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

Variant 1 easy 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 2 easy 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 3 easy 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 4 medium 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 5 medium 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 6 medium 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 7 hard 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 8 medium 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 9 hard 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

Variant 10 hard 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

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Aim for the largest product

#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.

2Pick the maximum

#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.

3Aim for the smallest product

#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.

4Pick the minimum

#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)

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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

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