Category Archives: authentic learning

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 🙂

Warning: The grade you have received may not reflect your actual level of learning in this class.

As some of you know this is my first year teaching math. I am currently teaching one section of 6th grade math. The rest of my assignment is 6th, 7th, and 8th grade technology which I have taught for six years. This new math class is the source of my focus on grades and grading. In my district we have standards-based report cards and in math we have district tests for our required assessments. For this post I will focus on my technology class and assessment (I will save math for the next post).

My Technology class flies under the radar. We have standards that reflect what the class was like five years ago but not what it is today. I and my fellow middle school technology teachers use project-based learning and computers to challenge kids with fun and relevant learning. I truly have more freedom than almost any other teacher in my district to teach whatever I want however I want. I am accountable to my principals who are happy to see children “doing” creative things. We build pop bottle rockets, balsa towers, hot air balloons, pneumatic devices, and egg drop vehicles. We use Lego Robotics and some math software games. Starting last year students blogged and use Google Docs. This year we are using programs such as Google Sketchup, Pivot, and Scratch.

I grade of course in Technology because I have to but my grades are either a rubric of checklists or once in a while based on reaching certain levels. So many of my grades reflect effort of students to complete the projects rather than measure learning. I am ok with that I guess.

For example some excellent students attempted some unique balsa tower designs (the picture is one of them) that totally failed when tested. By the way, most of the class voted their towers to be the strongest before we broke them. They did not receive an A but I would argue that they learned and taught the rest of the class more about good design that anyone else. Why, because they took a creative risk and tried something the rest of the class was unwilling to do. We do not have time or $ for materials to have students build multiple towers to improve their designs which would be the best way to show their level of learning.

Even more difficult for me was when my students made their own Pivots, Sketchup, and Scratch program. We created rubrics together as a class and then I made a grading form in Google that the students embedded on their blogs.

How does one grade creativity? What makes my opinion more valid than anyone else’s?

Therefore I was just going to count the students’ assessments of each other. This did not work out as planned as too many of them did not grade each others’ projects because some were completed late and some did not take the rating seriously. Basically I question the whole point of this grading as a waste of my time.

My ideal system would be to share these tools and projects with students and have them complete them as creatively as possible to the best of their ability. Their “reward” would be the learning that they experienced as a class. Would they all learn the same things or even the same amount? No, just like now they would learn based on the amount of thought and effort they put into their work. If they slacked off then they would learn less; if they worked hard then they would learn more.

I really don’t know how I can accurately measure their actual learning anyway. Grading feels like a game to me and some of the greatest projects like this and this happened for no grade or extra credit. I fail to see how the “grades” that I assign to them contribute to their learning experience and oftentimes do not adequately reflect their actual learning. I also know that grades do not motivate my at risk kids at all, and they motivate the “top” students just to perform for me not to actually learn.

The ultimate thing that matters to me is that students are challenged, given a chance to be creative, and explore in a hands-on way math, science, and technology. The rewards are internal for the students who give their best. Those who just go through the motions miss out no matter what their report card says.

Next post I will explain the conflict I feel comparing this class and grading to my math class.