Day 9: Going Irrational

Quote of the Day: “Let’s face it. This is the relationship age! The information age is over - there is too much of it, everyone has access to it, and much of it is not useful. Your success depends on the quality of your relationships.” - Greg Bell

Question of the Day: What is 0.16 with the 6 repeating times 100?

Regular Math Objective: Convert a repeating decimal to a fraction.

Regular Math Lesson Sequence:

1. QSSQ (I skipped the question part until after the number talk)
2. Number talk. 94 - 49. I had the students do a reflection in each class. I simply had them write down how many times that they have participated by either giving an answer, a strategy, asking a question of one classmate, or opening up a discussion for the entire class. I then had them make a goal of how many times they would participate in our next two number talks. In two out of four classes it had virtually no impact, in one there was a small impact, and in the other there was a huge impact. I think it all can be tied into social dynamics, but in any case some results are better than none. Some students are still answering it incorrectly at times, so we still have great conversations. I'm really surprised that people do not laugh at their classmates for wrong answers, but it is a complete none issue in this routine. 
3. My Favorite No. I asked a series of four questions. The first was to get students to extend 0.16 to 5 decimals places. The second was to get students to extend 0.16 with the 6 repeating to 5 decimals places. The third was for students to multiply 0.16 with the six repeating by 100. And finally I had students find x for the equation 90x = 30. The answer below was the most popular wrong answer I saw all day and I did not anticipate it at all until actually asking students to think before I spoon fed them the answer. 
   
   
   
   
   
4. After discussing popular mistakes for each of these four questions, we talk quick notes of how to go from .4 repeating to 4/9. I showed them this in five steps. First, set the repeating fraction equal to a variable. Next, multiply the variable a base ten number. We had to review what a base ten number was. In the future, it would be helpful to show students a chart with the 10s all put to different exponents. Third, students multiplied the variable by a different base number. Fourth, they found the difference between the two base ten numbers and the decimal equivalents of these values. Fifth, they solved the multiplication equation. 

Outcomes: I hate teaching in a stand and deliver mentality, so I sort of avoided. Students discovered some of the process on their own with My Favorite No. In essence it is the same thing in an ELA class as making predictions before reading to be more invested in what is being read. I also tried yesterday to get students to do this process on their own and had very limited luck and some classroom management dilemmas to confront as a result of students being over challenged. 

Overall students seemed ok with this concept. The mistake above is something that I'm barely concerned with when considering the bigger picture of everything else that needs to be done. 

On the surface this looks great. Perhaps I'm overanalyzing, but I'd be worried that this student would flop without notes and would have trouble thinking about this problem critically. What would happen for instance if the decimal was 0.121212 instead of a single repeating decimal? Could the student make the necessary changes to find the fraction equivalent? 

I have most confidence in an answer like this. The students recognizes how to simplify (which I did not ask students to do) which is nice, but they seem to recognize the entire process as one big entity rather than a series of steps. 

Honors Math Objective: Approximate square roots of irrational numbers

Lesson Sequence: 

1. QSSQ
2. We analyzed some of the pictures from the two-step equations of the previous class. I tried to boost student confidence by stating that most of their work was good. We just needed to clean up one flaws in a process that includes in some cases more than ten steps. 
3. Square Roots Go Rational. We began the discussion by looking at the distance in areas between each of the squares on this chart. One of the students pointed out that there was a distance of three numbers from 1 to 4, a distance of five numbers from 4 to 9, etc. In doing so, I then transitioned to the visual on the Illuminations lesson linked above ("Square Roots Go Rational"). I asked students to tell me how many X's were needed to go from a 3 x 3 square to a  4 x 4 square. Next, I showed them a 2 x 2 square with the X's and we discussed if I added three more X's if that would create a square. The class was quick to shoot down this notion. We had a brief discussion of rational versus irrational numbers. Then we were able to see that we had three out of five possible X's to get to the next perfect square. Using this number (2.6) we were able to approximate the square root. From there, students worked in groups of three to complete every irrational number's approximation from 1 to 40. 
4. Students did an exit ticket of the square root of 80. Intuitively they knew that it was between 8 and 9. I told students to solve it however they would like. Many of them guessed and checked. Some of them used fractions. And others drew the picture with the X's (although not all 80 X's). Overall there was solid understanding from the lesson.