## Open-ended problems in elementary school mathematics

I’m hoping to find (or potentially build, given how well my search is going) some open-ended problems appropriate for elementary school math classes. By open-ended problems, I mean problems which:

• do not have an obvious solution,
• require some time to figure out,
• have multiple solutions,
• may require some assumptions are made by the students,
• are extendable in some way,
• require that the solution be explained, rather than a single number given as the answer.

I’ve found that the definition of open-ended problem seems to vary quite a bit, with many sources that I’ve found using free-response or open-response as a synonym for open-ended.

Here’s a sample question (forgive the wording, it may need improvement).

Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?

The curriculum link here is either counting (likely to be a slow technique so I’d recommend reducing the number of friends if this is the strategy your students are going to use), addition, multiplication, or division. Note that if you do questions like this, it is important for students to explain their reasoning, and you may need to help some students do this. You may also have to point out that since Ellen doesn’t know exactly how many of her friends will attend, this problem is harder than it looks. Also, I may or may not give the actual diagram as this likely gives away too much of the problem to students. Once students have drawn a diagram though, one could turn this into a bit of a probability question (given the diagram above, how likely is it that one of Ellen’s friends will have to sit alone?).

Does anyone else know a source of questions which are this open-ended, and are designed for elementary school students?

Update:

Here are some resources I’ve been given or found so far:

• Matt Fletcher wrote:

This is exactly what we’re looking to do for our Grade 6 Math program next year. We’re trying to ditch our text and our ‘traditional’ way of teaching Math strand-by-strand, and instead focus on problem solving tasks involving multiple strands at one time and require the use of the Math Processes as outlined in the Ontario Curriculum.

We’ve found the same issue you have – that being there really isn’t a great resource out there. So far, it has involved a lot of searching on the internet and in some text books, and adapting problems which may come from a Secondary School, for our grade level.

One resource I have found useful is Math that Matters by David Stocker. While not always open-ended, the questions are based on Social Justice issues and are able to generate an incredible amount of discussion surrounding student answers.

• Gary Tupper wrote:

Having never taught elem, & being retired now for 12 yrs, the following may be a non-starter, but:

Little Johnnie is going to the store to buy a pair of Reeboks? which he knows will cost between \$65 and \$70.
How much money (which denominations & coinage) should he take in order to pay the exact amount? Ie the least that will be sufficient.

Extension: The following week little Johnnie and his friend Willis are heading off to the mall to buy some game cartridges. Johnnies from Walmart will be between \$20 and \$25; Willis’ from Zellers will be between \$35 and \$40. If they pool their money, what is the least amount they need in order to pay for both exactly, without change.

• David Wees wrote:

I think those are both good examples of open-ended problems which fit into the curriculum nicely. I believe that in grade 1 they start looking at adding coin values together, and in grade 2 they continue that process. Thank you Gary.

• Here are some resources I love using with my kiddos:

Just for the Fun of It! Book 1 and Book 2 – AIMS Education Foundation
In the Balance Algebra Logic Puzzles – Creative Publications
Get It Together Math Problems for Groups – EQUAL

I’m not sure if you can find this one but it is from Oregon Council of Teachers of Mathematics:
OCTM Intermediate Problem Box Problems from TOMT, 1980 to 1992
Here is the link to their publications: http://bit.ly/Ivrrvq

I use these resources with my kiddos.

• Thanks for sharing! I am a fan of open ended math questions for student to work on, because not only are they searching to solve the problem through critical thinking, but they are also creatively thinking in their own way, and I believe this type of example fits in well to a type of math examples students are personally engaged in as they are trying to find their own solution to the problem. Open ended math examples are great! Thank you for sharing!

• Thanks for sharing! I love open ended math questions as they give learners a chance to explore concepts for themselves and come to their own understandings and conclusions. As a tutor I find critical thinking to be one of the biggest skills kids lack across the board; this is a great task to help develop these skills.

• Thanks for the post! The thing i love about open-ended math questions is that it really promotes the exploration of critical thinking, as well as creativity. These concepts are so important for the intellectual development of students, and help with all subjects and ways of life; not just math. This is something that I believe needs to be focused on more, as I agree with what Moriah said about students seeming to really lack/struggle with critical thinking.