Category Archives: geometry

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.

Math from Canada

My student teacher, Kyle Webb, in Canada sent us a video problem to solve. Check it out on his blog.

We solved and sent a screencast back to him. This was my favorite thing that we have done this year so far.

Wood chips

I should have wrote this before my last post to better describe how my math class has been going. We are currently working on a basic geometry unit of area, perimeter, volume, and surface area. I was excited because this would be “easy” to teach with out a textbook.


I started off by explaining a problem I had of needing to know how much wood chips I needed to cover some landscaping that I did with a class last year. We discussed what we needed to know to figure this out and then went outside and measured the circle. They told me we needed to know the length and width of the circle so I had them measure where they told me those were. I did not try to correct their improper terms. We came up with 7 meters and 6.9 meters. One student noticed that they were approximately the same. We ran out of time for the day.

The next day we started in class and discussed the wrong use of terms. I had them search and find the area of a circle and we talked about what the formula means. Then I asked which of our measurements was right. They argued that 7 meters was correct because it was a whole number. We finally concluded that we did not know which one was right and that we had to go outside and take more measurements and then average them. We also talked about the fact that the circle was eye-balled when created and not perfect.

We ended up solving the problem and then solved two more wood chip problems for some rectangle gardens. Through out these lessons I asked lots of questions and guided their learning but did not give out any information. The students either came up with the answers themselves or surfed the web for them.

My evaluation of this teaching method was that I did not see the high engagement that I had hoped for by the class. My top students were with me and the bottom students seemed to be daydreaming or not really participating. I don’t have a great story of the student who always fails getting excited and being successful.

Next I needed to cover parallelograms and triangles which are harder shapes to find in the real world. So we did some visual proofs together in Geometer’s Sketchpad so they could play around and see why the area formulas work.

Again I have to give the district unit test so I gave them some practice problems with area and perimeter of parallelograms and triangles. They were totally lost. They could not remember the formulas or even use them when I gave the formulas to them. I ended up going around the room and individually teaching how to use the formulas.

We measured a bunch of food boxes and found their surface area and volume. I demonstrated how to use the formulas on the board and the majority of the class still needed me to re-teach individually.

So in response to the comment from Matt Townsley on last post about teaching at a deeper level. I have tried (I am not giving up!) but in the end I have to prepare the students for the district test. That is why I found ThatQuiz to be a useful tool for students to check their work on the basic problems that they need to know. I know it is not technology integration but doing the same old thing just on the computers. I do think the immediate feedback to students of whether or not they found the right answer is helpful. And unfortunately these are exactly the kinds of problems on the required tests.

All right push me back some more readers 🙂