Reference / Curriculum / 3B / Chapter 10

3B · Chapter 10 · Lesson 7

Adding and Subtracting Fractions — Part 2

Solve related-denominator fraction operations and word problems with models.

MATH-3B-C10-L08Independent and tutor-supported

Today's mathematical question

Find 1/3 + 2/6; rename parts and simplify.

Learning target

Solve related-denominator fraction operations and word problems with models.

Before starting: Prior lesson: Adding and Subtracting 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

    Calculate 2/6 − 1/6. Give the answer in simplest form.

    Reveal explanation

    Answer: 1/6

    Both fractions use 6ths. 2 − 1 pieces make 1/6 of a whole after simplification.

    2/61/6
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. P2

    Calculate 1/6 + 3/6. Give the answer in simplest form.

    Reveal explanation

    Answer: 2/3

    Both fractions use 6ths. 1 + 3 pieces make 2/3 of a whole after simplification.

    1/63/6
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. P3

    Calculate 3/5 − 2/5. Give the answer in simplest form.

    Reveal explanation

    Answer: 1/5

    Both fractions use 5ths. 3 − 2 pieces make 1/5 of a whole after simplification.

    3/52/5
    Each full bar represents the same-sized whole; shaded parts show the fraction.

Learn and understand

Sometimes fractional pieces have different names even though they come from the same whole. Convert one half into two fourths before adding one fourth, or rename thirds as sixths to combine with sixths. The model shows why the common denominator is necessary: you cannot count mixed-size pieces as though they were the same unit. For 1/2 + 1/4, divide each half into two equal quarters. Two quarters plus one quarter gives 3/4. Ask why adding the denominators would describe a different whole.

Represent it three ways

  1. Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
  2. Pictorial: Use two equal whole bars: subdivide larger fractional pieces until every piece has the same size, then compute and simplify.
  3. Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.

Important vocabulary

related denominators
one denominator is a multiple of another
equivalent fractions
equal quantities
common unit
same-size fraction piece

Fully worked examples and checks

Try each new understanding check before revealing the explanation.

Worked example 1

Explore, calculate, verify

Find 1/3 + 2/6; rename parts and simplify.

1/32/6
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. First rename 1/3 as 2/6. Add 2/6 to get 4/6 = 2/3.
  3. Check that the answer uses the right units and makes sense.

Answer: 2/3

Check your understanding 1

Find 1/3 + 3/6; rename parts and simplify.

Reveal answer and explanation

5/6 — First rename 1/3 as 2/6. Add 3/6 to get 5/6 = 5/6.

1/33/6
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 2

Explore, calculate, verify

Find 1/2 + 1/4; rename parts and simplify.

1/21/4
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. First rename 1/2 as 2/4. Add 1/4 to get 3/4 = 3/4.
  3. Check that the answer uses the right units and makes sense.

Answer: 3/4

Check your understanding 2

Find 2/5 + 2/10; rename parts and simplify.

Reveal answer and explanation

3/5 — First rename 2/5 as 4/10. Add 2/10 to get 6/10 = 3/5.

2/52/10
Each full bar represents the same-sized whole; shaded parts show the fraction.

Worked example 3

Explore, calculate, verify

Find 1/2 + 3/4; rename parts and simplify.

1/23/4
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. First rename 1/2 as 2/4. Add 3/4 to get 5/4 = 5/4.
  3. Check that the answer uses the right units and makes sense.

Answer: 5/4

Check your understanding 3

Find 1/5 + 1/10; rename parts and simplify.

Reveal answer and explanation

3/10 — First rename 1/5 as 2/10. Add 1/10 to get 3/10 = 3/10.

1/51/10
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

    Find 2/5 + 3/10; rename parts and simplify.

    Reveal explanation

    Answer: 7/10

    First rename 2/5 as 4/10. Add 3/10 to get 7/10 = 7/10.

    2/53/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. G2

    Find 1/2 + 2/4; rename parts and simplify.

    Reveal explanation

    Answer: 1

    First rename 1/2 as 2/4. Add 2/4 to get 4/4 = 1.

    1/22/4
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. G3

    Find 2/3 + 5/6; rename parts and simplify.

    Reveal explanation

    Answer: 3/2

    First rename 2/3 as 4/6. Add 5/6 to get 9/6 = 3/2.

    2/35/6
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. G4

    Find 2/5 + 5/10; rename parts and simplify.

    Reveal explanation

    Answer: 9/10

    First rename 2/5 as 4/10. Add 5/10 to get 9/10 = 9/10.

    2/55/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. G5

    Find 4/5 + 4/10; rename parts and simplify.

    Reveal explanation

    Answer: 6/5

    First rename 4/5 as 8/10. Add 4/10 to get 12/10 = 6/5.

    4/54/10
    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

    Find 2/4 + 4/8; rename parts and simplify.

    Reveal explanation

    Answer: 1

    First rename 2/4 as 4/8. Add 4/8 to get 8/8 = 1.

    2/44/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. I2

    Find 2/4 + 3/8; rename parts and simplify.

    Reveal explanation

    Answer: 7/8

    First rename 2/4 as 4/8. Add 3/8 to get 7/8 = 7/8.

    2/43/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. I3

    Find 1/5 + 9/10; rename parts and simplify.

    Reveal explanation

    Answer: 11/10

    First rename 1/5 as 2/10. Add 9/10 to get 11/10 = 11/10.

    1/59/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  4. I4

    Find 1/4 + 5/8; rename parts and simplify.

    Reveal explanation

    Answer: 7/8

    First rename 1/4 as 2/8. Add 5/8 to get 7/8 = 7/8.

    1/45/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  5. I5

    Find 3/4 + 3/8; rename parts and simplify.

    Reveal explanation

    Answer: 9/8

    First rename 3/4 as 6/8. Add 3/8 to get 9/8 = 9/8.

    3/43/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  6. I6

    Find 1/3 + 5/6; rename parts and simplify.

    Reveal explanation

    Answer: 7/6

    First rename 1/3 as 2/6. Add 5/6 to get 7/6 = 7/6.

    1/35/6
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  7. I7

    Find 3/4 + 7/8; rename parts and simplify.

    Reveal explanation

    Answer: 13/8

    First rename 3/4 as 6/8. Add 7/8 to get 13/8 = 13/8.

    3/47/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  8. I8

    Find 1/3 + 1/6; rename parts and simplify.

    Reveal explanation

    Answer: 1/2

    First rename 1/3 as 2/6. Add 1/6 to get 3/6 = 1/2.

    1/31/6
    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

    Without adding denominators, explain how to find 1/6 + 1/12.

    Reveal explanation

    Answer: 3/12

    Rename 1/6 as 2/12; then add 2/12 + 1/12 = 3/12.

  2. C2

    A student calculates 4/12 − 1/12 as 3/11. Find and explain the correct difference.

    Reveal explanation

    Answer: 3/12

    The number of equal parts in the whole stays 12; only the count of selected parts changes from 4 to 3.

Exit · 3 questions

Exit check

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

  1. E1

    Find 1/5 + 3/10; rename parts and simplify.

    Reveal explanation

    Answer: 1/2

    First rename 1/5 as 2/10. Add 3/10 to get 5/10 = 1/2.

    1/53/10
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  2. E2

    Find 2/4 + 1/8; rename parts and simplify.

    Reveal explanation

    Answer: 5/8

    First rename 2/4 as 4/8. Add 1/8 to get 5/8 = 5/8.

    2/41/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
  3. E3

    Find 2/4 + 5/8; rename parts and simplify.

    Reveal explanation

    Answer: 9/8

    First rename 2/4 as 4/8. Add 5/8 to get 9/8 = 9/8.

    2/45/8
    Each full bar represents the same-sized whole; shaded parts show the fraction.
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