Practice · Place digit cards for largest or smallest product · Sensim Math

Variant 1 answer: Largest product 324 (= 54 x 6); smallest product 90 (= 45 x 2)

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, 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 (the card 6) and the two-digit number large.

54 \times 6 = 324

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

#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.

45 \times 2 = 90

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. The largest, 324, puts the biggest card as the multiplier, and the smallest, 90, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 2 answer: Largest product 496 (= 62 x 8); smallest product 26 (= 26 x 1)

From the four number cards $1$ , $2$ , $6$ , and $8$ , 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 1, 2, 6, 8 (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 1, 2, 6, 8.
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 (the card 8) and the two-digit number large.

62 \times 8 = 496

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 62 times 8 gives the biggest product, so the largest possible product is 496.

62 \times 8 = 496

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 1) and the two-digit number small.

26 \times 1 = 26

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 26 times 1, so the smallest possible product is 26.

26 \times 1 = 26

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

Answer: Largest product 496 (= 62 x 8); smallest product 26 (= 26 x 1)

Review

Both products use exactly three of the four cards once. The largest, 496, puts the biggest card as the multiplier, and the smallest, 26, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 3 answer: Largest product 486 (= 54 x 9); smallest product 45 (= 45 x 1)

From the four number cards $1$ , $4$ , $5$ , and $9$ , 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 1, 4, 5, 9 (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 1, 4, 5, 9.
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 (the card 9) and the two-digit number large.

54 \times 9 = 486

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 54 times 9 gives the biggest product, so the largest possible product is 486.

54 \times 9 = 486

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 1) and the two-digit number small.

45 \times 1 = 45

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 1, so the smallest possible product is 45.

45 \times 1 = 45

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

Answer: Largest product 486 (= 54 x 9); smallest product 45 (= 45 x 1)

Review

Both products use exactly three of the four cards once. The largest, 486, puts the biggest card as the multiplier, and the smallest, 45, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 4 answer: Largest product 657 (= 73 x 9); smallest product 37 (= 37 x 1)

From the four number cards $1$ , $3$ , $7$ , and $9$ , 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 1, 3, 7, 9 (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 1, 3, 7, 9.
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 (the card 9) and the two-digit number large.

73 \times 9 = 657

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 73 times 9 gives the biggest product, so the largest possible product is 657.

73 \times 9 = 657

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 1) and the two-digit number small.

37 \times 1 = 37

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 37 times 1, so the smallest possible product is 37.

37 \times 1 = 37

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

Answer: Largest product 657 (= 73 x 9); smallest product 37 (= 37 x 1)

Review

Both products use exactly three of the four cards once. The largest, 657, puts the biggest card as the multiplier, and the smallest, 37, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 5 answer: Largest product 600 (= 75 x 8); smallest product 114 (= 57 x 2)

From the four number cards $2$ , $5$ , $7$ , and $8$ , 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, 5, 7, 8 (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, 5, 7, 8.
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 (the card 8) and the two-digit number large.

75 \times 8 = 600

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 75 times 8 gives the biggest product, so the largest possible product is 600.

75 \times 8 = 600

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.

57 \times 2 = 114

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 57 times 2, so the smallest possible product is 114.

57 \times 2 = 114

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

Answer: Largest product 600 (= 75 x 8); smallest product 114 (= 57 x 2)

Review

Both products use exactly three of the four cards once. The largest, 600, puts the biggest card as the multiplier, and the smallest, 114, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 6 answer: Largest product 684 (= 76 x 9); smallest product 268 (= 67 x 4)

From the four number cards $4$ , $6$ , $7$ , and $9$ , 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 4, 6, 7, 9 (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 4, 6, 7, 9.
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 (the card 9) and the two-digit number large.

76 \times 9 = 684

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 76 times 9 gives the biggest product, so the largest possible product is 684.

76 \times 9 = 684

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 4) and the two-digit number small.

67 \times 4 = 268

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 67 times 4, so the smallest possible product is 268.

67 \times 4 = 268

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

Answer: Largest product 684 (= 76 x 9); smallest product 268 (= 67 x 4)

Review

Both products use exactly three of the four cards once. The largest, 684, puts the biggest card as the multiplier, and the smallest, 268, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 7 answer: Largest product 301 (= 43 x 7); smallest product 68 (= 34 x 2)

From the four number cards $2$ , $3$ , $4$ , and $7$ , 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, 3, 4, 7 (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, 3, 4, 7.
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 (the card 7) and the two-digit number large.

43 \times 7 = 301

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 43 times 7 gives the biggest product, so the largest possible product is 301.

43 \times 7 = 301

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.

34 \times 2 = 68

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 34 times 2, so the smallest possible product is 68.

34 \times 2 = 68

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

Answer: Largest product 301 (= 43 x 7); smallest product 68 (= 34 x 2)

Review

Both products use exactly three of the four cards once. The largest, 301, puts the biggest card as the multiplier, and the smallest, 68, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 8 answer: Largest product 455 (= 65 x 7); smallest product 168 (= 56 x 3)

From the four number cards $3$ , $5$ , $6$ , and $7$ , 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 3, 5, 6, 7 (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 3, 5, 6, 7.
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 (the card 7) and the two-digit number large.

65 \times 7 = 455

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 65 times 7 gives the biggest product, so the largest possible product is 455.

65 \times 7 = 455

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 3) and the two-digit number small.

56 \times 3 = 168

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 56 times 3, so the smallest possible product is 168.

56 \times 3 = 168

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

Answer: Largest product 455 (= 65 x 7); smallest product 168 (= 56 x 3)

Review

Both products use exactly three of the four cards once. The largest, 455, puts the biggest card as the multiplier, and the smallest, 168, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 9 answer: Largest product 424 (= 53 x 8); smallest product 70 (= 35 x 2)

From the four number cards $2$ , $3$ , $5$ , and $8$ , 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, 3, 5, 8 (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, 3, 5, 8.
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 (the card 8) and the two-digit number large.

53 \times 8 = 424

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 53 times 8 gives the biggest product, so the largest possible product is 424.

53 \times 8 = 424

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.

35 \times 2 = 70

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 35 times 2, so the smallest possible product is 70.

35 \times 2 = 70

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

Answer: Largest product 424 (= 53 x 8); smallest product 70 (= 35 x 2)

Review

Both products use exactly three of the four cards once. The largest, 424, puts the biggest card as the multiplier, and the smallest, 70, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Variant 10 answer: Largest product 756 (= 84 x 9); smallest product 144 (= 48 x 3)

From the four number cards $3$ , $4$ , $8$ , and $9$ , 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 3, 4, 8, 9 (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 3, 4, 8, 9.
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 (the card 9) and the two-digit number large.

84 \times 9 = 756

A large single multiplier multiplies every part of the two-digit number, so giving the biggest card the multiplier role helps.

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

Among the candidates, 84 times 9 gives the biggest product, so the largest possible product is 756.

84 \times 9 = 756

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 3) and the two-digit number small.

48 \times 3 = 144

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 48 times 3, so the smallest possible product is 144.

48 \times 3 = 144

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

Answer: Largest product 756 (= 84 x 9); smallest product 144 (= 48 x 3)

Review

Both products use exactly three of the four cards once. The largest, 756, puts the biggest card as the multiplier, and the smallest, 144, puts the smallest card as the multiplier, matching the expectation.

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 the answers.

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!

Place digit cards for largest or smallest product · 10 practice problems

Generated variants — 10

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3

Understand

Plan

Execute

Review

Standards · min grade 3