Problem
The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 7; from the remaining cards 5, 4, 6 the largest valid numerator is 6. So the largest mixed number is 8 6/7.
A bigger whole part beats any fraction part, so maximize the whole number first.
4.NF.A.2Guess And CheckThe largest mixed number is made by putting the biggest card, 8, in the whole place and the biggest still-legal card, 6, in the numerator over 7, giving 8 6/7.
A mixed number is a whole-number part plus a fraction part that is less than one whole, so the whole part settles which mixed number is bigger before the fraction even matters, and 8 is the biggest whole part available.
The fraction part here is at most 6/7, and any number of sevenths below seven is less than one whole.
One whole splits into exactly seven equal sevenths, so having fewer than seven of them is having less than the whole.
When you compare two numbers you weigh the biggest place first, and one whole unit outranks any part of a unit the same way a ten outranks the ones.
Each higher place is a bundle of the place below it, so a single unit up in the whole place is worth more than anything sitting in the fraction below it.
With the whole part fixed as large as possible, the mixed number grows when the fraction grows, and among the leftover cards 5, 4, 6 the largest one still below seven is 6.
Over the same denominator 7, a bigger numerator names more of the same-size seventh-pieces, which is a larger fraction.
A fraction like 6/7 is just six copies of the piece 1/7 added together, and adding more equal pieces makes a larger total.
The numerator has to stay below 7 or the fraction part would be a whole or more, so 8 is not allowed as a numerator and 6 is the biggest card that is.
Seven sevenths already rebuild one full whole, so eight sevenths would run past a whole and stop being a proper fraction part.
Now make the whole part as small as possible: the smallest card is 4. For the smallest value the numerator should be as small as possible from the rest (5, 8, 6); but 8 is too big to be a numerator (it must be under 7), so the smallest valid numerator is 5. The smallest mixed number is 4 5/7.
A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 7.
4.NF.A.2Make A Systematic ListAdd 8 6/7 + 4 5/7. Whole parts: 8 + 4 = 12. Fraction parts: 6/7 + 5/7 = 11/7 = 1 4/7. Combine: 12 + 1 4/7 = 13 4/7.
Add wholes and like-denominator fractions separately, then regroup 11/7 into 1 4/7.
4.NF.B.3Guess And CheckThis only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 7!