Monday, April 18, 2016

Day 140: Approximating Irrationals & Discriminant Intro

Today's estimation was the Star Wares main title song. Some students elected to not use the clue and "go off experience." Those that thought 1/5 of the song had been played at 47 seconds got a close estimate. Once again I asked students to tell me how they interpreted the seconds as minutes and seconds. These discussions dig deep into place value understanding.

Students had background knowledge via notes on perfect squares, and a procedure for approximating an irrational number to the nearest tenth which brought up some amazing discoveries in 4th period. Read on below for that.

I also introduce a learning goal, and unfortunately did not develop the success criteria with students. The goal was to approximate, without a calculator, an irrational number to the nearest tenth. Another learning goal that I should have written was interpret the square root of a perfect square as well. We also did a participation quiz in each class to reinforce study team norms.
The number line method has students find the 2 perfect squares less than and greater than the radicand of the irrational number.
A student at table 5 was convincing his group that the number was between 4 and 5, and not 4 and 6. He said 4 squared was 16 and 5 squared was 25. Group 8 used notes, and one member used a calculator to approximate first. 1 team member continued to see if the group was following along. For Table 7 they had different answers so recorder reporter wasn't making sure students wrote it down.
Here a student successfully approximated the square root of 95. They put the square root of 81 and 100 on opposite sides of a number line and placed root 95 closer to the right. It was 81 and 95 are 14 spaces away, and the 2 perfect squares are 19 away, making it 9 14/19. This student did not convert it to decimal form after that. Interestingly, 100-81 is equal to 10+9.
Here is the example I did when directed by a student volunteer. the radicand of 40 is between the 2 perfect squares 36 and 49. They are 13 apart, while the square root of 40 is 4 away from the perfect square ON THE LEFT, meaning the square root of 40 is approximately 6 and 4/13. I'd like to make a desmos activity that scaffolded how to do this with students.
Table 5 had a group member that was difficult to deal with for the group, the class, and myself. We reinforced the study team norms. Group 6 asked me a question when the whole group had a try at answering it. In group 4 a student was working on the problem while his team mate was reading it. Group 1 stated that 24 is not a perfect square, which is why you couldn't make a square out of 24 square tiles.

A joke a student made up and showed you. "Four root punch." Get it?
Group 5 referred to their knows, group 6 checked to see what everyone got for the side lengths of perfect squares. Table 2 explained the number line method from the notes to a team member.
Here is where students noticed 49-36, is equal to +7.
This is my first attempt at wording it. I told a colleague at lunch and she found this resource, where I like the intro to multiplying binomials. There is a NEED to prove this:
My colleague Carrie found this resource for what Bing noticed. The dub it "Consecutive squares."
I tried to elaborate here.
I liked how group 5 checked on if others were going ahead, and the one who went ahead said they didn't want to hear about her drama. Nice job standing up! One group thought the square root of 49 was written as 7^2, and realized it couldn't. Group 4 had 2 members interpreting the notes. They got the idea, except the larger perfect square wasn't the closest one to that number.
Showing dividing 4 by 13. A student predicted you'd get a more accurate decimal if you made the mixed number an improper fraction. We checked this, and saw it gave the same decimal approximation.
In accelerated we reviewed quadratics in perfect square form (ax-b)^2=c^2. They realized if c was negative, you'd get no real solutions. If it was zero you'd get 1 solution. If it was greater than zero you got 2 solutions.

We expanded this to standard form when I asked student to solve 2x^2-3x+1/2 using the quadratic formula. I gave them time, I then had a student recite me the quadratic formula, and I stressed that it was "all over 2a" and if you forgot to underline the opposite of b it was completely wrong. I also stressed where I saw students make the most frequent mistakes and it was at -4ac. We talked about exact and approximate decimal form.

Basically the radicand, b^2-4ac, determines how many solutions there are students saw.

The lesson asked students to come up with a c value that made the equation have zero solutions, and for the equation to have no solutions. Students realized there'd be no solution if the radicand had the variable c and was less than 0. So, when solving that inequality, if c was greater than 9/8, there would be no real solutions. If it was 9/8, there would be one solution. Finally, if it was less than 9/8, there would be 2 solutions.

I asked students who took Kumon if they knew what b squared minus 4ac was. Moreen remembered it was called the discriminant. Therefore I asked students what was true about it for each situation. They said if it equals zero there's one solution, if it's less than 0, no real solutions, and if it's greater than 0 there are 2 solutions.
Investigating quadratics in depth. This makes me want to connect it to graphs with the Desmos activity I saw done by Shelly Caranza.
Annabel investigated the conjecture made by a student in an earlier class. Love how she rewrote the squaring as repeated multiplication and then as repeated addition to show how the difference between the numbers was 1, leaving you with 1 of the larger number left over, plus 1.
Nuri reviewed how to solve this absolute value equation. The extension was what would make this equation have no solution. They said if the expression on the right was a negative number. Basically, the absolute value of an expression will equivalent to 9 or -9 when the expression on the right is a positive 9.

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