← Place big digits high for largest product · Build the Largest or Smallest Value from Digit Cards

Place big digits high for largest product · 10 practice problems

3.OA.C.73.NBT.A.3

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 3763 (from 71 x 53)

Using the number cards $1$ , $3$ , $5$ , $7$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $1$ , $3$ , $5$ , $7$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 1, 3, 5, 7 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 1, 3, 5, 7, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (7 and 5) should be tens digits. Then test the few ways to place 3 and 1 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 7 and 5, in the tens places. That leaves 3 and 1 for the ones places.

7\square \times 5\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 3 and 1

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (3) to the factor with the larger tens digit (7), so it is multiplied by more. Compare 73 x 51 with 71 x 53.

73 \times 51 = 3723,\quad 71 \times 53 = 3763

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 71 x 53 gives the biggest result, 3763.

71 \times 53 = 71 \times 50 + 71 \times 3 = 3550 + 213 = 3763

Break 53 into 50 and 3, multiply each part, then add.

Answer: 3763 (from 71 x 53)

4 · Reviewdoes it hold up?

Both factors are in the 50s-70s, so the product should be roughly 3500. The answer 3763 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 1, 3, 5, 7; the maximum among all of them is 3763, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 2 medium answer: 5986 (from 82 x 73)

Using the number cards $2$ , $3$ , $7$ , $8$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $2$ , $3$ , $7$ , $8$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 2, 3, 7, 8 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 2, 3, 7, 8, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (8 and 7) should be tens digits. Then test the few ways to place 3 and 2 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 8 and 7, in the tens places. That leaves 3 and 2 for the ones places.

8\square \times 7\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 3 and 2

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (3) to the factor with the larger tens digit (8), so it is multiplied by more. Compare 83 x 72 with 82 x 73.

83 \times 72 = 5976,\quad 82 \times 73 = 5986

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 82 x 73 gives the biggest result, 5986.

82 \times 73 = 82 \times 70 + 82 \times 3 = 5740 + 246 = 5986

Break 73 into 70 and 3, multiply each part, then add.

Answer: 5986 (from 82 x 73)

4 · Reviewdoes it hold up?

Both factors are in the 70s-80s, so the product should be roughly 5600. The answer 5986 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 2, 3, 7, 8; the maximum among all of them is 5986, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 3 medium answer: 4212 (from 81 x 52)

Using the number cards $1$ , $2$ , $5$ , $8$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $1$ , $2$ , $5$ , $8$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 1, 2, 5, 8 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 1, 2, 5, 8, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (8 and 5) should be tens digits. Then test the few ways to place 2 and 1 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 8 and 5, in the tens places. That leaves 2 and 1 for the ones places.

8\square \times 5\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 2 and 1

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (2) to the factor with the larger tens digit (8), so it is multiplied by more. Compare 82 x 51 with 81 x 52.

82 \times 51 = 4182,\quad 81 \times 52 = 4212

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 81 x 52 gives the biggest result, 4212.

81 \times 52 = 81 \times 50 + 81 \times 2 = 4050 + 162 = 4212

Break 52 into 50 and 2, multiply each part, then add.

Answer: 4212 (from 81 x 52)

4 · Reviewdoes it hold up?

Both factors are in the 50s-80s, so the product should be roughly 4000. The answer 4212 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 1, 2, 5, 8; the maximum among all of them is 4212, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 4 easy answer: 5248 (from 82 x 64)

Using the number cards $2$ , $4$ , $6$ , $8$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $2$ , $4$ , $6$ , $8$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 2, 4, 6, 8 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 2, 4, 6, 8, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (8 and 6) should be tens digits. Then test the few ways to place 4 and 2 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 8 and 6, in the tens places. That leaves 4 and 2 for the ones places.

8\square \times 6\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 4 and 2

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (4) to the factor with the larger tens digit (8), so it is multiplied by more. Compare 84 x 62 with 82 x 64.

84 \times 62 = 5208,\quad 82 \times 64 = 5248

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 82 x 64 gives the biggest result, 5248.

82 \times 64 = 82 \times 60 + 82 \times 4 = 4920 + 328 = 5248

Break 64 into 60 and 4, multiply each part, then add.

Answer: 5248 (from 82 x 64)

4 · Reviewdoes it hold up?

Both factors are in the 60s-80s, so the product should be roughly 4800. The answer 5248 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 2, 4, 6, 8; the maximum among all of them is 5248, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 5 medium answer: 6225 (from 83 x 75)

Using the number cards $5$ , $7$ , $8$ , $3$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $5$ , $7$ , $8$ , $3$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 5, 7, 8, 3 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 5, 7, 8, 3, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (8 and 7) should be tens digits. Then test the few ways to place 5 and 3 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 8 and 7, in the tens places. That leaves 5 and 3 for the ones places.

8\square \times 7\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 5 and 3

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (5) to the factor with the larger tens digit (8), so it is multiplied by more. Compare 85 x 73 with 83 x 75.

85 \times 73 = 6205,\quad 83 \times 75 = 6225

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 83 x 75 gives the biggest result, 6225.

83 \times 75 = 83 \times 70 + 83 \times 5 = 5810 + 415 = 6225

Break 75 into 70 and 5, multiply each part, then add.

Answer: 6225 (from 83 x 75)

4 · Reviewdoes it hold up?

Both factors are in the 70s-80s, so the product should be roughly 5600. The answer 6225 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 5, 7, 8, 3; the maximum among all of them is 6225, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 6 easy answer: 4293 (from 81 x 53)

Using the number cards $3$ , $5$ , $8$ , $1$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $3$ , $5$ , $8$ , $1$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 3, 5, 8, 1 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 3, 5, 8, 1, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (8 and 5) should be tens digits. Then test the few ways to place 3 and 1 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 8 and 5, in the tens places. That leaves 3 and 1 for the ones places.

8\square \times 5\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 3 and 1

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (3) to the factor with the larger tens digit (8), so it is multiplied by more. Compare 83 x 51 with 81 x 53.

83 \times 51 = 4233,\quad 81 \times 53 = 4293

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 81 x 53 gives the biggest result, 4293.

81 \times 53 = 81 \times 50 + 81 \times 3 = 4050 + 243 = 4293

Break 53 into 50 and 3, multiply each part, then add.

Answer: 4293 (from 81 x 53)

4 · Reviewdoes it hold up?

Both factors are in the 50s-80s, so the product should be roughly 4000. The answer 4293 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 3, 5, 8, 1; the maximum among all of them is 4293, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 7 medium answer: 6808 (from 92 x 74)

Using the number cards $4$ , $7$ , $9$ , $2$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $4$ , $7$ , $9$ , $2$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

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.

Givens

The four number cards are 4, 7, 9, 2, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

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.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 9 and 7, in the tens places. That leaves 4 and 2 for the ones places.

9\square \times 7\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 4 and 2

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (4) to the factor with the larger tens digit (9), so it is multiplied by more. Compare 94 x 72 with 92 x 74.

94 \times 72 = 6768,\quad 92 \times 74 = 6808

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 92 x 74 gives the biggest result, 6808.

92 \times 74 = 92 \times 70 + 92 \times 4 = 6440 + 368 = 6808

Break 74 into 70 and 4, multiply each part, then add.

Answer: 6808 (from 92 x 74)

4 · Reviewdoes it hold up?

Both factors are in the 70s-90s, so the product should be roughly 6300. The answer 6808 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 4, 7, 9, 2; the maximum among all of them is 6808, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 8 hard answer: 7998 (from 93 x 86)

Using the number cards $3$ , $6$ , $8$ , $9$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $3$ , $6$ , $8$ , $9$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 3, 6, 8, 9 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 3, 6, 8, 9, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (9 and 8) should be tens digits. Then test the few ways to place 6 and 3 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 9 and 8, in the tens places. That leaves 6 and 3 for the ones places.

9\square \times 8\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 6 and 3

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (6) to the factor with the larger tens digit (9), so it is multiplied by more. Compare 96 x 83 with 93 x 86.

96 \times 83 = 7968,\quad 93 \times 86 = 7998

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 93 x 86 gives the biggest result, 7998.

93 \times 86 = 93 \times 80 + 93 \times 6 = 7440 + 558 = 7998

Break 86 into 80 and 6, multiply each part, then add.

Answer: 7998 (from 93 x 86)

4 · Reviewdoes it hold up?

Both factors are in the 80s-90s, so the product should be roughly 7200. The answer 7998 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 3, 6, 8, 9; the maximum among all of them is 7998, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 9 hard answer: 5824 (from 91 x 64)

Using the number cards $1$ , $4$ , $6$ , $9$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $1$ , $4$ , $6$ , $9$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 1, 4, 6, 9 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 1, 4, 6, 9, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (9 and 6) should be tens digits. Then test the few ways to place 4 and 1 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 9 and 6, in the tens places. That leaves 4 and 1 for the ones places.

9\square \times 6\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 4 and 1

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (4) to the factor with the larger tens digit (9), so it is multiplied by more. Compare 94 x 61 with 91 x 64.

94 \times 61 = 5734,\quad 91 \times 64 = 5824

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 91 x 64 gives the biggest result, 5824.

91 \times 64 = 91 \times 60 + 91 \times 4 = 5460 + 364 = 5824

Break 64 into 60 and 4, multiply each part, then add.

Answer: 5824 (from 91 x 64)

4 · Reviewdoes it hold up?

Both factors are in the 60s-90s, so the product should be roughly 5400. The answer 5824 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 1, 4, 6, 9; the maximum among all of them is 5824, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!

Variant 10 hard answer: 5888 (from 92 x 64)

Using the number cards $6$ , $2$ , $9$ , $4$ each exactly once, you want to form a (two-digit number) $\times$ (two-digit number) multiplication like the one on the right and compute it. What is the largest possible product?

The multiplication on the right has four blanks: a two-digit number $\square\,\square$ multiplied by a two-digit number $\square\,\square$ . Place each of the number cards $6$ , $2$ , $9$ , $4$ into the four blanks, using each card exactly once.

Show solution

1 · Understandwhat's really being asked

Place the four cards 6, 2, 9, 4 (each used once) into a two-digit times two-digit multiplication template to make the largest possible product.

Givens

The four number cards are 6, 2, 9, 4, each used exactly once.
The blank template (shown at right) is a two-digit number times a two-digit number.
Each blank cell holds one card.

Unknowns

The arrangement of the cards that gives the largest product, and that product.

Constraints

Both factors are two-digit numbers.
Each of the four cards is used exactly once.

2 · Planchoose the strategy

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

Digits in the tens place count for the most, so the two biggest cards (9 and 6) should be tens digits. Then test the few ways to place 4 and 2 in the ones places and pick the largest product.

3 · Execute3 carry out the plan

1Put the big cards in the tens places

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

A tens digit is worth ten times its ones-place value, so the largest product comes from putting the two largest cards, 9 and 6, in the tens places. That leaves 4 and 2 for the ones places.

9\square \times 6\square

Place value: a digit high up (tens) adds far more to a number than the same digit low down (ones).

2Test the two ways to place 4 and 2

#2 Make a Systematic List 3.OA.C.7

Give the larger remaining card (4) to the factor with the larger tens digit (9), so it is multiplied by more. Compare 94 x 62 with 92 x 64.

94 \times 62 = 5828,\quad 92 \times 64 = 5888

Only a few arrangements are possible, so listing and comparing them is quick and safe.

3Pick the largest product

#6 Guess and Check 3.OA.C.7

Of the candidates, 92 x 64 gives the biggest result, 5888.

92 \times 64 = 92 \times 60 + 92 \times 4 = 5520 + 368 = 5888

Break 64 into 60 and 4, multiply each part, then add.

Answer: 5888 (from 92 x 64)

4 · Reviewdoes it hold up?

Both factors are in the 60s-90s, so the product should be roughly 5400. The answer 5888 sits in that range, so it is reasonable.

Another way: Make a systematic list of every two-digit x two-digit arrangement of 6, 2, 9, 4; the maximum among all of them is 5888, confirming the place-value shortcut.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the two-digit factors to compare candidate products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Reasoning about place value to put the biggest digits in the tens places.

💡Takeaway. Big cards go in the tens spots -- that place-value trick from Grade 3 finds the largest product fast!