- explore non-perfect squares on geoboards to find their area, and then the side length
- prove the Pythagorean theorem in one of two ways
- investigate what makes a triangle acute, obtuse, right or a non-triangle
- view the beginning of the Wheel of Theodorus project from today as well as some finished projects from some 6th period students earlier in the year
- see what Desmos Marbleslides looks like on a Chromebook
- learn about scientific and decimal notation form of everyday objects from posters
- view posters detailing equations that are always, sometimes, and never true
- view posters on linear equations and whether they have one solution, infinite solutions, or no common solutions
Today students estimated the flight distance from New York, New York to Honolulu, Hawaii. Students that estimated it was twice the distance from Los Angeles to Virginia were pretty successful.
Then we started the Wheel of Theodorus project. I would eventually like to make a stop motion animation tutorial on how to do this, but I directed students how to start it with direct instructions under the document camera with choral response questions and explanatory questions.
I had students start by folding a sheet of copy paper in half, unfolding, then folding it in half the other way so you have it in quarters, and a center point. I had them draw a point in the center. Then they drew a vertical line segment that was exactly 1 inch long on the middle crease. Then they made a horizontal one inch line segment on the horizontal crease. They then connected those with 2 more inch long line segments to make a 1 by 1 inch perfect square.
It connected to yesterday's activity, Patterns in Prague, because the design was made out of square blocks. I asked what the other shape was in the design (triangles). They said to make a diagonal line to cut in half. We did, that, then erased one of the triangles. It created a right triangle with legs of 1 inch. I asked them how much the 3rd side was, estimating. They said it was about 1.5 inches, definitely more than an inch. I related this to the baseball diamond, where the throw from 1st to 3rd is longer than the throw from 1st to 2nd.
Then volunteers explained how to calculate the third side by using the Pythagorean theorem, step by step. You get a hypotenuse of the square root of 2. I asked students if that was rational or irrational. Without a calculator, I wanted them to see that the radicand was not a perfect square so it would be irrational. We labeled it (I) for irrational. Then we made a perpendicular 1 inch line segment to the hypotenuse to create another leg, and then made a new hypotenuse, connecting to the center point. They told me again how to calculate this.
This is a great discussion because you have to square the square root of 2 and add it to 1 squared to get 3. Then it's the square root of 3. Some students quickly realized squaring the square root of 2 would undo the square root leaving you with 2. I related it to c^2=2. To undo squaring c, we must square root both sides of the equation, leaving us with c = square root of 2.
The next one is great because you get a hypotenuse of the square root of 4, which they see can be simplified to 2. Then they realize this is irrational because 4 is a perfect square, leaving you with a rational number 2. We measured with rulers and it was pretty close to 2 inches, as it should be. They then continued and I worked with students individually.
In accelerated students continued work on their posters. Students were allowed to use their smart phones to download the Desmos app to quickly see how the value of k transformed functions for f(x) + k, f(kx), k*f(x), and f(x+k). The investigations were very fruitful and I can't wait to see how their posters come out. They will continue working on Friday, and with an assessment on Friday, the work will probably spill over into Monday, with some time for some Post it note gallery walk feedback.