Tag: group work

First-day-of-class Icebreaker

Yesterday was the first day of class! I’m teaching Calculus I for the third time, but this time as an active-learning style course. My students will be working in small groups every day in class, so I wanted to use an icebreaker to help them get used to working together.

A group’s first attempt at the icebreaker

The groups were randomly assigned (which is really important for making everyone feel included, and keeping people from just working with their friends every day). Each group of four had to share their names with their group, find something they all have in common, and find something that’s different about all of them. This part I’ll probably continue every day for the first several classes.

A group using a different building design than everyone else

Then their task was this: given 30 colored notecards and a ping-pong ball, build the tallest tower you can using the notecards that will support the ping-pong ball. The teams had 7 minutes to work. The tallest tower ended up being 5 notecards high (the taller towers couldn’t stand up for more than a few seconds at a time). This was a great activity for students who don’t think they’re math-minded to use some of the math/physics/building skills they have.

A group discussing if their design can be improved

After the activity, we talked as a class about applying lessons from this activity to daily group work. The students recognized the importance of

  • listening to everyone’s ideas
  • making sure no one was left out
  • accepting feedback and even criticism.

Overall the students really enjoyed the icebreaker, and I would do it again!

Emory Math Circle example lesson

Being a part of Emory Math Circle is one of my favorite things about being a grad student. Emory Math Circle is a free math-enrichment program for middle and high school students, that takes place on weekends at Emory. Lessons last around 75 minutes and are geared toward developing logical thinking, creativity, and skepticism in our students. I’ve been an instructor for the math circle since 2017 and have been its director since 2018.

It can be difficult to describe Emory Math Circle, so let me show an example lesson instead. One of my favorite lessons is Unfair Thrones. It’s designed for middle school students with some problem-solving experience.

First, the Story: An empress has two children who will inherit her empire. To prevent future war, she wants to build two thrones for them that are as even as possible (having one throne larger than the other sparks jealousy, and then war). So call down two students to sit in chairs in front of the blackboard. Put a fraction bar above each chair, and have students call out numbers from 1 to 4 to fill in the fractions. So you might get 2/3 over one chair and 1/4 over the other chair. You can’t reuse a number, and if you get an “improper fraction” (say, 3/2) declare, “This throne is upside down!” and flip the fraction. Now, what is the difference between the two fractions? In this example, it’s 5/12. Can we make the thrones more “fair” by shrinking the difference?

This is the main goal of the activity: to find the minimum difference between two proper fractions that use the numbers 1 through 4 exactly once. I love this activity because finding the solution (for this first question, at least) isn’t so hard, but justifying why it’s optimal is harder. Many students will say 1/3 and 2/4 are the closest you can get, but why? Force students to challenge their own answer! Can they enumerate all the possible combinations fractions? This would be one way to prove their answer.

Just as the students are all solving this first questions–and the whole time they are working together, practicing explaining their thinking–the empress has another baby! Joyful news! But now we need another throne. This time, we will have three thrones, three fractions, using the integers from 1 to 6. We want to minimize the maximum difference between each pair of thrones. For example, if we write 1/2, 3/6, and 4/5, then 1/2 and 3/6 have no difference–they’re equal!–but 1/2 and 4/5 differ by 3/10. This is the maximum difference, and we want to minimize it. Mini-max problems are likely new to every student in the room, and will be a struggle to them at first.

Of course, we can continue adding new children as long as the lesson last. Some extension questions we might ask are:

  • As the number of thrones increases, can we find a set of fractions that is more and more “fair,” or does the difference between thrones sometimes increase when a new child is born? (I.e., are the solutions monotonic decreasing?)
  • If the empress has n children, how many possible sets of fractions are there? Do we need to check all of these to find the solution, or are some set-ups never optimal?

As a warm-up to this activity, I give students this worksheet as they come in the door. It reminds them how to compare fractions, and how to find their sums and differences. The students should have these skills already, but it’s still good to spend some time discussing the warm-up activity. You may be surprised how many different correct ways students can compare the sizes of two fractions.

Calculus project – real-world data

I recently assigned and graded a take-home project for my Calc I class. Here is the assignment I gave them:

This is a group assignment.  You must find a partner in the class to complete this assignment with.  If you cannot find a partner, let me know and I will help you find one.
1. Find a real data-set that represents some quantity measured over time.  This can be population data, the position of some object/particle, height, weight, temperature, etc.  Look at the exercises from section 2.1 for examples of the type of data you’ll need.  You should provide the original data and a citation for where you found it.
2. Sketch a plot of the data over time.  This can be done by hand or on a computer, but it should be neat and legible.
3. Using the data, pick one of the times.  Then find the average rate of change for six different time periods, where the first time you pick is one of the end points for each time period.
For example, if your data uses the times 1900, 1910, 1920, 1930, 1940, 1950, and 1960, you might choose 1920 as your fixed time.  Then you’d find the average rate of change for 1900-1920, 1910-1920, 1920-1930, 1920-1940, 1920-1950, and 1920-1960.  Show your calculations for at least one of these, and create a table displaying all of the results.
4. Estimate the instantaneous rate of change at your chosen time, and justify where this number came from.  You should write a few sentences here explaining what the instantaneous rate of change represents and how you tried to find it.
5. Discuss briefly the difficulties in using discrete data to estimate instantaneous rates of change, and suggest in what ways the data collection could be changed to make your job easier.


If you have any questions about this assignment, please ask me, sooner rather than later.

I think my instructions for part 3 could have been more clear, but the class seemed to understand, and I was happy with most of their submissions. Several groups happened to pick “population of Atlanta” as their data set, but some were more creative, and this helped me get to know their personalities better. For example, one group did “number of YouTube followers PewDiePie has per month, so now I know these two students are gamers.

I was also happy to see different methods for answering part 4. Some students took the left and right estimates and averaged them (e.g. averaging the 1910-1920 and 1920-1930 rates), while others found a middle rate (e.g. calculating the 1910-1930 rate). People who chose an endpoint for their fixed time struggled here, so in the future I might recommend they always choose a middle time to fix, but it was nice seeing how these students justified their answers.

The answers to part 5 were of varied quality, but I generally saw the answer I wanted: having more frequent data points would help get a more accurate instantaneous rate of change. I think that it’s important for students to interpret their answers in words instead of just providing a number, which might be meaningless to them.

I liked this project for several reasons, not least of which being that now every one of my students knows someone in class, has their contact info, and knows where/when is a convenient time to meet with them. Studies show that one of the best indicators of student success in math classes is studying with peers. As I assign more projects throughout the semester, I’ll probably require students to have new partners each time, so they can continue to meet more classmates and hopefully find one they work best with, with whom they can study in the future.