Teach 180: Would It Be Possible To... (Day 60)

I love it when my students challenge me.  My favorite types of questions from students are the "What if..." or "Would it be possible to..." questions.  And then if the answer is yes, the natural follow-up is "Why does that happen?"

This line of questioning happened today in Calculus class.  The actual question was "Would it be possible for a function to have both a sharp point and a smooth part?" I should have asked the student to be more precise and reword his question.  But I knew the question he was really asking was: "Is it possible for a function to have both a relative max. or min. with an undefined derivative and a relative max. or min. with a defined derivative?"  Although I could have (and maybe should have) thrown the question back to my entire class, in a matter of about 15 seconds, I produced the following graph in Desmos and I asked "Why do we have cusps on this graph?"  The student recognized that it was due to the absolute value being used in the function.

Next, the student asked if it would be possible to have a cusp in the middle of the graph.  Within another few seconds, I produced the following graph.  To which I heard a student whisper, "that's cool".  Thanks again to Desmos for making it easier for me to keep me and my students curious.

(After class ended, I spent about 15 more minutes playing around with functions in desmos and have the beginnings of some pretty cool ideas for Christmas designs!  I can't wait to share my finished creation with my students.)

Teach 180: Starting to See Connections (Day 59)

One of the things I love about Desmos (besides the fact that it has better resolution than a graphing calculator and it is simple easy to use) is that it makes it easier for kids to see connections.  Today in Calculus we considered at the function f(x) = (x - 2)^2/3 + 1.  Prior to actually graphing the function in Desmos, we calculated the derivative and determined that there would be a critical point at x = 1, because the derivative was undefined at that point.   After we did a quick sketch of the graph by hand, we looked at both the function and its derivative in desmos.  The two graphs are shown on the same axes below.

We could easily see when the derivative was negative and when it was positive and how that corresponded to the left and right sides of the graph.  We could also see that the derivative had a vertical asymptote at x = 1 and that made sense since the derivative of the function was undefined at x = 1.   What was more interesting however was that the derivative had large positive values immediately to the right of x = 1, but then the derivative had smaller positive values as x got larger.  It was right around that moment that I could see the synapses in some students' brains firing as they were getting a better understanding - a visual understanding - of the derivative and its relationship to the graph of a function.  Thank you, Desmos!

Teach 180: Informal before Formal (Day 58)

Two years ago I created my very first Desmos Activity Builder lesson.  It was called "What is the Derivative, Anyway?"  This year I am teaching Calculus again and I was able to pull out this Desmos AB lesson to use in class today.  As I was reviewing it last night, I noticed several things that I think make this a good introductory AB lesson.

First, students are set up to be successful.  They have the foundation needed to do the lesson based on what they learned in our previous chapter.  And since we just had Thanksgiving break, it was a gentle way to get them thinking about math again.  Plus, I could easily use the teacher dashboard to identify student pairs who had "off" answers.  (One pair said "No" to a question that was not a "Yes/No" question and I was able to visit them for a quick discussion.)

Second, the screens build off of each other.  For example,  students are asked about intervals where the tangent lines have positive slopes and then are asked about intervals where the tangent lines have negative slopes.   

Third, it introduces vocabulary after students have had a chance to informally describe a concept for themselves.  On the screen below, students are asked "or is something else going on"?  Some students simply answered this with one word "constant" or horizontal".  Jack and Grayson said "there is a horizontal tangent, therefore the function is at a peak or valley".  To encourage more of a description of what is going on, I'm going to modify the directions on this slide slightly for future use. The informal idea of a peak or valley lead to formally talking about relative maximums and relative minimums. 

After we get through much of this current unit, I am hoping to create a Talkers and Drawers activity, like what is described by Job Orr in his blog post "Three New Desmos Activities: Talkers and Drawers".   I'll probably use it as a way to review concepts before our test in a few weeks, but it might also be a good activity to do as we ease back into things after Christmas break.

Teach 180: Working with Colleagues (Day 57)

I didn't teach until 1:15 today due to my schedule for the day.  However, I was still plenty busy.  I met with one my my colleagues to talk about students who were struggling and revise a test we are giving in PreCalculus tomorrow.  I also met with the Upper School director for about 20 minutes to discuss several items, including our new bell schedule and the challenges it is presenting with teaching in the math department.  The Upper School Director and I brainstormed some ideas for how we could deal with some of the difficulties.

https://pixabay.com/en/cooperate-collaborate-teamwork-2924261/

Next, I met with another colleague who had just come back from maternity leave.  I showed her how to create some formulas in Excel for her gradebook.  She had been using Rediker last year and we no longer use that system.  We also brainstormed some ideas relative to the schedule and course sequencing, which may lessen some of the problems in future years.  One of the best things about working with the people in my department is that they are willing to share ideas and if one of my ideas is a bad one, they let me know about it.  If they think of a better way to do something, they willingly share it. 

In a previous school (this was about 18 years ago), I wanted to work with students during study hall with their math.  They weren't my students, but I wanted to help them succeed and they were willing to work with me.  Unfortunately, a colleague at that school saw my help as a personal affront to his ability to teach.  At no point did I say to the student that their teacher didn't know what they were doing.  I simply offered alternative strategies to solving some problems.  Rather than working together with the teacher to come to an understanding, I was told that I could no longer help the students with their math.  I am very grateful that this is not the case at my current school.  The best outcomes in education are achieved with cooperation, negotiation and being willing to listen with empathy to the perspectives of our colleagues.  

Teach 180: Grandparents' Day Part 2 (Day 56)

(Note: Today (the day before Thanksgiving break) is a 1/2 day of school and many students choose to take the day off.  Rather than talking about that for Day 54, I will be talking about Grandparents' Day for both Day 53 and Day 54 of my blog.)

For Grandparents' Day in calculus students constructed boxes out of paper.  The task: cut squares of equal size out of the corners of an 8.5" x 11" piece of paper and fold up the sides to make a box without a lid.  The goal: make a box with the maximum volume.  You can see the variety of boxes here.

Next, we used calculus to answer the question.  What should the length, x, be for the side of the square to maximize the volume?  We created a function and then took the derivative.  Students knew that the maximum value would occur where the tangent was horizontal.  

I love that this question doesn't lead to an easily factorable quadratic.  For a purely calculus and algebraic approach, students actually needed to use the quadratic formula!  And there was one solution that we had to throw out.  It wasn't the negative solution, because both solutions were positive!  Why couldn't x be 4.915? Since the one side of the paper was 8.5 inches, cutting in 4.915 inches from both sides would mean there would be nothing to fold up.  The domain for x was 0 < x < 4.25.  Finally, we looked at the graph in desmos to confirm our solution.  

Who had the box that was closest to being the one with the maximum volume?  Caroline S. with squares that were 1.5 inches in length and the volume of her box was 66 cubic inches. 

Teach 180: Grandparents' Day Part 1 (Day 55)

(Note: The day before Thanksgiving break is a 1/2 day of school and many students choose to take the day off.  Rather than talking about that for Day 54, I will be talking about Grandparents' Day for both Day 53 and Day 54 of my blog.)

The Monday prior to Thanksgiving, we have an event called Grandparents' Day.  Students get to bring their Grandparents to two classes, attend chapel with them and have lunch with them.  Today I not only had the opportunity to teach my daughter in Probability and Statistics, but her three grandparents - my mother, my father and my mother-in-law.

In that class, we did the following activity involving probability and simulation.  I had 4 index cards; one card had a wave, one card had a circle, one card had a plus and once card had a star.  I asked for a student volunteer to guess what card I was holding to my head.  We did 10 trials and the student got 9 right!  Amazing!!!  Since the average number correct would be 2.5 in 10 attempts, getting 9 right seemed very unusual.  But just how unusual?  Was I giving Neo signals to increase the probability of getting it right?  Using dice and cards, the students worked with their grandparents to conduct a simulation to see how many they would get right in 10 trials if the probability of success was 1/4.  You can see the results from our dotplot below.

Next we used Fathom to run the simulation and we also calculated the binomial probability of getting 9 right by guessing.  The simulated results show that in 100 trials none resulted in 9 correct matches.  This makes sense since the theoretical probability is 2.86 x 10^-5.  Our Fathom results are shown below.

Teach 180: A Personal Day (Day 54)

I am not playing hookie today.  I took my second personal day to visit a college with my daughter, who is a senior.  (How did that time go by so quickly?) It has been an eventful trip. Our flight leaving Newark was delayed by 2+ hours, there was a scuffle that required a police escort of two men off our plane when we landed, and our rental car was reserved at the other Dallas airport.  We finally got into our hotel at 3 AM CST (meaning 4 AM in Easton, PA).  Despite the travel troubles, the visit has been informative for both my daughter and me.

What are my students doing while I am away? Today is E day in the 8 day rotation of our schedule and that means I would be teaching period B and F.  So, my sub is just teaching the first period of the day (8AM - 9 AM) and the last period of the day (2:15 PM - 3:15 PM).  The first period is Calculus and the students took at test and then watched the video "How Big is Infinity?".  After a 40 - 45 minute test, students are pretty spent and it doesn't make sense to try to move onto the next unit.  But I don't want them to sit around shooting the breeze either.  So, we either do a hands-on activity to introduce a topic or we watch a video like this one.

Period F was Probability and Statistics.  Because I am very concerned about falling behind with content due to our new schedule this year, I had the sub do the lesson I had planned for they day.  It included reviewing problems related to probability and having students work on a graded problem set.  Luckily, we have a math teacher that retired from my school a few years ago and having her as a math sub is a great asset. I know I can count on her to teach almost any content and she will do it well.

It looks like the weather for Sunday may cause us more travel headaches when we head home.  I hope not.  Monday is Grandparents Day and I have some special lessons and activities planned to have students interacting with their grandparents.

Teach 180: Rethinking Giftedness (Day 53)

A general perspective: I just finished watching Jo Boaler's film Rethinking Giftedness and although there are some valid things that the students say, I feel that the film is very one sided.  All of the students have a similar opinion - that being labeled as gifted was bad for them.  At no point did they mention a benefit to being labeled gifted.  Does this mean that we shouldn't label students as gifted?  What about labeling students as learning disabled?  The students labeled gifted saw it as a reason for why they couldn't learn something - "the gift running out".  Would those that are labeled learning disabled use it as an excuse for why they can't do well?  I would contend that we should label students (gifted, learning disabled, etc.) not to harm them or make them feel badly about themselves, but to make sure they get the help they need to maximize their potential.

Image from: jcoulter1992.wordpress.com/including-all-learners-2/gifted-and-talented/

My personal perspective: As a child I was labeled as gifted and I do recall the special test that was given to me at age 7 that the students in the film reference.  My parents didn't praise me for this label or expect more of me, because of it.  In fact to this day, I don't actually know the specific test scores or results. They did what all good parents do - encouraged me to learn a musical instrument, read for enjoyment, ride my bike and play with friends.  Because they saw my giftedness as just a part of who I was, I saw it as a part of me and it wasn't something that made me special or different.  It just made me, well, me.

After I was labeled gifted, I was put into a pull-out class with some of my other classmates.  If it wasn't for this class, I would have not learned the BASIC programming language (on a Radio Shack computer).  I would not have written and published a short story (after numerous rejection letters).  I would not have been exposed to logic puzzles and "thinking outside the box".  And I would not have experienced frustration that can sometimes happen in learning with supportive teachers to guide me through that frustration.  Perhaps if the students in the video had experienced their label in a similar way to the way I had experienced it, they would have seen their "giftedness" label differently.

Teach 180: Thoughts about Tracking in Math (Day 52)

Today at our department meeting we had a discussion about pre-requisites for certain courses, specifically Honors courses.  To remain in the Honors track, students need to maintain a B+ grade in the prior course.  Even with a grade that high, there are students who struggle when they are in Geometry Honors.  To lower the grade to a B would mean that we would have more students struggling in that course.  Alternatively, I would be forced to slow the pace of the course down for those students.  Although you may think that teaching material at a pace that is appropriate for the weakest student in the class is a good idea, it actually does a disservice to those students who catch onto concepts easily.  They become bored. Or they get into trouble. Or they do what my best friend and I did in fifth and sixth grade - complete their math work quickly so they can continue to silently read their favorite book.  To remove all Honors math courses completely or lower our current standards would be a mistake.  I don't have research or data to back my claim, but I do have twenty-five years of experience at six different schools in four different states and my experiences with tracking was the same at each school.  It was necessary to help those students who were struggling and important to keep top students challenged.

In my next blog, I'll critique "What Tracking Is and How to Start Dismantling It".  It is an article I found while composing this blog entry and I am curious as to what it says.  Plus I am interested in the short film by Jo Boaler called Rethinking Giftedness.  Her film is not so much about tracking, but more about labeling kids and the damage that can come with a label.  We wouldn't think labeling a student with a learning disability would be negative. So, why is "gifted" a negative label to use?

Teach 180: Math League (Day 51)

One of the things I wish I had more time to do during my day is to solve math problems.  I am not talking about writing answer keys, but problems that I do not initially know how to solve.  Problems that are novel and interesting.  Today I was given one of those opportunities.

Each month a handful of students get to solve challenging problems with the PA Math League contest.  It is a contest that consists of 6 questions and students have 30 minutes to solve those questions.  They span a variety of areas of mathematics including number theory, probability, geometry, algebra and trigonometry.  It is very challenging to get a perfect score on this contest.  When I took this contest in high school as a student, I would typically get 2 or 3 questions correct.

After the contest today, I had one senior who wanted to work with me at the whiteboard after the contest was over.  Since I had not been working on the contest when the students were taking it, (I was helping a student from one of my classes) I was initially stumped by the problem.  We discussed ratios of area of similar figures and between the two of us we generated some equations that eventually led us to the correct answer.  Our lively chatter about the solution brightened my day and made me wish for more moments like this.

As I was driving home in the car tonight after an 8th grade parent night, I was struck with another idea about how to solve the problem!  Since not all students have taken this contest across the country yet, I'll hold off on posting the problem and my solution for now.  In addition, I have some ideas that I want to try out in an effort to generalize a solution.  Generalizing solutions is the pinnacle of solving a math problem. As Fermat once said with his generalization, "I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain."

Teach 180: Pseudo-Context Problems are Dull (Day 50)

I'll admit it.  I sometimes pull things from my file and say to myself, "This is what I did last year and this is what I will do this year."  And then afterwards I say to myself, "What was I thinking last year? That was awful."

Well, today was one of those days.  In PreCalculus we are studying function composition.  Here is the problem I pulled last year from the textbook to model a problem that students would see in the homework. 

There really is no context here.  Why 1000?  Where did 5 come from?  What is it that we are selling?  The problem itself involves a bunch of algebraic manipulation and there is no real reason for doing it.  There is no question to solve.  At one point, I may have even yawned while going over this question.   No, I am pretty sure I yawned.   Why did I drag my students through this pseudo-context?  Actually, I am not sure I can even call it pseudo-context.

Tomorrow is another day and I have made notes for next year to scrap this problem and replace it with a problem grounded in real context and not pseudo-context. 

Teach 180: Connecting Inverses (Day 49)

Anytime you can help students to connect new learning to previous learning, I believe that the understanding of the new concept is stronger. Today I set out to purposefully connect the concept of function inverse to the familiar concept of inverse operations. I began by having each table find f(g(x)) and g(f(x)) for the two cards at their table. Then, I had each table report out on their results. (You can see the pairs of functions at the left.) Me: "What????? Wait a minute. You mean each of you got that f(g(x)) = x and g(f(x)) = x. I wonder if that always happens. Hmmmm. Let's switch g(x) cards and see if that will happen again." So, I had the student groups swap their g(x) functions and calculate f(g(x)) and g(f(x)) again. This time no groups got a result of x. Me: "Well, that's thoroughly disappointing. I was hoping we would get x again. I wonder why we didn't. OK, let's look at the initial pairs we had." Next, we wrote the original pairs of functions on the board and I asked students what they noticed about each pair of functions. One student quickly realized that one pair of functions had squaring and square rooting and she described those operations as "opposites". After that other students noted the other inverse operations they saw, like multiplying by 2 and dividing by 2, and adding 3 and subtracting 3. I pointed out that not only did each pair of functions have inverse operations, but that they were done in an inverse or opposite order.  For example, in the first pair of cards for f(x) we would add 3 and then square the result.  For the inverse function g(x), we would do the inverse of those operations in the opposite order.  This means add 3, then square becomes square root, then subtract 3. Our next class will meet on Tuesday (there is no school on Friday due to Parent-Teacher conferences and no class on Monday due to the new bell schedule).  For now, I am going to ask students to create inverse functions by using inverse operations.  We'll graph our functions and inverses in desmos to see that the composition of a function and its' inverse is the identity function. We will also note that if a and b are numbers that are defined for both functions that f(a) = b and g(b) = a.  This will also be easy to show on desmos. I will give my students "easy" inverses to create initially, but then they will be given ones that are more challenging, such as find the inverse of f(x) = (1 + x)/x.  This will lead to a need for the formal way to find an inverse - replacing f(x) with y, switching x and y, and solving for y.

Teach 180: The Calculus Toolbox (Day 48)

One of the things I love about teaching at a small school is that students find it easier to take risks and ask questions during class.  Today, we were working on learning the Chain Rule in Calculus.  One of the problems we reviewed involved using both the Chain Rule and the Quotient Rule.  However, one of the students in my class raised her hand and asked if we could rewrite the quotient as a product and use the product rule instead.  At this point, I told my students that they had a "Toolbox of Calculus Tools" at their disposal and that they could choose to do many of the problems in a variety of ways.  In fact, one of the things that I love about math is that there are often a variety of approaches to solving a specific problem. We ended up working through the problem using both the Quotient Rule and the Product Rule and saw it resulted in equivalent forms of the solution.

I am curious as to which tools students would prefer to use when given the option.  So, for our class opener on Monday (that's our next day of class - 5 days from today), I plan on giving students a few problems that lend themselves to using different tools in their Calculus Toolbox.  It will be interesting to see which tools they choose. 

Teach 180: Bad Test Grades (Day 47)

(Warning: This blog entry is being written after spending almost 10 hours at school and working 3 hours at home.  There may be some ramblings in this particular entry.  You have been fairly warned.)

I just finished grading a Pre-Calculus test.  Sometimes students do poorly on a test.  It happens. It could be for a variety of reasons or combination of reasons, such as:

1) The student didn't study.
2) The student wasn't paying attention in class.
3) The student stayed up late the previous night.
4) The student just broke up with his/her girlfriend or boyfriend.
5) The student didn't have a good foundation from previous classes.
6) The student has test anxiety.
7) The student has senioritis and doesn't feel like learning.
8) The student is waiting to turn 18 for his trust fund to kick in. (This was true one year.)
9) The student didn't meet with the teacher to review prior to the test.
10) The student didn't understand the material and/or had major misconceptions.

And there are probably many others that I didn't even consider that could be on this list.

A test is designed to assess student understanding of concepts (Reason #10) and not other things that can impact a student's performance (Reasons #1 - 9).  I know many of my math teacher colleagues at other schools allow for re-testing.  I even re-test from time to time.  But at some point students need to be held accountable for what they learn.  And if they don't learn it, they re-take the course in high school or take a remedial course in college.

If I constantly allow re-testing and it is because of reasons like #1, #2, or #3, then I am just reinforcing bad habits.  If it is because of Reason #6, then the student should get help from the learning specialist.  If it is a reason like #7 or #8, then at what point am I the fool for wasting my time and energy?  Time and energy are finite resources and I would rather give them to a student who is willing to work and learn.

Teach 180: Always Plot Your Data (Day 46)

Today in my Probability and Statistics class we did one of my favorite activities.  We used the Anscombe quartet to learn the lesson that summary statistics are only part of data analysis, and the fact that it is very important to always plot your data!  The Anscombe quartet has 4 sets of bivariate data.  You can see the data below.

Each of the four data sets have the same correlation coefficient of about 0.816.  They also have the same least-squares regression line.  But is a linear model appropriate in each situation?  What do the graphs tell you?

You could see the initial disbelief on my students faces when we looked at the results of each group on the board.  First, they couldn't believe that there could be data sets that could have such a high correlation coefficient that were clearly not linearly related.  Second, they realized that the only way they could see the relationship was non-linear was to look at the graphs.

For more on Anscombe's quartet, I invite you to read this interesting blog post.

Teach 180: Irony and Chocolate Consumption (Day 45)

We finally came to a place in my schedule with my Probability and Statistics classes where I could actually teach the students the same content on the same day! (This year I teach 2 sections of Prob/Stat, 1 section of Calculus and 1 section of PreCalculus.)  I was so excited to have a day where I would only had to prep for teaching 3 different classes, instead of 4!

However, after teaching my first class of Probability and Statistics, I realized that there would be no way to get the new content taught in just 20 minutes.  (Students who needed extended time on the quiz in my first class could use part of the lunch period to finish the quiz.  There was no lunch period in my afternoon class and this led to 10 minutes less instructional time.)  Plus, the three big ideas to be taught with my second class were to be taught between 2:55 PM and 3:15 PM on a Friday afternoon.  Would my students remember these ideas when we had class again on Tuesday? Unlikely.

So...did I plow through the content?  Did I plan for this and "flip my classroom", having all students watch video explanations of the content after the quiz? I did neither of these things.  Instead, I had students read an article from The New England Journal of Medicine called "Chocolate Consumption, Cognitive Function and Nobel Laureates".  It is a brief article that shows that there is a strong positive linear correlation between chocolate consumption (in kg/yr/capita) and the number of Nobel Laureates per 10 million people.  The article suggests three possible reasons for the association, but clearly we cannot assume that creating a  law that requires people to eat more chocolate will increase the number of Nobel Laureates within a given country.  The brief, but important, lesson students learned on a Friday afternoon is that correlation does not imply causation.

Teach 180: The Document Camera (Day 44)

Today we did more work on Pre-Calculus with transforming functions.  When several transformations are involved at once, it can be quite challenging for students to draw the image from the preimage.  Today, I decided it was best to use my HoverCam document camera to work through several problems with students. This allowed students to better see how each of the points was being transformed.  A screenshot of a completed problem is shown below.  The original function is in black and the transformed function is in purple.  The red annotation was added using the HoverCam software to show a pair of corresponding points.  I also had several students come to the front of the room and they also used the document camera to demonstrate how to transform various functions.

Depending on the assignment, I have found the HoverCam to be a helpful way for showcasing and critiquing student work.

Teach 180: Celebrating Students (Day 43)

                                                                                                                                          

One of the aspects of my school that I really like is that as a whole school we celebrate student successes.  We aren't just talking sports, but also successes in the arts and academics.  Today, I was able to announce in front of the entire student body that our Math Madness team won its first round of bracket competition with a score of 21-15. Plus, we have a student that is ranked #72 out over 20,000 students nationwide in Math Madness for the fall season.  The students cheered and clapped just as enthusiastically for this as they do for winning a district sports championship.

And speaking of district sports, good luck to the field hockey team and boy's and girl's soccer teams on their district games today and tomorrow. Go MA Lions!