You will learn
- Test whether a value works.
- Justify a claim with examples.
- Generalise a pattern.
- Find a counterexample.
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
- Test whether a value works by substituting.
- Give examples to support a claim.
- One counterexample shows a claim is false.
- Generalise patterns carefully.
Words to know
Give reasons.
State a rule for all cases.
An example that shows a claim is false.
A statement to test.
1. Discover: always true?
Ada says “the sum of two odd numbers is always even.” How can she check?
2. Understand: reasoning
- 3 + 5 = 8, 7 + 9 = 16, 11 + 1 = 12: all even. This supports the claim.
- Is x = 4 a solution of 3x + 2 = 14? 3 × 4 + 2 = 14. Yes.
- Claim: “every number times 10 ends in 0.” True: 10 × 7 = 70.
- Claim: “every multiple of 3 is even.” False: 9 is a counterexample.
Idea 1: Test
Is 5 a solution of 2x + 1 = 11? 2 × 5 + 1 = 11, so yes. Is 6 a solution of x − 2 = 5? 6 − 2 = 4, so no.
Idea 2: Counterexample
“All primes are odd” is false because of 2.
Idea 3: Generalise
7 + 9 = 16, even. Even + even = even is true.
Why one example? A single counterexample breaks a general claim.
3. Worked examples
Example 1 (I show you): Test
Is 5 a solution of 2x + 1 = 11?
- 2 × 5 + 1.
Answer: yes
Example 2 (We do it together): Counter
All primes are odd
- A prime that is even.
Answer: 2
Example 3 (You try): Sum
7 + 9
Show the steps and answer
- Odd + odd.
Answer: even (16)
Did you know?
Mathematicians prove claims; many examples alone are not proof.
Why this matters
Reasoning is how we know a rule is always true.
4. Explore: test
You need paper.
- Test three pairs of odd numbers.
- Check the sums.
- Try a counterexample search.
- State your claim.
Think about it: what did you notice?
Always even.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- I substitute ___ and get ___.
- So it is / is not a solution.
- A counterexample is ___.
- The claim is false because ___.
Guided practice
Question 1. Is 4 a solution of 3x + 2 = 14?
Yes.
Question 2. A counterexample to “multiples of 3 are even”?
9.
Question 3. 3 + 5?
8, even.
Independent practice
Level 1: Start
- Is 5 a solution of 2x + 1 = 11?
- Counterexample: all primes are odd
- 7 + 9 even or odd?
- Is “even + even = even” true? (yes/no)
- Is 6 a solution of x − 2 = 5?
Level 2: Grow
- Is 4 a solution of x + 3 = 7?
- A counterexample to “all numbers ending in 5 are even”
- 7 + 3 is odd or even?
Level 3: Stretch
- Is 2 a solution of 3x − 1 = 5?
- A counterexample to “all multiples of 4 are multiples of 8”
- Ben says odd + odd is odd because 3 + 5 = 8 is odd. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
Bayo says all primes are odd. Is he right?
Show a sample answer
No; 2 is prime and even.
Another one. A claim says any number ending in 3 is prime. Give a counterexample.
Show a sample answer
33 = 3 × 11
7. Common mistakes
Fix: One counterexample disproves.
Fix: Look for a counterexample.
Fix: Try different values.
8. Lesson summary and revision card
- I can test whether a value works.
- I can justify a claim with examples.
- I can generalise a pattern.
- I can find a counterexample.
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
- Test and justify.
- Counterexamples disprove.
- Generalise carefully.
11. Next lesson
Next lesson: Two-dimensional figures. 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 5 a solution of 2x + 1 = 11? Explain. [2 marks]
- Find a counterexample to “all primes are odd”. [2 marks]
- Is 7 + 9 odd or even? [2 marks]
- Is 6 a solution of x − 2 = 5? Explain. [2 marks]
- Ali says all multiples of 3 are odd because 3, 9 and 15 are odd. Is he right? Explain. [3 marks]
Total: 11 marks.
Marking guide and model answers
- 2 × 5 + 1 = 11 [1]; yes [1]
- 2 is prime [1]; 2 is even [1]
- 16 [1]; even [1]
- 6 − 2 = 4 [1]; no [1]
- No [1]. 6 is a multiple of 3 and is even [1]. One counterexample is enough [1].
Score guide: 10 to 11 = secure. 7 to 9 = nearly there, revise the parts you missed. Under 7 = work through the lesson again with a grown-up.
Printable worksheet
- Is 5 a solution of 2x + 1 = 11? (yes/no)
- Counterexample: all primes are odd
- 7 + 9 even or odd
- even + even = even? (yes/no)
- Is 6 a solution of x − 2 = 5? (yes/no)
- Is 4 a solution of 3x + 2 = 14?
- A counterexample to “multiples of 3 are even”
- Sum of two odd numbers is…
- Is 5 a solution of 2x + 1 = 11?
- Is 6 a solution of x − 2 = 5?
Homework
Make a claim about numbers and test it with five examples.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Is 5 a solution of 2x + 1 = 11? (yes/no) | yes | 2 |
| Counterexample: all primes are odd | 2 | 2 |
| 7 + 9 even or odd | even | 2 |
| even + even = even? (yes/no) | yes | 2 |
| Is 6 a solution of x − 2 = 5? (yes/no) | no | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 yes; 2 2; 3 even; 4 yes; 5 no.
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; 7 9; 8 even; 9 yes; 10 no.
Independent practice answers. Level 1: 1 yes; 2 2; 3 even (16); 4 yes; 5 no. Level 2: 1 yes; 2 15; 3 even. Level 3: 1 yes; 2 12; 3 8 is even, so odd + odd is even..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Test whether a value works | 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 Justify a claim with examples | 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 Generalise a pattern | 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 Find a counterexample | 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: Thinking examples prove a claim for all. One counterexample disproves.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Test simple claims.
- 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 tests claims and finds counterexamples. You do not need to be good at mathematics to help.
- Challenge each other with claims.
- Ask: can you find a counterexample?
- Test with different numbers.
- 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: test whether a value works; justify a claim with examples; generalise a pattern; find a counterexample.
Prerequisite knowledge. Lessons 1 to 6.
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. Does it work? Is there a counterexample?
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. Thinking examples prove a claim for all. One counterexample disproves.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 11-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. Test simple claims. 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): Grade 5 » Operations & Algebraic Thinking. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core MP3 and 6.EE.B.5; Cambridge Primary Mathematics Stage 6 (reasoning). These sources informed the lesson.