# 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.

# How I laid out the square

Here goes my attempt at explaining how we laid out a square parallel to our school around a circle with diameter of 21′. I must confess that I did the math and started by having some students help me in class. I ended up having two students stay after school with me for about an hour to finish the layout.
First,I may have been unclear in the previous post but the circle already existed that the square needed to go around. So we measured 12′ 6″ off the building in two spots outside of the circle to establish the east line of the square parallel to the building. We put stakes in and ran a string line. Next we measured over 21′ from that line to establish the west side of the square also parallel to the building. That was the easy part. The tricky part was finding the corners.

(The building is on the east side and west is up on this sketch)

Our reference points were the furthermost northern and southern points of the circle. These points are the midpoints of the north and south sides of the square. But how did we establish  another point to create a line that it is perpendicular to the east line? If you are thinking Pythagorean Theorem that is part of it, but how do you find the corner? You guess, of course. Mathematicians, in the real world sometimes you have to estimate.

We multiplied a 3-4-5 triangle by 3 to get the dimensions 9′-12′-15′. We measured 12′ off from the east string in line with the midpoint. Then we held a string from the east string to the west string so that it barely touched the stake and we “eyeballed” it square with those lines. We marked the intersection between this string and the east string with a sharpie on the east string. We measured over from this mark 9′ and put another mark on the east string. Next we measured from this mark to the stake (the hypotenuse) and it should have been 15′. Of course we guessed on our corner point so we were off one inch and it measured 15′ 1″.  We corrected this by shifting the corner and this point north 1″. Now the hypotenuse measured exactly 15′ and we put a stake in at the northeast corner.

Now it gets easier. We pulled a string from the established northeast corner to the west line so that it barely touched the stake at 12′. This allowed us a very good estimate of the northwest corner. Next we measured over 21′ from both northern corners on our string lines to establish the two southern corners.

Our last step was to measure the diagonals which should be congruent. We ended up about an inch off so we double checked our overall dimensions and found one stake was leaning in so it was not quite 21′. We straightened it and our diagonals were within 1/2″.

The next day we measured over 2′ 6″ from all four sides of our square and pulled lines to find their intersections which gave us the four corners of our outer square. I really do not think that this problem was solvable by my 8th grade class but I wish I would have taken the time to walk them all through it.I think it was a great opportunity to show my students that math does apply to real problems in blue-collar jobs not just at universities or in a lab somewhere.

PS If I was doing it over I would have just “picked” center point on the east line pulled off the building where it touched the circle. Then I could have measured 10′ 6″ both ways and established both of the east corners and used Pythagorean theorem to find perpendicular lines off from both corners. It would have been much easier 🙂

# Construct a square

My landscaping class has been full of interesting math. So I thought I would try my hand at a version of WCYDWT We created this outside garden space last year:

Unfortunately over the summer some vandalism left this:

So this year’s landscaping class had the task of re-designing in Google Sketchup with the following goals/restrictions:

• Design a concrete path using a maximum of 2 cubic yards of concrete
• The thickness is 3″
• All of the broken and whole tiles need to be embedded into it.
• Width must be a minimum of 2′ 6″
This open ended problem using a new tool (Sketchup) was very challenging for my 8th graders. We went outside, measured, and sketched the dimensions on graph paper. We then drew the school and existing landmarks in 2D using the tape measure too in Sketchup to make our model exact. Then the students drew their ideas for creating a new space. Their first attempts were much too large. We spent the next day figuring out the maximum square footage for two cubic yards of concrete (I did not plan this out well enough and I ended up doing most of the work-too helpful). Then they re-drew. I then tried to get the students to find an approximate area of their designs by breaking them into estimated rectangles (Again I should have planned better and been more clear in my expectations). This was the winning design:

(Not my favorite, others were a bit more creative/artistic, but I wanted to give the students the right to pick)
The next day we went outside to stake it out. So how would you “construct” a square that is parallel to the building around an inscribed circle with a diameter of 21′ ? Remember you are not drawing it, but actually staking out the four corners. Give me your solution and is this problem accessible to the average 8th grader?

My solution in the next post.

Spoiler alert: I was way too “helpful.” Unfortunately that is a theme of this project but that does not mean you can not learn from it. In my defense I had a different half of my class for multiple days due to the non-stop pullouts for Earth Day movie, Relay for Life, a trip to Michigan State University, students planting trees, chicken pox, suspensions, etc. and I need to get to the digging and building phase before we run out of days in the school year.

# Better "homework" practice

I was just venting Friday about how when I teach a new concept in my math class the majority of the students do not seem to be listening very well. When I have them start working on their own problems, too many of them need me to re-teach to them. I enjoy doing this but I find that I run out of class time before I can help them all. So my first solution is to pair them up and have a few of my students that “get” concepts quickly help those that tend to struggle.

Then I found a great resource to help on twitter. That Quiz is a math site (and some geography, science, and vocabulary in English, Spanish, German, and French too) that I learned about from @karlyb. It covers many of our math topics and is designed for teachers to make and give quizzes. My purpose will be a bit different. First of all I can have students select specific problems related to our current unit. But what I like about the site is it immediately gives feedback on whether they got a problem right or wrong. This will serve the same purpose as giving students the answers to their homework ahead of time as recommended by Matt Townsley. The problem I have with just giving them the answers ahead of time is that this unit (area, perimeter, volume, surface area) is so easy that it is really just memorize the formula and plug and chug.

Therefore I can have students practice on this site and they can self-assess the areas that they understand and those where they need help. I think I will be using this site as a review tool from now on. Then I can spend my time helping re-teach the concepts that they tell me they need help on.

# Math History and Pre-tests

On Friday I had my students write their “math history” and then take a pre-test on factors and multiples. I know this is probably not revolutionary to many of you, but it was new for me and I am very happy with the results.

First I had my students tell me their “math history.” I got the idea from Glenn Kenyon’s blog. I just had them write one paragraph of how they did in math class in elementary school and how they feel about math as a subject. They were very honest about their past performance and grades. Many really like math and say that they are good at it. A few others have struggled and not surprisingly do not like math very much. None of them sounded like they have given up though, and were hopeful that they could be successful this year.

The pre-test was simply 7 questions covering the basic topics of our unit: prime and composite numbers, factors, multiples, LCM, GCF, and prime factorization. We did not review anything ahead of time so it was cold turkey. I could now compare my students self-assessment of their math abilities with their knowledge on our first unit. Most of their abilities matched what they said about themselves.

I also found out the overall strengths and weaknesses of the class. They did best on primes, composites, and factors. Some did not know multiples and only one student got the correct LCM of 4 and 5 being 20.

Based on their self-assessment and pre-test I know now where to spend my focus in this unit. Furthermore I have a good starting point of their math abilities whereas beforehand I had no clue of their level. I am now able to divide them into pairs for this week by their abilities.

I highly recommend both of these strategies to start a new class or new unit.