# Unauthentic, Imaginary World Math

 Photo by Cowboytoast

“Real world” and “authentic” are two of many educational buzzwords overused right now. What if instead of making sure that everything has a truly “real” context we give students a creative opportunity to explore the “unreal.”

The inspiration for this post comes from a new blog by Randall Munroe, author of xkcd, called What If?. In this blog he answers hypothetical questions by doing the actual math to answer them. So far he has shown things such as how much force does Yoda have? and what would happen if you gathered a mole (unit of measurement) of moles (the small furry creature) in one place? These questions are not real or authentic but the math and science is.

But these questions are fun and interesting! Students love to talk about fantasy and science fiction such as zombies and vampires.

So why not expose your students to a few of these kind of questions and have them try to “prove” their answer. Afterwards show them what Randall Munroe came up with. Then have students come up with their own questions and write out their reasoning and solutions. This activity would tap into their creativity but also demonstrate their mathematical computations and more importantly their mathematical reasoning. It also would be a literacy task in math. Finally and most important in my opinion it may also be an avenue to engage a student’s passions in math class that Jeff de Varona has been asking about.

# 10-10-10

So Binary Day is a cool day. It is a lucky day to the Chinese to get married. Why not have a baby?

You may have to click to my post to see the video. It is not showing up in my Google Reader.

Turns out locally (story) that one couple has had a baby each of the last three years. The birthdays are 08-08-08, 09-09-09, and now 10-10-10. An interesting story and the first thought I had are what are the odds of that?

I think this leads into an interesting problem/ discussion. I would ask students: “How random was this or do you think that they cheated?” Students could decide if they were “aiming” for these dates and if this is a truly “random” problem.

What do you think? Is this worthy question for your class and how would you use it?

# Results of Math Icebreaker

Normally I start off the year with a boring going over the rules/syllabus the first day. I decided not to torture the students this year but to start with a hands on activity. So I tried the making the rectangle activity on the first day with my two 8th grade technology classes. By the way I changed it into a theme for a complete geometry unit if you are interested. I also made a chart for students to fill in with their data.

The students were interested in the pictures. They really had no idea about the construction processes. When we went outside most of the students immediately made the mistake of confusing area for perimeter. I asked lots of questions to re-direct them and to get them to re-think about the difference. Then the groups staked out quadrilaterals, some more square than others. One of my favorite parts was watching a student lay down on the ground to estimate six feet instead of using the tape measure. I challenged them to consider if their shape was a “perfect” rectangle.

At the end of the hour the students measured the sides and the diagonals. I had to go to each group and help with this. At first every group just wrote down the measurements that they thought they had measured. I told them that they must measure what their sides actually were and write down their imperfect measurements not what they meant them to be.

The next day we discussed how to determine whose was the best rectangle. We ended up having to define a rectangle which was a good activity for the students. I then showed them how the diagonals of a true rectangle are congruent and we talked about Pythagorean Theorem. I then showed them this video of an area proof:

We then talked about 3-4-5 triangles and I had students come to the front of the room and create a perfect 90 degree corner by holding three tape measures at 3,4, and 5 feet.

I was disappointed in the student’s prior knowledge about the Pythagorean Theorem. None of them had heard of it even though a few of them had Algebra last year. I will repeat this on the first day of my class for the next three quarters. I will be curious to see if students do better later in the year after they have been exposed to more background knowledge in math class.

# Math Icebreaker

Lots of teachers use icebreakers to start off the year. I use some with mixed results. Sometimes I feel like they are a waste of time. I like group problem-solving activities better than “find someone with the same birthday month as you” type.  Well I had an idea for a way to start off this year at my construction job this week.This idea is in the WCYDWT philosophy but is more hands on than video based. I also spell out the problem as a challenge.

At my job we are forming walls on a funky house right now. It is two rectangles crossed in an X. The angle of the intersection is 60 degrees. It was not laid out by surveyors so we had to set stakes to find the exact dimensions and angles. It was quite a challenge. (I had to laugh when the homeowner, a computer programmer, pulled out his compass app on his iPhone 4 to check our pins. He also wanted the main part of the house parallel to a fence row 300 feet away with hills and trees in between.)

So my “icebreaker” is to give groups of students six wooden stakes, a tape measure, and a hammer. I will also allow them to use calculators, textbooks, and perhaps their phones as resources. The problem is for them to lay out a perfect rectangle with an area of 48 square feet.

I like this problem because it is cooperative, real world, outside, hands-on, and has multiple ways to solve it. I picked 48 square feet because students can use a 6-8-10 (3-4-5) triangle to find 90 degrees. They also can check the diagonals for congruency to find out if it is square. Students can also just use trial and error to try to make their rectangles better.

I see this as a 9th grade level or higher problem. It allows for review of Pythagorean Theorem and properties of rectangles. Of course discussion of different solution strategies and how to “prove” a rectangle is perfect at the end are a critical part of making this activity successful.