You will learn
- Test whether a value works.
- Justify a claim with examples.
- Generalise a pattern.
- Find a counterexample.
Version 0.1 · published for early use · expert review pending Standards alignment: pending verification. Not yet reviewed against the Courseversity scorecard.
Student lesson
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.
3. 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.
4. Practise
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
- 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?
Answers are in the answer key below. Try first, then check.
5. Apply: everyday mathematics
Bayo says all primes are odd. Is he right?
Show a sample answer
No; 2 is prime and even.
6. Common mistake
Fix: One counterexample disproves.
7. Check and master
Answer all questions. You will see feedback after each one.
Your result appears here after you answer every question.
8. Key takeaways
- Test and justify.
- Counterexamples disprove.
- Generalise carefully.
9. Next lesson
Next lesson: Two-dimensional figures. Keep going with the next idea in this course.
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)
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, challenge C.
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 |
Proposed Courseversity thresholds for this page: Bronze = at least 3 of 5 core questions correct; Silver = all 5; Gold = all 5 plus the challenge. 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.
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. 35–45 minutes.
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.
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 10-mark scheme and all answers are in the answer key 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
Alignment is pending verification and no external 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. Specific standard codes will be added once checked against the official documents.