You will learn
- Use clues to name a shape.
- Justify a classification.
- Test a claim about shapes.
- 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
- Use clues to name a shape.
- Justify with properties.
- Test claims about shapes.
- One counterexample disproves a claim.
Words to know
A feature of a shape.
Give reasons.
An example that disproves.
Sort.
1. Discover: who am I?
I have 4 sides, all equal, and 4 right angles. Who am I? What if I have 4 equal sides but no right angles?
2. Understand: geometric reasoning
- 4 equal sides and 4 right angles: square.
- 4 equal sides, no right angles: rhombus.
- Claim: “every rectangle is a square.” False — a 2 by 3 rectangle is a counterexample.
- Claim: “every square is a rectangle.” True.
Idea 1: Name
3 sides all different: scalene triangle. 4 sides with only one pair parallel: trapezium. 5 equal sides and angles: regular pentagon.
Idea 2: Claim
“Every rectangle is a square” is false.
Idea 3: Counterexample
A 2 by 3 rectangle has four right angles but unequal sides.
Why one example? It shows the claim cannot always be true.
3. Worked examples
Example 1 (I show you): Name
3 sides, all different
- No equal sides.
Answer: scalene triangle
Example 2 (We do it together): Test
Every rectangle is a square
- Find a rectangle that is not.
Answer: false
Example 3 (You try): Counter
Rectangle claim
Show the steps and answer
- A 2 by 3 rectangle.
Answer: 2 by 3 rectangle
Did you know?
Squares are special rectangles, but not all rectangles are squares.
Why this matters
Clear reasons make geometry arguments convincing.
4. Explore: puzzle
You need paper.
- Write three clues.
- Name the shape.
- Test a claim.
- Find a counterexample.
Think about it: what did you notice?
e.g. 2 by 3 rectangle.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- It has ___ sides.
- Its property is ___.
- So it is a ___.
- A counterexample is ___.
Guided practice
Question 1. 4 equal sides, 4 right angles?
Square.
Question 2. 4 equal sides, no right angle?
Rhombus.
Question 3. Is every square a rectangle?
Yes.
Independent practice
Level 1: Start
- 3 sides, all different
- 4 sides, only one pair parallel
- Every rectangle is a square? (yes/no)
- Counterexample to that claim
- 5 equal sides and angles
Level 2: Grow
- 4 right angles, opposite sides equal, not a square
- 4 equal sides but no right angles
- Is every square a rhombus?
Level 3: Stretch
- A counterexample to “all quadrilaterals have a right angle”
- True or false: a shape with 4 equal sides is a square
- Ben says a triangle can have two obtuse angles. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A shape has 4 sides, opposite sides equal, 4 right angles, not all sides equal. Name it.
Show a sample answer
A rectangle.
Another one. A shape has 4 sides; two are parallel and two are not. Name it.
Show a sample answer
trapezium
7. Common mistakes
Fix: One counterexample disproves.
Fix: Look for counterexamples.
Fix: A square also has right angles.
8. Lesson summary and revision card
- I can use clues to name a shape.
- I can justify a classification.
- I can test a claim.
- 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
- Use properties.
- Justify.
- Counterexamples.
11. Next lesson
Next lesson: Metric conversions. 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.
- Name a shape with 3 sides, all different. [2 marks]
- Name a 4-sided shape with only one pair of parallel sides. [2 marks]
- Is every rectangle a square? Explain. [3 marks]
- Name a shape with 5 equal sides and angles. [2 marks]
- Ali says all rhombuses have right angles. Is he right? Explain. [3 marks]
Total: 12 marks.
Marking guide and model answers
- 3 sides [1]; scalene triangle [1]
- 4 sides [1]; trapezium [1]
- No [1]. A rectangle needs only right angles [1]. A 2 by 3 rectangle is a counterexample [1].
- 5 sides [1]; regular pentagon [1]
- No [1]. A slanted rhombus has no right angles [1]. It is a counterexample [1].
Score guide: 11 to 12 = secure. 8 to 10 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.
Printable worksheet
- 3 sides, all different
- 4 sides, one pair parallel
- Every rectangle is a square? (yes/no)
- Counterexample
- 5 equal sides and angles
- 4 equal sides, 4 right angles:
- 4 equal sides, no right angles:
- Only one pair of parallel sides:
- Every rectangle is a square.
- 5 equal sides and angles:
Homework
Write three “who am I” riddles about shapes.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 3 sides, all different | scalene triangle | 2 |
| 4 sides, one pair parallel | trapezium | 2 |
| Every rectangle is a square? (yes/no) | no | 2 |
| Counterexample | 2 by 3 rectangle | 2 |
| 5 equal sides and angles | regular pentagon | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 scalene triangle; 2 trapezium; 3 no; 4 2 by 3 rectangle; 5 regular pentagon.
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 square; 7 rhombus; 8 trapezium; 9 false; 10 regular pentagon.
Independent practice answers. Level 1: 1 scalene triangle; 2 trapezium; 3 no (false); 4 a 2 by 3 rectangle; 5 regular pentagon. Level 2: 1 rectangle; 2 rhombus; 3 yes. Level 3: 1 a slanted rhombus; 2 false; 3 Two obtuse angles are over 180°, but a triangle has only 180°..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Use clues to name a shape | 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 classification | 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 Test a claim about shapes | 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 one example proves a claim. One counterexample disproves.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Name shapes from properties.
- 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 names shapes from clues and tests claims. You do not need to be good at mathematics to help.
- Play 20 questions with shapes.
- Ask: how do you know?
- Look for counterexamples.
- 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: use clues to name a shape; justify a classification; test a claim about shapes; find a counterexample.
Prerequisite knowledge. Lessons 1 to 7.
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 properties do I know? 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 one example proves a claim. One counterexample disproves.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 12-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. Name shapes from properties. 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 » Geometry. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 5.G.B.3 and 5.G.B.4; Cambridge Primary Mathematics Stage 5 and 6 (geometric reasoning). These sources informed the lesson.