Problem
Same numerator: smaller denominator is larger
Order the three fractions , , and from greatest to least.
Fractions
Your answer
Watch out It's easy to think a bigger denominator means a bigger fraction (like comparing whole numbers), which flips the order to , , — but for same-numerator fractions, the smaller denominator makes the bigger piece.
How to solve
Strategy Organize Information in More Ways — Rewrite each fraction as 1 plus a unit fraction. Then the comparison becomes comparing unit fractions with the same numerator (1), where the smaller denominator means the larger value.
1STEP 1
Split each fraction into 1 plus a small part
Each fraction's top is one more than its bottom, so it's 1 plus a unit fraction: =1+, =1+, =1+.
Each fraction is just a little over 1, so what matters is which 'little extra piece' is biggest.
3.NF.A.3Organize Information In More Ways2STEP 2
Compare the unit fractions by their denominators
Fractions with the same numerator grow as the denominator shrinks, so is greatest, next, least.
Cutting a whole into fewer pieces makes each piece bigger, so is the biggest extra piece.
3.NF.A.3Look For A PatternWhen two unit fractions share the numerator 1, the one with the smaller denominator is the larger fraction, so is bigger than .
Why?
Each of these is one equal piece of the very same whole, where cuts that whole into 143 pieces and cuts it into 278 pieces.
Why?
Cutting the same whole into fewer equal pieces leaves each piece holding more, and 143 pieces is fewer than 278.
Why?
The equal pieces together make up exactly one whole with nothing left over, so when fewer of them must fill that same whole, each single piece has to be bigger.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Write the fractions in order
The biggest extra piece makes the biggest fraction. So is greatest, is next, and is least.
Order the wholes-plus-pieces by the size of the extra piece.
3.NF.A.3Look For A PatternAnswer
, ,
All three are a tiny bit more than 1. is about 1.007, is about 1.0036, and is about 1.0028, which matches the order greatest to least.
Takeaway
This only needs the Grade 3 idea that same-numerator fractions get bigger as the denominator gets smaller!
- Split each fraction into 1 plus a small part
- Compare the unit fractions by their denominators
- Write the fractions in order
Where next?
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Step back and solidifysuggested