## Saturday, April 16, 2016

### Day 138: Geoboard Irrational Intro & 2 way frequency tables

Day 130 of Estimation 180 brought us to James Brown "I Feel Good." The most common method for reasoning about this problem was estimating the song was 3/4 of the way through. Below a student thought it was 2/3, and divided the song length by 3. The common theme was finding the unit rate by dividing by the numerator and then adding it on to the song. Students must fluently work between seconds and mixtures of minutes and seconds like 2:06. I really like listening to a clip of music in class.

Then students investigated different sized squares on 5 by 5 Geoboards and rubber bands. Each class found between 1 and 3 of the non-perfect squares with irrational side length.
 This student thought of it as percent. I liked how this student estimated each 1/4 of the song is about 30 seconds. I ask why we can't divide 11 seconds by 2:48, and the answer part of the ratio must be converted to the same units of all seconds.
I had students setup a chart with side length, area, and perimeter like NCTM Illuminations had suggested. We began with the smallest square. I also instructed students to draw 4 5 by 5 dot arrays to represent different squares found.

 I introduced students to irrational side lengths with the top left square. Students saw it in many different ways. One class even split the square horizontally into 2 triangles. Then they multiplied the base and the height of the one triangle and realized they didn't have to multiply by 1/2. So 4*2=8.
 This was the most common method, dividing 3/4 by 3, and then adding the quotient to the length of 3/4 of the song.
 The biggest jump students had to make was to realize that the diagonal was longer then the horizontal and vertical distance between pegs. This student asked a great question: What about the non-perfect square formed by overlapping these 2 squares of area 2? I was so intrigued and would like to explore this further. In the few years I've done this, no student had asked this brilliant question before.
 The thinking is by working with familiar perfect squares and seeing that you square root the area to get the side length, by determining the area of the non-perfect square you could square root the area to get the side length. Students reasoned why the area wasn't 4, because there was a perfect square of 4 in the middle. They said the perimeter was 4 times square root of 8, and I told them they could shorten that to "4 root 8."
 In the picture above you can see the non-perfect square with area 5. We are missing 1 in the bottom right of the picture. Alex N in 6th period was the first student to say they multiplied the time after 3/4 of the song by 1 and 1/3. I didn't think all students saw how he did, and showed the relationship between the 2 methods. By multiplying by 1 and 1/3 you saved the step of adding 1/3 of 3/4. I illustrated with a number line on the left.
In accelerated students worked on different methods for solving equations, from undoing an operation on both sides, to multiplying it out and solving. At the conclusion of class I decided to review one of the homework problems that dealt with 2 way frequency tables because it was the next skill they'd be assessed on. We did not answer part C as a class. What is the probability someone takes the Sunday paper? We will revisit that.
 I also introduced vocabulary involved in these tables. The data outside the table is called the "marginal frequency."
 Dan Meyer, who makes the activity Taco Cart that we will be using soon, MC'd the Desmos trivia night Thursday night.
 I got to meet 2 of my favorite Canadian math teachers. Mr. Orr on the right and Mr. Pearce in the middle. Mr. Pearce made the candle burning task we used in 6th period.
 And everyone should recognize this guy. Met one of my heroes, Mr. Stadel, who made all of the estimation warm ups we use.
 Here is a sample of the goodies I got from the Wednesday Desmos conference. I'll be wearing the socks, shirt, and sunglasses to school next Friday.