# Open ended problem: Boarding passengers onto planes

This might be an interesting conversation starter for a class: what’s the best way to board an airplane?

The students will probably have lots of ideas to discuss, and best of all, every single one of them will be better than what the airlines actually do.

# Open-ended problem: Do groups of people follow predictable patterns?

My sister and I were walking our kids back from a trip to Science World, when we passed a park sparsely filled with people. My sister looked at the people sitting in the park, and wondered aloud, “I wonder if you can use mathematics to figure out how far apart people will sit on a lawn?” I looked carefully at the park too, and noticed that everyone seemed to be carefully at an maximal distance apart from anyone else on the field. I am particularly excited about my sister’s question, because she has always described herself as “not a math person.”

I decided to generalize her question, to “do groups of people follow predictable patterns?” This would allow for exploration in a wide variety of ways, for example:

• Do people tend to follow the same paths when crossing open-space, like a field or in the meeting room of a train station?
• How random is the motion of people as they sit waiting in a theatre?
• Can you track use of phrases of language through groups of people?
• What similarities exist, if any, between the networks of relationships each person has?

# Open-ended project: Mathematical modelling with student-collected data

For this project, students collect data on a phenomena in which they are interested, with teacher guidance, and then look for patterns in this data. They can use a variety of different means to collect their data, attempt to find a model which matches the data, and then present what they have found. Ideally students would go through multiple revisions in their data collection model to collect as clean data as possible, particularly when they end up with major problems with their results.

Some specific things which students might model:

• The growth of plants over time;
• Population growth of bacteria in a petri dish;
• Temperature during the day;
• See more ideas here.

If you want more information on what mathematical modelling looks like, see this document or this document.

# What is my area?

I love this activity by Bruce Ferrington with his students.

What Is My Area?

Wait – before you say anything – I know your body is 3-dimensional and area is a measurement of two dimensions – so I probably should say, “What is the area of my silhouette?”

So…

What is the area of my silhouette?

In the past week we have measured height and width, talked about Vitruvian Man, combined our heights, compared them with other classes.

Time to have a look at comparing areas.

My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don’t have to cover their entire body in 1cm grid paper and count out each square.

# Open-ended investigation: The Math of Voting

Students could look at different ways of voting, and see what impact each of these ways has on a local election in their own school. They could compare the different methods, decide on ways to check for “fairness” of the election results, and even attempt to come up with their own system of voting. This project is likely to work better and have more interest from the students if they use actual data from things they are voting on for their experiments, as well as data from other sources.

# Open ended investigation: Exploring fractions in different bases

This project is related to the repeating decimals project I’ve outlined before (see this post) but with a twist.

What do fractions look like in bases other than 10 (for example: base 7)?

Instead of decimals, what would septimals look like? When we write 10.356, what we really mean is $10 + \frac{3}{10} + \frac{5}{100} + \frac{6}{1000}$. In septimal notation, $7 + \frac{4}{7}$  would be written 10.4. Can you extend this example?

Here’s a trickier question: how is 0.5 in decimal notation represented in septimal notation?

Is there a relationship between the number of digits it takes a fraction to repeat when it is written in expanded (in base ten, we call these decimals) form and the base in which it is being represented? Is it possible to write numbers in any rational base, for example 1.5? What about base $\pi$?

# Open ended problem: Dots and lines

In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.

This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?

# Collaborative Mathematics project

The Collaborative Mathematics project, created by Jason Ermer, looks like another excellent source of rich mathematical tasks for students. I recommend following the Problem a Day blog. Jason encourages the problems to be done collaboratively, hence the name of the project.

Here is a sample:

Notice how Jason takes a closed form question (What are the four digit numbers that can be flipped when multiplied by four?) and converts it into a much more open-ended investigation simply by making the restraints less restrictive. This is a useful general strategy you can use to make closed-form problems more open-ended.

Thanks to the Math Munch for sharing this project.

# Excellent resource for math problems

A colleague of mine at work shared this excellent resource with me for interesting and perplexing mathematics problems. The Galileo project looks like it has about 100 interesting mathematics problems for students to do for a variety of different age levels.

How many parents do you have?
How many grand-parents do you have?
How many great grand-parents do you have?
How many great-great-grand-parents do you have?
How many great-great-great-grand-parents do you have?
….

Wait a minute! Do you see a problem with this?

# Open-ended investigation: Fold and cut

If you can fold a piece of paper anyway you like and as many times as you like, and then take a pair of scissors and make just one straight cut, what possible shapes can you make?

For example, I’ve folded a piece of paper (shown below) three times.

Now I make the following cut…

What shape will this result in when I unfold the paper and lay it flat?

* I first learned about this idea from the movie “Between The Folds.” Here are a few more examples from Dr. Erik Domaine’s website.