Category Archives: problem solving

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 Should I Not Plan My Class?

This short post by Shelly Blake Plock has been on my mind since school let out. How can I not plan my class better? It ties into the post about Girls in Technology too. I want to make my class more student-centered and problem-based. I already teach the majority of my class as problem-based, but currently I create the problems for the students. I am thinking of ways to get students involved in creating the problems themselves.

Problems are supplies, unmotivated students,

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.

Tinkering Part II

Can’t get tinkering off my mind. Here are some other ways that we tinker in my technology class. This year I taught my first semester-long class in 8th grade. We did the same hands-on projects that I have always done in a 9 week class, but added some new computer applications. Three programs that we used were Pivot, Google Sketchup, and Scratch.

My teaching method was to have them download the program and play with it for a day. The next day we created a rubric of requirements together and then they went to work. We used Sketchup first and the students struggled with it. Since then I had a group of repeat 7th graders watch some tutorials on YouTube about Sketchup first and they have been more successful. Pivot is a much simpler program and they did very well tinkering with it.

I have used Scratch a little bit before with students and learned from the Sketchup experiment that students would need some support. I found four Google Slideshow instructions from Simon Haughton that taught students how to make an etch-a-sketch, race car maze, pong, and pacman games. Students followed these instructions and created the games. Those who finished early were challenged to make their own game. Only two students actually made something of their own. But to be fair it was the last week of the class so motivation to work was not super high.

Things I learned are that all of the students are willing to play at the beginning. But it is important that the task is at their skill level or that adequate support is provided. Students that struggle academically are often used to spoon-feeding or failure and give up quickly when not supported. The amount of support needed is difficult to judge and may be different for each student (Check out this John Spencer TAD talk video for a good explanation). I try to point students to resources first rather than helping them directly. I also have the students teach each other (and me) as much as possible.

One thing that seems to help is to start the first tutorial together as a class up to a certain point. It helps every student “get their feet wet” and builds important confidence in those that are unsure. Another technique I use is to announce to the class a problem that a particular student is having and ask if anyone can help them with it. A third thing that helped was to show examples of the best work from a previous class. My repeat 7th graders were not giving me much of a story line with their Pivots until I showed them some of the best 8th grade examples and they improved theirs immediately.

Students that are used to success in school often care too much about grades rather than creativity. They will faithfully complete the “lessons” and then help others, stall, or just sit there rather than try to create their own game in Scratch. I am now seriously considering a class with no grades to get rid of this problem. There would be no questions of “Does this count toward my grade?” or “How many points is this worth?” The class would be pass/fail based on did you attempt to learn? Experimenting and failure would be encouraged. We would talk about learning, not grades. Now to sell that idea to my principal…