# Engaging students: Combinations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Taylor Vaughn. Her topic, from probability: combinations.

How could you as a teacher create an activity or project that involves your topic?

One interesting way that I thought about introducing combinations is bringing in combinations that students do often, but do not really think about. When it is known that the family is going on vacation, as a girl, the first thought is “what am I going to wear?” Being a girl, I was always told that I cant pack as much as I wanted to because I also wanted to bring extra clothes just in case I didn’t want to wear what I had planned for that day. One activity I thought bout is actually bring in a suitcase to class with clothes and try and plan a 3 day vacation and figure out how we, as a class, was going to pack this suitcase. I could include different scenarios such as, if the hotel has a laundry room, and how would being able to wash clothes and put them back in the suitcase change how we pack. Also, what happens if we add shoes and socks? How would this change affect the number of combinations we can have? I think it would be really cool for students to touch and play and bring in ideas that they don’t necessarily think has anything to do with math.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Everyone listens to music, but there are so many different types of genres, artists, and songs. Have you ever thought, “Will we ever run out of new music?” Well someone by the name of Michael has. He has done the research of what others say about the math of the order of the notes and how many combinations of these notes can we get that will create a new song

One activity that could be done after the video is given 8 notes, how many different measures could students in the class come up with. Then the whole class could see how many people got the same measure or did everyone get something completely different. Then you could also ask “Did we cover all the possibilities? How do we know? How can we show this mathematically?” Lastly, if there are so many possibilities, why are there so many songs with the melodies? There is a video that has one melody and sings a lot of songs to that one melody. (PG-13 Warning: gratuitous cursing near the end of the video.)

The one thing I didn’t like about the first video is the length and he makes connections about songs that are really outdated. SO this video has songs that will relate closer to this generation of students.

How can this topic be used in your students’ future courses in mathematics or science?      In school, students that didn’t like math the way I did always asked, “Well when will we ever use this again?” Well even though we use combinations more than we think, it can also be used in later math classes. Ever thought that combinations had anything to do with Pascal’s triangle? What is Pascal’s triangle, you may ask? Well according to Math Is Fun, it is a pattern of numbers where the starting number is 1 and “each number is the two numbers above it added together (except for the edges, which are all “1”).” Well what if you are asked to find entry 20 of the triangle. The one thing I would do would keep the pattern of the triangle and write out all the entries until I got to that number, but using combinations you can get any entry you would like without writing all the entries out. The formula is where n is the row and k is where it is in the nth row.

$\displaystyle {n \choose k} = \frac{n!}{k! (n-k)!}$

While teaching this, I would definitely talk about factorial and how it relates to the lesson.
References

Stevens, Michael. “Will We Ever Run Out of New Music?” YouTube. YouTube, 20 Nov. 2012. Web. 02 Sept. 2015.

The Axis of Awesome. “4 Chords.” YouTube. YouTube, 20 July 2011. Web. 02 Sept. 2015.

Pierce, Rod. “Pascal’s Triangle” Math Is Fun. Ed. Rod Pierce. 30 Mar 2014. 3 Sep 2015 http://www.mathsisfun.com/pascals-triangle.html