## Friday, October 20, 2017

### Teach 180: Dynamic & Crowdsourced Learning w/ Desmos (Day 33)

## Thursday, October 19, 2017

### Teach 180: The Dollar Store Tip (Day 32)

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!

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 31)

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.

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 30)

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!

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 29)

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.

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 28)

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.

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.

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 27)

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 26)

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.

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 25)

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 -> a

^{- }versus 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 24)

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

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.

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 23)

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.

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 22)

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?

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 21)

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.

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.

## Friday, September 29, 2017

### Teach 180: Limits Aren't Just for Calculus (Day 20)

One of the things I love about being where I teach is that teachers are welcome to be a part of the student community. A few years ago, I dusted off my violin and played with the string ensemble at school. That doesn't work in my schedule this year, because I am teaching a class when strings is scheduled. However, I am able to sing with chorale and singing as part of a larger group is something I really enjoy. It helps me to see students as more than "just a math student".

The Processing Speed of the Fuzzy Brain |

Near the end of chorale today students were starting to loose focus. I know my mind was starting to get fuzzy, too. Keep in mind that classes are 1 hour long this year due to a change in our bell schedule. The director of chorale recognized we had reached our limit. To switch things up he had us sing an acapella tune that he made up on the spot. He started with the bases, giving them a line to repeat. Next came the altos, then the tenors and finally the sopranos. Once all the groups were singing together there was clapping and swaying and smiles on the faces of many of the students. It was just the break we needed before singing through a piece one last time.

There are times when I see students are at their limits of focus, but too often I power though rather than taking a momentary break or detour. What the teacher did in chorale today was clearly valuable and allowed the class to finish strong. I'll need to think back to that moment when I encounter a similar loss of focus with my students in the future.

## Thursday, September 28, 2017

### Teach 180: A Distribution of Pennies (Day 19)

Today in Prob/Stat students worked in groups of 3-4 to create a distribution for the ages of pennies. One group even organized their pennies in a dotplot of pennies.

Once we combined all the data we

had the following dotplot at the front

of the room. (The right tail

ended up at 59, but I erased part of it and then snapped this picture.) There are more younger pennies than older pennies, but it was clear to see that creating a dotplot like this by oneself would not have been very practical. Our final dotplot had 180 observations!! Clearly dotplots are better for smaller data sets. What would be a better way to organize the data? A histogram!

First we created the histogram by hand and then we used our TI-84's to make the histogram. Students set the window manually to match the original histogram we made by hand. Then, we used the Zoom Stat feature. This allowed the calculator to determine the bar width. But what was the bar width? 6.875 years? Since penny ages were recorded in years, on a quick glance the histogram at the left makes more sense than the histogram at the right. Next up? Measures of center and spread and boxplots.

## Wednesday, September 27, 2017

### Teach 180: Learning More About Card Sorts (Day 18)

Today we did a card sort activity in Calculus to review some basic parent functions and to remind students about creating "offspring" by using basic transformations of parent functions. As a teacher, what did I learn from today's activity?

1) The teacher dashboard can be displayed to give feedback to students. Students could see if their matches were correct and fix their errors. This also allowed me to touch base quickly with the few pairs that were struggling. Correct matches are in green and incorrect matches are in red. It was interesting to note on the second slide that not all students used the same strategy, but there were two or three main strategies.

2) The end of activity "How did you do?" shows me that I need to circle back to one specific pair of students tomorrow. But it also shows me that even though all students completed most of the activity correctly, they admitted to not easily remembering all of the parent functions. This is something I need to keep in mind when I do anything with students that is supposedly review. Review and recall does not come as easily for some students.

3) The "offsprings" of a vertical shift down 5 units or a horizontal shift right 5 units were reversed by some students. This will be something I need to pay extra close attention to in future units.

1) The teacher dashboard can be displayed to give feedback to students. Students could see if their matches were correct and fix their errors. This also allowed me to touch base quickly with the few pairs that were struggling. Correct matches are in green and incorrect matches are in red. It was interesting to note on the second slide that not all students used the same strategy, but there were two or three main strategies.

2) The end of activity "How did you do?" shows me that I need to circle back to one specific pair of students tomorrow. But it also shows me that even though all students completed most of the activity correctly, they admitted to not easily remembering all of the parent functions. This is something I need to keep in mind when I do anything with students that is supposedly review. Review and recall does not come as easily for some students.

3) The "offsprings" of a vertical shift down 5 units or a horizontal shift right 5 units were reversed by some students. This will be something I need to pay extra close attention to in future units.

## Tuesday, September 26, 2017

### Teach 180: A Different Card Sort (Day 17)

Yesterday in Calculus we did a card sorting activity in Desmos that focused on linear and non-linear equations and linear and non-linear tables. We began class today by talking about the importance of using precise language in math. We critiqued the first answer seen here that was given by one pair of students. The summary question was "How can you determine if an equation is linear or non-linear?"

A few students hadn't been in class the previous day and wanted to know what a reciprocal function was. Several students knew that a reciprocal function's graph wasn't linear, but couldn't recall what the graph looked like. Rather than pulling up desmos, we created a quick sketch by substituting a few values into 1/x.

Next, we critiqued the word "exponent". One student said that it wasn't clear that the exponent is to be on a variable. He gave an example of y =3

^{2}x and the class agreed that this was linear, even with the exponent on the 3. A second student pointed out that even linear equations have an exponent; it is a 1. At this point, we discussed the form of the line that was used in the card sort activity, which was slope-intercept form. Many students used this form to identify the linear equations.
After a discussion of domain and range, we did a different card sort activity. There are 16 students in my class and each student was given an index card. The white index cards had a domain and range listed and the blue index cards had a function listed. So, there were 8 pairs of cards. Students were told that they had to get up and move around and find their partner. Once students found each other, they had to write the information found on their index cards on the board. When we were done, we had 8 examples of functions and their domains and ranges displayed around the room to discuss. We reviewed each of them and found one error. The range of y=1/x and y=1/x

^{2}were reversed! This gave us a 5 second penalty.
The table below shows the 8 functions I used and their domains and ranges. Tomorrow Mr. Galitsky's Calculus class will be doing this activity. Will they be able to beat our time of 2:06.91?

## Monday, September 25, 2017

### Teach 180: Desmos Card Sort (Day 16)

It's still early in the year and we are continuing to review concepts in Calculus. Our lesson today focused on functions in general and for part of the lesson students worked in pairs on a Desmos Card Sort activity. In this activity, students worked with a partner to sort equations into piles of linear and non-linear equations and tables into piles of linear and non-linear tables.

Most students had no difficulty identifying the equations as linear or non-linear. Where there was some discussion between students was when they were sorting the tables. Within each pair, it seemed like one student knew that the slope had to be constant (or the same) when looking at how y changed in relation to how x changed.

Where many students need a little more work is on using mathematically precise language. The response below shows an example of an answer to the two summary questions. The questions were: 1) How can you determine if an equation is linear or non-linear? and 2) How can you determine if a table of values represents a linear or non-linear relationship?

I think we will begin class tomorrow by critiquing this response to the two summary questions. I'll blog about how that went along with a different sorting activity tomorrow.

## Saturday, September 23, 2017

### Teach 180: One Simple Idea (Day 15)

Some days we do Desmos activity builder lessons and work with Fathom. Other days we collect data as a class and ask "What do we notice?" These classes often fly by for me and I am guessing my students find it that way, too. Why? Because they are engaged in what they are learning.

However, there is sometimes a need for direct instruction. Within a class where the primary teaching method is direct instruction, I ask students questions and also have them work at their desks to solve problems. Under our previous bell schedule, a class might have 10 - 25 minutes of direct instruction. And within those 40 minute class periods, there was also time for a warm-up, reviewing homework questions and time for individual work on problems similar to the ones done together in class. Students are also engaged in this class, but 25 minutes of direct instruction is the upper limit for my students with this more passive learning style.

Created by Freepik |

*Zoning out identification technique: Ask a question. Call on student. Student answers a totally different question or admits outright "I wasn't paying attention."*

A few days ago I read a brief blog post on doing board work, called "Should Math Students Be at the Board Working?" A simple idea really. Give students problems to do in small groups at the board. Since I normally have students do problems at their desks or have individual students put homework problems on the board, this was a quick and easy fix to my problem of keeping students engaged.

After reviewing one problem from the previous day and comparing its solution to a similar problem, we went over the homework by having students put the problems on the board. One problem was assigned per table and I already had the problems put on the board. I told the students that after about 20 or 30 seconds, I would say "switch" and someone else would have to write on the board. This made sure everyone participated in getting the problem on the board and there was definitely more engagement by all students in the learning. I could hear students in each group discussing the solution and correcting each other's errors. After the problems were up, we reviewed the strengths seen in each of the solutions. The energy of putting the problems up on the board and having students discuss them carried us through into the direct instruction component of the lesson which lasted about 20 minutes. Not too bad for a Friday afternoon at 3 PM.

## Wednesday, September 20, 2017

### Teach 180: The Students Can Teach You, Too (Day 14)

One of the best things about where I teach is the inquisitive nature of the students. They aren't afraid to ask questions. Today I was giving a quiz in PreCalculus and some of the problems involved solving equations with rational exponents. For these equations, we would remove the lowest power of the variable to get a quadratic and then solve the quadratic. Here is an example of one of these types of questions.

I thought when I first became a teacher that I wouldn't be a learner anymore, that I would know everything about what I was to be teaching. After 25 years of teaching, I know that is wrong and days like this remind me that we never stop learning or making new connections within our disciplines, even teachers. Thanks to the student who stopped to make me think and learn today.

When you solve this equation, it appears that it has three solutions: 0, -3 and -5. However, -3 and -5 are extraneous solutions. We had shown solutions were extraneous by substituting the values into the original equation and finding it simplified to two different results for each side of the equation. For this particular problem, we also know the negative solutions are extraneous, because the exponents have 2 in the denominator and taking the square root of a negative number leads to a non-real result.

Right before the quiz one of my students approached me and showed me the following two screens on his calculator. On the left, we can see that he has entered both the right side and left side of his equations in the calculator. On the right, we can see a table of values. He asked if the error message is showing that these (meaning -3 and -5) are not solutions. He also asked if the table shows that 0 is a solution, since both y1 and y2 showed the same function value of 0. I told him that he was correct and then it dawned on me that I had not thought of checking for extraneous solutions this way. Now I will need to think of how to incorporate this connection between tables of values and extraneous solutions into lessons in the future.

Of course you can do something similar in Desmos, and in my opinion Desmos wins. It doesn't report an "ERROR" when evaluating the function. Desmos calls it like it is "undefined".

## Tuesday, September 19, 2017

### Teach 180: What Is Going On Here? (Day 13)

When you solve the following equation, you get 2 solutions. At least that is what it looks like initially. But then when you substitute the values into the equation, only one works! Why does that happen? What is going on here?

In years past, I would have said something to the effect that squaring in this particular problem made a new equation with two solutions and yet one of the solutions was not a solution to the original equation. Although this was true, I would often get blank stares from my students.

Today someone asked me this question and in an instant I was able to show the following two screenshots from Desmos.

By graphing both sides of the original equation, we can see that there is only one solution, x = 6. Notice that squaring both of these expressions turns the red line into a parabola and turns the blue square root function into a line. We can see that the line and the parabola have two points of intersection, when x = 3 AND when x = 6. Transforming the equation by squaring both sides of the equation changed the graphs in such a way that an additional solution was created that was not a solution to the original equation.

Connecting ideas in mathematics was something that I don't recall from my days as a high school student or from the early days of my teaching career. Now, it is so quick and easy to help students to form those connections with tools, like Desmos.

### Teach 180: Plickers (Day 12)

Today in Calculus we did some review for our test on Probability by using Plickers. I use it as a type

students aren't on a computer with other tabs or windows open to distract them. Second, it is easy to encourage them to work together. When we use Kahoot, it is about competition and getting your answer in first, even if that means guessing and not really understanding the problem.

Finally, I can print off individual student results based on the plicker number I have assigned to him or her. And here is a partial summary of my results from today. Notice that some students did very well (their names are cut off the left side of the screenshot, but you can see the scores) and some questions were easier for the class as a whole than others.

Tomorrow I'll give students their individual one page report that shows how they they did on each of the seven questions. They can use these results as they study individually for the upcoming test.

## Friday, September 15, 2017

### Teach 180: Games Day (Day 13)

Today there is just one class. It is D period and I don't teach D period. So, today I have no classes. What am I doing today? Today is

What happens after lunch? This year I have mandatory child abuse recognition training from 1 PM to 4 PM. (There is required training for teachers once every 5 years.) Students will be participating in various activities including sports practices, scholastic scrimmage, working in the community garden, coda red music rehearsal, working in the art studio and my favorite - Who Wants to be a Mathematician qualifying contest!

**Red**and**Gold**Games Day. It is a tradition that started in 2000 as the Millennial Games. The entire student body (grades 1 - 12) are split into two teams - red and gold. We play field day type games, like sack races, tug-of-war and 100 yard dash. First grade students are paired with twelfth grade students and the twelfth grade students cheer on their first grader buddies as they race and help them with getting lunch.What happens after lunch? This year I have mandatory child abuse recognition training from 1 PM to 4 PM. (There is required training for teachers once every 5 years.) Students will be participating in various activities including sports practices, scholastic scrimmage, working in the community garden, coda red music rehearsal, working in the art studio and my favorite - Who Wants to be a Mathematician qualifying contest!

### Teach 180: Dealing with Student Absences (Day 12)

One of the most challenging parts of teaching is handling
student absences. Although it can be challenging for a student to get caught up on new content that was missed, it is even more challenging to get them to make-up a missed assessment. Students can be absent
for a variety of reasons – a dental appointment, a college visit, being sick or
an early sports dismissal.

On Tuesday, I
gave a quiz in PreCalculus on solving quadratic equations. All of the content onthe quiz was review from Algebra 2. There were 3 students who were dismissed early for sports. One student took the quiz early, one student took the quiz during a free period the next day and a third student did not have any free periods to take the quiz. Rather than have him wait another day to take the quiz, I had him take it during class on Thursday. (Note: With our new bell schedule this year, we did not have the PreCalculus class on Wednesday.) This was less than ideal. The student missed going over a review of the content he missed after the Tuesday quiz. In addition, he did not do well on the quiz despite the fact that I told him that the most commonly missed question was the one on solving a quadratic equation by completing the square. He earned 1 of 5 points on that question, because he got the right answer by factoring, but he showed no understanding of how to complete the square to solve the equation.

After 25 years of teaching, you would think I would have developed a way to effectively deal with absences for tests and quizzes. But the truth is that I usually have a list of three or four students that need to make up assessments at the end of a grading period. Often these students have waited over a month to make-up the assessment. At that point, they have forgotten much of the material that is on the quiz or test and usually don't do well. Our school has a make-up policy that allows me to give them a grade of "0", but that doesn't sit well with me either.

If anyone has suggestions on dealing with student absences, please post them in the comments below. Thanks in advance.

## Thursday, September 14, 2017

### Teach 180: The Birthday Paradox (Day 11)

Due to our new schedule, I only taught two classes today –
Probability/Statistics and Calculus. In
Calculus we are actually studying some elementary counting and probability
ideas. For about 25 minutes today, we
explored the birthday paradox. If you
don’t know what that is, you can view it on this TED-Ed video. I showed the first 30 seconds or so of this
video and paused it when it got to the question.

The main question being “How many people do you need to have
in a room for there to be more than a 50% chance that at least two people share
the same birthday?” This problem
incorporates complementary events and independent events.

I started by having each of the 16 students write two dates
on a single slip of paper. Their
birthday and the birthday of someone they know.
This doesn’t quite match the scenario, because it is very unlikely for a
student to have his or her two listed dates be the same. However, in reality two people standing
beside each other could actually share the same birthday. We did this twice and both times we found a
match, which students found surprising.

Next, we used technology (TI-84) to randomly generate a list
of 32 integers from 1 to 365. We stored
these numbers in a list and then sorted the list to make it easier to identify
if we had a match. Unfortunately, only
10 people had calculators on them, but we had 80% have a match the first time
we used the technology. We had 80% with
a match on our second trial and 100% match on our third trial.

To help students understand why this probability was so
high, I played the remaining 4 minutes of the TED-Ed video. So, how many people do you need to have in a
room for there to be more than a 50% chance that at least two people share the
same birthday? The answer is about 23.

## Tuesday, September 12, 2017

### Teach 180: Why Randomization Matters (Day 10)

Today in Probability and Statistics we reviewed the article The Immediate Impact of Different Types of Television on Young Children's Executive Function from The American Academy of Pediatrics. This article looked at four different tasks to measure executive function across the three treatment groups: fast-paced television, educational television and drawing. Children were randomly assigned to one of the treatment groups and a table from the article (seen below) shows that the groups were fairly similar. But does randomization always matter? Will randomization tend to create groups that are similar?

To see if this was the case, we worked through an activity in the CollegeBoard's Curriculum Module on Random Sampling and Random Assignment. There are 14 subjects listed on cards. Each card has the subject's name, gender and IQ score. Students shuffled the cards and split the cards into two piles of 7 cards. Cards in the left pile represented the students that were randomly assigned to watch an episode of Sponge Bob Squarepants. Cards in the right pile represented the students that were randomly assigned to the drawing group.

Next, students calculated the difference in the proportion of females in the two piles (Sponge Bob group - Drawing group) and the difference in the average IQ scores in the two piles (Sponge Bob group - Drawing group). Each pair did this a total of five times and then we recorded our results on a dotplot. Students noted that there was variability from sample to sample, but that the dots tended to center around 0 for both the difference in the proportions of females and the difference in the mean IQ. Here you can see our class dotplot.

We finished the activity by watching a video done by Doug Tyson which uses Fathom to do the simulation. If students weren't convinced by our dotplots that randomization matters, they were definitely convinced by the end of the video.

To see if this was the case, we worked through an activity in the CollegeBoard's Curriculum Module on Random Sampling and Random Assignment. There are 14 subjects listed on cards. Each card has the subject's name, gender and IQ score. Students shuffled the cards and split the cards into two piles of 7 cards. Cards in the left pile represented the students that were randomly assigned to watch an episode of Sponge Bob Squarepants. Cards in the right pile represented the students that were randomly assigned to the drawing group.

Next, students calculated the difference in the proportion of females in the two piles (Sponge Bob group - Drawing group) and the difference in the average IQ scores in the two piles (Sponge Bob group - Drawing group). Each pair did this a total of five times and then we recorded our results on a dotplot. Students noted that there was variability from sample to sample, but that the dots tended to center around 0 for both the difference in the proportions of females and the difference in the mean IQ. Here you can see our class dotplot.

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