Thursday, April 28, 2016
Day 147: Pythagorean Theorem Proof 1 & More Systems
Students worked on their first proof of the Pythagorean Theorem. You start with 4 congruent right triangles with sides labeled a, b, and c. They arrange them with the 90 degree angles matched up with the corners of a square. They are asked why is the middle, empty, unshaded area c^2? I had to poke and prod students to go further with their explanations. Each triangle had its hypotenuse, c, facing in, to form 4 congruent sides of a square with 4 90 degree angles and 4 sides of length c. To find the area, they said length times width or base times height to get c*c giving them c squared.
The text prompts them to reason if the area within the square will change if you move the triangles around. They reasoned that the area of the triangles hadn't changed and neither had the square, so the area wouldn't change. They then rearranged it like the second photo below this one...
They were asked why the empty or unshaded area was a^2+b^2. I liked when students reasoned that the hypotenuses were facing each other so the sides of b length were exposed to form the smaller square of area b^2. The smaller square had side lengths of a, making it a^2. Therefore they concluded, that the original square, c squared, was equal to the sum of a^2 plus b^2. I asked students if they were like terms, or were the same variable. They said no, which is why we can't add them together as variables.
In accelerated students grappled with a challenging system that was an exponential equation and a linear equation. It couldn't be solved algebraically, though some students were determined and tried figuring it out with a TI calculator. I encouraged students to graph both functions to find the intersections. I saw that some students graphs of 2^x-3 were looking like parabolas, so I stopped the class to review negative exponents by reviewing exponents of base 2.
I wanted to make sure they remembered the relationship between 2 squared and 2 to the -2nd power. They are reciprocals of each other.
Then students worked on sketches of possible intersections. As you can see below, one situation called for 2 parabolas with 1 point of intersection. Many students thought of a parabola and an upside down reflected parabola. Students were impressed that Aaron came up with 2 side by side parabolas that intersected only once.
One student pointed out that a line and a parabola that intersect only once would be called a tangent line.
Then later that evening I tutored an Algebra 2 student on Graphing the Sine function. He challenged himself to do the same function except with the cosine function. This is a great activity by Shelly Carranza. Go to teacher.desmos.com and search "Graphing the Sine Function."