Multiplying by one and by zero

3.OA.B.53.OA.C.7 · take · grade 3

Archetype: Multiplication as Equal Groups · step in a 2-type progression

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 $?$ ).

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

List every pair product from {5, 3, 0, 1}: 15, 5, 3, 0, 0, 0; the maximum is 15, confirming the answer.

Standards · min 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.

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