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# Largest Number Game

Alignments to Content Standards: 2.NBT.A.1

Dona had cards with the numbers 0 to 9 written on them. She flipped over three of them. Her teacher said:

If those three numbers are the digits in another number, what is the largest three-digit number you can make?

1. First Dona put the 8 in the hundreds place. Is this the right choice for the hundreds place? Explain why or why not.
2. Next, Dona said, “It doesn’t matter what number I choose for the other places, because I put the biggest number in the hundreds place, and hundreds are bigger than tens and ones.” Is she correct? Explain.

## IM Commentary

It is important that students be asked to explain well beyond saying something like “She should choose the $8$ because it is the biggest.” They should be asked to think through the other possibilities and then draw on their ability to compare three digit numbers (as developed in 2.NBT.4) to complete the task.

In the second part, students are presented with an incorrect statement supported by a correct one. It is worth pausing to ask students to carefully sort this through, since attending to reasoning that is partially true and partially false lends itself to the SMP.3: Constructing viable arguments and critiquing the reasoning of others.

One can ask students if they know how to build the biggest three-digit number given any three numbers between 0 and 9 to use as digits. If students can’t explain the best strategy at the greatest level of generality, one could have them play the game and explain how their method works in examples.

## Solution

1. Dona is correct in putting the 8 in the hundreds place. If the 8 is in the hundreds place, the number will be bigger than 800. If she puts the 5 in the hundreds place, the number must be smaller than 600. If she puts the 1 in the hundreds place, the number must be smaller than 200.
2. Dona is not correct; all the digits matter. Tens are greater than ones, so she needs to choose the next largest number for the tens place. If she chooses 1 for the tens place and then 5 for the ones place the result is 815. The only other possibility is if she chooses 5 for the tens place and then 1 for the ones place, yielding 851, which is greater than 815. So the choice matters (and 851 is the “winning” total).