You will learn
- Partition a shape into equal parts.
- Name halves, thirds and quarters of a shape.
- See that a shape can be cut in more than one way.
- Check that parts are equal.
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
- Partition means divide into parts.
- Equal parts of 2 are halves, of 3 are thirds, of 4 are quarters.
- A shape can be cut in more than one way.
- Check the parts are equal.
Words to know
Divide into parts.
One of 2 equal parts.
One of 3 equal parts.
One of 4 equal parts.
1. Discover: sharing a cloth
Two friends share a square cloth. How can they cut it so each gets the same? Is there more than one way?
2. Understand: partitioning shapes
To partition a shape is to cut it into parts. Equal parts have the same size and shape.
- A rectangle cut across the middle makes 2 halves.
- A rectangle cut into 3 same strips makes thirds.
- A square folded twice makes 4 quarters.
A square can be cut in half in more than one way: along a side fold or along a diagonal. Each way gives two equal halves.
Idea 1: Halves, thirds, quarters
To partition a shape is to divide it into parts. A rectangle cut into 2 equal parts gives halves. Into 3 equal parts gives thirds. Into 4 equal parts gives quarters.
Idea 2: More than one way
A square can be cut in half by a line down the middle, across the middle, or along a corner-to-corner line.
Idea 3: Check
Lay one part on top of another. If they match, the parts are equal.
Why check? Fractions need equal parts.
3. Worked examples
Example 1 (I show you): Name the parts
A rectangle is cut into 2 equal parts. What are they called?
- Two equal parts are halves.
Answer: Halves
Example 2 (We do it together): Cuts
How many cuts make a bar into 3 equal strips?
- Two cuts make three strips.
Answer: 2 cuts
Example 3 (You try): Check
How do we check parts are equal?
Show the steps and answer
- Lay them on top of each other.
Answer: They match
Did you know?
When you fold paper in half twice, you get four quarters.
Why this matters
Partitioning is how we make fair shares of shapes.
4. Explore: fold and cut
You need paper squares and scissors.
- Fold a square into 2 equal parts and cut.
- Fold another square into 4 equal parts.
- Cut one square along a diagonal into 2 triangles.
- Lay the parts on each other to check.
Think about it: what did you notice?
Each way gives equal halves or quarters.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- I cut the shape into ___ equal parts.
- Each part is a ___.
- I can cut it another way by ___.
- I check by ___.
Guided practice
Question 1. A square cut in 2 equal parts: each part is a…
half.
Question 2. A rectangle in 3 equal strips: each part is a…
third.
Question 3. A square folded twice: each part is a…
quarter.
Independent practice
Level 1: Start
- Draw a rectangle cut into halves.
- Draw a square cut into quarters.
- Draw a rectangle cut into thirds.
- Can a square be halved in two ways?
- Name one part of a shape cut in 4 equal parts.
Level 2: Grow
- A cake is cut into 2 equal parts. What are they called?
- A bar is cut into 4 equal parts. What are they called?
- How many pieces do 2 cuts make in a bar?
Level 3: Stretch
- Show two ways to cut a square into 4 equal parts.
- A circle is cut in half, then each half is cut in half. What are the parts?
- Ali cuts a rectangle into 3 parts of different widths and says thirds. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
Two children share a rectangular farm equally. How many equal parts?
Show a sample answer
2 equal parts (halves).
Another one. Fold a paper square into 4 equal parts. Describe the folds.
Show a sample answer
Fold in half, then fold in half again.
7. Common mistakes
Fix: Check that each part matches.
Fix: Check that parts match.
Fix: Try other cuts that make equal parts.
8. Lesson summary and revision card
- I can partition a shape into equal parts.
- I can name halves, thirds and quarters.
- I can show more than one way.
- I can check the parts are equal.
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
- Partition into equal parts.
- Halves, thirds and quarters.
- Check by laying parts together.
11. Next lesson
Next lesson: Introduction to symmetry. 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.
- A rectangle is cut into 2 equal parts. What are the parts called? [1 mark]
- Draw or describe a square cut into 4 equal parts. [2 marks]
- How many cuts make 3 equal strips? [1 mark]
- Show a second way to cut a square into halves. [2 marks]
- Ben cuts a bar into 4 parts of different sizes and calls them quarters. Is he right? Explain. [3 marks]
Total: 9 marks.
Marking guide and model answers
- Halves [1]
- 4 parts [1]; equal [1]
- 2 [1]
- A different cut [1]; equal parts [1]
- No [1]. Quarters must be equal [1]. Lay them on top of each other to check [1].
Score guide: 8 to 9 = secure. 6 to 7 = nearly there, revise the parts you missed. Under 6 = work through the lesson again with a grown-up.
Printable worksheet
- Rectangle cut into 2 equal parts: each is a …
- Square cut into 4 equal parts: each is a …
- Rectangle cut into 3 equal strips: each is a …
- Can a square be halved two ways? (yes/no)
- How to check parts are equal?
- 2 equal parts are called ___.
- 4 equal parts are called ___.
- 3 equal parts are called ___.
- Can a square be cut in half in more than one way?
- How do we check parts are equal?
Homework
Fold a paper square into halves, then quarters. Show it to a family member.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Rectangle cut into 2 equal parts: each is a … | half | 2 |
| Square cut into 4 equal parts: each is a … | quarter | 2 |
| Rectangle cut into 3 equal strips: each is a … | third | 2 |
| Can a square be halved two ways? (yes/no) | yes | 2 |
| How to check parts are equal? | lay them on top of each other | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 half; 2 quarter; 3 third; 4 yes; 5 lay them on top of each other.
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 halves; 7 quarters; 8 thirds; 9 Yes; 10 Lay them on top of each other.
Independent practice answers. Level 1: 1 a rectangle with one cut making 2 equal parts; 2 a square with a cross making 4 equal parts; 3 a rectangle with 2 cuts making 3 equal strips; 4 yes; 5 one quarter. Level 2: 1 halves; 2 quarters; 3 3. Level 3: 1 For example a cross, or three vertical lines; 2 quarters; 3 Thirds must be equal parts..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Partition a shape into equal parts | 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 Name halves, thirds and quarters of 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 |
| O3 See that a shape can be cut in more than one way | 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 Check that parts are equal | 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: Cutting into parts of different sizes. Check that each part matches.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Fold a rectangle in half first and open it.
- 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 cuts shapes into equal parts and names halves, thirds and quarters. You do not need to be good at mathematics to help.
- Fold paper into halves, thirds and quarters.
- Cut a chapati into equal parts.
- Ask: how do you know the parts are equal?
- 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: partition a shape into equal parts; name halves, thirds and quarters of a shape; see that a shape can be cut in more than one way; check that parts are equal.
Prerequisite knowledge. Unit 5: fractions.
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 squares and scissors. 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. How many equal parts? What are they called? Are they equal?
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. Cutting into parts of different sizes. Check that each part matches.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 9-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. Fold a rectangle in half first and open it. 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 2 » Geometry. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 2.G.A.3 (partition shapes); Cambridge Primary Mathematics Stage 2 (geometry). These sources informed the lesson.