Scaling up on a number line

Overview

In this lesson, students learn to represent division on a number line and understand “scaling up” (using repeated jumps of a unit) to move from a smaller known step to a larger one. They will justify their reasoning using active explanations, consistent with KS2 number and division expectations.

Learning intentions

• Students will understand division as repeated jumps along a number line.
• Students will scale up their jumps when the divisor is larger (e.g., moving from jumps of 2 to jumps of 5, or from 10 to 20).
• Students will solve division problems using a number line and explain their thinking.
• Students will check their answers by reasoning about what each jump represents.

Success criteria

• I can show division as equal jumps on a number line.
• I can scale up the jumps and explain why the jump size changes (or stays equal).
• I can use number line language (jump, land on, equal intervals) to justify my method.
• I can check if my answer makes sense using estimation or inverse reasoning.

Curriculum links

• Number — multiplication and division (Year 4): use place value and known facts to multiply/divide mentally and solve problems using number facts.
• Use place value, known and derived facts to multiply and divide mentally, including dividing by 1 and scaling with known facts.
• Solve problems involving multiplying and adding, including harder correspondence problems (connected to interpreting “n objects connected to m objects”).
• Aligns with White Rose Maths: Division as scaling using structured number line representations.

Lesson structure (30 minutes)

1. 0–4 min · Hook (number line surprise). Teacher draws a short number line 0–30 and asks: “If I hop in equal jumps of 3, where will I land after 4 jumps?” Students answer and justify using verbal explanation. AfL: Observe whether pupils talk about “equal jumps” and can describe landings (not just guess).

2. 4–10 min · Model + active reasoning (prove it question). Teacher introduces the division idea: “40 ÷ 5.” Draw a number line and label equal jumps of 5. Then add a “scale up” layer: “What if I didn’t know 40 ÷ 5 yet—how could I use what I know?” Teacher guides: use 40 = 20 + 20, or count groups of 5 by reaching 25 then 30 etc. Use model reasoning with a timer:

• 0:00–0:20: Teacher reads a reasoning stem aloud.
• 0:20–0:40: Students use the stems verbally to explain their reasoning to a partner.
• 0:40–1:00: Students write one sentence justification. Sentence stems (display):
• “Each jump represents ___, so I start at ___ and land on ___ after ___ jumps.”
• “I scaled up by using what I know about ___ groups of ___, so ___ jumps of ___ reach ___.” AfL: Listen for correct interpretation of “jump = group size” and for verbal justifications that match the diagram.

3. 10–14 min · Targeted whole-class questions (layered, reasoning-first). Teacher uses questions that require justification; pupils answer with mini reasoning and can refer to the number line.

Q1 (Remember/Understand): “What does one jump of 5 mean in 20 ÷ 5?” Q2 (Explain/Apply): “If one jump is 5, how many jumps to reach 35? How do you know?” Q3 (Apply/Analyse): “If you can do 30 ÷ 5, how can you ‘scale up’ to 40 ÷ 5 on the same style of number line? What changes?” Q4 (Evaluate): “A pupil says, ‘I used jumps of 4 because 40 ÷ 10.’ Do you agree or disagree? Prove it using the number line idea.” AfL: Use cold call + “show me on the number line” and “say it first, then write” checks. Track misconceptions listed below.

4. 14–22 min · Guided practice (copy example, then solve). Students copy the worked example, then complete a structured set of problems that increase in complexity.

Copy this example (in books): “Solve 24 ÷ 6 using a number line.”

• Draw number line from 0 to 30.
• Mark 0.
• Divide means equal groups, so each jump represents 6.
• Jump 6 each time: 0 → 6 (1 jump), → 12 (2 jumps), → 18 (3 jumps), → 24 (4 jumps).
• So, 24 ÷ 6 = 4.
• Check: 4 groups of 6 make 24.

Guided task (layered increase):

• A (easy): 18 ÷ 6 (one number line, count jumps).
• B (medium): 30 ÷ 6 (scale up by reaching a known checkpoint like 24 then add one more jump).
• C (harder): 36 ÷ 6 (scale up again; justify how you used a smaller known part). Teacher circulates with a checklist: “Did they draw equal jumps of the divisor? Do they state what each jump represents? Do they justify the number of jumps?”

AfL:

• Mid-task quick oral check: “Stop! Tell me: what does one jump represent?”
• Marking focus: reasoning sentence, not just final number.

5. 22–28 min · Independent reasoning (division on number line + scaling up). Students solve one more problem individually: 42 ÷ 6. They must include: (1) a number line, (2) a written justification sentence. AfL: Use a 1-minute “verbal first” routine before writing: students explain their plan to a partner using stems, then write.

6. 28–30 min · Plenary (prove it / misconception check). Teacher shows two incorrect statements (spoken or written quickly) and students choose which is correct and prove why:

• Statement 1: “In 24 ÷ 6, each jump should be 4.”
• Statement 2: “To scale up from 30 ÷ 6 to 36 ÷ 6, you add one more equal jump of 6.” Students vote (thumbs/boards), then two pupils justify using sentence stems. AfL: End-of-lesson oral summary: “What does scaling up mean on a number line for division?”

Resources

• Whiteboards or mini-slates for student responses
• Printed or teacher-drawn number line templates (0–60 range)
• Example worked solution poster for copying
• Sentence stem strip cards (for verbal justification)
• Timer (visible to students during verbal-reasoning moments)
• Dividing counters or arrow cards (optional, for those who need a concrete entry)

Assessment

• During modelling: listen for correct explanation of “jump size = divisor” and count of equal jumps.
• During whole-class questions: check for reasoning quality (they must justify, not only answer).
• Exit check in plenary: pupils prove which misconception statement is wrong and why.

Differentiation

• Support: provide a partially drawn number line with endpoints and one or two jumps; give sentence starters with missing words (“Each jump represents ___”).
• Support: allow using counters/arrows to mark jumps before drawing intervals.
• Challenge: give a correspondence-style twist: “A game lasts 42 seconds. Each round takes 6 seconds. How many rounds?” Students must link to “equal groups” and justify scaling.
• EAL/SEN: sentence stems and a word bank (equal, jump, groups, land on, scale up) plus pair work before writing.

Common misconceptions and difficulties

• Confusing jump size with the answer (e.g., using jumps of 4 for 24 ÷ 6).
• Miscounting equal jumps because of uneven spacing; students must remember “equal intervals.”
• Thinking “scaling up” means changing the jump size; clarify that scaling up is about reaching a larger total by extending the same equal-jump pattern (or using a known checkpoint plus extra jumps).
• Writing a correct final answer without a justification sentence (assessment focus on reasoning, not only result).