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3B · Chapter 10 · Lesson 5

Comparing Fractions — Part 2

Handle harder comparisons and explain why a strategy is valid.

MATH-3B-C10-L05Independent and tutor-supported

Today's mathematical question

Compare 11/12 and 6/9. Write > or < and justify using equal parts.

Learning target

Handle harder comparisons and explain why a strategy is valid.

Before starting: Prior lesson: Comparing Fractions — Part 1. Try the warm-up independently.

Prerequisite · 3 questions

Check earlier skills first

These are earlier ideas you need for today; revisit them if they are difficult.

  1. P1

    Compare 4/5 and 5/6. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 24/30 and 25/30. Compare numerators: 4/5 < 5/6.

    4/55/6
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. P2

    Compare 1/4 and 1/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 3/12 and 1/12. Compare numerators: 1/4 > 1/12.

    1/41/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. P3

    Compare 1/6 and 3/4. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 2/12 and 9/12. Compare numerators: 1/6 < 3/4.

    1/63/4
    Each full bar represents the same-sized whole; shaded parts show the fraction.

Learn and understand

For less obvious comparisons, choose a method you can justify. Renaming fractions with a common denominator is always valid for positive denominators. You can sometimes compare distances to one: 7/8 is one eighth short of a whole, while 5/6 is one sixth short. Since one eighth is smaller than one sixth, 7/8 is closer to one. To compare 5/8 and 2/3, partition both equal wholes into 24 equal pieces: 15/24 and 16/24. The extra twenty-fourth decides the comparison.

Represent it three ways

  1. Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
  2. Pictorial: Place both fractions on the same unit interval or subdivide identical strips into common-sized parts. Name the shared unit before comparing.
  3. Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.

Important vocabulary

benchmark
familiar reference such as 1/2 or 1
equivalence
same fraction amount with different numbers
unit interval
length from zero to one

Fully worked examples and checks

Try each new understanding check before revealing the explanation.

Worked example 1

Explore, calculate, verify

Compare 11/12 and 6/9. Write > or < and justify using equal parts.

11/126/9
Each full bar represents the same-sized whole; shaded parts show the fraction.
  1. Identify the whole, the units, and what the question asks.
  2. Rename as 33/36 and 24/36. Compare numerators: 11/12 > 6/9.
  3. Check that the answer uses the right units and makes sense.

Answer: >

Check your understanding 1

Compare 3/5 and 9/12. Write > or < and justify using equal parts.

Reveal answer and explanation

< — Rename as 36/60 and 45/60. Compare numerators: 3/5 < 9/12.

3/59/12
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 2

Explore, calculate, verify

Compare 2/8 and 1/8. Write > or < and justify using equal parts.

2/81/8
Each full bar represents the same-sized whole; shaded parts show the fraction.
  1. Identify the whole, the units, and what the question asks.
  2. Rename as 2/8 and 1/8. Compare numerators: 2/8 > 1/8.
  3. Check that the answer uses the right units and makes sense.

Answer: >

Check your understanding 2

Compare 1/6 and 7/9. Write > or < and justify using equal parts.

Reveal answer and explanation

< — Rename as 3/18 and 14/18. Compare numerators: 1/6 < 7/9.

1/67/9
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 3

Explore, calculate, verify

Compare 4/5 and 2/7. Write > or < and justify using equal parts.

4/52/7
Each full bar represents the same-sized whole; shaded parts show the fraction.
  1. Identify the whole, the units, and what the question asks.
  2. Rename as 28/35 and 10/35. Compare numerators: 4/5 > 2/7.
  3. Check that the answer uses the right units and makes sense.

Answer: >

Check your understanding 3

Compare 2/3 and 1/5. Write > or < and justify using equal parts.

Reveal answer and explanation

> — Rename as 10/15 and 3/15. Compare numerators: 2/3 > 1/5.

2/31/5
Each full bar represents the same-sized whole; shaded parts show the fraction.

Guided · 5 questions

Guided practice

Keep the model and earlier examples visible while practising.

  1. G1

    Compare 2/3 and 1/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 16/24 and 3/24. Compare numerators: 2/3 > 1/8.

    2/31/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. G2

    Compare 3/12 and 1/9. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 9/36 and 4/36. Compare numerators: 3/12 > 1/9.

    3/121/9
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. G3

    Compare 1/3 and 8/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 4/12 and 8/12. Compare numerators: 1/3 < 8/12.

    1/38/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. G4

    Compare 2/12 and 4/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 2/12 and 4/12. Compare numerators: 2/12 < 4/12.

    2/124/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. G5

    Compare 2/4 and 2/9. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 18/36 and 8/36. Compare numerators: 2/4 > 2/9.

    2/42/9
    Each full bar represents the same-sized whole; shaded parts show the fraction.

Independent · 8 questions

Independent practice

Work these without hints and explain the method used. Write answers on paper.

  1. I1

    Compare 1/8 and 9/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 3/24 and 18/24. Compare numerators: 1/8 < 9/12.

    1/89/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. I2

    Compare 5/8 and 4/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 5/8 and 4/8. Compare numerators: 5/8 > 4/8.

    5/84/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. I3

    Compare 4/10 and 7/9. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 36/90 and 70/90. Compare numerators: 4/10 < 7/9.

    4/107/9
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. I4

    Compare 3/4 and 1/7. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 21/28 and 4/28. Compare numerators: 3/4 > 1/7.

    3/41/7
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. I5

    Compare 4/10 and 1/9. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 36/90 and 10/90. Compare numerators: 4/10 > 1/9.

    4/101/9
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  6. I6

    Compare 2/8 and 11/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 6/24 and 22/24. Compare numerators: 2/8 < 11/12.

    2/811/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  7. I7

    Compare 1/8 and 3/5. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 5/40 and 24/40. Compare numerators: 1/8 < 3/5.

    1/83/5
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  8. I8

    Compare 2/3 and 4/7. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 14/21 and 12/21. Compare numerators: 2/3 > 4/7.

    2/34/7
    Each full bar represents the same-sized whole; shaded parts show the fraction.

Challenge · 2 questions

Challenge and error analysis

Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.

  1. C1

    Two fractions have the same whole and denominator 10: 2/10 and 5/10. Which is larger, and why?

    Reveal explanation

    Answer: 5/10

    The pieces are equal-sized, so five of them exceed two.

  2. C2

    Without converting to decimals, explain why 3/9 is greater than 3/10 for equal-sized wholes.

    Reveal explanation

    Answer: 3/9 > 3/10

    Each piece is larger when the whole is divided into fewer equal parts; taking three larger pieces gives more.

Exit · 3 questions

Exit check

Solve independently, then return on another day to check retention.

  1. E1

    Compare 3/4 and 1/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 6/8 and 1/8. Compare numerators: 3/4 > 1/8.

    3/41/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. E2

    Compare 10/12 and 8/9. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 30/36 and 32/36. Compare numerators: 10/12 < 8/9.

    10/128/9
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. E3

    Compare 2/5 and 2/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 24/60 and 10/60. Compare numerators: 2/5 > 2/12.

    2/52/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
Continue →Complete answer key