Day 1:
I gave the students the book’s exam to do as a group review assignment – yes, I made sure to get the exam back from everyone. Since I lost my voice, I just had them do it in groups and graded it that night so that I could go over it the second day.
Day 2:
The students didn’t do as well as I hoped. I got the online textbook on the projector and showed them how many times they’ve done questions like number one on the practice exam (eight), how many times they’ve done question two but with different numbers (four), and that the fractal for numbers three and four was on page 7 in the textbook. I underscored the importance that the homework and classwork problems were to be used to learn, not just to do.
I then went over those four problems and promised that they would see them again. I gave them the rest of the period to work on the review problems in the textbook, questions 1 – 7. I told them that they should do the problems that they were unsure of first so that I could answer any questions before they went home.
0.1 – The Same Yet Smaller
0.2 – More and More
0.3 – Shorter yet Longer
0.4 – Going Somewhere?
0.5 – Out of Chaos (I didn’t do the ones that are crossed out)
I did the investigation in class, but no kids were able to see any pattern that would develop because it would take so many points to see anything. This is the lesson where we could use an overhead projector – if each kid was given the triangle handout on transparencies, we could stack the transparencies and (maybe!) they would show what the kids need to see. But I’m not so sure that this is a good use of time of the class period.
During my last class, I modeled what they were to do with the investigation and then passed out the dice and rulers and let them go. After about 7 or 8 minutes, I walked around and picked the dice and rulers up. I then went over a problem of the type from section 0.4 – say 0.2 × ⃞ – 4 and gave my calculator an initial value of a huge number such as 78893478. I then hit enter three times and asked if the students knew what the attractor value was (no). Then I hit enter a few more times and asked the same question (no) and repeated the process a couple more times. I then explained how sometimes, even though the data might seem quite random, there would be an attractor value and hit enter on the calculator enough for them to see it.
Then I asked if any of the groups could see the “attractor value” of the investigation. But even with the best kids getting 30 or 40 points, there is nothing to see. So then I showed them the program “Chaos” on the TI-84 program and they got to see many, many dots. Once they saw this, I asked them to see if they could tell that there would be a big empty triangle in the middle of their investigation. Some kids did see that and liked it.
But like I wrote above, I’m not sure that the kids doing the investigation as is was a good use of time.
Vocabulary:
Due to a shortened class period the previous Friday for pep-fest, I had no homework to go over today, so I began the lesson right away.
I changed the investigation a bit to make it move a bit more quickly….I did the first part myself, and then let the students do the last two steps on their own, which worked well with the number lines.
When going over example A on page 23, I made sure that I evaluated each expression just like they did in the table…horizontally, although I hate it when kids do it this way. But they need to be able to read it horizontally for homework question #6.
I’m not exactly sure what they use the term “attractor” rather than “limit” since I don’t understand why attractor is any more easily understood conceptually than limit and the importance of the term limit later on.
After working on the first section for three days, I found that the investigation went fairly well, although many students are still not seeing the patterns that result with the number of segments and length of each segment in the fractals for this section. So I did have to lead the students through the investigation at some points.
I did the example on page 15 with the students, and it became a bit more clear, although I was quite surprised to find that there were plenty of kids that didn’t see the pattern needed.
Since I wasn’t going to assign questions 7 and 8, I did those for examples in class as well. Finally, I had a majority of kids that could at least mimic what I was doing, although I still feel that I could have done a better job of explaining why the patterns work.
I did give a quiz the following day but didn’t include any questions such as the tables on the homework from this section.
A lot of what is done in this section was already done in section 0.1. I think that I may have gone over too much(?) material from section 0.1 that could have waited until now, especially the work with the exponents.
For this section, I had the students do the investigation but only gave them five minutes to do so. After that, I assigned questions 1-6 and 9 – 12 and gave them the class period to work on it. I will write a quiz to give them the following day.
First Day:
I talked a bit about recursion and gave some examples with numbers of what recursion does.
The students weren’t really good at understanding how to do an investigation, so I went up through Step 3 with them. I didn’t want to give too much away as far as seeing patterns is involved, only stating that there are lots of patterns to find.
I think next time I’ll have to go over fractions some, particularly
By the end of the class period, most students had not completed the investigation. I think that this one should be typed out to help kids through it.
Second Day:
Disappointed with the way the first day went – the kids weren’t seeing the patterns – I put the four stages on the board and went through two lists with them:
For each of these, I took time showing them why these had to be the case, and then we found the number and areas of the triangles out to stage 7 without drawing them.
I found more success talking about areas of shapes (base times height) and how a triangle that would fit across the large triangle n times would mean that the area of the triangle was 1/n.
I then did numbers 1 through 4 with them in class and assigned question 5 for homework. I gave them copies of stage 3 so that they could complete the exercise more quickly.
Third Day:
I went over question 5 and then had them do questions 8 – 13. At the end of the class, I gave them a quiz on what they have learned.
I’m not too happy with how the students did on this section….it seems that there is something missing. The third day was on a Monday, which didn’t help, but the kids just aren’t getting some of the basics, but I can’t justify a fourth day on it. Section 0.2 is more of the same, particularly with writing expressions for finding the “rules”.
