## Wednesday, April 20, 2016

### Day 141: Rational, Irrational Numbers & Proving the Quadratic Equation

Students estimated Jimi Hendrix All Along the Watchtower, day 133 of estimation 180. I didn't take a picture of sample reasoning.

I read the introduction of the section to students detailing how rational numbers can be written as a ratio, and since 1/3 is a repeating decimal, it is also concerned a rational number. I taught students how to square root a number using the TI 83 calculator and I also had them take out their iPhones or Androids to use the square root button. On the Android you have to press the radical or square root button first before the radicand. On an iPhone you can do it either way.

We also discussed how 5/9 showed up as .555555556 and I asked students why that was and what it meant. They said it was repeating and the calculator was rounding the last digit.

For some reason students questioned if the negative sign outside the radical affected it being rational or irrational. Some students pressed the negative sign before the square root button and there calculators gave them an error. One student knew that it was impossible to square root a negative number.

Some students mistook pi squared as pi times 2, and students were quick to correct that 2 was an exponent so it was pi times pi. I was relieved that a few students remembered scientific notation with the last expression thrown in. Some students reasoned you move the decimal to the left twice. Others said that 10^-2 is like .01 or 1 one hundredth so they multiplied that by the decimal.

This also re-introduced fractions with 9's in the denominator as repeating in a certain pattern.

Students summarized that when you square root a non-perfect square, the result goes on forever in a non-repeating pattern. Therefore the square root of 9 is rational because it is equal to 3 (3/1).
 We reviewed the fact that consecutive whole numbers when summed are equal to the larger number squared minus the smaller number squared. Students explained how I should foil, and then cancel out the x squared terms, leaving the 2 expressions equivalent. I demonstrated and reinforced the generic rectangle for multiplying (x+1) squared.
 A student in 6th period was investigating growth of quadratics and the relationship to derivatives and slope.
In accelerated students copied the steps to solving the quadratic formula from the standard form of the quadratic equation. The biggest road block was the 3rd step of completing the square. Owen said that when you complete the square you add c to both sides, which in a perfect square trinomial is (b/2)^2. In the quadratic equation step, b is b/2, so students have to make the jump to see b/a divided by 2 is b/2 multiplied by 1/2, which is b/2a. Then it is (b/2a)^2. Which equal b^2/4a^2 added to both sides of the equation.

CPM has students complete a generic rectangle for this step which they started with x^2 in the bottom left corner. The top right corner is b^2/4a^2 and the other diagonal is the middle terms in a perfect square trinomial: (b/2a)+(b/2a).

Then you write it as a perfect square trinomial. I would personally reverse steps 4 and 5. I think of completing the square as adding a term to both sides. I didn't take a picture of the rest of the steps, but next is commutative property, followierd by making c/a a like fraction by multiplying it by 4a, leaving you with b^2-4ac/4a^2. When you square root both sides the plus or minus sign is introduced and the denominator simplifies to 2a. The numerator is the opposite of b plus or minus the square root of b squared minus 4ac. We discussed yesterday how this radicand, the discriminant, determines if there's no real solutions, one solution, or two solutions.
 Wrestling with proving the quadratic equation.
 I want to make a sign out of math is supposed to make sense and "in this class mistakes are expected, inspected, and respected!"
I liked this answer to D for a test question, but apparently the percentage of students wearing a backpack is highest of the whole school.
 2 way table assessment question