Courseversity

Multistep fraction problems

Primary EducationGrade 6MathematicsCV-PRI-G06-MATH-FRA-U01-L0945–60 min

You will learn

  • Plan a multistep solution.
  • Combine operations.
  • Check against the story.
  • Write a clear answer.

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

The lesson in 60 seconds

  1. Break the story into small steps.
  2. Use fraction of, add, subtract or divide.
  3. Check the answer makes sense.
  4. Write a clear sentence answer.

Words to know

multistep
More than one step.
remainder
What is left.
whole
The full amount.
reasonable
Makes sense.

1. Discover: flour in the shop

A baker has 12 kg of flour. She uses 1/3 for bread and 1/4 for cake. How much is left?

2. Understand: multistep problems

  • Bread: 1/3 of 12 = 4 kg. Cake: 1/4 of 12 = 3 kg.
  • Used 4 + 3 = 7 kg, so left 12 − 7 = 5 kg.
  • Check: 1/3 + 1/4 = 7/12 used, 5/12 left, and 5/12 of 12 = 5.
  • A 5 m ribbon cut in 1/2 m pieces gives 5 ÷ 1/2 = 10 pieces.

Idea 1: Steps

2/5 of 30 = 12. Then 1/3 of 12 = 4. Do one step at a time.

Idea 2: Pieces

5 m cut into 1/2 m pieces: 5 ÷ 1/2 = 10 pieces.

Idea 3: Tank

3/4 full, 1/3 of that used: 1/3 × 3/4 = 1/4 used. Left: 3/4 − 1/4 = 1/2.

Why check? A quick check catches wrong steps.

Remember: Break into steps and check.

3. Worked examples

Example 1 (I show you): Two steps

2/5 of 30, then 1/3 of that

  1. 2/5 of 30 = 12.
  2. 1/3 of 12 = 4.

Answer: 4

Example 2 (We do it together): Divide

5 m into 1/2 m pieces

  1. 5 ÷ 1/2 = 5 × 2.

Answer: 10 pieces

Example 3 (You try): Left

3/4 full, 1/4 used

Show the steps and answer
  1. 3/4 − 1/4.

Answer: 1/2 left

Did you know?

Drawing a bar model often shows the steps.

Why this matters

Real problems have several steps, like shopping, cooking and building.

4. Explore: plan

You need paper.

  1. Read the story.
  2. Plan the steps.
  3. Calculate.
  4. Check the answer.
Think about it: what did you notice?

5 kg.

5. Practise

Say it in a sentence

Use these starters when you talk about the lesson. Say them out loud.

  • First I find ___.
  • Then I ___.
  • The answer is ___.
  • It makes sense because ___.

Guided practice

Question 1. 1/3 of 12?

4.

Question 2. Total used?

7 kg.

Question 3. Left?

5 kg.

Independent practice

Level 1: Start

  1. 1/4 of 12
  2. 2/5 of 30, then 1/3 of that
  3. 5 m cut into 1/2 m pieces
  4. A tank 3/4 full; 1/3 of that is used. How much used?
  5. Left in the tank

Level 2: Grow

  1. 1/3 of 18
  2. 2 m cut into 1/4 m pieces
  3. 3/4 of 8

Level 3: Stretch

  1. 1/2 of 3/4 of 16
  2. A tank is 2/3 full. 1/4 of that is used. How much used?
  3. Ben says 2/5 of 30, then 1/3 of that is 12. What is wrong?

Answers are in the answer key below. Try first, then check.

6. Apply: everyday mathematics

12 kg flour, 1/3 for bread, 1/4 for cake. Left?

Show a sample answer

5 kg.

Another one. A farm has 40 hens. 1/4 are brown. Of the rest, 1/3 are white. How many are white?

Show a sample answer

10 hens

7. Common mistakes

Mistake: Stopping after step one.
Fix: Finish every step.
Another mistake: Stopping after step one.
Fix: Re-read what is asked.
Another mistake: Using the wrong whole.
Fix: Find 1/3 of the new amount.

8. Lesson summary and revision card

I can…

  • I can plan steps.
  • I can combine operations.
  • I can check the story.
  • I can write a clear answer.

9. Check and master

Answer all 14 questions. You will see feedback after each one.

1. 1/4 of 12 kg is:

2. 2/5 of 30, then 1/3 of that:

3. 5 m cut into 1/2 m pieces:

4. A tank 3/4 full uses 1/3 of that:

5. Left in the tank:

6. 3/4 of 20, then 1/5 of that

7. 8 m ribbon cut into 1/4 m pieces

8. A class of 30 has 2/5 girls. How many boys?

9. Ama has 40 sweets, gives away 1/4, then eats 1/3 of the rest. How many left?

10. A tank is 5/6 full. 1/2 of that is used. How much of the tank is used?

11. A cake has 12 slices. 1/4 are eaten, then 1/3 of the rest are given away. How many given?

12. Kofi reads 1/3 of a 90-page book on Monday and 1/2 of the rest on Tuesday. Pages read?

Challenge 1. 12 kg, 1/3 bread, 1/4 cake; flour left:

Challenge 2. A shop has 60 kg of rice. It sells 1/3 on Monday and 1/4 of the rest on Tuesday. How much is left?

Your result appears here after you answer every question.

10. Key takeaways

  • Plan before calculating.
  • Do each step.
  • Check the story.

11. Next lesson

Next lesson: Decimal place value. 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.

  1. Find 1/4 of 12. [1 mark]
  2. Find 2/5 of 30, then 1/3 of that. [3 marks]
  3. A 5 m rope is cut into 1/2 m pieces. How many pieces? [2 marks]
  4. A tank is 3/4 full. 1/3 of that is used. What fraction is used? [2 marks]
  5. Ali says 3/4 of the tank is left. Is he right? Explain. [3 marks]

Total: 11 marks.

Marking guide and model answers
  1. 12 ÷ 4 = 3 [1]
  2. 2/5 of 30 = 12 [1]; 1/3 of 12 [1]; 4 [1]
  3. 5 ÷ 1/2 = 5 × 2 [1]; 10 pieces [1]
  4. 1/3 × 3/4 [1]; 1/4 used [1]
  5. No [1]. 1/4 was used [1]. 3/4 − 1/4 = 1/2 is left [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

  1. 1/4 of 12
  2. 2/5 of 30 then 1/3 of that
  3. 5 m cut into 1/2 m pieces
  4. Tank 3/4 full, 1/3 used
  5. Left in the tank
  6. 1/4 of 12
  7. 2/5 of 30, then 1/3 of that
  8. 5 m into 1/2 m pieces
  9. 3/4 full, 1/3 of that used: used
  10. Left in the tank

Homework

Make up a recipe problem with two fractions.

Answer key and marking scheme

Question Expected response Marks
1/4 of 12 3 2
2/5 of 30 then 1/3 of that 4 2
5 m cut into 1/2 m pieces 10 2
Tank 3/4 full, 1/3 used 1/4 2
Left in the tank 1/2 2
Total 10

Practice answers: Worksheet: 1 3; 2 4; 3 10; 4 1/4; 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; 7 4; 8 10; 9 1/4; 10 1/2.

Independent practice answers. Level 1: 1 3; 2 4; 3 10 pieces; 4 1/4; 5 1/2. Level 2: 1 6; 2 8 pieces; 3 6. Level 3: 1 6; 2 1/6; 3 He stopped after step one. 1/3 of 12 is 4..

Mastery and remediation

Objective Bronze (Developing) Silver (Proficient) Gold (Advanced)
O1 Plan a multistep solution 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 Combine operations 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 Check against the story 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 Write a clear answer 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

  1. Find the misconception: Stopping after step one. Finish every step.
  2. Explain another way: use real objects first, then pictures.
  3. Show a simpler example: One-step fraction problems.
  4. Guided practice: work through one example together and let the learner try the next one.
  5. Reassess: use the quiz again or the worksheet.
  6. Move on when the learner shows Silver-level evidence independently.

For parents and guardians

In this lesson your child solves problems with more than one step. You do not need to be good at mathematics to help.

  • Use shopping or cooking problems.
  • Ask: what do we find first?
  • Draw a bar.
  • 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: plan a multistep solution; combine operations; check against the story; write a clear answer.

Prerequisite knowledge. Lessons 4–8.

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 is asked? What is the first step?

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. Stopping after step one. Finish every step.

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. One-step fraction problems. 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 6 » Ratios & Proportional Relationships; Grade 6 » The Number System. 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.6 and 6.NS.A.1; Cambridge Primary Mathematics Stage 6 (problem solving with fractions). These sources informed the lesson.

Keep learning

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