Area of a Triangle Equals Base Times Height Divided By 3
A few weeks ago, I asked my son, who is in 7th grade, what are you doing in mathematics? He responded with a tone of non-questionable flatness, “area of a triangle”. Let me first say that my son’s teacher is absolutely amazing — buckets of empathy and even weaves Fortnite ideas into her classroom. However, I knew where this lesson on triangle area would go — and not go.
Like thousands of students, my son was given the formula for the triangle, and then spent two days doing worksheets plugging into what shouldn’t have been an inert transmission of information. Once the area of a triangle is established, the lesson is over. There is nothing left to be done besides some problems of dry algebra in which students have to find height or base instead.
Much of mathematics still starts like this in many classrooms all over the world — with the punctuation mark. This is where the idea of direct instruction feels so much like a rotary dial phone in the world of math education.
The lesson is over. How the lesson should have started with is something like this…
The entire article from 2014 can be found here. A simple mathematical relationship, but with two completely different outcomes. One is a cliched, dry, and narrow excursion through the nightmare of worksheets. The other is a natural, historical, and playful romp through what mathematics really is — bustling, open-ended creativity.
Our presentation of mathematical relationships/formulas is so sterile, that we could easily change the constants, and there would be no effect on the outcome other than a different answer. I mean, what if we taught the area of a triangle with a wrong formula? How long would it take for a student to comeback to us and say that was incorrect? I am scared to think of the proposition that nobody would challenge the idea, and even if they did, not from a mathematical one — just on getting the facts wrong.
Over the holidays, I plan on introducing The Pythagorean Theorem to my kids — but, in the most roundabout, “weirdo” way! The Geoboard is a wonderful resource for kids to play with shapes. And, as you can see from the above picture, even the digital ones feel playful and real.
I will ask my kids to construct as many different size squares as possible on the Geoboard. This is the answer I am kind of expecting.
These answers can be verified visually and backed up with actual answers of 1, 4, 9, 16. Doing this activity on a Geoboard also automatically creates the “backward L’s” — consecutive odd numbers — that are added to get larger squares. It is a nice ancillary benefit!
Of course I will naturally ask them if there are any more different size squares. There will be probably some humming and hawing, some trial and error, maybe even duplicates created, but I am not expecting a different size square — although anything can happen!
The next thing I will show them is this:
I am hoping for some push back here! “That’s not a square” or maybe “That’s the same size as the…ummm….one of the others”…
I am hoping that my son remembers Pick’s Theorem, something I showed him two years ago. I will mention that we can verify that this square is indeed a different size(different area) with Pick’s Theorem:
I usually use “e” for exterior dots. So using Pick’s for our polygon, which happens to be square, we get A = 1 + 4/2 subtract 1 = 2.
The area of this square is 2!
I will now ask my kids if the area of square is 2, what is the side of the square? What number multiplied by itself gives…2? While they are chewing on that, I will construct this.
Game on!
The two yellow squares have an area of 1. The larger square is equal to two. Hmmm…:)
This now becomes my portal into The Pythagorean Theorem. The formula takes a few seconds to write on a board. Doing a few examples takes a few minutes. Then, students are asked to do 50 more. In that period of time, a story like this could have been woven.
Until mathematics is no longer enslaved by tests and performance on these tests, then there will be little time for mathematical hikes likes this.
Fortunately, more and more math educators are changing the way their students intersect and explore mathematics. Yes, tests are still required. But, these are now feeling the pressure of the burgeoning open-ended ideas that are infiltrating classrooms all over.
Hopefully, in the not to distant future, something like the area of a triangle will be the end of the lesson — not the beginning.
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Sunil Singh is the author of Pi of Life: The Hidden Happiness of Mathematics(2017, Rowman and Littlefield) and Math Recess: Playful Learning for The Age of Disruption(2019, IMPress)
