Mixed fractions, parts & whole

Overview

In this lesson, students explore mixed fractions by linking the “whole(s)” to the “part(s)”. They learn to represent and explain what mixed numbers mean using diagrams, making clear how numerator, denominator, and the whole amount relate.

Learning intentions

• Students will recognise that a mixed fraction contains whole(s) and a fraction part.
• Students will explain how many equal parts make one whole and how this affects the mixed fraction value.
• Students will represent mixed fractions using diagrams and connect the diagram to the numeric form.
• Students will justify reasoning using sentence stems when comparing and reasoning about mixed fractions.

Success criteria

• I can show a mixed fraction as whole(s) plus a fraction part on a diagram.
• I can explain what the denominator tells me about the size of the parts.
• I can explain what the numerator tells me about how many parts I have.
• I can tell whether two mixed fractions are equal or which is larger, and justify my thinking.

Curriculum links

• Number – fractions (including decimals): solve problems involving increasingly harder fractions to calculate quantities, including non-unit fractions where answers are whole numbers.
• Number – fractions (including decimals): recognise and show families of common equivalent fractions (used when justifying “equal whole” ideas).
• Fractions addition/subtraction with the same denominator (as a justification step if needed when counting parts).
• Mixed fractions connect to the idea that dividing into equal parts creates fractions (including hundredths/tenths ideas, used for understanding “one whole” in parts).

Lesson structure (35 minutes)

1. 0–5 min · Hook & reasoning introduction

• Teacher shows a clear picture: 2 whole rectangles and an extra rectangle split into 4 equal strips, with 3 shaded.
• Teacher asks: “What fraction of one more whole is shaded? What fraction of the total amount is that? What mixed number matches this picture?”
• Students think-pair-share, then a few verbal explanations.
• Targeted whole class questions (Layered, explanation-required):
• Recall/Explain: “How do you know this is three quarters and not something else?”
• Apply: “If one whole is made of 4 equal parts, what does the denominator tell us about the size of each part?”
• Analyse: “Why does the answer include 2 wholes and not just a fraction?”
• Justify: “Can you point to the part that represents the fraction, and the part(s) that represent the wholes?”
• AfL: Listen for correct “one whole = denominator parts” reasoning; note misconceptions on a class chart.

2. 5–12 min · Modelling (copy + prove it)

• Teacher models “Convert diagram to mixed fraction” with a worked example on the board (large, slow, annotated).
• Example children copy into books (exact model):

“Example: 1 3/4 I see 1 whole shaded. The next square is split into 4 equal parts (denominator = 4). 3 parts are shaded (numerator = 3). So the diagram shows 1 whole and 3/4 of another whole → 1 3/4.”

• Prove it reasoning with timer + sentence stems (verbal first, then write):
• Teacher says: “Prove the mixed fraction matches the diagram.”
• Set 30 seconds verbal reasoning in pairs using stems, then 60 seconds to write.
• Sentence stems on board:
• “The denominator tells me that one whole is split into ___ equal parts.”
• “The numerator tells me I have ___ of those parts.”
• “Because there are ___ complete wholes shaded, the mixed number is ___.”
• Teacher uses a different diagram briefly and asks students to predict the mixed fraction before revealing it.
• AfL: Check that students use both denominator and numerator language, not just naming the mixed number.

3. 12–20 min · Guided practice with layered whole-class questioning

• Teacher projects 3 short prompts (no answers shown yet). Students hold mini-whiteboards.
• Prompts increase in complexity by adding reasoning demands:
• Prompt A (easy): one whole plus 2/3 shaded.
• Prompt B (medium): 2 wholes plus 1/4 shaded (students must justify why it’s not 2/5).
• Prompt C (harder): 1 whole plus 5/6 shaded (should be expressed as 1 5/6, and students can also reason about “more than half”, or explain why it isn’t 1 1/6).
• Targeted whole class questions (Bloom’s progression):
• Understand: “What does the denominator mean in this picture?”
• Apply: “How many equal parts make one whole?”
• Analyse: “Why does the shaded part represent the numerator amount?”
• Evaluate: “Which is bigger: 1 2/3 or 1 3/4? Explain your reasoning.”
• Create (extension inside task): “Change one part on the diagram so it becomes 1 3/4—what changes?”
• AfL: Use cold-calling for explanations; tally misconceptions (below).

4. 20–29 min · Independent task (gradual complexity)

• Students complete a worksheet with 4 items:

1. Draw a diagram for 1 2/3 and label whole(s) and parts.
2. Write the mixed number shown in a diagram (1 whole + 3/5).
3. Compare two mixed fractions using diagrams (e.g., 2 1/4 vs 1 7/4). Students must justify in words.
4. Prove a claim: “This diagram shows 1 3/4.” Students must accept/reject and explain using denominator/numerator/wholes.

• AfL during task: Teacher circulates with a quick checklist:
• Can pupils link “numerator = shaded parts” and “denominator = parts per whole”?
• Do they mention “wholes” as complete groups?
• Are comparisons justified, not guessed?
• Quick verbal checkpoint (2 minutes): Teacher selects 3 students to explain item 3 using sentence stems before writing their final answer.

5. 29–35 min · Plenary (active reasoning + misconceptions revisit)

• “Prove it” lightning round: Teacher shows one diagram and asks one final reasoning question.
• Students respond by speaking first (10 seconds prep), then writing final answer in 1 line.
• Plenary question:
• “Is this mixed fraction the same as a different mixed fraction shown? How do you know?” (Aim for justification using parts-per-whole.)
• AfL: Collect boards or written responses to confirm correct mixed fraction interpretation and reasoning.

Common misconceptions to address (surface, then fix)

• Students think the denominator is the number of wholes (denominator = parts per whole only).
• Confusing numerator with denominator when counting shaded parts.
• Ignoring the “whole(s)” and treating only the fraction part.
• Saying “bigger denominator means bigger fraction” without considering which numerator is shaded.
• Treating 1 5/6 as if it were 1 + 5/6 but failing to recognise it is still “one whole and five equal sixths”.

Resources

• Board/presentation with large mixed fraction diagrams (2–3 examples).
• Mini-whiteboards and pens.
• Worksheet with 4 questions (draw, write, compare, prove).
• Sentence stem cards (denominator/numerator/wholes).
• Timer (projected or on board) for verbal reasoning moments.

Assessment

• Formative during hook and guided practice: listen for “denominator = parts per whole” and “numerator = shaded parts” language.
• During modelling: check written “Example” copy includes all three parts (wholes, denominator, numerator).
• During independent task: teacher checklist + brief verbal checkpoint; note whether comparisons are justified.

Differentiation

• Support: sentence stems partially completed (e.g., “One whole is split into ___ parts.”) and a pre-labelled diagram template with circles/strips count markers.
• Support for SEN/EAL: allow diagram-only answers for first two items, then require a short sentence justification using one stem.
• Challenge: add an extra compare-and-explain question where students must justify both “why it’s larger” and “why the other is smaller”.
• Extension within task: ask students to create a new diagram that matches a given mixed fraction but with a different visual grouping (still showing equal parts).