Reference / Curriculum / 3B / Chapter 10

3B · Chapter 10 · Lesson 4

Comparing Fractions — Part 1

Use equivalent fractions, benchmarks, and common units to justify comparisons.

MATH-3B-C10-L04Independent and tutor-supported

Today's mathematical question

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

Learning target

Use equivalent fractions, benchmarks, and common units to justify comparisons.

Before starting: Prior lesson: Simplifying Fractions. 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

    Simplify 12/18 fully.

    Reveal explanation

    Answer: 2/3

    Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 2/3.

    12/18
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. P2

    Simplify 12/18 fully.

    Reveal explanation

    Answer: 2/3

    Divide numerator and denominator by their greatest common factor 6. The equivalent simplest fraction is 2/3.

    12/18
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. P3

    Simplify 12/20 fully.

    Reveal explanation

    Answer: 3/5

    Divide numerator and denominator by their greatest common factor 4. The equivalent simplest fraction is 3/5.

    12/20
    Each full bar represents the same-sized whole; shaded parts show the fraction.

Learn and understand

Fractions with different numerators and denominators require a common way to compare. You may rename them using equal-sized pieces, compare each with one half, or place both on the same number line. Any method is valid if it preserves the fraction values. Place 1/2 and 2/5 on equal-length strips. Rename one half as 5/10 and two fifths as 4/10. The comparison now uses equal-sized tenths.

Represent it three ways

  1. Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
  2. Pictorial: Draw both strips with identical whole lengths and choose a common subdivision when needed. Explain where the shaded amounts end.
  3. Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.

Important vocabulary

common denominator
same number of equal parts in the whole
benchmark
familiar comparison value
equivalent fractions
same amount

Fully worked examples and checks

Try each new understanding check before revealing the explanation.

Worked example 1

Explore, calculate, verify

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

1/128/12
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 1/12 and 8/12. Compare numerators: 1/12 < 8/12.
  3. Check that the answer uses the right units and makes sense.

Answer: <

Check your understanding 1

Compare 6/10 and 5/6. Write > or < and justify using equal parts.

Reveal answer and explanation

< — Rename as 18/30 and 25/30. Compare numerators: 6/10 < 5/6.

6/105/6
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 2

Explore, calculate, verify

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

1/31/12
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 4/12 and 1/12. Compare numerators: 1/3 > 1/12.
  3. Check that the answer uses the right units and makes sense.

Answer: >

Check your understanding 2

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

Reveal answer and explanation

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

1/34/6
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 3

Explore, calculate, verify

Compare 1/8 and 9/10. Write > or < and justify using equal parts.

1/89/10
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 5/40 and 36/40. Compare numerators: 1/8 < 9/10.
  3. Check that the answer uses the right units and makes sense.

Answer: <

Check your understanding 3

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

Reveal answer and explanation

> — Rename as 5/8 and 1/8. Compare numerators: 5/8 > 1/8.

5/81/8
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 3/8 and 2/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

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

    3/82/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. G2

    Compare 3/10 and 4/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 12/40 and 20/40. Compare numerators: 3/10 < 4/8.

    3/104/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. G3

    Compare 2/4 and 1/10. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 10/20 and 2/20. Compare numerators: 2/4 > 1/10.

    2/41/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. G4

    Compare 2/4 and 10/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 6/12 and 10/12. Compare numerators: 2/4 < 10/12.

    2/410/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. G5

    Compare 1/6 and 9/10. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 5/30 and 27/30. Compare numerators: 1/6 < 9/10.

    1/69/10
    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 5/8 and 9/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 15/24 and 18/24. Compare numerators: 5/8 < 9/12.

    5/89/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. I2

    Compare 5/12 and 1/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 5/12 and 1/12. Compare numerators: 5/12 > 1/12.

    5/121/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. I3

    Compare 2/4 and 1/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 6/12 and 1/12. Compare numerators: 2/4 > 1/12.

    2/41/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. I4

    Compare 9/12 and 2/8. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 18/24 and 6/24. Compare numerators: 9/12 > 2/8.

    9/122/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. I5

    Compare 5/10 and 8/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 30/60 and 40/60. Compare numerators: 5/10 < 8/12.

    5/108/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  6. I6

    Compare 4/5 and 1/4. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 16/20 and 5/20. Compare numerators: 4/5 > 1/4.

    4/51/4
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  7. I7

    Compare 2/4 and 8/10. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 10/20 and 16/20. Compare numerators: 2/4 < 8/10.

    2/48/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  8. I8

    Compare 4/5 and 1/6. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 24/30 and 5/30. Compare numerators: 4/5 > 1/6.

    4/51/6
    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 9: 2/9 and 5/9. Which is larger, and why?

    Reveal explanation

    Answer: 5/9

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

  2. C2

    Without converting to decimals, explain why 3/8 is greater than 3/9 for equal-sized wholes.

    Reveal explanation

    Answer: 3/8 > 3/9

    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 4/5 and 3/4. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 16/20 and 15/20. Compare numerators: 4/5 > 3/4.

    4/53/4
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. E2

    Compare 3/10 and 8/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: <

    Rename as 18/60 and 40/60. Compare numerators: 3/10 < 8/12.

    3/108/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. E3

    Compare 2/3 and 5/12. Write > or < and justify using equal parts.

    Reveal explanation

    Answer: >

    Rename as 8/12 and 5/12. Compare numerators: 2/3 > 5/12.

    2/35/12
    Each full bar represents the same-sized whole; shaded parts show the fraction.
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