The M&Ms Problem

The following problem was shared at the CGI institute I attended two weeks ago. It was originally given to a group of second grade students who had not been taught how to multiply using the traditional algorithm.

Todd has 6 bags of M&Ms. Each bag has 43 M&Ms. How many M&Ms does Todd have?

I decided to give this problem to my 9-year-old and see if he is able to show multiple ways to solve it.

He immediately recognized it as a multiplication problem, wrote 43 x 6 vertically, and found 258 as the answer. When I asked him if he could solve it a different way, he proceeded to make a list, showing the number of M&Ms in one bag, two bags, three bags, and so on. Notice how in order to create the list, he needed to set up a multiplication vertically to find each answer.

So is this really using a different way to solve the problem?

I asked him if there was any way that he could use the number of M&Ms in one bag and in two bags to help him find the number of M&Ms in three bags. There was silence. I made the following sketch to help him:

"If you know how many M&Ms are in one bag, and you know how many M&Ms are in two bags, can you find the number of M&Ms in three bags?" I asked. "Yeah," he said. But still, he was staring at the paper, so I made a diagram to show him that three bags is just one bag plus another two bags. "Did we get the same answer you wrote in the list?" I asked. "Yes," he replied.

"Now, do you think we can use what you know about three bags of M&Ms to figure out the number of M&Ms in four bags?" I asked. "Yes, add the answer from 3 bags to one bag of M&Ms," he said. " I helped him record in a diagram.

"Is there another way?" I asked. "Yes, two bags plus two bags," he said.

"How about six bags of M&Ms? Can we use what we already figured out so far to help us find the number of M&Ms in six bags?" I asked. "Two bags plus four bags of M&Ms," he replied.

"Is there another way?" I asked. "Three bags plus three bags," he said. Notice how he wrote 129 x 2 and then wrote 129 + 129. His mind might already be conditioned to think multiplicatively when a number is repeated.

You might be wondering why I want my child to go through all this hassle when he already recognizes that, given the number of groups and the number in each group, all he has to do is multiply to find the total number. I wondered about this while I was at the training, and I guess I want him to have the tools to be able to solve problems in more than one way. I think the following problem does a good job of illustrating why I want to encourage non-traditional ways to solving a problem.

There are 258 M&Ms in 6 bags. How many M&Ms are in 12 bags?

When I posed this problem to my son, his response was: "You have to divide first and then multiply." "Can you explain what you mean?" "I don't know. I'm confused. I don't know how many M&Ms in one bag, so I can't do this problem," he said. How interesting! First, I was surprised that he didn't even recognize that 258 is the exact same answer we just found for six bags and therefore there must be 43 M&Ms per bag! And, after we've solve the problem using combinations of bags, he still couldn't make the connection. In his mind, I think he wanted to divide so he could find the number of M&Ms per bag, and then multiply by 12 to get the answer. (In hindsight, I probably didn't allow enough time for him to fully process and explain his thoughts.)

"Do we need to know how many M&Ms are in each bag? What is the relationship between 6 bags and 12 bags?" I asked. "I need twice as many bags," he answered. "Can you show me a picture?" I asked.

OK, so I'm cautiously optimistic that he is catching on now. So I give him yet another problem.

There are 258 M&Ms in 6 bags. How many M&Ms in 18 bags?

He got it this time. I didn't even have to teach him how to set up a proportion to solve!