Reasoning · Grade 3-1 Applying Fractions
Problem
Under 1/2 needs the denominator past twice the numerator.
If you cut a whole into b parts and take a of them, you have less than half only when there are more than twice as many parts as you took.
3.NF.A.3Guess And CheckA fraction is below one half exactly when the whole was cut into more than twice as many pieces as you took.
Taking the pieces you hold and doubling them tells you what a half of that cut would be, so comparing that double with the bottom number settles it.
Doubling the top and doubling the bottom would leave the fraction unchanged, which is why comparing twice the top with the bottom is a fair test.
Numerator 1 allows denominators 3 to 8: 6.
Holding the numerator fixed and sweeping the denominator from its smallest allowed value to its largest catches every fraction in order.
3.NF.A.3Make A Systematic ListNumerator 2 allows 5 to 7: 3.
Same sweep with a bigger numerator: the window of allowed denominators is narrower because the sum limit bites sooner.
3.NF.A.3Make A Systematic ListAt numerator 3 the conditions clash — none.
Once the 'more than double' window opens past the sum limit, no denominator can fit, so we can safely stop.
3.NF.A.3Guess And CheckTogether that is 6 + 3 = 9.
Adding the per-numerator counts gives the grand total because the lists never overlap.
3.NBT.A.2Make A Systematic ListA fraction beats 1/2 only when the bottom is more than double the top - list numerator by numerator and you find just 9, using Grade 3 fraction-comparing!