← Multiplying by one and by zero · Multiplication as Equal Groups

Multiplying by one and by zero · 9 practice problems

3.OA.B.53.OA.C.7

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Variant 1 easy

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $3$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $2$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 2, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 3 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 2 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 3 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 3 must come from 3 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 2 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 3

#3 Eliminate Possibilities 3.OA.C.7

A product of 3 from single digits must be 3 x 1. Card 4 is already 1, so the other hidden card must be 3 (since 2 cannot make 3).

3 \times 1 = 3

3 is only 3 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 3.

3Find the greatest product

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

The four cards are 3, 2, 0, 1. Try the largest pairs: 3 x 2 = 6 is the biggest, beating 3 x 1 = 3 and any pair using 0.

3 \times 2 = 6

To make the biggest product, multiply the two largest cards, 3 and 2.

Answer: 6

4 · Reviewdoes it hold up?

The cards 3, 2, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 3 (3 x 1), so they fit the clues, and 3 x 2 = 6 is the largest pairing.

Another way: List every pair product from {3, 2, 0, 1} and take the maximum, confirming the answer of 6.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 3 = 3 x 1 and computing 3 x 2 = 6 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 2 easy

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $5$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $3$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 3, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 5 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 3 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 5 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 5 must come from 5 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 3 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 5

#3 Eliminate Possibilities 3.OA.C.7

A product of 5 from single digits must be 5 x 1. Card 4 is already 1, so the other hidden card must be 5 (since 3 cannot make 5).

5 \times 1 = 5

5 is only 5 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 5.

3Find the greatest product

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

The four cards are 5, 3, 0, 1. Try the largest pairs: 5 x 3 = 15 is the biggest, beating 5 x 1 = 5 and any pair using 0.

5 \times 3 = 15

To make the biggest product, multiply the two largest cards, 5 and 3.

Answer: 15

4 · Reviewdoes it hold up?

The cards 5, 3, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 5 (5 x 1), so they fit the clues, and 5 x 3 = 15 is the largest pairing.

Another way: List every pair product from {5, 3, 0, 1} and take the maximum, confirming the answer of 15.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 5 = 5 x 1 and computing 5 x 3 = 15 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 3 easy

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $5$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $2$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 2, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 5 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 2 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 5 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 5 must come from 5 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 2 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 5

#3 Eliminate Possibilities 3.OA.C.7

A product of 5 from single digits must be 5 x 1. Card 4 is already 1, so the other hidden card must be 5 (since 2 cannot make 5).

5 \times 1 = 5

5 is only 5 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 5.

3Find the greatest product

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

The four cards are 5, 2, 0, 1. Try the largest pairs: 5 x 2 = 10 is the biggest, beating 5 x 1 = 5 and any pair using 0.

5 \times 2 = 10

To make the biggest product, multiply the two largest cards, 5 and 2.

Answer: 10

4 · Reviewdoes it hold up?

The cards 5, 2, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 5 (5 x 1), so they fit the clues, and 5 x 2 = 10 is the largest pairing.

Another way: List every pair product from {5, 2, 0, 1} and take the maximum, confirming the answer of 10.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 5 = 5 x 1 and computing 5 x 2 = 10 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 4 medium

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $5$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $4$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 4, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 5 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 4 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 5 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 5 must come from 5 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 4 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 5

#3 Eliminate Possibilities 3.OA.C.7

A product of 5 from single digits must be 5 x 1. Card 4 is already 1, so the other hidden card must be 5 (since 4 cannot make 5).

5 \times 1 = 5

5 is only 5 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 5.

3Find the greatest product

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

The four cards are 5, 4, 0, 1. Try the largest pairs: 5 x 4 = 20 is the biggest, beating 5 x 1 = 5 and any pair using 0.

5 \times 4 = 20

To make the biggest product, multiply the two largest cards, 5 and 4.

Answer: 20

4 · Reviewdoes it hold up?

The cards 5, 4, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 5 (5 x 1), so they fit the clues, and 5 x 4 = 20 is the largest pairing.

Another way: List every pair product from {5, 4, 0, 1} and take the maximum, confirming the answer of 20.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 5 = 5 x 1 and computing 5 x 4 = 20 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 5 medium

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $7$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $2$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 2, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 7 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 2 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 7 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 7 must come from 7 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 2 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 7

#3 Eliminate Possibilities 3.OA.C.7

A product of 7 from single digits must be 7 x 1. Card 4 is already 1, so the other hidden card must be 7 (since 2 cannot make 7).

7 \times 1 = 7

7 is only 7 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 7.

3Find the greatest product

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

The four cards are 7, 2, 0, 1. Try the largest pairs: 7 x 2 = 14 is the biggest, beating 7 x 1 = 7 and any pair using 0.

7 \times 2 = 14

To make the biggest product, multiply the two largest cards, 7 and 2.

Answer: 14

4 · Reviewdoes it hold up?

The cards 7, 2, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 7 (7 x 1), so they fit the clues, and 7 x 2 = 14 is the largest pairing.

Another way: List every pair product from {7, 2, 0, 1} and take the maximum, confirming the answer of 14.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 7 = 7 x 1 and computing 7 x 2 = 14 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 6 hard

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $7$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $4$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 4, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 7 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 4 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 7 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 7 must come from 7 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 4 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 7

#3 Eliminate Possibilities 3.OA.C.7

A product of 7 from single digits must be 7 x 1. Card 4 is already 1, so the other hidden card must be 7 (since 4 cannot make 7).

7 \times 1 = 7

7 is only 7 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 7.

3Find the greatest product

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

The four cards are 7, 4, 0, 1. Try the largest pairs: 7 x 4 = 28 is the biggest, beating 7 x 1 = 7 and any pair using 0.

7 \times 4 = 28

To make the biggest product, multiply the two largest cards, 7 and 4.

Answer: 28

4 · Reviewdoes it hold up?

The cards 7, 4, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 7 (7 x 1), so they fit the clues, and 7 x 4 = 28 is the largest pairing.

Another way: List every pair product from {7, 4, 0, 1} and take the maximum, confirming the answer of 28.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 7 = 7 x 1 and computing 7 x 4 = 28 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 7 hard

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $7$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $5$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 5, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 7 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 5 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 7 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 7 must come from 7 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 5 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 7

#3 Eliminate Possibilities 3.OA.C.7

A product of 7 from single digits must be 7 x 1. Card 4 is already 1, so the other hidden card must be 7 (since 5 cannot make 7).

7 \times 1 = 7

7 is only 7 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 7.

3Find the greatest product

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

The four cards are 7, 5, 0, 1. Try the largest pairs: 7 x 5 = 35 is the biggest, beating 7 x 1 = 7 and any pair using 0.

7 \times 5 = 35

To make the biggest product, multiply the two largest cards, 7 and 5.

Answer: 35

4 · Reviewdoes it hold up?

The cards 7, 5, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 7 (7 x 1), so they fit the clues, and 7 x 5 = 35 is the largest pairing.

Another way: List every pair product from {7, 5, 0, 1} and take the maximum, confirming the answer of 35.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 7 = 7 x 1 and computing 7 x 5 = 35 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 8 hard

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $7$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $6$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 6, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 7 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 6 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 7 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 7 must come from 7 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 6 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 7

#3 Eliminate Possibilities 3.OA.C.7

A product of 7 from single digits must be 7 x 1. Card 4 is already 1, so the other hidden card must be 7 (since 6 cannot make 7).

7 \times 1 = 7

7 is only 7 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 7.

3Find the greatest product

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

The four cards are 7, 6, 0, 1. Try the largest pairs: 7 x 6 = 42 is the biggest, beating 7 x 1 = 7 and any pair using 0.

7 \times 6 = 42

To make the biggest product, multiply the two largest cards, 7 and 6.

Answer: 42

4 · Reviewdoes it hold up?

The cards 7, 6, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 7 (7 x 1), so they fit the clues, and 7 x 6 = 42 is the largest pairing.

Another way: List every pair product from {7, 6, 0, 1} and take the maximum, confirming the answer of 42.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 7 = 7 x 1 and computing 7 x 6 = 42 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Variant 9 medium

You will choose $2$ of the $4$ number cards and multiply the two numbers. Yunji's product is $7$ , and Seungmin's product is $0$ . Of all the products you can make, find the greatest one.

The $4$ number cards lie in a row. The second card shows $3$ and the fourth card shows $1$ ; the numbers on the first and third cards are hidden (covered by a $?$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

Four number cards lie in a row: the second is 3, the fourth is 1, and the first and third are hidden. Picking 2 cards and multiplying, one pair gives 7 and another gives 0. We must find the greatest product possible from any 2 of the four cards.

Givens

Card 2 is 3 and card 4 is 1; cards 1 and 3 are hidden.
Some pair of cards multiplies to 7 (Yunji).
Some pair of cards multiplies to 0 (Seungmin).

Unknowns

The two hidden card numbers.
The greatest product of any two cards.

Constraints

Each card is a single-digit number.
A product of 0 means one chosen card is 0.
A product of 7 must come from 7 x 1.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #2 Make a Systematic List

Use the special facts about 0 and 1 to deduce the hidden cards from the genuinely limited possibilities, then list all pair products to pick the largest.

3 · Execute3 carry out the plan

1Use the zero product

#3 Eliminate Possibilities 3.OA.B.5

A product of 0 only happens when one factor is 0. The visible cards are 3 and 1, neither is 0, so one hidden card must be 0.

\square \times 0 = 0

Zero times any number is zero, so a product of 0 forces a 0 card to exist.

2Use the product of 7

#3 Eliminate Possibilities 3.OA.C.7

A product of 7 from single digits must be 7 x 1. Card 4 is already 1, so the other hidden card must be 7 (since 3 cannot make 7).

7 \times 1 = 7

7 is only 7 x 1 among one-digit products, and a 1 is already on a card, so the partner card is 7.

3Find the greatest product

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

The four cards are 7, 3, 0, 1. Try the largest pairs: 7 x 3 = 21 is the biggest, beating 7 x 1 = 7 and any pair using 0.

7 \times 3 = 21

To make the biggest product, multiply the two largest cards, 7 and 3.

Answer: 21

4 · Reviewdoes it hold up?

The cards 7, 3, 0, 1 do allow a pair giving 0 (with the 0 card) and a pair giving 7 (7 x 1), so they fit the clues, and 7 x 3 = 21 is the largest pairing.

Another way: List every pair product from {7, 3, 0, 1} and take the maximum, confirming the answer of 21.

Standardsmin grade 3

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Using the zero property to identify the hidden 0 card.
3.OA.C.7 Fluently multiply and divide within 100 — Recognizing 7 = 7 x 1 and computing 7 x 3 = 21 as the greatest product.

💡Takeaway. Zero makes 0 and one keeps a number the same -- those facts unmask the hidden cards, pure Grade 3 thinking!

Where next?

← Back to the problem: Multiplying by one and by zero