full cost?”
• Pupils must reason using “part-to-whole” and interpret context.
• Layered questioning while circulating (verbal mini-checks):
• “Explain what your diagram shows. Why is it divided into those parts?” (analyse)
• “How does your calculation connect to the real-life story? Where is the ‘whole’ in your workings?” (evaluate)
• “If someone gets 2/5 as 0.4, what assumption are they making? Do you agree? Show why.” (justify)
• “What would you do if the fraction wasn’t simplified? Would your method change?” (apply)
• AfL:
• Use a quick checklist: diagram present, correct operation, decimals to 2 d.p., justification written.
• Select 3 pupils for “instant share”: one strong model, one common misconception, one needs precision fix.

4. 26–33 min · Address misconceptions (targeted whole class)

• Choose misconceptions to address based on live observations:
• Misconception A: dividing instead of “of” (e.g., 3/5 of 2.40 mistaken as 2.40 ÷ 3).
• Questions:
• “Bloom: apply—If ‘of’ means ‘take a fraction of’, what operation should we write? What should we divide by instead?”
• Misconception B: confusing denominator with total number of parts (e.g., thinking denominator means numerator parts).
• Questions:
• “Bloom: analyse—Why does the denominator tell us how many equal parts the whole is split into?”
• Misconception C: decimal place errors (e.g., giving £3.8 instead of £3.80).
• Questions:
• “Bloom: evaluate—How can we check we’ve rounded/recorded to two decimal places correctly for money?”
• AfL: quick thumbs/mini-whiteboards for method choice (not just answers).

5. 33–38 min · Independent task (complexity increase)

• “Challenge Ladder” (same concept, harder context):
• Ladder step 1: “3/4 of 5.60 kg is how much?” (quantity)
• Step 2: “You sell 2/3 of a £18.00 item price. What is the selling price?” (money)
• Step 3: “After using 2/5 of the water from a 1.20 L bottle, you refill with 0.40 L. What fraction of the original bottle is now full?” (requires reasoning about remainder then compare)
• Teacher supports with sentence stems if needed.
• AfL: teacher marks explanations for reasoning and accuracy.

6. 38–40 min · Plenary (reasoning-based exit check)

• Whole class: show one completed solution with one subtle mistake (e.g., 3/5 of 2.40 = 1.20 instead of 1.44).
• Questions:
• “What is wrong with the reasoning? Point to the step.”
• “What would you write to prove it correctly?”
• Exit ticket (short): “Explain in 2–3 sentences how to find 2/5 of £30.00 and why your answer to two decimal places is correct.”
• AfL: teacher collects to identify next lesson focus.

Resources

• Fraction/bar model template (A4 or on boards)
• Whiteboards + pens
• Timer (phone or classroom timer)
• Worked example sheet (for reference during guided practice)
• Common misconception cue cards (optional teacher prompt)
• Calculator allowed/limited? (teacher decision—focus is reasoning, not tool use)

Copy example (to model and pupils copy into books)

Problem: A ribbon is 2.40 m long. You use 3/5 of it. How much ribbon is used?

1. Represent the whole:

• Split 2.40 m into 5 equal parts (each part is 1/5 of 2.40 m).

2. Find one fifth:

• 2.40 ÷ 5 = 0.48 m

3. Find three fifths:

• 3 × 0.48 = 1.44 m

4. Answer (2 decimal places):

• You use 1.44 m.

Reasoning sentence stems (write in full sentences):

• “I know 3/5 means three equal parts out of five, so the ribbon is split into 5 equal parts.”
• “I found 1/5 by dividing 2.40 by 5, which gave 0.48 m, then multiplied by 3 to get three fifths.”
• “This makes sense because 3/5 is more than half, so the answer should be more than 1.20 m.”

Assessment

• During modelling: listen for correct verbal justification using stems (especially meaning of numerator/denominator).
• During guided/independent tasks: check diagrams, operation choice, and correct rounding to two decimal places.
• Exit ticket: reasoning quality (not just final value) and correct method for “of” problems.

Differentiation

• Support:
• Provide a partially completed bar model template and sentence starters on the table.
• Offer a “step checklist” card: Split → Find 1 part → Multiply by numerator → Write to 2 d.p. (when required).
• Challenge/extension (still within same lesson):
• Ask pupils to create their own real-life story that matches a given calculation (e.g., “2/5 of a £x purchase”).
• Add a “prove it” requirement: add one sentence explaining why the answer is reasonable in context.
• EAL/SEN:
• Encourage verbal rehearsal before writing; use stems and model language.
• Allow extra time for writing explanations, but ensure they complete at least one justified explanation sentence.