You will learn
- Multiply numerators and denominators.
- Simplify products.
- Use area models.
- Find a fraction of a fraction.
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
- Multiply the numerators.
- Multiply the denominators.
- Simplify the product.
- A product of two proper fractions is smaller than each.
Words to know
Answer.
Top.
Bottom.
Lowest terms.
1. Discover: half of three quarters
You have 3/4 of a cake and eat half of it. What fraction of the whole did you eat?
2. Understand: fraction × fraction
- 1/2 × 3/4 = 3/8.
- 2/3 × 3/5 = 6/15 = 2/5.
- 3/4 × 8/9 = 24/36 = 2/3.
- 1/3 of 1/4 = 1/3 × 1/4 = 1/12.
Multiply tops, multiply bottoms, then simplify.
Idea 1: Straight
1/2 × 1/5 = 1/10. 1/4 × 2/3 = 2/12 = 1/6.
Idea 2: Simplify
2/3 × 3/4 = 6/12 = 1/2. 3/5 × 5/6 = 15/30 = 1/2.
Idea 3: Model
4/5 × 5/8 = 20/40 = 1/2. A fraction of a fraction is a smaller piece.
Why top × top? We are counting parts of parts.
3. Worked examples
Example 1 (I show you): Multiply
1/2 × 1/5
- 1 × 1 and 2 × 5.
Answer: 1/10
Example 2 (We do it together): Multiply
2/3 × 3/4
- 6/12.
Answer: 1/2
Example 3 (You try): Multiply
1/4 × 2/3
Show the steps and answer
- 2/12.
Answer: 1/6
Did you know?
You can cancel common factors before multiplying to make it easier.
Why this matters
Multiplying fractions finds a part of a part.
4. Explore: multiply
You need paper.
- Draw a rectangle.
- Shade 3/4 across.
- Shade 1/2 down.
- Count the overlap.
Think about it: what did you notice?
3/8.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The numerator is ___ × ___.
- The denominator is ___ × ___.
- The product is ___.
- In lowest terms it is ___.
Guided practice
Question 1. 1/2 × 3/4?
3/8.
Question 2. 2/3 × 3/5?
2/5.
Question 3. 1/3 × 1/4?
1/12.
Independent practice
Level 1: Start
- 1/2 × 1/5
- 2/3 × 3/4
- 3/5 × 5/6
- 1/4 × 2/3
- 4/5 × 5/8
Level 2: Grow
- 1/3 × 1/2
- 2/5 × 1/4
- 3/4 × 2/3
Level 3: Stretch
- 5/6 × 3/5
- 2/3 × 9/10
- Ben says 1/2 × 1/3 = 2/5 by adding. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
You eat half of 3/4 of a cake. What fraction of the whole?
Show a sample answer
3/8.
Another one. Ana has 3/4 of a cake and eats 1/3 of it. What fraction of the whole cake does she eat?
Show a sample answer
3/4 × 1/3 = 3/12 = 1/4
7. Common mistakes
Fix: Just multiply across.
Fix: Multiply them.
Fix: Divide top and bottom by the common factor.
8. Lesson summary and revision card
- I can multiply numerators and denominators.
- I can simplify products.
- I can use area models.
- I can find a fraction of a fraction.
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
- Multiply across.
- Simplify.
- Area models help.
11. Next lesson
Next lesson: Dividing unit fractions by whole numbers. 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.
- Find 1/2 × 1/5. [2 marks]
- Find 2/3 × 3/4. [3 marks]
- Find 3/5 × 5/6. [3 marks]
- Find 1/4 × 2/3. [3 marks]
- Ali says 4/5 × 5/8 = 20/13. Is he right? Explain. [3 marks]
Total: 14 marks.
Marking guide and model answers
- 1 × 1 = 1 and 2 × 5 = 10 [1]; 1/10 [1]
- 6/12 [1]; divide by 6 [1]; 1/2 [1]
- 15/30 [1]; divide by 15 [1]; 1/2 [1]
- 2/12 [1]; divide by 2 [1]; 1/6 [1]
- No [1]. Denominators are multiplied: 5 × 8 = 40 [1]. 20/40 = 1/2 [1].
Score guide: 12 to 14 = secure. 9 to 11 = nearly there, revise the parts you missed. Under 9 = work through the lesson again with a grown-up.
Printable worksheet
- 1/2 × 1/5
- 2/3 × 3/4
- 3/5 × 5/6
- 1/4 × 2/3
- 4/5 × 5/8
- 1/2 × 3/4 =
- 2/3 × 3/5 =
- 1/3 × 1/4 =
- 3/4 × 8/9 =
- 2/3 × 3/4 =
Homework
Multiply six pairs of fractions.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 1/2 × 1/5 | 1/10 | 2 |
| 2/3 × 3/4 | 1/2 | 2 |
| 3/5 × 5/6 | 1/2 | 2 |
| 1/4 × 2/3 | 1/6 | 2 |
| 4/5 × 5/8 | 1/2 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 1/10; 2 1/2; 3 1/2; 4 1/6; 5 1/2.
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 3/8; 7 2/5; 8 1/12; 9 2/3; 10 1/2.
Independent practice answers. Level 1: 1 1/10; 2 1/2; 3 1/2; 4 1/6; 5 1/2. Level 2: 1 1/6; 2 1/10; 3 1/2. Level 3: 1 1/2; 2 3/5; 3 Multiply tops and bottoms: 1/6..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Multiply numerators and denominators | 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 Simplify products | 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 Use area models | 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 fraction of a fraction | 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: Finding a common denominator. Just multiply across.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Multiply fractions by whole numbers.
- 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 multiplies fractions by fractions. You do not need to be good at mathematics to help.
- Find half of half of a snack.
- Ask: parts of what?
- Draw a rectangle and shade.
- 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: multiply numerators and denominators; simplify products; use area models; find a fraction of a fraction.
Prerequisite knowledge. Lesson 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. What do I multiply on top and on the bottom? Can I simplify?
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. Finding a common denominator. Just multiply across.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 14-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. Multiply fractions by whole numbers. 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 » Number & Operations—Fractions; Grade 5 » Number & Operations in Base Ten. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 5.NF.B.4 and 5.NF.B.6; Cambridge Primary Mathematics Stage 6 (multiplying fractions). These sources informed the lesson.