3B · Chapter 10 · Lesson 1
Equivalent Fractions
Show identical quantities using different equal partitions and number-line positions.
Today's mathematical question
Draw or explain why 4/5 and 16/20 represent the same amount.
Learning target
Show identical quantities using different equal partitions and number-line positions.
Before starting: Chapter 9: equal whole fractions and number lines. 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.
- P1
A same-sized whole is split into 6 equal parts and 1 part is shaded. What fraction is shaded?
Reveal explanation
Answer: 1/6
The denominator counts all 6 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/6.
Each full bar represents the same-sized whole; shaded parts show the fraction. - P2
A same-sized whole is split into 6 equal parts and 1 part is shaded. What fraction is shaded?
Reveal explanation
Answer: 1/6
The denominator counts all 6 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/6.
Each full bar represents the same-sized whole; shaded parts show the fraction. - P3
A same-sized whole is split into 4 equal parts and 3 parts are shaded. What fraction is shaded?
Reveal explanation
Answer: 3/4
The denominator counts all 4 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/4.
Each full bar represents the same-sized whole; shaded parts show the fraction.
Learn and understand
Two fractions are equivalent if they land at exactly the same amount of the same whole. Halves and fourths can show the same region after each half is split into two smaller equal pieces. This creates twice as many chosen pieces AND twice as many total pieces without adding any area. Divide one identical rectangle into two halves and a second into four quarters. Shade one half and two quarters. Their shaded ends line up, even though the number of pieces differs.
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Draw 1/2 on a strip. Draw an identical strip divided into fourths; shade 2/4. Compare the right ends of the shaded regions.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- equivalent fractions
- different names for the same quantity
- partition
- divide into equal parts
- whole
- complete reference amount
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
Draw or explain why 4/5 and 16/20 represent the same amount.
- Identify the whole, the units, and what the question asks.
- Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/5 = 16/20.
- Check that the answer uses the right units and makes sense.
Answer: They are equal
Check your understanding 1
Draw or explain why 4/6 and 16/24 represent the same amount.
Reveal answer and explanation
They are equal — Subdivide each of the 6 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 4/6 = 16/24.
Worked example 2
Explore, calculate, verify
Draw or explain why 2/12 and 6/36 represent the same amount.
- Identify the whole, the units, and what the question asks.
- Subdivide each of the 12 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/12 = 6/36.
- Check that the answer uses the right units and makes sense.
Answer: They are equal
Check your understanding 2
Draw or explain why 4/6 and 12/18 represent the same amount.
Reveal answer and explanation
They are equal — Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/6 = 12/18.
Worked example 3
Explore, calculate, verify
Draw or explain why 3/12 and 9/36 represent the same amount.
- Identify the whole, the units, and what the question asks.
- Subdivide each of the 12 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/12 = 9/36.
- Check that the answer uses the right units and makes sense.
Answer: They are equal
Check your understanding 3
Draw or explain why 2/4 and 8/16 represent the same amount.
Reveal answer and explanation
They are equal — Subdivide each of the 4 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 2/4 = 8/16.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
Draw or explain why 7/8 and 14/16 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 8 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/8 = 14/16.
Each full bar represents the same-sized whole; shaded parts show the fraction. - G2
Draw or explain why 4/6 and 8/12 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 6 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 4/6 = 8/12.
Each full bar represents the same-sized whole; shaded parts show the fraction. - G3
Draw or explain why 7/12 and 14/24 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 12 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 7/12 = 14/24.
Each full bar represents the same-sized whole; shaded parts show the fraction. - G4
Draw or explain why 4/5 and 12/15 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/5 = 12/15.
Each full bar represents the same-sized whole; shaded parts show the fraction. - G5
Draw or explain why 1/5 and 3/15 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 5 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 1/5 = 3/15.
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.
- I1
Draw or explain why 8/12 and 24/36 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 12 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 8/12 = 24/36.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I2
Draw or explain why 1/8 and 3/24 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 1/8 = 3/24.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I3
Draw or explain why 3/6 and 9/18 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 6 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 3/6 = 9/18.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I4
Draw or explain why 4/8 and 12/24 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 4/8 = 12/24.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I5
Draw or explain why 5/6 and 20/24 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 6 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 5/6 = 20/24.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I6
Draw or explain why 2/3 and 6/9 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 3 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 2/3 = 6/9.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I7
Draw or explain why 2/3 and 4/6 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 3 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/3 = 4/6.
Each full bar represents the same-sized whole; shaded parts show the fraction. - I8
Draw or explain why 2/10 and 4/20 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 10 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/10 = 4/20.
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.
- C1
A student changes 2/5 to 4/5 and says the fractions are equivalent. Is this correct? Give the correct denominator if the numerator doubles.
Reveal explanation
Answer: 4/10
No. Both numerator and denominator must be multiplied by 2 to preserve the same portion: 2/5 = 4/10.
- C2
Explain why reducing 6/12 by dividing numerator and denominator by 3 preserves the value.
Reveal explanation
Answer: 2/4
Grouping numerator and denominator into threes gives 2/4; both represent the same share of the same whole.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
Draw or explain why 1/5 and 4/20 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 5 equal parts into 4 pieces. That multiplies shaded parts and total parts by 4, so 1/5 = 4/20.
Each full bar represents the same-sized whole; shaded parts show the fraction. - E2
Draw or explain why 2/4 and 4/8 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 4 equal parts into 2 pieces. That multiplies shaded parts and total parts by 2, so 2/4 = 4/8.
Each full bar represents the same-sized whole; shaded parts show the fraction. - E3
Draw or explain why 5/8 and 15/24 represent the same amount.
Reveal explanation
Answer: They are equal
Subdivide each of the 8 equal parts into 3 pieces. That multiplies shaded parts and total parts by 3, so 5/8 = 15/24.
Each full bar represents the same-sized whole; shaded parts show the fraction.