Courseversity

Mixed-number operations

Primary EducationGrade 6MathematicsCV-PRI-G06-MATH-FRA-U01-L0845–60 min

You will learn

  • Convert mixed and improper.
  • Add and subtract mixed numbers.
  • Multiply mixed numbers.
  • Divide mixed numbers.

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. Convert between mixed and improper forms.
  2. Add and subtract wholes and fractions.
  3. Regroup when the fraction is too small.
  4. Change to improper fractions to multiply or divide.

Words to know

mixed number
Whole plus fraction.
improper fraction
Top bigger than bottom.
convert
Change form.
regroup
Rename to subtract.

1. Discover: recipe amounts

A recipe needs 1 1/2 cups of flour and 2 1/4 cups of sugar. How much altogether?

2. Understand: mixed numbers

  • Add wholes and fractions: 1 1/2 + 2 1/4 = 3 + 2/4 + 1/4 = 3 3/4.
  • 3 1/3 − 1 1/2 = 10/3 − 3/2 = 20/6 − 9/6 = 11/6 = 1 5/6.
  • 1 1/2 × 2 = 3/2 × 2 = 3.
  • 2 1/2 ÷ 1/2 = 5/2 × 2 = 5.

Idea 1: Add

2 2/3 + 1 1/6: wholes 2 + 1 = 3; fractions 4/6 + 1/6 = 5/6. Total 3 5/6.

Idea 2: Subtract

3 1/3 − 1 1/2: 3 2/6 − 1 3/6. Regroup to 2 8/6 − 1 3/6 = 1 5/6.

Idea 3: Multiply and divide

1 1/2 × 2 = 3/2 × 2 = 3. 2 1/2 ÷ 1/2 = 5/2 × 2 = 5. 7/2 = 3 1/2.

Why improper? It lets us multiply and divide in one step.

Remember: Convert to improper for × and ÷.

3. Worked examples

Example 1 (I show you): Add

2 2/3 + 1 1/6

  1. Wholes: 3.
  2. Fractions: 5/6.

Answer: 3 5/6

Example 2 (We do it together): Subtract

3 1/3 − 1 1/2

  1. Denominator 6.
  2. Regroup.

Answer: 1 5/6

Example 3 (You try): Convert

7/2

Show the steps and answer
  1. 7 ÷ 2 = 3 remainder 1.

Answer: 3 1/2

Did you know?

A mixed number is a whole number added to a fraction.

Why this matters

Mixed numbers appear in measuring, cooking and building.

4. Explore: mixed

You need paper.

  1. Convert if needed.
  2. Do the operation.
  3. Convert back.
  4. Check size.
Think about it: what did you notice?

3 3/4.

5. Practise

Say it in a sentence

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

  • As an improper fraction: ___.
  • The common denominator is ___.
  • I regroup to get ___.
  • The answer is ___.

Guided practice

Question 1. 1 1/2 + 2 1/4?

3 3/4.

Question 2. 1 1/2 × 2?

3.

Question 3. 2 1/2 ÷ 1/2?

5.

Independent practice

Level 1: Start

  1. 2 2/3 + 1 1/6
  2. 3 1/3 − 1 1/2
  3. 1 1/2 × 2
  4. 2 1/2 ÷ 1/2
  5. Write 7/2 as a mixed number

Level 2: Grow

  1. 3/2 as a mixed number
  2. 1 1/4 + 1 1/4
  3. 2 × 1 1/2

Level 3: Stretch

  1. 4 1/4 − 1 1/2
  2. 2 1/2 × 1 1/5
  3. Ben says 7/2 = 3 1/7. What is wrong?

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

6. Apply: everyday mathematics

1 1/2 cups flour and 2 1/4 cups sugar. Total?

Show a sample answer

3 3/4 cups.

Another one. A plank is 3 1/2 m long. Kofi cuts off 1 3/4 m. How long is the rest?

Show a sample answer

1 3/4 m

7. Common mistakes

Mistake: Multiplying wholes and fractions separately.
Fix: Convert to improper first.
Another mistake: Subtracting the smaller fraction from the larger.
Fix: Regroup when the top fraction is smaller.
Another mistake: Multiplying wholes and fractions separately.
Fix: Convert to improper fractions first.

8. Lesson summary and revision card

I can…

  • I can convert mixed and improper forms.
  • I can add and subtract mixed numbers.
  • I can multiply mixed numbers.
  • I can divide mixed numbers.

9. Check and master

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

1. 1 1/2 + 2 1/4 =

2. 2 2/3 + 1 1/6 =

3. 3 1/3 − 1 1/2 =

4. 1 1/2 × 2 =

5. 2 1/2 ÷ 1/2 =

6. 2 1/3 + 1 1/6

7. 4 3/4 − 2 1/4

8. 3 1/2 − 1 3/4

9. 2 1/2 × 2

10. 1 1/3 × 3

11. 3 ÷ 1 1/2

12. 11/4 as a mixed number

Challenge 1. 7/2 as a mixed number:

Challenge 2. 1 1/2 × 2 1/3

Your result appears here after you answer every question.

10. Key takeaways

  • Convert for × and ÷.
  • Regroup to subtract.
  • Check by estimating.

11. Next lesson

Next lesson: Multistep fraction problems. 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. Work out 2 2/3 + 1 1/6. [3 marks]
  2. Work out 3 1/3 − 1 1/2. [3 marks]
  3. Work out 1 1/2 × 2. [2 marks]
  4. Work out 2 1/2 ÷ 1/2. [2 marks]
  5. Ali says 7/2 = 3 1/7. Is he right? Explain. [3 marks]

Total: 13 marks.

Marking guide and model answers
  1. Add wholes 2 + 1 = 3 [1]; 4/6 + 1/6 [1]; 3 5/6 [1]
  2. 3 2/6 − 1 3/6 [1]; regroup to 2 8/6 − 1 3/6 [1]; 1 5/6 [1]
  3. Write as 3/2 × 2 [1]; 3 [1]
  4. 5/2 × 2 [1]; 5 [1]
  5. No [1]. 7 ÷ 2 = 3 remainder 1 [1]. The fraction part is 1/2, so 3 1/2 [1].

Score guide: 12 to 13 = secure. 8 to 11 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.

Printable worksheet

  1. 2 2/3 + 1 1/6
  2. 3 1/3 − 1 1/2
  3. 1 1/2 × 2
  4. 2 1/2 ÷ 1/2
  5. 7/2 as a mixed number
  6. 2 2/3 + 1 1/6
  7. 3 1/3 − 1 1/2
  8. 1 1/2 × 2
  9. 2 1/2 ÷ 1/2
  10. 7/2 as a mixed number

Homework

Calculate 1 3/4 + 2 1/2 and 2 1/4 × 2.

Answer key and marking scheme

Question Expected response Marks
2 2/3 + 1 1/6 3 5/6 2
3 1/3 − 1 1/2 1 5/6 2
1 1/2 × 2 3 2
2 1/2 ÷ 1/2 5 2
7/2 as a mixed number 3 1/2 2
Total 10

Practice answers: Worksheet: 1 3 5/6; 2 1 5/6; 3 3; 4 5; 5 3 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 5/6; 7 1 5/6; 8 3; 9 5; 10 3 1/2.

Independent practice answers. Level 1: 1 3 5/6; 2 1 5/6; 3 3; 4 5; 5 3 1/2. Level 2: 1 1 1/2; 2 2 1/2; 3 3. Level 3: 1 2 3/4; 2 3; 3 7 ÷ 2 = 3 remainder 1, so 3 1/2..

Mastery and remediation

Objective Bronze (Developing) Silver (Proficient) Gold (Advanced)
O1 Convert mixed and improper 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 Add and subtract mixed numbers 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 Multiply mixed numbers 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 Divide mixed numbers 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: Multiplying wholes and fractions separately. Convert to improper first.
  2. Explain another way: use real objects first, then pictures.
  3. Show a simpler example: Mixed to improper.
  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 works with mixed numbers. You do not need to be good at mathematics to help.

  • Measure ingredients like 1 1/2 cups.
  • Ask: do we need to regroup?
  • Convert before multiplying.
  • 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: convert mixed and improper; add and subtract mixed numbers; multiply mixed numbers; divide mixed numbers.

Prerequisite knowledge. Lessons 4–7.

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. Mixed or improper? Do I need to regroup?

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. Multiplying wholes and fractions separately. Convert to improper first.

13–15. Assessment questions, answer keys, marking scheme

Use the quiz and worksheet on this page. The 13-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. Mixed to improper. 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.A.1 and 6.NS.A.1; Cambridge Primary Mathematics Stage 6 (mixed numbers). These sources informed the lesson.

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