Teach 180: The Golden Ratio (Day 114)

Today I gave a chapel talk in the middle school called "The Golden Ratio: A Common Pattern".  The bigger theme for the chapel was to look for common things we share with others rather than focusing on the differences.  My talk focused on generating the Golden Ratio from the Fibonacci sequence.  I then showed how the golden ratio is found in architecture, art, insects and the human body.  Did you know that the ratio from the tip of the index finger to the wrist and the wrist to the elbow is in the golden ratio?  Or that the width of the ear to the height of the ear is in the golden ratio?  In fact, the ratio of the length of a minor groove to a major groove in DNA is the golden ratio!

So, this ratio is seen many places.  But is it really that common?  Take any two integers and form a Fibonacci-like sequence from them.  As you find ratios of consecutive terms, it approaches the golden ratio.  The golden ratio truly is a common pattern.

Teach 180: A Visit to a French Market (Day 113)

Bonjour!  Je m'appelle Leigh Nataro y je ne parle pas le français tres bien.  But today I spent about 15 minutes visiting a French market in one of the French classrooms.  The students had items for "sale" and were practicing their speaking skills relative to buying and selling food items at a market.  Several of these students will be traveling to France over spring break and this activity helped them prepare for the trip.  Faculty were invited to the classroom to participate in the activity.

I said, "J'ai soif", and a student poured me a delicious (raspberry, I think) sparkling soda.  Next I told one of my current AP Statistics students, "Je voudrai un meringue."  It was a tasty cookie, but a bit overcooked.  I also paid way too much for deux carottes.  I think it was 2 Euros for the baby carrots??  Hmmm, perhaps next time I should study the Quizlet cards a bit more.  In any event, I enjoyed seeing my students in a different setting and this was a fun way to show them that teachers can be students, too.

Teach 180: Teaching the Wrong Way (Day 112)

Today I did a disservice to my period A class.  I lectured for the entire class period, which was nearly an hour long.  I could tell many of their brains were full after about thirty minutes, but I plowed ahead. So, why did I lecture for a full hour?  The answer is the AP exam; it is on May 17th.  This is the first time in fifteen or so years of teaching that the date of the AP exam has been a concern to me.

In years past, I had plenty of time to review for the AP exam.  I split my content up into bit-sized, easily digestible 40 or 50 minute chunks.  Some days were entirely for students to work through problems.  Other days were driven by a data collection activity.  Some days included lecture for part of the class period. 

In years past, we had about three weeks to review prior to the start of AP exams.  This year AP exams start about one week later and you would think that would allow me more time for my students to review for the AP exam.  Unfortunately, that is not the case.  Based on my estimates, we will have the equivalent of 5 one-hour classes to prepare for the AP exam.  My AP Stat colleague from souther states will have 20-30 hours.  I fear my five hours of review will be woefully insufficient.

In order to protect those five minimal review days, I chose to take two 30 minute lectures today and put them in one 60 minute lecture.  It is true that 2 x 30 = 60 and technically, the timing equates.  We talked about null and alternative hypotheses, p-values, the wording of conclusions, type I and type II errors, and quickly touched on power of a test.  Basically, three big ideas that should have been separated into 2 or 3 separate class periods.   Yes, it would have been better to separate these ideas, but spending an extra day on content now means one less review day in the future.

Teaching in longer periods will still be my reality next year at my new job.  So, what will I do differently?  Mostly, I'll plan better.  No time was allocated for teachers to do this at the beginning of the year or to pay them for this work over the summer.  Had we been given time to work within our departments, we would not be scrambling to determine what content will be left out of our curricula for the remainder of this year.  Or better yet, we could have reviewed the math curriculum that was over 12 years old to determine what content really matters the most.

Teach 180: Specific Student Feedback (Day 111)

If you ever ask a teacher what they dislike the most about their job, it is likely that they will say grading.  It never seems to end and when I taught in the public school, I could easily spend 10 hours a week or more grading tests, quizzes and problem solving tasks.  One of the purposes of grading should be to provide students with feedback.  I am not just talking about a percentage or number correct.  I am talking about enough feedback that the student can understand where he or she fell short and how to improve.

Feedback on assessments is usually quick, things like "be careful of your signs" or "don't forget the Such-and-Such property".  In my mind the feedback is clear.  But perhaps it isn't as clear as I think.  Today a student came back and asked me about the following feedback I gave on his Calculus quiz.

He was wondering what my feedback was showing.  What did he do wrong?  Had he not asked me about it, my feedback to him would have been meaningless.  Would writing sentences like these have helped?  "If you don't have the parentheses, only the 4 is multiplied by 10e^x.  You need the parentheses, because we want the entire denominator to be multiplied by the derivative of the numerator."  Maybe it would have been helpful. Or maybe not.  It would certainly take me about 10 times longer to write that than what I wrote.

So, how can I get students to understand my feedback and not use up the ink of a dozen pens?  Would giving the students a copy of the answer key help?  Perhaps.  But I am guessing most students would see that their answer to a question as wrong, but not fully understand why.  Would doing test or quiz corrections help?  Perhaps.  But that leads to more grading for me.  Would having students consult with their neighbor about quiz or test errors work?  Maybe.  But student grades should be private. 

So, what is the answer to providing more specific student feedback AND not drowning in red ink?  I am hoping to find the answer to this in the new book I starting to read "Grading Smarter, Not Harder" by Myron Dueck.  Any ideas I try, I'll be sure to share them in my blog.

Teach 180: Smelling Parkinson's (Day 110)

Today students were introduced to hypothesis testing with an activity developed by Doug Tyson.  The lesson begins by showing students a video about Joy Milne, the woman who can smell Parkinson's disease.  (If you don't like this particular video, there are many others online.)  Students learn that Joy smelled t-shirts and correctly identified the shirt as being from a Parkinson's patient or not in 11 of 12 t-shirts.  We then wonder how likely it is to get 11 of 12 identifications correct.

We set up the null and alternative hypotheses (Ho: p = 0.50 and Ha: p > .5) and then did a simulation by hand to determine how likely it is for 11 of 12 identifications to be correct by chance alone.  The pictures below show students smelling index cards that have either a P or NP on the back of the card.  (It is quite fascinating how excited students get when the guess correctly.)

We plotted the results of 40 simulations on a dotplot and got an estimated p-value of 0.025.  (One student got 11 of 12 correct by guessing.)  Finally, we did 10,000 runs of the simulation using  statkey.  Based on these simulated results, our p-value was 0.0045.   This simulated p-value was definitely small enough for us to reject the null hypothesis.

Incidentally, one of the patients that Joy had identified as not having Parkinson's returned to the lab several months later to report that he, in fact, had Parkinson's.  Joy was actually 100% accurate in all of her identifications!!

Teach 180: The In-Service Day (Day 109)

It's not a teaching day today.  It's an in-service day.  Thinking back to my in-service days over the past 25 years of teaching I would say that there is really only one that I remember with lasting importance.  When I taught in Ames, Iowa, there was a 3-day cooperative learning seminar that the entire school was required to attend.  We had to create sub plans and 1/3 of the school was out for 3 consecutive school days. By the end of day 9, we all had been trained by the authors of the book Learning Together and Learning Alone.  Some teachers were definitely more into learning than others, but having a common language and understanding common strategies for cooperative learning was very beneficial.  Language like "think-pair-share" and "jigsaw method" were quickly understood by the students, because so many teachers were implementing cooperative learning methods.  We had follow-up meetings throughout the year to discuss successes and setbacks.  I must have learned something about cooperative learning, because I still use some of the structures in my classes.

Today's in-service day was nothing like that.  It was one day that was fairly fragmented.  We had much to "cover".  We had a presentation about a travel grant. Then, the Head of School shared the new school vision, and data on student engagement that had been collected last spring.  Next, we met in cross-divisional groups to continue to work on group projects.  We have been given 5 hours to complete a group project on a topic of interest and prepare a presentation or document to share in April.  My group has come to the consensus that Project-Based Learning would not work well on a large scale at our school due to the lack of time needed to collaborate on a weekly basis.

Finally, math and science teachers met with a consultant for 3.5 hours to talk about concerns we have about the new schedule and its impact on the math and science curriculum.  We touched on curriculum, assessment, and strategies for teaching in longer periods.  We could have easily spent 3.5 hours or more on each of these topics individually.  The consultant had a mathematics background and certainly understood our concerns, but we didn't really have enough time to make any substantive changes for the immediate future (say next week) or for the fall.  We all agreed that more time was needed to make curricular changes and that we needed to investigate that as a group and not in isolation.

The day ended with announcements relative to enrollment, finances and the school calendar for next year.  We got through our agenda for the day and we certain did "cover" much.  However, we did not accomplish much and what I learned today will unfortunately have little impact on what happens in my classroom.

Teach 180: Log Properties by Discovery (Day 108)

When I was in high school, I was content with being told certain rules in math.  I could either see that they were true for myself or I willingly accepted them as true.  We rarely, if ever, discovered rules or were asked to make and test conjectures.  Discovery, reasoning, and sense-making is more common in my classroom now than when I first started teaching, but it can be easy to fall back into pulling notes out of a file cabinet and just working through examples of concepts with the students.  If discovery is better for learning, why not just create lessons based on noticing, wondering and discovery?  Two main reasons: time and collaboration. With inadequate time to plan and lacking collaboration with a colleague, the chances of moving closer to discovery are greatly reduced.

This year I decided to make the time to work with a colleague and change our lesson on logarithm properties.  I had reached out via twitter asking for ideas and got a response back from Ralph Pantozzi (a.k.a.@mathillustrated).  He gave my colleague and me something he had used with his students and that was where we started.

These were the objectives for the lesson:
1) Students will make observations about numbers from a table of values for x and f(x), where f(x) was log₂x.
2) Students will use the observations to create the product, quotient and power property for logarithms.
3) Students will practice using the properties to combine and expand logarithms.

Page 1

Page 2

Page one shows part of the full table of values we gave our students.  Page two was what we had planned to give to our students, if they had trouble coming up with some ideas.  But they had so many ideas!!! We didn't need to give them the page with the questions.  They just took off on their own and our full class discussion led to observations about each of the log properties. Photos of two of the boards from our discussion are shown below.



At one point, my students asked if what we were doing would work for all bases.  For some reason, I didn't anticipate that the students would ask this.  I didn't have a spreadsheet set up where we could change the base to be a different value.  Was there a way to test to see if what we were doing with the sum and product would work for all bases?

Desmos to the rescue!  I instructed students to type in two expressions in Desmos: log_bM + log_bN and log_bMN, where b was a base of their choosing and M and N where numbers of their choosing.  Students could see that the results of both calculations were the same!  I made expressions with sliders for M and N to show that changing values for M and N produced the same result. The screenshot here shows something similar with the quotient property.

Based on our results from the product property, we derived the power property. We also reviewed the values of log_bb and log_b1.  Next, I had the students type f(x) = b^log_bx into desmos. Students were instructed to choose a base for b and a value for x. Why did it make sense that the f(x) always equal to x?  Why was the input value the same as the output value?  Does this happen for all values of base b? I created a table of values and a slider and when the function b^log_bx was plotted, it resulted in a line.  (See the short screencast below.) This led to a discussion of the fact that b to a power and log base b were inverse functions.  So, it made sense for them to undo each other and for f(x) to equal x.

So, what lessons have I learned about moving closer to having students learn by discovery, by noticing, by wondering?  Two things: First, it's ok to not plan for everything.  I didn't think that my students would ask about other bases, but we were able to quickly examine a multitude of bases.  Second, don't be afraid to let the students take control of the lesson.  They have the ability to notice and wonder, if you let them.  Their insights might surprise you.

Discovery can come naturally.  Learning logarithm properties by discovery made the lesson more fun for me to teach and it made it more interesting for them to learn.

Teach 180: Contest Day! (Day 107)

Today our contingency plan went into effect.  We had a snow day the previous week and that meant we had to give the American Mathematics Competition (AMC) Form B contest, instead of the Form A contest.  This year we held the contest during an assembly period and students only missed half of one class period.  (In previous years students missed two class periods.  However, I think it is important for students to be in class, because you can't replicate the learning that happens as students interact with each other.)

The contest lasts 75 minutes and consists of 25 multiple choice questions.  Students cannot use a calculator and the questions are quite challenging.  After the contest was over, the room was buzzing with chatter, such as "How did you solve question #8?" and "How many did you actually put down an answer for?"  Tomorrow students can get their test books back from me and I look forward to discussing some of the problems with them.

Teach 180: Mathograms (Day 106)

There are times when the school day is over and I just don't feel like working.  Sure I could plan for future lessons or grade the test I just gave.  I could even organize the desktop of my computer. Instead I chose to get lost in some math.  Yesterday I saw a tweet about Desmos Mathograms. I found the tweet again today and opened up the link  http://mathogram.desmos.com/  Then I sent Mathogram Valentines to my daughter and husband. (My husband responded about a minute later with "Geek. Luv ya too.")                          

This is the one I sent my husband. 

This is the one I sent to my daughter.

One of the things I love about both of these is that Desmos lets you see the "Behind the Scenes" of each graph.  Seeing the flower transform to a heart, I was instantly curious how it happened.  And I was surprised to see that it happened in just 9 lines of code! The Sierpinski triangle, just 4 lines of code!! Now I need to take some time to break it down and see just how it works.  I wonder if my tests will get graded tonight.

                               

Teach 180: Helping Students (Day 105)

One of the aspects of my job that I enjoy the most is working with students individually.  When I taught in the public school, there was little time to give students the additional help they needed.  If they were free, I would usually have a class.  Or if I was free, they would usually have a class.  This would mean coming to school earlier than the start of the school day or sticking around after school was over to offer individual help.  Either before school or after school was an impossibility for many students due to bussing, part time jobs, or caring for siblings.

At my current job, I teach 4 classes and this allows me to have 3 periods where I am available to work with colleagues and meet with students.  Some days I don't work with any students and other days I might work with five different students.  Today I met with two students.  One had been absent several days in AP Stat and was trying to get caught up.  She had the basic understanding of what she was doing relative to confidence intervals, but needed some encouragement.  She had contacted me via email to arrange for a meeting time.

The second student had done poorly on a quiz and realized that she really didn't understand the basic properties of logarithmic and exponential functions.  She sought me out and found me talking to a colleague about the recent Calculus test.  This student wants to do well, but she often misplaces items, like her homework.  Even though I have time to meet with students during the school day, there are still times I need to meet with students before school or after school.  I have been meeting with one student once or twice a week since the beginning of October.  He recognizes that without this accountability he probably wouldn't get his work done for PreCalculus.

Some may think that these students should be able to figure math out on their own.  After all, the students I have described are seniors.  Doesn't giving the students help make them weaker?  First, the fact that they are willing to ask for help is a sign of strength.  It can be challenging to admit that you need help and are struggling.  If the first time a student asks for academic help is in college, it is likely to be even more challenging.  It is good to learn this skill now.  Second, I ask the students that I work with questions that they should be asking themselves.  How is this like the problem we just did?  How is it different?  Does your answer seem reasonable, and why or why not?  What do we know and what are we trying to figure out?  Modeling the thinking and questioning that students should be doing will ultimately help the student to help themselves.

Teach 180: The Derivative of e^x (Day 104)

I had to have a substitute cover my last period class today and I never count on a sub to know math, especially Calculus.  Less damage control needs to be done the next day, if I go with that assumption.  Today's lesson was finding the derivative of f(x) = e^x and using the chain rule to find the derivative of g(x) = e^f(x) .  In years past, I had students use their graphing calculators to find the value of the derivative of f(x) = e^x  at various values of x.  Next, we would plot points such as (0, 1), (1, 2.718), (2, 7.4) and so on, and students would see that these points lined up on the function f(x) = e^x.

Since I wasn't going to be in class, I couldn't easily lead the students through this activity, and I had no expectation that my sub would be able to do this.  So, I created a short 8 minute video instead and posted it to my YouTube channel to show students why the derivative of f(x) = e^x is f(x) = e^x by using the limit definition of the derivative and a little help from desmos.  With desmos we could easily see the value of the following limit was 1.

To convince the students further, I used desmos to create a table of values for f(x) = e^x

and g(x) = f ' (x).  Behold, they have the exact same values!  Why?!?  Because they are the exact same function!

It's hard to believe that I started my YouTube channel back in November of 2008, almost 10 years ago.  Prior to Desmos, YouTube and Screencast-o-matic my sub plans would have consisted of students reading the book and working through examples.  Technology has certainly made my math lessons richer and has allowed for me to help students draw more connections among various representations. 

Teach 180: The Calculus Test (Day 103)

Today I gave a calculus test on two sections - applications of the derivative, such as maximizing revenue or minimizing amount of material to make a box, and implicit differentiation, which included related rates.  As students took the test, I became worried.  A few of them approached my desk to ask a question or two.  Several of them were confusing surface area and volume.  Others were are trying to solve the problems without using any Calculus!  The title of the class is Calculus and we have been taking derivatives for the past 4 months!!!!

(Fast forward to Saturday) I graded the tests and there was 1 A, 1 B, 2 Cs, 6 Ds and 2 Fs.  I think the lack of continuity over the past few weeks is partly to blame.  Our bell schedule is new this year and we don't meet each day.  Normally, we meet 6 out of every 8 school days.   But here is how things have worked out for the past 3 weeks. (Note that classes are 1 hour in length.)  If the students had been doing some Calculus each of the days we didn't have class, they would have had reviewed for the test for the equivalent of five school days.  As a point of reference, we only reviewed for a comprehensive midterm for three consecutive school days in January.

So, how do I rectify this situation?  Do I simply curve the test and move on?  Clearly that would be the easiest thing to do.  But that would mean that the curved grades would not be a representation of the true level of understanding of the material.

It looks like re-teaching and re-testing is in order.  I am not quite sure how this will be accomplished yet, because I need to consult with the other Calculus teacher.  Luckily, this class is an elective class and we are not bound by AP testing dates.  We don't have to push the students through the course.  

Teach 180: Students Lack of Sleep (Day 102)

Students are naturally more engaged when the topic is of interest to them.  Today's topic in AP Statistics was a review of confidence intervals for a population proportion and our data set was the number of hours of sleep my students got on the first night of school.  I had gathered data from my students on the first day of school using a google form.  I saved the sleep hours column as a .csv file and imported the data into Fathom for us to review.

Next we created a column called "Less_Than_8_Hours" and created the formula at the left to change the quantitative variable to a categorical variable.  If the variable Sleep was less than 8 hours, a "Y" was displayed.  Otherwise an "N" was displayed.  So, what is the point estimate for the proportion of the 47 students surveyed slept less than 8 hours?  And what is the confidence interval?  After we calculated it by hand, we used Fathom to confirm our results.  Our conclusion is that we are 95% confident that the interval from 0.6702 to 0.9042 contains the true proportion of students who slept less than 8 hours the night before school started.  It's crazy to think that students were lacking in sleep even before the school year started!

Teach 180: Teaching on Snow Day #4 (Day 101)

Although it is a snow day, we must keep moving ahead in AP Statistics.  Other schools in the south start their review on April 1.  Because our school year starts later and we typically have snow days, we are lucky to have review begin by April 20th.  Losing one hour of clas today means one less hour of review for the AP exam.  Due to our new bell schedule this year, my one AP Statistics class has missed 5 one-hour classes due to snow, delays and early releases.  I believe my other AP Statistics section has only missed one class.

To try to keep some momentum going, my AP Statistics students will be greeted with an email from me at some point today. (It's still before 9 AM and I am pretty sure they are still sleeping.)  In the email there is a link to one of two videos.  Period A will be learning the lesson that Period F did yesterday - determining sample size and confidence level for a given margin of error for a confidence interval for proportions.  And Period F will be learning about the t-distribution and constructing confidence intervals for a mean.  Luckily for me, I had these videos ready to go on my YouTube channel thanks to snow events in 2014 and 2015.

Happy Snow Day everyone!

Teach 180: You Know You are Winning When...(Day 100)

https://chehockey.files.wordpress.com/2014/06/winner2.jpg

At the beginning of the school year, you quickly identify those students who you think will struggle.  They often have weak backgrounds in a previous math class and have a low self-esteem relative to mathematics.  (At my school, the students often cite "Hot Pencils", a timed math facts activity, as a pivotal moment that changed their view of mathematics.)  Sometimes these students will even leave questions blank on a test or quiz rather than risk putting something down that they think is wrong and in their minds, makes them look stupid. 

At the beginning of the year, I identified a few students like this in my PreCalculus class.  Today, I knew I had reached some of them when two of these students, independent of each other, asked me, "Why did log(-4) on my calculator give me an error message?"  It was at the end of class and it was totally unsolicited.  It's a question I had planned to ask my class that day, but we ran short on time.  The bell had rung and these two students stayed after the bell to ask me their question.  The fact that they had the curiosity to ask that question and not dart like Pavlov's dog at the sound of the passing bell means that I have won.  It means that they, too, are now winning.

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.