Teach 180: Planning to Discover Log Properties (Day 99)

Today we had our third (or maybe it was our fourth?) two-hour delay.  We were scheduled to originally have periods A, B, C, D and E.  I teach periods A, B, F and G.  With the two-hour delay, I had no classes today.  You may wonder why I even came into school.  Why not call in sick?  If you know me, you know that is not my style.  I had a meeting with a student planned and I also had a meeting with a colleague planned to discuss PreCalculus.

When I met with my colleague, we talked about the upcoming unit on logarithms in PreCalculus.  We wanted students to discover the properties for themselves, but we weren't quite sure how to do it.  For example, we want student to discover that log_b(MN) = log_bM + log_bN.  We could have them create a table of values like the one seen at the left on a spreadsheet.  But would they be able to see that log(2) + log(3) = log(6)?  I felt like there were too many numbers for the students to see that relationship.  Perhaps if we showed the values rounded to the nearest hundredth, it would be easier for the students to see that relationship.  Would it work?

My colleague had a suggestion for using index cards and placing log(M) on one side and the decimal approximation on the other side. Perhaps this might work.  We could tell students to look at the side with the decimal values and find three cards such the sum of two cards equals the third card.  Then, they flip the cards over and see if they can form a relationship with the numbers on the other sides of the cards.  We could also do this to also allow them to discover that log_b(M/N) = log_bM - log_bM.  Clearly, we are still in the planning stages on this.  But that's ok.  This lesson is for NEXT Monday, assuming no more two-hour delays or snow days between now and then.  If you have a way for students to discover log properties, let me know in the comments below.

Teach 180: Flipping Kisses (Day 98)

Today AP Statistics students constructed their very first confidence interval based on data they collected.  The data came from tossing or flipping Hershey kisses.  Students flipped 10 kisses out of a cup and repeated that 5 times to have a sample of n = 50 flips.  Students counted the number of times they landed bottom side down and recorded that as a sample proportion.  From there, we looked at the general "recipe" for creating a confidence interval for their specific set of data.  Intervals were plotted on chart paper and their calculations confirmed with their graphing calculators. 

It's unfortunate that half of the class was absent.  Although learning can happen by watching a video or reading a book, nothing can replace a hands-on activity and face-to-face discussions to build understanding of a concept. Being able to eat the chocolate is, of course, an added bonus!

Teach 180: The Confidence Interval (Day 97)

In AP Statistics today, we looked at how changing confidence level and sample size impacts the width of a confidence interval.  We also saw how the confidence level is related to the long run capture rate of confidence intervals.  The website that we used to investigate this was http://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html, but rather than having my students type this in, I told them to go to bit.ly/ConfidenceIntervalBasics. (Creating a shorter, custom link is easy to do at bit.ly or tinyurl.com and makes it easier for the students to correctly enter the web address in their browser.)

First, I asked students what they noticed about the two distributions that were shown on the screenshot above.  They commented on the fact that they had the same normal distribution shape and same mean, but that the sampling distribution was less variable than the population distribution.  Then, we used the applet to take several samples of size n = 20 and created 95% confidence intervals from each sample.  I asked them what they noticed about the intervals.  Students commented on the symmetry of the intervals, but weren't entirely sure if the dot in the center was the median or the mean.  They also noted that the one interval was red.  Why was that?  Noticing the sample under the interval gave them a sense that this sample had many values that were above the mean and that the entire interval was bigger than the population mean. Seeing this in action with several intervals helped the students to begin to understand that the probability that an individual interval captures the population parameter is 0 or 1, and not 95% (or whatever the confidence level is).  The 95% level refers to the long run capture rate of all possible intervals constructed by this method.

Next, I had half the room keep the confidence level at 95% and investigate the change of sample size on the width of the interval.  The other half of the room kept the sample size the same and investigated what happened as the confidence level changed.  If I had students change both variables at the same time, they may not have fully understood the relationship between confidence level and interval width and sample size and interval width.  Focusing on just one variable at a time was an important part of their discovery of the relationships.  Below are screenshots of the applet and what students discovered. 

Changing Sample Size, Keeping Confidence Level Constant : Larger Sample, Narrower Intervals

Changing Confidence Level, Keeping Sample Size Constant: Larger Confidence Level, Wider Intervals

I believe that building an informal understanding without any formulas leads to a better understanding of how the formulas work to create wider or narrower intervals.  Tomorrow we toss Hershey Kisses to create our first intervals for the proportion of kisses that would land bottom side down when tossed.