Tuesday, April 26, 2016

Day 145: Pythagorean Theorem Learning Goals / Success Criteria & Intersections of Quadratic & Linear Functions

Today was the second day of estimating plane flight distances. The benchmark was a cross country flight from Los Angeles to Virginia. That was 2227 miles. Students reasoned the flight to Chicago would be one fourth the distance so they divided the previous distance by 4. 

Students were continuing practicing using the Pythagorean theorem. So I used what I learned from my online class and introduced learning goals and developed success criteria with students using questions and creating buy in with the students. 

I began by drawing a triangle. I labeled one angle with a box denoting it as a right triangle. Then I referenced our investigation of acute obtuse and right triangles. I said that when you have a right triangle the side lengths have special names. They said the diagonal side is the hypotenuse. A few students asked if that's the longest side and other students chipped in that it was. 

Students volunteered the sides to the left and right of the right angle were calls the legs. I decided to review this from Friday because students in second period were solving for the hypotenuse, c, when given c and a leg. So, I made it more explicit in later periods.

To address the first learning goal I developed success criteria with the students that encouraged students to show work at an 8th and 9th grade level.

That began with writing the Pythagorean theorem. Next was substitution. Then simplifying by squaring. Then either adding if solving for the hypotenuse, or subtracting to isolate a variable if solving for a leg. Then square root both sides, to undo the squaring of the variable.

The second learning goal was to identify whether there answer was rational or irrational. I asked students how we could know if the square root of a number was irrational. Most students say it is rational if the square root results in a whole number. We have to be careful here though because the square root of 0.25 is rational because the square root is 0.5, which is a decimal, not a whole number. I moved students to realizing that the radicand must be a perfect square for it to be rational.

Then students practiced on a few problems, as well as interpreting the distance between 1st and 3rd base in baseball when the distance between the bases is 90 feet.
Sample work after developing success criteria.
This student clearly was following the success criteria, showing work superbly.
An interesting problem was 9-124. It gave a triangle on the coordinate plane and you had to find the perimeter. I and most other students saw it as a rectangle with triangles missing out of it, leaving you with 2 pythagorean theorem problems. Amir in 4th period saw it in a much simpler way: draw a horizontal line to split the triangle into 2 triangles. Then use the Pythagorean theorem to solve for the missing side lengths. I liked that elegant method.
After finding the missing sides and perimeter, students were pushed to prove if it was a right triangle. They realized it was acute. This was the first time they saw a square root of a number being squared, which was unfamiliar to them.
In accelerated students identified the difference between intercept and intersect. Two functions intersect when they overlap. A function has an intercept when the function crosses the an axes (x or y). I asked them what those coordinates had in common. They said there was at least 1 zero in the x or y coordinate.

Students used substitution to find points of intersection between quadratics and linear functions. Then Robert showed how once you solved for the roots, these were the x coordinates of the intersection points. He then said to substitute the x values into the linear equation since it would be much easier than substituting into the quadratic.
Find the intersections of functions.
My friend invited me to my first Giants game of the season. We ended up winning, 5-4! Went on too late, so had to leave early.

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