Equivalent fractions: one whole, different partitions
Why do 3/4 and 6/8 name the same number?
Align fraction bars, split every part equally, and see why multiplying numerator and denominator by the same whole number preserves the shaded length.
Learning goals
- Compare two fractions using bars of the same length.
- Explain why multiplying numerator and denominator by the same non-zero whole number gives an equivalent fraction.
- Generate and verify equivalent fractions using multiplication.
Before you begin
- Know that the denominator counts equal parts in one whole.
- Use whole-number multiplication and read fractions on a bar or number line.
Equivalent fraction bars
- 1Fix the whole
- 2Split each part
- 3Change multiplier
- 4Write the equation
Observe, then explain
Work through four steps. Each button moves the interactive model to the matching observation state.
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Both bars must have the same total length. The upper bar has denominator equal parts, with numerator of them shaded.
Your task: Choose 1/2 and confirm the shading covers exactly half the bar.
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Split every original part into 2 equal pieces. The whole now has twice as many parts, and every shaded part also splits, so the shaded count doubles too.
Your task: Compare the right endpoints of 1/2 and 2/4.
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The multiplier can be 2, 3, or another whole number. If every original part is split in the same way, the shaded boundary does not move.
Your task: Start with 2/3, set the multiplier to 3, and read the new fraction.
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The new numerator and denominator are each multiplied by the same factor. Each piece becomes smaller while the piece count grows in the same ratio, so the value stays unchanged.
Your task: Complete your own example: “__/__ = (__×__)/(__×__) = __/__”.
Build an explanation
The denominator changes piece size
More equal parts in the same whole means smaller individual parts. Comparing denominators alone does not compare fraction values.
Another way to multiply by 1
For non-zero k, k/k equals 1. Multiplying a/b by k/k preserves its value while multiplying both terms by k. The bars make that symbolic rule visible as splitting.
Equal value, different notation
The numerators and denominators in 2/3 and 6/9 differ, but both reach the same point on the same whole. Equivalent means equal in value, not identical in notation.
Check your understanding
Printable worksheet
Complete the worksheet first. Answers and explanations are on a separate page.
Mathematics
Equivalent fractions: one whole, different partitions
Why do 3/4 and 6/8 name the same number?
- What must be true before fraction bars can show equivalence?
A. The bars represent wholes of the same size B. Both denominators must match C. The shaded cell counts must match
- Split each part of 2/3 into 2. Which equivalent fraction results?
A. 3/4 B. 4/6 C. 2/6
- Why is 3/10 not equivalent to 3/5 when only the denominator is doubled?
A. The pieces got smaller but the shaded count did not increase B. 10 is not even C. Fractions cannot contain 10
- Which equation is always true for a non-zero whole number k?
A. a/b=(a+k)/(b+k) B. a/b=(a×k)/(b×k) C. a/b=(a×k)/b
Answer page
Equivalent fractions: one whole, different partitions
- The bars represent wholes of the same size
Shaded lengths are directly comparable only when the wholes match; equivalent fractions may use different denominators and cell counts.
- 4/6
Multiply both terms by 2: 2×2=4 and 3×2=6, so 2/3=4/6.
- The pieces got smaller but the shaded count did not increase
When the whole has twice as many smaller parts, every shaded original part must also split. The equivalent result is 6/10.
- a/b=(a×k)/(b×k)
Multiplying both terms by the same non-zero number multiplies the fraction by k/k=1, preserving its value.