Problem
Compare the sizes of fractions
Eight fraction cards sit along the bottom of a knockout bracket. Neighbouring pairs play and the larger one moves up. Eight become four, four become two, two become one. Fill in all seven empty boxes.
Reasoning · Grade 5-1 Fraction Calculations
Your answer
Watch out A big denominator looks small, but if the numerator grows too it can be larger.
How to solve
Strategy Organize Information in More Ways — Each pair has been chosen so that one particular way of re-writing it makes the comparison obvious. Rather than forcing every pair through one big common denominator, I look at each pair and ask which re-organisation it is begging for: is one of them exactly one half? do they share a numerator? can they be made to share a numerator? do numerator and denominator differ by the same amount? Choosing the right lens per pair turns seven hard comparisons into seven one-line ones, and the bracket itself is the diagram that keeps track of who has won.
1STEP 1
Round 1, left pair: 5/12 versus 8/16 - compare against one half
5/12 is under one half, so 8/16 goes up.
8/16=1/2, 5 × 2 = 10 ＜ 12 → 5/12 ＜ 1/2
One half is the easiest landmark there is: doubling the numerator and comparing it with the denominator tells you instantly which side of a half a fraction is on, without any common denominator.
4.NF.A.2Solve An Easier Related Problem2STEP 2
Round 1, second pair: 8/11 versus 20/23 - equal gap between numerator and denominator
By shortfall from 1, 20/23 is larger.
8/11=1-3/11, 20/23=1-3/23, 3/23 ＜ 3/11
When two fractions are the same number of pieces away from a whole, the one made of smaller pieces is closer to the whole - and pieces get smaller as the denominator grows, which is Grade 3 fraction sense.
4.NF.B.3Look For A Pattern3STEP 3
Round 1, third pair: 26/34 versus 52/64 - make the numerators match
Matching numerators, 52/64 is larger.
26/34=(26×2)/(34×2)=52/68 ＜ 52/64
Matching numerators is often far cheaper than matching denominators - here a single doubling does it - and once the numerators agree, the fraction with the smaller denominator always wins.
4.NF.A.1Organize Information In More WaysMaking the two numerators match lets the fractions be compared by their denominators alone.
Why?
Multiplying a fraction's top and bottom by the same number renames it without changing its size, so the numerators can be forced to agree.
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
Why?
With the same number of pieces taken from each, the fraction cut into fewer pieces has bigger pieces and is therefore larger.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
4STEP 4
Round 1, right pair: 13/21 versus 13/19 - the numerators already match
Same numerator, so 13/19 wins.
13/21 ＜ 13/19
Picture two identical chocolate bars, one cut into 21 strips and one into 19: taking 13 strips from the 19-cut bar clearly gives you more chocolate.
3.NF.A.3Draw A Diagram5STEP 5
Round 2, left: 8/16 versus 20/23 - compare against one half again
Round 2 on the left sends up 20/23.
8/16=1/2 ＜ 20/23 (20 × 2 = 40 ＞ 23)
The same one-half landmark settles this pair as fast as it settled the first one - a benchmark you can reuse is worth more than a trick that works once.
4.NF.A.2Solve An Easier Related Problem6STEP 6
Round 2, right: 52/64 versus 13/19 - make the numerators match again
Round 2 on the right sends up 52/64.
13/19=(13×4)/(19×4)=52/76 ＜ 52/64
Because 52 = 13 x 4, the numerator of the other fraction is already a multiple of 13 - noticing that saves you from ever finding a common denominator of 64 and 19.
4.NF.A.1Organize Information In More Ways7STEP 7
The final: 20/23 versus 52/64 - equal gap once more
In the final 20/23 wins.
52/64=13/16=1-3/16, 20/23=1-3/23, 3/23 ＜ 3/16
Simplifying first is what exposes the shared gap of 3; the same 'how far from a whole' comparison used in round 1 then finishes the bracket.
4.NF.B.3Look For A PatternAnswer
20/23
8/16, 20/23, 52/64, 13/19
As decimals the eight fractions are about 0.417, 0.500, 0.727, 0.870, 0.765, 0.813, 0.619 and 0.684, so the round-1 winners really are 0.500, 0.870, 0.813 and 0.684; the round-2 winners are 0.870 and 0.813; and the champion is 0.870, which is 20/23 - the largest of all eight, as a knockout bracket must produce. Every box holds a fraction copied unchanged from directly below it, and each fraction written is between 0 and 1, matching the sizes on the cards.
Takeaway
Look at each pair before you calculate - one is half, one shares a numerator, one is the same distance from a whole - and every comparison becomes a single line.
- Round 1, left pair: 5/12 versus 8/16 - compare against one half
- Round 1, second pair: 8/11 versus 20/23 - equal gap between numerator and denominator
- Round 1, third pair: 26/34 versus 52/64 - make the numerators match
- Round 1, right pair: 13/21 versus 13/19 - the numerators already match
- Round 2, left: 8/16 versus 20/23 - compare against one half again
- Round 2, right: 52/64 versus 13/19 - make the numerators match again
- The final: 20/23 versus 52/64 - equal gap once more