## Tuesday, October 31, 2017

### Teach 180: A Little Competition Can Be a Good Thing (Day 42)

Today was our first day in Probability and Statistics for analyzing bivariate data.  One of the concepts that students sometime struggle with is estimating the strength of a linear relationship from a scatterplot.  They think there is no association when there is a weak, negative linear association.  Or they think there is a strong, positive linear association when it is more of a moderate, positive linear association.

Today, we used the Rossman-Chance applet called Guess the Correlation to improve our estimation skills with correlation.  Student were arranged in a bracket-style competition.  Each student was shown a randomly produced scatterplot with 25 dots and the student had to guess the correlation coefficient.  The student in the pair that was closest to the correct value moved on to the next round of competition.  An example is shown below.  You can see that I did fairly well. But not as well as the winner of the tournament, who was within 0.005 with his estimate!

### Teach 180: Sometimes I Don't Listen (Day 41)

One of the things I love about working with the other math teachers at my school is the fact that we enjoy collaborating.  If we had a common office space, I am guessing it would be a challenge for us to get anything accomplished individually, because we would be sharing ideas all the time.

Even though we share ideas frequently, I don't always listen.  Marilyn told me that this one activity took her longer than she planned and I didn't listen.  In my plans, I had even made a note that I thought the activity would only take about 15 minutes.  Then, 20 minutes elapsed, then 25 minutes and finally the last group completed the activity in about 30 minutes.  Students were to match a function with its graph, domain & range and characteristics. You can see a grouping of four such cards below.

It was a valuable activity and students did really well working together.  Why did it take so long?  Part of it was the fact that there were 40 cards in front of them and I gave them no guidance in what might be easiest to match first.  In addition, I let students debate which cards were to be matched together and did not step in when there was a disagreement.  Usually, the student with the right answer prevailed and convinced the other students at the table as to why the answer was correct.

So, what would I do differently?  I might do eight groups instead of ten and I might model the thinking process for completing one match.  This would allow us to review key ideas, like open and closed intervals, prior to having them work on the activity in their groups.

## Sunday, October 29, 2017

### Teach 180: The German Tank Problem (Day 40)

I spent about two hours on Friday finalizing what I needed to do to prep for a talk that I was giving on Saturday.  The talk was given at the Association of Math Teachers of Philadelphia and Vicinity (ATMoPAV) fall conference.  The title of my talk was "The German Tank Problem: Simulating a Statistic".  For those of you who teach AP Statistics, this is an activity you likely do to introduce sampling distributions.  When I first did this activity, it bombed, because I wasn't comfortable with using Fathom to create a simulation of a sampling distribution.

Then in 2009, I took an online course to learn how to use Fathom.  I still wasn't comfortable with using it entirely and spent about a year screencasting demos to show Fathom in class.  Then in 2010, I started to use Fathom with my students.  It is a great visualization tool and it is great for helping students understand the difference between the distribution of a population, the distribution of a sample and the distribution of a sampling statistic. (Note: Fathom has a free trial version and a 1-year subscription is \$5.95.)

For the German Tank problem, students are given a bag with N "tanks" numbered in their bag consecutively from 1 to N.  They randomly select 7 tanks and use the numbers on the tanks to create a statistic to estimate the total number of tanks. The eight participants at my workshop on Saturday worked in pairs and spent a good fifteen minutes to create their statistics.  At the right you can see the common statistics students often create.  The Partition Method is the one that was used by the mathematicians in WW II to estimate the number of tanks.  At the workshop on Saturday, participants created several of these common statistics.

I had been doing this problem for several years and last year I had a pair of students create a statistic that was good as (or perhaps better than) the Partition statistic.  I named the statistic after the two students, Cecily Redfern and Neelam Ferrari.  It's called the Redfern-Ferrari statistic.  Two hundred trials of the simulation can be seen on the screenshot at the left.  Note that 344 is the true number of tanks in the bag and that both Partition and Redfern-Ferrari are centered near 344 with similar variability.

What actually is the Redfern-Ferrari statistic?  The formula for the statistic is Max. + Range/6.  The students found the average difference between two consecutive numbers by calculating Range/6. Then, they added it to the maximum number they drew to get the approximation for the value of N.

If you have tried this activity and abandoned it, I would encourage you try it again. If you are interested in the Fathom files to do this activity and/or the handout I give to my students to do the activity, send an email to me at mathteacher@ptd.net or leighnataro@gmail.com.

## Thursday, October 26, 2017

### Teach 180: Marbleslide Fun! (Day 39)

Today I filled in for one class of Algebra 2 and I was asked to lead a Marbleslides activity on lines .  Students primarily used slope intercept form and worked on restricting the domain of the function.  Although the teacher I was subbing for had shown students graphs in Desmos, the students did not have much experience using Desmos for themselves.  So, I used the teacher pacing feature to lead the students though the first few slides of the activity.  The few screenshots seen here show what the teacher pacing looks like on the teacher dashboard.

When you select the Pacing icon on the left side of the screen, you see this on the dashboard.

When you click on a slide, it highlights it and then you click on the orange "Restrict to Screen X" button to place all students on the chosen screen.

Once on that screen you can advance or go back one screen at a time by moving left and right with the < and > buttons.  But did you notice the plus sign?  Do you wonder what happens when you hit that?

What?!?! You can select multiple screens!  Now I can assign certain tables to one problem, if I wish and then we can have a breakout discussion after a few minutes.

The students were fully engaged and working with a partner for about 45 minutes.  Two girls did not want to work together and I allowed them to work alone.  I soon noticed that both of them struggled and did not persevere for as long as those working in pairs did.  In addition, I spent most of my time working with the two of them instead of checking in with the other students in the class.  This definitely showed me the value of a 2:1 ratio of students to computers.  In the future, I'll use my "random assignment by playing card" method to pair all students up and require students to work together.

## Wednesday, October 25, 2017

### Teach 180: Sometimes They Need Time (Day 38)

Some days I create review games, but today I simply gave my Calculus students time to work and check their own understanding in preparation for their test. Too often students look to the teacher as the one to give them to say "OK, you got this."  However, I think it is important for students to be able to gauge this for themselves, especially seniors.  In addition, I believe it is valuable for students to have time in class to work together and help each other to construct their own understanding of concepts.  This proved to be very helpful for students in Prob/Stat yesterday.  When I couldn't get around the room fast enough to answer individual questions, they turned to each other.  Hopefully, I'll have some time this year to be more purposeful in teaching my students some of the metacognitive strategies seen here.

Note: The above infographic came from the website blog.innerdrive.co.uk/.

## Tuesday, October 24, 2017

### Teach 180: Gallery of Parent Functions (Day 37)

In PreCalculus class today, we took our time with graphing parent functions and as a result, we didn't get through the entire lesson I had planned.  However, I think it was important for us to not rush.  Plowing through content for the sake of "coverage" can lead to shallow understanding by students. The students were engaged the entire time and accurately created what needed to be done for their given functions.  They worked in pairs to graph their given function, made a table of values, identified the domain and range and determined if their function was even or odd.

Posted above is a picture of one parent functions that we hung in our gallery and the second picture shows the entire gallery.  Having these posted in the back of the room will be a quick reference for us as we start to talk about graphing transformations from the parent functions, or as I like to say graphing "offspring" of the parent functions.  Having all the functions up at once had the added benefit of allowing students to compare and contrast each of the parent functions.  Understanding why certain functions are alike and different helps students to get a better grasp of the structure of the underlying mathematics.  Without this understanding, students see math as a disconnected set of ideas or rules to be memorized.

### Teach 180: What Happened to Monday? (Day 36)

Monday was such a crazy day that I didn't end up posting about it on Monday.  (It is now Tuesday evening.)  After making copies and finalizing some items for the day, I taught four classes that were an hour long and led a 45-minute advisory period discussion on identity. During the one period that I didn't teach, I met with four other teachers to discuss proposals that had been submitted by twenty students for independent study projects.  At 3:30, I proctored a student for the Who Wants to Be a Mathematician Round 2 Qualifying Contest and had a student make up a test.

Whew! The day flew by.  What happened to Monday?  This was one of those days where if you asked me to reflect on my teaching and what I might do differently I might be at a loss.  The one thing I can say is that my PreCalculus students definitely did not recall how to find the domain of various simple functions, like y = x2 after 3 consecutive calendar days of no math class.

## Friday, October 20, 2017

### Teach 180: Dynamic & Crowdsourced Learning w/ Desmos (Day 35)

Today I put the finishing touches on my presentation for the Careers in Mathematics Conference that will be at Millersville University tomorrow.  (Even as I am blogging, I am making a few more changes to my presentation slides!)  I am hoping it goes well, but I always get nervous when it comes to presenting.  Although I had put the finishing touches on my presentation last night, I still did more work on it at school today.  A 50-minute session with time for reflection and hands-on learning on the part of the participants is very challenging.  I have a co-presenter with me, Bob Lochel, and I am very grateful that he will be there for back-up.  What's planned for the presentation?  Quite a bit.  Reflecting on the role or purpose of technology in math class.  Crowdsourcing some data to make some predictions with the help of desmos.  Letting participants see Desmos Activity Builder from a student and teacher perspective.  And finally, encouraging participants to take steps to incorporate desmos into their teaching.

## Thursday, October 19, 2017

### Teach 180: The Dollar Store Tip (Day 34)

I have purchased large gridded chart paper from Staples in the past.  It's not cheap at \$1 per page!  I use it from time to time, but sparingly because it is so expensive.  Yesterday a colleague and friend, Marilyn, said that she thought rolled wrapping paper might work, because it has the grid lines that we would need.  I said perhaps, but I would be concerned about the waxy-ness of the paper.  Would the marker rub off or smudge when writing on it?

If you use Sharpie markers, the answer is no.  Here you can see the graph she drew and the fact that it is on the backside of some colorful wrapping paper.  The roll of wrapping paper cost her \$1 at the local dollar store. I am guessing the approximate cost for her to use this with her math students is 5 or 10 cents per page.  And the bonus?  She can still use it to wrap her presents when the unit on parent functions and transformations is done!

### Teach 180: Sub Plans Thwarted (Day 33)

It was 11:50 AM and I had just finished putting together my sub plans for the next day.  It was a task that took two hours.  I had recorded a 30 minute screencast using screencast-o-matic and posted the video to my youtube channel.  The screencast was an introduction to normal distributions and using the table for a standard normal distribution.  This was not something I was going to expect my sub (a former history teacher) to be able to do.  My other classes had work to review concepts from previous classes.  Then, I saw the text shown at the right from our technology coordinator.

Initially, I was upset.  Should I redo my plans and do something different tomorrow?  It was then that I remembered that I would be missing about 10 students in total due to a field trip.  For those students, the screencast would allow them to stay caught up.  Maybe it wasn't so bad that I did the screencast after all.

Note: I highly recommend screencast-o-matic for doing screencasting.  Screencast-o-matic is free for the 15 minute version.  If you want to make screencasts longer than 15 minutes and you want to be able to use the cool editing features (like splicing and annotating), it is worth it to pay the extra \$30 per year fee.

## Tuesday, October 17, 2017

### Teach 180: Anyone, Anyone (Day 32)

We had a good turnout for our Math Madness team this morning and our first Math League contest was successful - triple the number of students in previous years.  But I felt like it was a sub-par day for teaching.  I had good activities planned, but I felt myself rushing in my Period A Probability & Statistics class.  I wanted to get things done quickly to make sure the students had time to work on a mini project in Fathom.  I found myself asking basic questions and then answering my own questions!  Ugh!

At one point, I think my students got lost in the technology and missed the point of the lesson.  Did they really understand how transforming data (adding a constant or multiplying by a constant) impacts the results of summary statistics?  I will be teaching this same concept tomorrow to my other Probability & Statistics class and they will not be using Fathom.  Will things be better?  I hope so.  Stay tuned!

## Monday, October 16, 2017

### Teach 180: Playing Cards for Quick Grouping (Day 31)

If I allowed my students to choose the people to be in their group, they would choose the same group almost all the time.  And there might be some students that wouldn't get chosen - the mean kid or the bossy kid, for example.  The quickest way to randomly put students into groups is with playing cards.  Today I had 14 students in Calculus.  I used 15 playing cards (3 Aces, 3 Twos, 3 Threes, 3 Fours and 3 Fives) and once the cards were distributed there was one card left over.  It was a five.  That meant that the group based on the cards with Fives only had two people in the group.  Student then worked on solving problems at the board in their groups.  Quick and easy!

Bonus tip: If you are looking for a bunch of playing cards, I have been told by a workshop attendee that you can often get used cards from casinos.  Apparently they drill holes in the deck once they are done using it.  The cards are still good for use in class and the casinos give them away for free.

## Sunday, October 15, 2017

### Teach 180: 6 of 13 (Day 30)

6 out of 13 or 46%.  What do you think that number represents?  The percent of students who turned in their homework.  Nope.  The percent of students who were late to class due to traffic on Route 22.  Guess again.  If you said, the percent of students who took the Probability and Statistics test that was scheduled on Friday, you would be a winner!

Why were about half of my students not taking the test?  There were 3 reasons.

Reason #1: Three students were required to be downtown at the lower school campus to sing for Chamber Singers at the Alumni Chapel service.  There was no time for them to take the test prior to getting on the bus around 1 PM.  Even if they were prepared to take the test, they need to wait until Monday or Tuesday and take it when they have a free class period.

Reason #2: Two students had missed the previous two days of class due to golf tournaments and weren't prepared for the test.  Technically, I could have required them to take the test, but I did not.  I would rather that they have a chance to learn and review the material first.

Reason #3: One student was on a college visit and had a pre-arranged absence.  This will happen with each of my seniors about twice this year.  That is about 47 seniors x 2 absences/senior or 94 absences.

If I tried to plan my testing around all of these different events, I would only be giving the final exam and midterm exam or testing on the first or second day of school.  At the end of the day, I looked at the post-it note on my desk and realized that I had somehow over the past two days got the students to commit to a make-up time.  Unfortunately, I need to be available for 5 different make-up times for the 7 students.

## Thursday, October 12, 2017

### Teach 180: The Importance of Sleep (Day 29)

Today I talked to a student about the fact that he was a bit tired.  He had asked to go to the restroom and when he returned he admitted that he really was just walking around to wake up a bit. Then, he went on to describe his very busy day the previous day and mentioned that he did not get to sleep until 2 AM, which was not that much later than the 1 AM he was accustomed to.  It doesn't matter how engaging my lesson is. Sleepiness will win - always.

This student was not necessarily an atypical student.  Below you will see a graph showing the hours of sleep my students said they got on the night before school started.  Yes - BEFORE school started.  There was no reason for them to be up late - no tests to study for and no homework to complete.  (Note: This graph is a compilation of several years of data.)

According to Nationwide Children's Hospital, teens need 9 to 9.5 hours of sleep each night to function and the average that teens actually get is closer to 7 to 7.25 hours.  How did my students do?  The mean for the distribution is marked on the plot at about 6.97 hours. In addition only 11 of 206 students got 9 or more hours of sleep.  That's about 5%!

I see students caffeinated and drinking Monster energy drinks frequently at school.  Some skip school to catch up on their sleep.  Others take naps in the library.  You may think the student that got 14 hours of sleep is doing well.  However, I know that the student that gave this value is the same student who is often seen with an energy drink in his hand at 8 AM.  The 14 hours was probably to make up for sleep deprivation.

Clearly something needs to be done to make students aware of the damage that sleep deprivation can cause.  Many think that staying up late studying will help them perform better, when in fact it is just the opposite.  This is not a problem that I can solve alone, but it is a problem that needs to be addressed schoolwide.  If you have addressed this problem at your school successfully (or even unsuccessfully), please let me know what you have tried by posting a comment about it.

Now it is time for me to take my own advice and get some rest.

### Teach 180: Speed Dating (Days 28)

Note: Although I was to blog each day for my attempt at a Teach 180 blog, I got home on October 18th from visiting Baylor University in Texas at 3 AM and was at school at 7:20 AM.  Sleep trumped blogging last night.  So, this is essentially a blog for Wednesday, October 11th.

Students often learn best when they need to explain ideas to others.  They also learn best when they have the opportunity to struggle with questions themselves.  You get the best when you review with "Speed Dating", which I have used several times to review this year for tests.  On October 11th, I used it in my Probability and Statistics class.  The set-up is to have students sit in two circles facing each other.  (See diagram below.) Initially a student sits across from another person in the class and those two students become experts on the problem or problems they have been assigned.  While students are working on their assigned problems, I circulate around the room and check in on each pair as they finish solving their problem. (In the diagram, you can see that two students - denoted a circle and a square - are assigned the same problem.)

Then, the "dating" begins.  I have the students in the inner circle move one seat to their left.   (See diagram below.) Now each student is seated across from a different student and the students share the problems they worked on with each other. Some students like to work individually on the questions first and then confirm the solution to their problems with their "date".  Others like to chat about the problems they have solved from the beginning.  Each round of sharing lasts about 2-3 minutes and then students in the inner circle move to their left again.  This continues until the students in the inner circle end up back at their original seat.
What if students finish early?  I usually have 2 or 3 questions that aren't assigned to anyone that I encourage students to work on if they finish early.  This keeps everyone engaged, even during the down time.  What if there are an odd number of students?  I partner up with a student and participate in the activity.  It is enjoyable for me, because I also have a few seconds to talk with each student about how their senior year is going so far.

## Saturday, October 7, 2017

### Teach 180: DeltaMath (Day 27)

Often the best ways to learn something is to learn from our mistakes.  A few years ago, I discovered www.deltamath.com.  I don't remember how I discovered it, perhaps it was at a conference.  But more than likely, it was through twitter or a suggestion posted by someone in the chat room at a Global Math Department webinar.  DeltaMath is free and allows teachers to create individual practice assignments for students.

On Friday, I was teaching students limits for the first time in Calculus.  Understanding the notation of x -> aversus x -> a can be confusing for students.  After going over an example or two as a class, I could tell that some students were having trouble understanding the concept.  I had anticipated this and had created an assignment in DeltaMath for my students to use.  This was the first time students had to log in to DeltaMath this year and it was very easy for students to enter my teacher code and get started.  Within ten minutes, each student had correctly answered ten questions like the following.  Some students needed fifteen questions and individual instruction from me, but all students showed understanding of the concept of one-sided limits fairly quickly. Thank you DeltaMath!

## Thursday, October 5, 2017

### Teach 180: Using Student Feedback (Day 26)

Teaching is filled with moments of on the spot decisions based on feedback from students.  The feedback can be the sound of crickets when asking a question.  The feedback can also be in the form of students asking questions which make you realize that a concept needs more clarification.  And from time to time, you do a lesson with one group of students in the morning and change it up a bit for the afternoon, based on something that didn't go according to plan.

Today I decided to do something a little differently than I had in the past.  I did a lesson with desmos activity builder called AP Stats: Matching Boxplots, Histograms and Summary Statistics.  The original activity was written by Sandi Takis and adapted from Activity Based Statistics.  The activity was revised and reformatted by Kathy Fritz and then adopted for Desmos Activity Builder by Robert Peterson.  I modified Bob's Desmos activity to include some question slides after students had created the matches in the card sort.  (So, this is the 5th iteration of the activity, if you are trying to keep track.)  Here is one of the matches from that activity.

Eight sets of 3 matches was a bit much to fit on one tiny computer screen.  Even though students worked in groups of three, they struggled with getting the correct trio of boxplot, histogram and summary statistics.  Students seemed to be more concerned about getting the answer right and checking it against the red and green cards in the front of the room than understanding why it was right.

For my afternoon class, I decided at the last minute to pull out the laminated cards version of the activity.  There were more animated discussions and even some arguments within the groups as to how to match the cards.  The groups finished at about the same time and we were able to finish the activity with a rich whole class discussion.  This important de-briefing was lacking in the earlier class, because we ran out of time.

During my twenty-five years of teaching, I always felt bad for my first class to get a lesson.  The lesson that was taught to them was good, but for my other sections of that class later in the day, it was always better.  Using student feedback and modifying lessons based on student feedback is what makes accomplished teaching a craft and not a trade.

### Teach 180: Working with Individual Students (Day 25)

When I worked in a public school, I typically had 25-30 students per class and I taught 5 classes.  If a student struggled, I would want to work with him or her during a planning period. However,  that often didn't work due to the student being in class when I had my planning period.  This would require me to work with the student before school or after school.  With about 4-5 students per class needing help, you don't need to be a math teacher to see that this is just not sustainable.  But let's confirm this.  Here's the math. Working with 20 students for 15 minutes once a week was 5 hours of individual help.  If the students actually needed 30 minutes of help, that became 10 hours per week.  Arriving at school at 7:15 and leaving at 5:15 would have been necessary to accomplish 10 hours of helping students outside the school day.  With an hour long commute each way...well, you get the picture.  Definitely not sustainable.

I couldn't help students the way I knew worked best - individually.  So, I decided for the sake of sanity to change jobs.  Instead of teaching in a public school in New Jersey, I now teach at Moravian Academy, an independent school in Pennsylvania.  With 4 sections and only a total of 53 students, I have fewer students that need help and I have the time to help them individually.

For the past few days I have been working with one student in particular.  I can already see that this student is gaining confidence in his ability to do math and that student is starting to participate in class again.  Even better, this student has not said "I am stupid," when working on math this week.  Individual help for students matters.

## Tuesday, October 3, 2017

### Teach 180: The Twenty-Sided Die (Day 24)

Today students worked in groups in class to solve problems related to circles, distance and midpoints.  I could have told students to work on these questions together at their tables.  However, when you can add points to something to make it a competitive game, you get more buy-in and students have more fun.  And let's face it, it's more fun for the teacher, too.

Because students were working together, there was a high probability of each team getting the questions right.  Having all teams get one point for a correct answer seemed a bit boring.  So, I introduced a 20-sided large foam die to our game.  We had four teams and the first team rolled the die.  It was 18.  That meant that particular team could get 18 points, if they got the question right and all other teams would get 1 point.  Basically, each team had a chance to get up to 20 points when they rolled the die. This definitely added to the excitement of answering the questions and you could hear a collective groan when one team rolled only a 3!  Who says learning can't be fun?

### Teach 180: Automatic Recall (Day 23)

From time to time, we (the teachers in my math department and me) require students to do a portion of an assessment without a calculator.  Just like we don't expect students to use a calculator to multiply 8 x 3 in elementary school, we don't expect students to need a calculator to do basic arithmetic with rational exponents. Today in Calculus, I reviewed rational exponents with my students.  Below, you can see some of the types of questions we did.

To review these we played a game called "Exponents Around the World".  There are 32 questions and one question is on each slide.  A student goes in head-to-head competition against another student for a question.  Whoever says the answer first moves on to compete against the next student, trying to get "around the world" of the classroom.  It was very competitive as one student had just three students to beat to make it around the world.  The class is primarily seniors, but there are two juniors.  This one junior boy lost his only head-to-head competition against the junior girl and did not quite make it "around the world".

Although this is a fun game, not all students thrive with intensive competition or time pressures.  In fact, according to the book Neuroteach, if the pressure or stress is too high learning stops completely.  Since this was a topic students had seen before, I thought it was acceptable to use this highly competitive game to review.  However, I would not have used this game as students were just starting to learn the concept of rational exponents.