Place digit cards for largest or smallest product

3.NBT.A.33.OA.B.5 · take · grade 3

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

From the four number cards $2$ , $4$ , $5$ , and $6$ , choose $3$ of them and use each chosen card once to build a single multiplication of the form (two-digit number) $\times$ (one-digit number). Find the product when it is as large as possible, and the product when it is as small as possible.

Show solution

Understand

Using three of the four cards 2, 4, 5, 6 (each used once), build a (two-digit number) times (one-digit number). I need the largest possible product and the smallest possible product.

Givens

The available cards are 2, 4, 5, and 6.
Exactly 3 cards are chosen and each chosen card is used once.
The expression has the form (two-digit number) times (one-digit number).

Unknowns

The largest product that can be made.
The smallest product that can be made.

Constraints

One card is left unused; only 3 of the 4 cards appear.
Two cards form the two-digit number and one card is the single multiplier.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

There are only a handful of sensible arrangements, so make a short systematic list of the strong candidates and check each product. For the largest product, big digits in high place values matter; for the smallest, small digits do. Testing the top few candidates each way pins down the extremes with certainty.

Execute

#6 Guess and Check 3.NBT.A.3

To make the product large, the one-digit multiplier should be large and the two-digit number large. Test the strongest candidates: 65 times 4, 64 times 5, and 54 times 6.

65 \times 4 = 260,\quad 64 \times 5 = 320,\quad 54 \times 6 = 324

A large single multiplier multiplies every part of the two-digit number, so giving 6 the multiplier role wins.

#2 Make a Systematic List 3.OA.B.5

Among the candidates, 54 times 6 gives the biggest product, so the largest possible product is 324.

54 \times 6 = 324

Comparing the listed products directly shows which arrangement is greatest.

#6 Guess and Check 3.NBT.A.3

To make the product small, the one-digit multiplier should be small (the card 2) and the two-digit number small. Test 45 times 2 and 46 times 2.

45 \times 2 = 90,\quad 46 \times 2 = 92

A small multiplier and a small tens digit both shrink the product.

#2 Make a Systematic List 3.OA.B.5

The smallest of these is 45 times 2, so the smallest possible product is 90.

45 \times 2 = 90

Putting the two smallest digits where they matter most makes the product as small as possible.

Answer: Largest product 324 (= 54 x 6); smallest product 90 (= 45 x 2)

Review

Both products use exactly three of the four cards once. 324 is near the top of what these cards allow (under 70 times 6 = 420 but using only valid cards), and 90 is small as expected with the 2 as multiplier. A quick scan of other arrangements (65 times 4 = 260, 56 times 4 = 224) stays between 90 and 324, confirming the extremes.

Make a full systematic list (tool 2) of all 24 ways to pick an ordered (tens, ones, multiplier) from the four cards; the maximum and minimum of that complete list are again 324 and 90.

Standards · min grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about how the tens digit and multiplier scale the product when placing digits.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Comparing candidate arrangements to choose the largest and smallest products.

💡 Big digit as the multiplier for the most, small digits for the least: Grade 3 place-value thinking!