Problem
Building fractions from number cards
Pick one of the cards 0, 1, 2, 3 as numerator and another as denominator. Neither may be 0. A fraction is under 1 exactly when the numerator is smaller. Count the fractions under 1.
Reasoning · Grade 3-1 Comparing Fraction Sizes
Your answer
Watch out Four cards suggest twelve fractions, but 0 is barred top and bottom, leaving only 1, 2 and 3.
How to solve
Strategy Make a Systematic List — The card set is tiny, so we can list every numerator-denominator pair in an organized way (by numerator) and keep only those where the numerator is smaller than the denominator. We eliminate any pair with a 0 numerator or 0 denominator before counting.
1STEP 1
Translate 'less than 1' into a card rule
Under 1 means the numerator is the smaller card.
a/b ＜ 1 ⇔ a ＜ b
Comparing a fraction to 1 is just comparing the numerator and denominator, a Grade 3 fraction idea.
3.NF.A.3Make A Systematic List2STEP 2
List by numerator = 1
Numerator 1 takes 2 or 3 below: 1/2, 1/3.
1/2, 1/3
Holding the numerator fixed and listing larger denominators makes sure nothing is missed.
3.NF.A.3Make A Systematic ListHolding the numerator at 1 and sweeping the denominator upward catches every fraction with that numerator, in order and with none missed.
Why?
Each fraction has one definite numerator, so sorting by numerator puts every fraction in exactly one sweep and never in two.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
A fraction is below 1 exactly when the pieces taken are fewer than the pieces the whole was cut into, so the numerator alone sets where the sweep may start.
🧱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.
3STEP 3
List by numerator = 2
Numerator 2 leaves only 3: 2/3.
2/3
Only the card 3 is bigger than 2, so just one fraction works here.
3.NF.A.3Make A Systematic List4STEP 4
List by numerator = 3, and discard 0
Numerator 3 has nothing bigger below — none.
3 is the biggest card, so nothing can be below 1 with 3 on top.
3.NF.A.3Eliminate Possibilities5STEP 5
Count the survivors
Altogether that is 3.
1/2, 1/3, 2/3 → 3 fractions
A short organized list is easy to count exactly.
3.OA.A.1Make A Systematic ListAnswer
3 fractions
1/2, 1/3, 2/3
Only fractions with a small numerator (1 or 2) can be less than 1 here, since 3 is the largest card. We found exactly 1/2, 1/3, 2/3 and no others, so 3 is a sensible, small count.
Takeaway
A fraction is less than 1 whenever the top number is smaller than the bottom number, so you just list the small-over-big pairs!
- Translate 'less than 1' into a card rule
- List by numerator = 1
- List by numerator = 2
- List by numerator = 3, and discard 0
- Count the survivors