Problem
Place big digits high for largest product
Place the four cards 4, 7, 9, 2 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.
OperationsBase-ten numbers
Your answer
Watch out It's tempting to give the bigger leftover card to the bigger tens digit (making 94 x 72), but the closer-together pairing 92 x 74 actually gives the larger product.
How to solve
Strategy Guess and Check — Digits in the tens place count for the most, so the two biggest cards (9 and 7) should be tens digits. Then test the few ways to place 4 and 2 in the ones places and pick the largest product.
1STEP 1
Put the big cards in the tens places
A tens digit counts ten times as much as a ones digit, so the two biggest cards, 9 and 7, go in the tens places.
9□ × 7□
Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).
3.NBT.A.3Guess And CheckThe largest product comes from placing the two biggest cards, 9 and 7, in the tens places.
Why?
A card in a tens place adds ten times as much to its factor as the same card would in a ones place, so the biggest cards do the most good when they sit in the tens places.
Why?
A two-digit number is a pile of tens named by its left card plus a few ones named by its right card, so the left card gets counted ten times over.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
We want each factor to be as large as it can be, because making a factor larger always makes the product larger.
Why?
A product is just one number added over and over, so making that number bigger makes the running total bigger too.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Test the two ways to place 4 and 2
The two smaller cards, 4 and 2, can go either way -- compare 94 x 72 with 92 x 74 to see which is bigger.
94 × 72 = 6768, 92 × 74 = 6808
Only a few arrangements are possible, so listing and comparing them is quick and safe.
3.OA.C.7Make A Systematic List3STEP 3
Pick the largest product
Of the candidates, 92 x 74 gives the biggest result, 6808.
92 × 74 = 92 × 70 + 92 × 4 = 6440 + 368 = 6808
Break 74 into 70 and 4, multiply each part, then add.
3.OA.C.7Guess And CheckAnswer
6808
92 × 74 = 6808
Both factors are in the 70s-90s, so the product should be roughly 90 x 75 = 6750. The answer 6808 sits right in that range, so it is reasonable.
Takeaway
Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!
- Put the big cards in the tens places
- Test the two ways to place 4 and 2
- Pick the largest product
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