# What Makes A Circle, A Circle?

In the seventh grade curriculum, geometry studies are few and far between. Sometimes students will stumble upon a textbook question regarding angles, triangles, area, or perimeter, but there is little exposure in isolation of geometric properties in our grade level. (Of course, all mathematical concepts are connected, but I digress). In our school, 6th grade is the year for circles, so we were advised to insert a review or to backfill for any or all students who possibly did not receive exposure to the concept. This makes it tricky when some students have lots of experience and other students have none or simply no memory of it. And although this might sound like an unusual problem to someone without 25 or more different personalities in class, it is often par for the course for every teacher and student each day. Some concepts are completely mastered by some students, others are sort of remembered, a few have surpassed curriculum expectations through independent study, and some have no recollection, regardless of the topic at hand. The reasons are plentiful. We as teachers all do our best to expose students to topics that they will claim they never saw, even if we previously taught it to them in the same year! The question is: how do we as teachers prevent this scenario?

We are supposed to be able to differentiate for all levels and reach all students in every lesson. Although I try via various modalities and methods, I have yet to figure out how to accomplish this with fidelity each day. However, one method that I find differentiation is embedded, is in well-developed three act tasks. Today, I used Dan Meyer’s Best Circle task. Here is a link: http://threeacts.mrmeyer.com/bestcircle/.

I am sure there are lots of different ways to utilize this lesson, which I would love to hear about from you if this fact applies. All I can share is what I did with it.

In the first act, four different gentlemen draw a circle. I paused the video clip and asked students to vote on which of the four circles was the best circle. I wrote the names of the circle creators (Chris, Timon, Andrew, and Nathan) on the board. Immediately, Chris received most of the votes. Students let me know it looked the least like an oval. I did not agree or disagree with their comment, but simply repeated, “Ok, less like an oval than the rest, got it.” I then asked, “Is that a true defense?” Students shrugged and expressed that they thought it was as good a reason as they could collectively argue at that point in time. Other than that, students did not have much to sustain their reasoning for choosing Chris. Although they would find out that the majority of their initial guesses was the correct one, a good math teacher always asks, “but why?” A student excited about math wants to be able to answer that question. And boy, were they interested.

Next, I assigned a new task to the students by asking a question: What makes a circle, a circle? I gave them about 10 minutes to determine the answer to this question. They were permitted to use their Chromebooks, textbooks, and each other to come up with research and data to support their opinions.  Students were allowed to revise their original hypothesis based on any information they found.

Work options were offered. Some students worked independently, others partnered with one student, and some collaborated with a group of three or four peers. One student spotted compasses on my desk and asked if he could use it. He went up to the board where the frozen screen shot with the circles remained. He slowly used the compass on each circle. As students researched and revisited words like equidistant, radius, and diameter, I was asked if they could go to the board and use a ruler to take measurements of the circles. When one student witnessed another student trying it out, she commented, “Oh, that is a good idea.” She joined the student, and then another and before my eyes, students formed a small cohort within the class in front of the screen shot with the four circles. Students began debating each other what part of the circle they should measure and whether or not they could make determinations from their measurements. Some were discussing the midpoint, others radius, hemispheres, quadrants, circumference, etc. A few were cheering themselves for their initial guess and others were disappointed that new information changed their previous prediction. When the 10 minutes was up, I shared act 2 with the additional information provided by the lesson. Students were asked whether the area, circumference and coordinate points of the circle were offered for each circle would provide a clue to help solve the problem. This is where it came out that students recalled very little in the way of formulas and the inner workings of a circle from previous learning. I showed them the resolution (the third act), but I quickly realized we weren’t done with the task today. There is so much left to dissect, I did not even want to enter into the explanation Dan provided in the teacher guide. They weren’t ready.
I do have a bit of formal review to do with students, perhaps a day or two of lessons, but we will return to this three act lesson to see if students can in fact determine why the best circle was in fact the best circle. The quick review will come…ahem…full circle. Maybe, just maybe at this time in the next school year, students will still remember some details about what makes a circle a circle.

# Flipping my Teaching, Not Just my Classroom

My teaching approach is getting flipped upside down…repeatedly.

It all started with my on-line introduction to Yeap Ban Har’s discussion on number bonds. Here is the link for anyone interested:  Number Bonds . This was the first time my mathematical mind was blown. Throughout my years teaching elementary school, I had stumbled across multiple approaches in computation, but never had the pitfalls of memorizing procedures and algorithms without context been succinctly explained. This is literally a 2 minute 50 second video!

This one youtube video launched my researching life. Don’t get me wrong, I had always tried to search for great lessons, etc., but this was the first time I felt like I was (for lack of better explanation) doing everything wrong in my teaching.

The timing for this epiphany was not super as I was pregnant with my second child and about to take the majority of the school year off to take care of my baby. In between changing diapers, cleaning spit up, and a very snowy winter trapped in the house, any spare moment was spent investigating better ways to teach math. Fast forward through 10 months massive sleep deprivation, the trials and tribulations (and wonder) of having two children instead of one, and intermittent mathematical research, I was back in my classroom wondering what to change first.

I have written a post about my first foray into 3-act math, as the great Dan Meyer was also a new discovery to me during my maternity leave/initial research period. Not only did I “meet” Dan Meyer, I also was “virtually” introduced to Andrew Stadel, Robert Kaplinksy, Jo Boaler, and of course, the DESMOS and MTBOS communities. Although I have never actually met any of these mathematicians in person, this growing group of educators provides me with daily inspiration.

Throughout this year, many 3 act lessons have made their way into my classroom. One that I recently completed, Robert Kaplinsky’s Zoolander had me questioning if what I was doing was working. Were these lessons as amazing I thought? Did they provide students with a context that made the experience and math meaningful? Were students making connections in their brains? Was I providing enough structure? In short:  effective or not?

Whenever I try something new, it is normal for me to question myself. Acknowledging this fact, I can see that this has been a wonderful transformation for my teaching and math learning for my students. These lessons have had a major impact and I know this from events in the last few weeks. Several weeks after the Starburst lesson by Dan Meyer and the Zoolander lesson by Robert Kaplinsky, my students were referencing them in math conversations in the hallway and classroom. You read that right, the hallway!!! Apparently there was a question on the standardized test about scale models and the students were discussing how easy it was compared to the work they had to do in the Zoolander lesson. Another student commented that the Zoolander lesson helped them really understand the concept better than any book and that was why the question was so easy. I rest my case.

The Starburst lesson initiated a debate about sample space. A passionate debate! When does this happen from a textbook example? I have no reference for that. In short, these lessons make a difference.

At the same time I have felt success achieved in my teaching and by my students, it has also been an immense struggle for me professionally. This is especially poignant with my lower performing students. How do I convince them to believe in themselves and see the beauty in mathematics? If they don’t know the basics, can they still participate in these lessons with confidence? How often will they give the line of, “I don’t understand” in lieu of a rigorous debate with their peers and investigative excitement?

In all honesty, I have experienced both ends of the participation specturm from lower achieving students. Although I had read numerous and convincing articles by Jo Boaler, I only just obtained a copy of her wondrous book Mathematical Mindsets. As I am reading it, I am shouting, “YES! Oh my goodness, I agree! And then in the next minute I am asking, “How can I do this every day? When does procedural math come in to play, does it?” What does this look like lesson by lesson, day by day? Does it transform the students the way she says it does? I am so IN and can’t get enough, period.

As I was researching youcubed, I noticed an opportunity for the summer to attend a workshop with Jo Boaler in California. At this time, I cannot afford to fly to California, pay for the workshop and a hotel room, not to mention the childcare issue, but oh to dream. I am going to take Boaler’s courses through youcubed and finish her book soon. Every free moment I have is spent reading, taking notes, and rereading it. It is my current math bible.

I do not have a neat and tidy way to wrap up this blog post. Once again, I am asking the mathematical world for a conversation about balancing the math classroom. Have you read Jo Boaler’s books? Have you tried 3-act math? What were your successes? What were your failures? How can we work together to keep the math conversation evolving and growing? Anyone else in? Leave a comment, tweet me at @drpolakmath, or send me an e-mail at mpolak@ridgefield.org. The larger our community, the greater our collective success in helping all students achieve in mathematics. Who is with me?

# Sweet Math: Dan Meyer’s Starburst Lesson and Probability

When I first discovered that Dan Meyer’s lessons could bring math to life in a new way last summer, I took the time to investigate the three-act math options he created. One that struck me as extremely engaging was his lesson on Starbursts. I saved it in a probability folder knowing full well that probability was slated for the end of our school year. A glimpse of it was so memorable; I actually had the wherewithal to incorporate it into my lesson plan this past week, in April. That might not seem earth shattering, but trust me, it is.

The first act of the lesson launches with the opening of Starburst two packs. The pack reveals one yellow and one pink. A skull and crossbones image appears over the yellow Starburst and an audible yuck is heard in the background.  A second pack is then opened, revealing two yellow Starbursts, which leads to two skull and crossbones over the Starbursts and an even louder yuck sound. Clearly, Meyer does not seem to like the yellow Starbursts. The camera then focuses on a large pile of Starbursts two packs.

That is the end of act one. Immediately, students began to debate the merit of each flavor of Starbursts and began to wonder aloud. I let them question and debate each other for a minute. Alerting them at this time I would not provide them with additional information, I asked them to make a prediction that was both too high and too low regarding how many yellow Starbursts they believed were in the pile. We wrote several too high and too low predictions on the board, and then I asked them if I could provide them with any information to help them solve the problem, what they would like to know.

Immediately students’ hands shot up and the first student I called on asked, “How many packages of Starbursts are there in that pile?” Another asked how many flavors there were. Several students scoffed at the second question and, somewhat exasperated commented, “FOUR!!! Have you never eaten Starbursts before?”  One asked to find out how many double flavor packs there were in the pile. The rest of the students loved that idea and complimented the thinking involved behind that one. And of course, inevitably, one student just wanted the answer. Sigh; there is always the need for that request!

The next two slides I shared were images from Dan’s lesson (Act 2) that revealed that there were 287 packages in the pile and the four flavors of (not by flavor, color) yellow, red, orange, and pink.

Now that students had a bit of information at their disposal, I asked them the following questions:  “In those two-packs, how many packages do you think have two yellow Starbursts? How many do you think have one yellow Starburst? What do you believe the overall percentage of yellow Starbursts is in the pile? Use what we have learned in our probability studies to make a prediction.

Students walked around the room and worked with anyone and everyone to try to figure out the answer. I was amazed as I witnessed the thinking displayed. Many students immediately wrote the total possible outcomes of Starbursts such as yellow-yellow, yellow-red, yellow-pink, yellow-orange, etc. They then used total possible combinations to convert to favored outcomes. With that, they used ratios and came up with their predictions. They found a way to apply the procedural math we had been studying for the previous two days in class on their own accord.

Students shared their predictions and many were close to each other, a few, not so close…Funny enough, many students who had different answers, upon hearing their peers’ strategies recognized probability mistakes that they made. When it was time to reveal Act 3, students were cheering. I love to hear cheering in my class, over, yes, MATH!!! They quickly calculated their percent error and found out how very close (and far) each was in their work.

A specific feature of Dan Meyer’s lessons is that he leaves them quite open for interpretation.  In my mind, he recognizes that teachers are not robots in the classroom and deserve the flexibility to interpret and customize to our heart’s content. This gave me an idea for an extension at the end of the lesson.

I pulled out a bag of Starbursts and had each student grab two. We recorded the flavors of the Starbursts pulled from the bag and made a frequency table displaying the sample space on the board. Unfortunately, we ran out of time, but I recorded our data on a frequency table so we could do a follow up the next day. My first question I plan to ask is:  What type of questions and answers can be generated with this information?

For those who might be wondering, students were granted permission to eat the two Starbursts they selected. After all, I wanted to make sure that this math lesson left everyone with sweet memories.

# Customizing Andrew Stadel’s Sweet Snacks

Sweet Snacks: A video and concept by Andrew Stadel, customized by Dr. Polak!

I spend a big chunk of my weekends in search of grand lesson ideas. My summers are also preoccupied with this obsession, but the problem with the summer is that I have to wait so long to actually use the lessons I discover, I often forget that I found them. It is not that I am disorganized; my organization skills are pretty top notch. However, that old saying if you don’t use it, you lose it, applies to me in this situation.

One of those lessons I found (was unable to use immediately) and then promptly forgot about was Robert Kaplinsky’s calorie lesson for inequalities. This was serendipitous because this year, as I was hopelessly searching, tweeting, i.e. begging for inequality inspiration, I had come across a three act math lesson called Sweet Snacks by Andrew Stadel. After tweeting him for some guidance with his lesson, he promptly tried to sway me away from his and towards Kaplinsky’s calorie lesson as Kaplinsky’s lesson was superior in Stadel’s opinion. His tweet reminded me that I had found and planned to use Kaplinsky’s lesson, but I saw something spectacular in the Sweet Snacks lesson as well. I am so happy that I found a way to use them both!

I utilized Kaplinky’s lesson for equations instead of inequalities (see previous blog post for details), and I then got to work to customize Stadel’s Sweet Snacks. I started the lesson with my class by telling them that I was going to share a short video with them of a math teacher hero of mine named Mr. Stadel. They were instructed to think about any questions that came to mind and anything in particular they wondered about. Of course, we would discuss immediately following the video.

As this is a three act math lesson, the first act involved Stadel’s son sitting in a shopping cart as Stadel pulled 8 boxes of Teddy Grahams into the cart. The video then switched to Stadel pulling 8 bags of “Circus Animals” (Animal crackers,) into the cart. The video zooms in closely on the young boy in the cart as he pulls out a 20 dollar bill out of a wallet. The wallet clearly has no additional cash.

My students wanted to know why anyone would buy so many boxes of snacks , how much did each box cost, what the two Teddy Graham flavors were, and whether \$20 was enough for all of the sweet snacks in the shopping cart (JACKPOT). Of course, off topic but hilarious and worth a share in my opinion, was “wondering” why a cut out statue of Guy Fieri appeared in the video and whether or not his hair was ridiculous.

I asked them what kind of information could help them with the math questions. Students quickly agreed they needed to know the prices of the sweet snacks. And with that, I shared Act 2.

It was immediately revealed that the Teddy Grahams were \$2.49, while the price of Circus Animals was \$3.49 .The students asked me to rewind the video several times so they could count how many bags of each were put in the cart. It was obvious to them quite quickly that \$20 was not enough money. Mr. Stadel could only purchase items that were \$20 or less. I was silently cheering my students’ recognition of the inequality example without me needing to articulate its existence in a direct instruction type of way.

This is where I prompted students to translate the scenario into an algebraic inequality. Students had already solved basic procedural inequality problems in the previous lesson. Sweet Snacks, provided a context for the types of examples they had seen. They were all sure that Mr. Stadel could not afford all of the bags of Teddy Grahams and Circus Animals with only \$20, so I asked them to write the math language to demonstrate that fact. It was not easy for them, but eventually, students came up with:  8(\$2.49+\$3.49)=\$47.84 >\$20

The next question, naturally, was, what are the combinations of sweet snacks he can afford? Students were instructed to write an algebraic inequality using the prices given of Teddy Grahams and Circus Animals. They had the option to write an inequality with one sweet treat or both sweet treats. In addition to writing the inequality, they were asked to solve it and graph all the results. If able to finish quickly, they were asked to write a second or third alternative algebraic inequality and/or help out a neighbor.

When we shared out and compared, we talked about the constraints of budgets. Every household has one, even if students were not privy to the information that gave details about the restraints their parents must use to control their purchases. We all have to stay within the range of some sort of budget. Do not spend more than x amount; do not let your bank account fall below x amount. This is life for all of us!

They had calculated combinations of sweet treats that were possible to purchase with \$20 and possible combinations that would have exceeded the \$20 within this short amount of time.

Keeping this in mind, I asked students to write an inequality to represent how many Teddy Grahams Mr. Stadel would be able to purchase if he had \$20, but also had to pay an additional 6.33% sales tax on his total purchase. Sometimes there is tax on snacks, depending on where you live. This was another off topic, but interesting conversation from the perspective of 7th graders. Believe me, I could write a separate blog on off topic comments. This is not that blog! Once again, they were asked to write, solve and graph their results.

In the next scenario, I presented a circumstance where there was no tax on this type of food. So obviously, a new algebraic inequality needed to be written, but I didn’t stop there…Students were asked to write an algebraic inequality to represent the Teddy Grahams Mr. Stadel would be able to purchase if he had \$50 (woohoo, more money), AND had a 20% coupon off the price of his total purchase.  They did great with synthesizing all of the different math concepts in this particular problem.

At this point, I was convinced students were more than ready for the manufactured inequality problems from our textbook and they proved my hypothesis quickly. Let’s be honest, textbook questions are about a dime a dozen, but they do serve a purpose. The problem with most textbook lesson ideas is that they offer instructional inspiration at a very superficial level. That is why I am always in search of a way to bring the level of instruction to a deeper and more meaningful place with my students. This, of course, is why I continue to be an enormous fan of Andrew Stadel, Robert Kaplinsky, Dan Meyer, and the magnificent math community that allows me to become a better teacher every day.

# A Review of Mr. Stadel’s Rolling Tires

The teachers in my grade level were instructed to insert a circle unit (just the basics folks) into the scope and sequence for the advanced classes this year. The idea of teaching circles made me happy because I loved teaching circles to my fifth graders; back when I taught at the elementary level. One activity based on the now defunct Growing with Mathematics program involved students’ self-discovery of the pi ratio. Although I used the activity as a brief introduction, I knew I needed to up my game for my seventh graders.

Upon my searching of all circle lessons (wow there are so many!), I stumbled across Andrew Stadel’s 3-Act Math plan for Rolling Tires. Essentially, Mr. Stadel is rolling two different sized tires towards a tower of toilet paper on a makeshift table. That is the premise:  toilet paper and tires. Naturally the students were captivated.

I had rehearsed my 3 act math questioning technique with a few Dan Meyer lessons already, so I felt prepared. I showed act one and asked the students to discuss the questions that came to mind. Oh my goodness, they couldn’t get their questions out fast enough. “Will the tire knock down all of the toilet paper? Will the tire hit the target? Is the tower of toilet paper sitting on a table or something else? How quickly are the tires rolling? What is the diameter of each tire? What is the circumference? How many times does each tire need to rotate to get to the target?”

Many students asked me to replay the video over and over again. I was instructed to pause it at certain points, and conversation exploded in the room. The conversations were all about math, geometry, and calculations in real life and it must be said, the students and I were happy.

I asked my student investigators, before showing act two, to consider what specific pieces of information they would want to be provided with in order to satisfy their own varying levels of curiosity. It was the moments following that question which proved Mr. Stadel created a genius lesson. Without saying another word, my math family began to debate each other about the merits of having the radius, diameter, or circumference of each circle. In the middle of the debate, one of the students (I will call her Jan) interjected, “Hey, it doesn’t matter which piece of information we have, if we have the radius, we can find the area, circumference, and diameter. We already know pi is a constant ratio.”

Boom. Just like that, that one student changed the course of the classroom conversation. I asked the other students to clarify Jan’s comments to each other as I walked around and listened in on their dialogue. As I checked in with the students, it became clear that they all understood what she meant and I had this lesson to thank for the reinforcement.

I then showed act 2, which presented the diameter of each tire, the distance of the tire track, and the distance from the tire to the target (if it actually hit the bullseye). The question Mr. Stadel presented, which many students had asked in their own inquiry, was how many rotations would it take for each tire to hit the target and would it actually hit the target?

The students worked furiously and shouted out numbers to each other regarding the circumference. One student started to calculate the area and her partner corrected her with a visual. She actually rolled her water bottle on its side and demonstrated why circumference (kind of like the perimeter) would be important information to answer the question.  She asked if I still had string left over from the other lesson and showed her how the length of the string would be helpful to see the distance each rotation covered on the ground.

When most students arrived at their own conclusions (most saw no reason that both tires would not hit the target, but questioned whether or not the force would be powerful enough from the smaller tire to knock all of the toilet paper over), I asked if they were ready to view act three and most responded yes. However, one student protested, “No, don’t show it yet, wait!!! I am not done!” This is a math teacher’s dream.

We all agreed to wait a few more minutes, and then I played act three. The students screamed when the big tire hit the target and complained endlessly when the small tire missed the target. Questions continued to abound.

“Why did he miss? Was his aim bad? Did it have something to do with the tire being smaller? Why did he highlight the central angle of the circle? Why is that significant?”

I don’t need to describe this lesson any further because it is clear that it had students going beyond any math textbook exercise and yet still provided so much actual understanding of a concept. As I continue to search for lessons that produce results like these I must give Mr. Stadel a huge and grateful shout out for this lesson. Mr. Stadel, I am officially a big fan!!!

# My First Foray Into Three Act Math

I recently returned to teaching after an extended maternity leave. As much as I love my girls, it was tough for me to be out of the classroom for almost a year. A lot can happen in a year and a lot did happen in a year in our math world. We adopted a Singapore inspired program, embraced the mathematical practice standards, and had Yeap Ban Har train us in a better way to teach math. It was at Ban Har’s workshop where my mind truly experienced a renaissance, if a mind can experience such a phenomenon. Of course, by the time Ban Har reshaped my focus I had already been trained via staff developers of the Math in Focus program. Every lesson structure we discussed and I witnessed allowed little light bulbs in my head to flicker. When I returned to my classroom this past August, I was determined to change everything.

Anyone who has been in teaching will tell you that changing everything, for lack of a better term, is stupid. As true as that may be, I knew the type of math teacher of which I was aspiring to become, so I attempted such a transformation. I furiously researched my new textbook topics and scoured the Internet for lessons that were already brilliantly designed and would complement the objectives I knew I must meet. It was this search that led me back to Dan Meyer.

I had seen Mr. Meyer’s Ted Talk discussing how math instruction must change. But like most people, I need to be introduced and reintroduced to something multiple times before I truly embrace and understand it. I re-watched Dan Meyer’s Ted Talk and then went further to watch examples of his 3-Act Math. In a nutshell (apologies to Dan Meyer here for not doing this explanation justice), three act math includes a conflict/hook, a problem where students must develop ways to overcome the obstacles presented, and a resolution. There are various ways to get that hook, and it is our job as instructors to find it and lead our students’ interest in our direction.

Upon searching for something to do with integers and absolute value, I came across one of his lessons that had students guessing ages of celebrities. Like all teachers out there, I “stole” his idea, and modified it to make it my own. I spent hours debating which celebrities to use in my presentation and how many I needed. Then, realizing that I would be at a different pace with each of my classes; I figured I needed to make at least two versions of this lesson so the celebrities would be different for each class.

Here is what happened in my first class.

I asked the students how good they thought they would be at guessing somebody’s age. The responses varied from, “I am so good at that, to, I am the worst.” After they polled each other quickly on their anticipated success or failure at such a task, I posted a slide with lots of celebrity pictures with the challenge:  Let’s examine your talents. On the slide, I showed Barack Obama, Daniel Radcliffe, Donald Trump, Oprah Winfrey, Selena Gomez, Serena Williams, Michael Strahan, and Tom Brady.

Next, I distributed a table with the following categories:  Name, Age Guess, About, Difference.

I then posted one slide at a time of each celebrity and the students had about 30 seconds to record their age guess.

The students were excitedly shouting out their guesses and arguing with each other as each celebrity was shown on the screen. After they recorded their guesses, I posted the actual age of each celebrity in a table that matched the one I created for them. After they filled it out, they were instructed to determine the difference between the actual age of the celebrity and their age guess. We figured out who was the best in the class at this game and who was the worst guesser. Both students were celebrated by a round of vigorous applause. The discussion led to the fact that there was never a consideration as to whether the guesses were too high or too low, just the distance from the actual age.

Eventually, students figured out that this was an example of absolute value because they were measuring the distance from the actual age.

Students were then asked the following question:  “If provided the exact birthday of any of the celebrities from the previous slides, how could you find a more precise difference between your guess and the actual age? For example, Donald Trump’s actual birthday is June 14, 1946. Use this information to find a more precise difference between your guess and the actual age of Donald Trump. Discuss, explain, and problem solve.

Students soon realized that they were dealing with rational numbers. Some students decided to post the information in fraction form out of 12 months, others used 365 as the denominator and were showing the age difference to the exact day, and still others tried to tie in hours! They were all on a mission to be the most precise and my classroom was alive.

Since that lesson, I have referred to the age activity when reminding students about the concept of absolute value. This lesson became one of those lessons. The lessons teachers dream about. Students were inspired to perform all of the problem solving, research, and data gathering independently and collaboratively, without much from me. Most of my work was in the lesson structure and observation. The rest was up to them.

That was a good day.