Chapter 14: Hockey Team Task

This week in class, we examined the "Data Management" topic of combinations and permutations. During the debriefing section of the presentation, we discussed different ways we have learned, taught, or observed this topic being taught. One of the methods discussed involved the use of baseball cards and creating teams/batting orders out of the cards.

This concept really resonated with me, and I used it as inspiration to create my own lesson on combinations and permutations using hockey cards. It would go something like this:

In groups of 3-4, provide students with 15 hockey cards (7 forwards, 5 defense-men, and 3 goalies). Tell the students that a starting lineup consists of 3 forwards, 2 defense-men, and 1 goalie.

Ask the students, how many different "starting forward lines" could you create with the players in your deck? 

Ask the students, how many different "starting defensive-pairs" could you create with the players in your deck? 

Ask the students, how many different "starting goalie options" would you have with the players in your deck? 

These questions could all be answered using combinations, as the order of selections doesn't matter.

Next, tell students that if a hockey game remains tied after regulation and over-time minutes, they conclude the game with a shootout. A shootout is where 3 (or more if required) players (typically forwards) are selected to shoot on the opposing team.

Ask the students, how many different 3-man "shootout lineups" could you create with the forwards in your deck? 

Ask the students, how many different 5-man "shootout lineups" could you create with the forwards in your deck? 

Here, order matters, so students would need to use permutations in order to answer the question.

This task could be delivered as an introduction to combinations/permutations as outlined above, however, it could easily be extended into a task to be completed once some of these skills have been further developed. 

For instance, students could be asked to create as many different complete starting lineups (3F, 2D, 1G) as possible. This would required them to treat forwards, defense, and goalies individually before creating a full line-up. 

Or, you could break down the forward groups into left-wing, center, and right-wing positions, and the defense into left and right positions and ask students to create lineups according to this order. This would require students to treat each of those positions as individual groups, much like the scenario above. 

Or, you could spiral the activity to incorporate the statistics of players in order to create a team with the best potential of scoring or stopping a goal. There are plenty of free websites that track the statistics of players, such as http://stats.hockeyanalysis.com/. 

The potential for extension of this task is almost limitless. 

Again, the idea of using hockey cards really resonated with me. The traditional "playing card" tasks associated with permutations and combinations always seem to lack context that is relevant to high school students. Hopefully a task such as this would get students genuinely interested in the math they are exploring. 

Chapter 13: Spaghetti Trig

The spaghetti task is designed to help students better their understanding of the unit circle, as well as develop a relationship between the unit circle and the graphs of trig functions. Essentially, students either develop, or are provided with a unit circle. They are then asked to wrap a length of string around the circumference of the unit circle. At each of the ordered pairs of each special angle, students should place a "dot" on the string using a marker. Once this is completed, the students would straighten the string and glue (or tape) it to a piece of paper (see the image below).

Next, students will measure and break pieces of spaghetti that run from each ordered pair point back to the baseline (horizontal axis) of the circle. These pieces would be glued (or taped) to their corresponding point on the string. The pattern created would either be a sine or cosine wave (see image below) depending on whether students measured the distance of an ordered pair to the vertical or horizontal axis.

https://www.pinterest.com/lesfleur87/trig/

I found this task to be a super refreshing take on a unit circle activity. The context of this sort of task almost always involves a Ferris wheel, which means the activity is typically completed on paper, and best-suits theoretical thinkers (most schools do not typically have a Ferris wheel on hand...). This activity however, is highly differentiated; I believe that students who have more success with visual/hands on thinking will be able to grasp the innately theoretical trig concepts presented in this unit. Another benefit of this activity would be the fact that students actually create a product that can be kept or displayed in order to "recall" the concepts learned.

These are the types of tasks that mathematics educators should aim to present as often as possible. Students want to create things, they want to discover relationships, and they want to have fun! I truly believe that we can tap into this intrinsic desire to learn math through this type of lesson.

Thanks for reading!

Chapter 12: Utilizing the Context of Data

What percentage of your country's population is foreign-born? According to the data provided by "Our World in Data", if you are an American citizen, you probably over estimated the "correct" value. In fact, this data set suggests that many major countries (such as the Netherlands, France, and the United Kingdom) believe that the percentage of foreign-born citizens within their nation is higher than it actually is. This, and many other relevant data sets, can be found here.

There are three very important and beautiful facts about data: 

1) It is literally everywhere. 

2) Whatever the context, data can always be interpreted in a mathematics setting. 

3) Mathematics educators can utilize the context of data to extend dialogue well outside of the classroom.

Referring back to the question "what percentage of your country's population is foreign-born?", we can generate all sorts of math inquires. Let's take a look at the data chart provided in class:

The data modeled in this chart can be analyzed in many ways. For instance, students could be asked to determine which countries overestimated the share of immigrants in their population the most? The least? Why do you think each of those countries are where they are on this chart? Where do you think Canada would fall on this chart? The context of this data is relevant to current world issues; having students analyze this data should lead to important conversations that would have been difficult to facilitate in a mathematics classroom otherwise.

As mathematics educators, we need to shift away from using data sets such as "height vs age" or "shoe size vs height" in favour of those that explore issues related to class, race, gender, ability, age, faith, heteronormativity, language, social economic status, and mental health. Challenging "normal" ways of being and knowing within the mathematics classroom may seem like a daunting task, however, analyzing data with relevant context seems like a great way start rich conversations on equity issues. I look forward to practicing an exercise of this nature in a classroom of my own.

Have any other ways of incorporating/facilitating discussions related to social justice in the mathematics classroom? Comment below!

Chapter 11: Cup Stacking

This week in class, we looked at a seemingly simple question...

How many cups tall am I?

The thing I loved about this question, is how many ways a student would be able to look at this question, and how many ways an educator can facilitate the activity according to the information they provide. 

For instance, giving them absolutely no information, you could ask students what they would need to know to solve the problem. I would expect that they would want to know things such as:

-the teachers height

-the height of a cup

-how they are stacking them

You could then given them all of that information, however it still isn't enough to solve the problem if they are to stack the cups conventionally (see the image below).

         https://crazedmummy.wordpress.com/category/useful-for-teaching/

Concluding to the class that the cups must be stacked conventionally, you could then ask, what further information would you need to solve this problem? This conversation should lead to the height of the base and the lip of the cup. Given this information, students should be able to work through the problem, either numerically, graphically, or using a data table. Regardless of how the students solve the problem (assuming they are correct), they are exploring linear relations.

This is only ONE approach to the problem. Another great approach would be providing students with the height of two different stacks of cups. At this point, they would have to find the slope of a line using two points to solve the problem. Or, you could extend the problem by having students use different size cups. Or, you could actually just give the cups to the students and have them test their answers. Or, you could given them only a few cups and have them develop a solution on their own....

There are so many different ways this question could to presented, all of which should result in a unique learning experience. I find that as mathematics students/educators, we tend to value math questions based on their complexity, however, "seemingly simple" tasks such as this allows students to become more invested in the math. Stacking cups is tangible, and its "within reach" of students regardless of their academic achievement levels. They are not scared of the question. I think this is the key to unlocking the potential of students that identify as being "bad at math". 

Thanks!