We have had a question we have used for years in Title One Math that continually stumps students in division situations because of the context of the problem.  It is an excellent way to get students to think about the remainder in a division problem as it applies to different situations.  We lovingly refer to this as the package problem.  The question is, I have 34 cupcakes. They are packaged in packages of 6.  How many packages will I need?

When I was making and packaging my annual Christmas pretzels this year, I took a few pictures and thought this might be an excellent example of the package problem.

There are many different scenarios for this task which is what I think makes a really rich task.  You can ask, “How many packages of three will I be able to give out as presents?” or “The pretzels are packaged in packages of 3.  How many packages will I need to package all of the pretzels?”  The first question ignores the remainder because it is irrelevant to the situation.  The second question requires students to look at the remainder as a necessary part of the problem, but allows them to recognize that the last package will not be complete.  If I pair the first question with a third question, “How many will not be packaged?” attention is called to the remainder and can lead to a discussion such as, “Why did I make 2 extra?” or “How would I package them if I wanted to give a bag to each of 6 friends?” or “If I didn’t want to have any pretzels remaining, how could I package them?”

Problem solving is all about context.  In my opinion, the reason students don’t do well with word problems is because they are not exposed to multiple contexts in mathematics when skills are taught in isolation.  Hence, the power of 3 Act Math Tasks (thanks Mr. Meyer).  I hope you enjoy this task and would love your feedback as always!

When I was taking pictures for this task, I was elbows deep in melted chocolate so I forgot to take some of the pictures I needed for the different contexts.  My plan is to make another batch soon and add those:)  So for now, I have the third act addressing the first question.

I’ve added a link on the 3 Act main page of my site to the google folder with all of my tasks, videos and pictures.  That way you can choose the pics you’d like. I love to use Nearpod for my 3 Act Tasks when I teach for the awesome opportunities for student input and data analysis:)  However, every time I run a Nearpod session it breaks the link to my blog, so I thought Google might be a better way to share these:)

Act 1: 

What do you notice?  What do you wonder?

How many packages will I need to package all of the pretzels?  Write a too high and a too low estimate.

Act 2:

There are 26 pretzels.  Pretzels are packaged in bags of three. (Insert preferred question here) Act 3 addresses the question, “How many packages of 3 will I be able to give out as presents?” (Students might count the one package that is already shown in the picture and come up with 9)

Act 3:

or this depending on whether they count the first package in Act 1.

Extensions:

“How would I package them if I wanted to give a bag to each of 6 friends?” or “If I didn’t want to have any pretzels remaining, how could I package them?“