Problem
To make the product large, the multiplier should be large and the two-digit number large. Test 65 × 4, 64 × 5, and 54 × 6.
A large single multiplier multiplies every part of the two-digit number, so giving 6 the multiplier role wins.
3.NBT.A.3Guess And CheckTo make the product as large as possible, the biggest card should be the one-digit multiplier rather than a digit sitting inside the two-digit number.
The one-digit multiplier is applied to the whole two-digit number, so it multiplies both the tens part and the ones part.
Multiplying the two-digit number by a card is the same as multiplying its tens part and its ones part separately and then adding the two results.
The front digit of the two-digit number stands for whole tens, so the number it multiplies is already large.
A larger multiplier makes more copies of that already-large two-digit number, so the total grows fastest when the biggest card takes the multiplier role.
Multiplying the two-digit number by a card means taking that many equal copies of it and adding them up.
Among the candidates, 54 times 6 gives the biggest product, so the largest possible product is 324.
Comparing the listed products directly shows which arrangement is greatest.
3.OA.B.5Make A Systematic ListTo make the product small, the multiplier should be small (the card 2) and the two-digit number small. Test 45 × 2 and 46 × 2.
A small multiplier and a small tens digit both shrink the product.
3.NBT.A.3Guess And CheckThe smallest of these is 45 times 2, so the smallest possible product is 90.
Putting the two smallest digits where they matter most makes the product as small as possible.
3.OA.B.5Make A Systematic ListBig digit as the multiplier for the most, small digits for the least: Grade 3 place-value thinking!