You will learn
- Show working.
- Write a reason.
- Justify a choice.
- Use mathematical words.
Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- Show your working step by step.
- Give a reason for each claim.
- Use words like because and therefore.
- A full explanation has an answer, working and a reason.
Words to know
Say how and why.
Give reasons.
Gives a reason.
Shows a conclusion.
1. Discover: prove it
Is 0.4 greater than 0.35? How can you explain your answer?
2. Understand: explaining
- Write 0.4 as 0.40.
- Compare tenths: 4 tenths is more than 3 tenths.
- So 0.40 > 0.35 because 40 hundredths is more than 35 hundredths.
- A good explanation has the answer, working and a reason.
Idea 1: Compare
Is 2/3 greater than 3/5? In fifteenths: 10/15 and 9/15, so yes. 3/5 is less because 9/15 is less than 10/15.
Idea 2: Reason
25% of 80 is 20 because 25% is one quarter and 80 ÷ 4 = 20. 12 is a multiple of 4 because 4 × 3 = 12.
Idea 3: Words
Because gives a reason. Therefore shows a conclusion.
Why explain? A reason shows you understand, not just guessed.
3. Worked examples
Example 1 (I show you): Compare
2/3 and 3/5
- Fifteenths.
- 10/15 and 9/15.
Answer: 2/3 is greater
Example 2 (We do it together): Explain
25% of 80
- A quarter.
- 80 ÷ 4.
Answer: 20
Example 3 (You try): Justify
12 is a multiple of 4
Show the steps and answer
- 4 × 3.
Answer: because 4 × 3 = 12
Did you know?
Mathematicians justify every step so others can check.
Why this matters
Explaining deepens understanding and earns exam marks.
4. Explore: explain
You need paper.
- State the answer.
- Show working.
- Give a reason.
- Read it back.
Think about it: what did you notice?
0.4 > 0.35.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- I think ___ because ___.
- First ___, then ___.
- Therefore ___.
- This is correct because ___.
Guided practice
Question 1. Is 0.4 > 0.35?
Yes.
Question 2. 0.4 as hundredths?
40.
Question 3. Why?
40 > 35.
Independent practice
Level 1: Start
- Is 2/3 greater than 3/5? (work in fifteenths)
- Why is 3/5 less?
- Explain 25% of 80
- Is 12 a multiple of 4? Why?
- Write a reason word
Level 2: Grow
- Is 3/4 greater than 2/3?
- Which word gives a reason?
- Which word shows a conclusion?
Level 3: Stretch
- Is 15 a multiple of 4? Why?
- Explain why 0.5 is greater than 0.45.
- Ben says 3/5 is greater than 2/3 because 3 is greater than 2. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
Explain why 0.4 is greater than 0.35.
Show a sample answer
0.40 has 40 hundredths, 0.35 has 35.
Another one. Explain why 40% of 50 is 20.
Show a sample answer
10% of 50 is 5, so 40% is 4 × 5 = 20
7. Common mistakes
Fix: Add the working and reason.
Fix: Add working and a reason.
Fix: Use exact maths words.
8. Lesson summary and revision card
- I can show working.
- I can write a reason.
- I can justify a choice.
- I can use mathematical words.
9. Check and master
Answer all 14 questions. You will see feedback after each one.
Your result appears here after you answer every question.
10. Key takeaways
- Answer, working, reason.
- Use maths words.
- Read back.
11. Next lesson
Next lesson: Checking reasonableness. Keep going with the next idea in this course.
Exam practice
These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.
- Is 2/3 greater than 3/5? Show your working. [3 marks]
- Explain why 3/5 is less than 2/3. [2 marks]
- Explain why 25% of 80 is 20. [3 marks]
- Is 12 a multiple of 4? Give a reason. [2 marks]
- Ali says 3/5 is greater than 2/3 because 3 is greater than 2. Is he right? Explain. [3 marks]
Total: 13 marks.
Marking guide and model answers
- Common denominator 15 [1]; 10/15 and 9/15 [1]; 2/3 is greater [1]
- 3/5 = 9/15 [1]; 9/15 is less than 10/15 [1]
- 25% is one quarter [1]; 80 ÷ 4 [1]; 20 [1]
- 4 × 3 = 12 [1]; yes [1]
- No [1]. Compare using the same denominator: 9/15 and 10/15 [1]. 2/3 is greater [1].
Score guide: 12 to 13 = secure. 8 to 11 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.
Printable worksheet
- 2/3 vs 3/5, bigger
- 3/5 less because
- 25% of 80 because
- 12 multiple of 4 because
- A reason word
- Is 2/3 greater than 3/5? (work in fifteenths)
- Why is 3/5 less?
- Explain 25% of 80
- Is 12 a multiple of 4? Why?
- A reason word
Homework
Explain how you solved one problem this week.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 2/3 vs 3/5, bigger | 2/3 | 2 |
| 3/5 less because | 9/15 < 10/15 | 2 |
| 25% of 80 because | a quarter | 2 |
| 12 multiple of 4 because | 4 × 3 | 2 |
| A reason word | because | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 2/3; 2 9/15 < 10/15; 3 a quarter; 4 4 × 3; 5 because.
Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.
Extra worksheet answers (items 6 onward): 6 yes, 10/15 compared with 9/15; 7 9/15 is less than 10/15; 8 a quarter of 80 is 20; 9 yes, 4 × 3 = 12; 10 because.
Independent practice answers. Level 1: 1 yes: 10/15 compared with 9/15; 2 9/15 is less than 10/15; 3 25% is one quarter of 80, so 20; 4 yes, 4 × 3 = 12; 5 because. Level 2: 1 yes, 9/12 is greater than 8/12; 2 because; 3 therefore. Level 3: 1 no, 4 × 3 = 12 and 4 × 4 = 16; 2 0.50 is greater than 0.45; 3 Compare 9/15 and 10/15: 2/3 is greater..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Show working | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O2 Write a reason | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O3 Justify a choice | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O4 Use mathematical words | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: Giving only the answer. Add the working and reason.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Showing working.
- Guided practice: work through one example together and let the learner try the next one.
- Reassess: use the quiz again or the worksheet.
- Move on when the learner shows Silver-level evidence independently.
For parents and guardians
In this lesson your child explains and justifies solutions. You do not need to be good at mathematics to help.
- Ask your child to explain how they know.
- Ask: why does that work?
- Praise clear reasons.
- Ask “how do you know?” after each answer, and listen.
- Praise effort, not only right answers. Mistakes are how we learn.
- Ten to fifteen minutes a day is better than one long session.
Teacher package
1–5. Objectives, prerequisites, materials, preparation, duration
Lesson objectives. Learners will: show working; write a reason; justify a choice; use mathematical words.
Prerequisite knowledge. Lessons 1–4.
Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.
Preparation. Gather the counters and draw the example pictures on the board before the lesson.
Suggested duration. 45–60 minutes. It can be split into two shorter sessions.
6–9. Teaching procedure, explanation notes, guided and independent activities
Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.
Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.
Guided activity (10 min). You need paper. Learners follow the Explore steps in pairs.
Independent activity (10 min). Learners complete the independent practice and the worksheet.
Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.
Key questions. What is my answer? What is my reason?
Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.
10–12. Differentiation, formative assessment, common misconceptions
Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.
Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.
Misconceptions. Giving only the answer. Add the working and reason.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 13-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.
16–19. Remediation, enrichment, homework, teacher reflection
Remediation. Showing working. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.
Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.
Sources and standards
This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.
Official standards for this topic area. Common Core State Standards for Mathematics (official site): Standards for Mathematical Practice. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core MP3; Cambridge Primary Mathematics Stage 6 (reasoning and communication). These sources informed the lesson.