Problem
A proper fraction nears 1 when the gap between top and bottom is small, so 4/5 and 7/9 lead.
Comparing fractions to the whole 1 is a Grade 4 idea: the smaller the missing piece, the closer the fraction is to 1.
4.NF.A.2Look For A PatternA proper fraction sits closest to 1 when its numerator falls only a little short of its denominator.
A fraction cuts the whole into as many equal pieces as the denominator says and then takes as many as the numerator says, so what is missing from 1 is exactly the pieces not taken.
Leaving fewer pieces untaken therefore leaves less of the whole uncovered, which puts the fraction nearer to 1.
Each untaken piece matches one piece-worth of the shortfall from 1, so fewer of them means a smaller shortfall.
Those two use four different cards and share none, so they can be added together.
Listing only the largest proper fractions keeps the search short — the biggest sum must come from the biggest pieces.
4.NF.A.2Make A Systematic ListTo add 4/5 and 7/9, rewrite both with the common denominator 45. Then 4/5 = 36/45 and 7/9 = 35/45, and I add the numerators.
Adding unlike fractions needs the same-size pieces (Grade 5), so I switch both to forty-fifths before adding.
5.NF.A.1Guess And CheckThe next-best pairing 3/4 + 7/9 comes to 55/36, smaller than 71/45, so the first pair wins.
Checking one rival guess is enough because any pair with a smaller proper fraction must give a smaller sum.
4.NF.A.2Guess And CheckA proper fraction is biggest when the top number is just a little smaller than the bottom number, so pick the cards that leave the smallest gap.