# Rolling Cups: Modeling in Geometry

The Rolling Cups task from the Shell Center is a perfect geometry modeling project. I use some form of this task every year for the last three days of the first semester as a final project. This task incorporates constructions, similarity, functions and modeling while pre-assessing students readiness for next semester’s content: solids and circles.

Here is this link to the Formative Assessment Lesson

I’ve discovered that this activity is most effective when I have students produce something each of the 50 minute class periods. Over the years, I have been collecting a wide variety of cups from Goodwill, which I keep in a giant tote in the basement of my school.

With finals complete and students’ motivation dwindling, Rolling Cups is the perfect way to make the most of the days right before break. I move all of the chairs and desks out of the way that we have a big open space in the middle of the class. I also make sure students can easily access the dry erase boards on all of my walls to encourage teamwork and collaboration (more on vertical dry erase boards).

Here is how I break up this activity-

Day 1: Experiment

Here are 4 cups. What will happen when I roll them? Which one will make the biggest circle? Which one will make the smallest? Try it.

I hand each student a different cup and this sheet below to guide their thinking and keep them on task, and then I get out of the way.

At some point during this class period I also show students the Rolling Cup Calculator. I put a link in Google Classroom for easy access. Most students use it on their cell phones to try to find patterns.

Day 2: Develop an equation

This is the formula derivation day. I start by not mentioning the cups at all and just playing a quick whiteboard game reviewing similarity.

During this task, I usually have a few students suddenly yell out:

“This is the cup thing!”…and then start sketching cups on their whiteboards and they begin to use similar triangles to determine the roll radius.

Here is a discussion I had with a student, trying to support their thinking:

Other students look at the first group like they are crazy and we just carry on.

Then I have a few students summarize what they’ve noticed from the previous class.  Next I hand out this sheet: side 1 is the original task and side 2 is for students to write a few sentences summarizing their findings, and score themselves. My school has been working to develop a structure and rubric to elicit quality student writing about mathematics. Below is the current format.

As I present this sheet to students and summarize the expectations for the day, I also tell them that I made a deal with another teacher on Twitter and that I will be scanning their work and sending it to this teacher in Ohio.

I make a big deal about them not writing their names on the back of the sheet where they describe their thinking because that is the side I am going to send to Ohio and I want to preserve their anonymity. In return, I explain, the class in Ohio will be sending me their work, which we will look at tomorrow.

The quality of students work is so much better when they think it will be analyzed by someone else. Of course, this is a big fat lie. I don’t have any plan to send their work to another teacher. But they do so well with this added piece of motivation.

Day 3: Critique other students work

This day goes pretty much as described in the original task. Students review the included samples of other students work and analyze it answering the well written questions from the original task linked above.

It is fun for students to see that other students in other parts of the country approached the problem the way they did. They get excited and genuinely interested. They begin to reflect on their approach and compare it to the student work provided. You can see their confidence grow a little when they recognize that their (alternative school) work is just as good, if not better than the work of typical students in Ohio.

# Pencil sharpeners from Classroom Friendly Supplies

A few weeks ago, what I think is my 10th electric pencil sharpener stopped working. I decided I was done with electric pencil sharpeners and began a search to find a reliable, quiet, affordable pencil sharpener.

I found this pencil sharpener from Classroom Friendly Supplies, and I discovered that they would send me one in exchange for an honest review – so, here it is!

The positive:

1. It is QUIET!
2. No mounting needed, no electricity / batteries / plugs required.
3. It get pencils super sharp.
4. If a piece of led beaks off in the sharpener, it is super easy to take apart and repair.

The negative:

1. Students are not sure how to use it at first and they may interrupt class to ask how it works.  Fortunately, Classroom Friendly Supplies also provides a nice little printable step by step to post for students.

I think this video from one of my students sums it up perfectly:

# What even IS good teaching?

This has been a weird school year.

I have no idea what I am doing. Most of my class periods end with me thinking about what a mess it was and how it could have gone better.

I am very unorganized. I feel like I’ve tricked others into thinking I have it together, when in reality, I am barely keeping my head above water.

I’m teaching precalculus for the first time in a long time this year. Each student in my class has a unique path, they came to our alternative high school from a variety of other high schools, and they each have different gaps in prerequisite knowledge. It is a 80 minute class that runs from 2:45 to 4:05 pm. It is long and rough. If I am ever observed during this period, my administrator would wonder how I am even employed.

Yet, when I ask my students in the precalc class how it is going, they say they’ve learned more math in the past few months than they have in any other high school class.

How is this even possible?

For the past two years I have been voted as Teacher of the Year at my school. In August I was awarded the Presidential Award for Excellence in Math and Science Teaching. In September, the students at my school voted me as Teacher of the Month, a new recognition developed by our student council.  I must be doing something right…but what?

Here is what I’ve realized:

There is too much focus on what is observable in a class period (bulletin boards, lesson plans, time management) and not enough on the culture and connections that teachers develop with and among  their students.

For my first 8 years of teaching, I created a rigid plans of what I would teach each day for the quarter and then further developed scripted daily lesson plans and specific homework from the textbook. I then implemented the plan with fidelity. I made no effort to get to know my students, partially because I was overwhelmed making these scripted to the minute plans and then I was concerned about following the schedule during each lesson. While my administrator observations looked good on the surface, students were not really learning much.

Now, beginning with the first day of school, I intentionally work at building a unique relationship with each student. I make sure to find reasons to genuinely value each  of them. This starts with weekly “How is it going?” type questions on their warm up sheets and continues by using their mistakes on “Find the flub Friday” and through feedback quizzes. I also share a lot of myself with them. When we understand each other, my classes are more productive. I still make plans, but I allow flexibility to meet my students where they are. I pre-assess their understanding and readiness through warm ups and then use this to direct our units.

There are (many) days where I have just a terrible lesson, but through listening to my students and leaving space to adapt to their feedback, it appears messy to an observer, but we all learn anyway.

# My NCTM regional Phoenix session

I presented at NCTM’s regional conference in Phoenix a few weeks ago on reasons to develop and use effective warm up routines. Here is the blurb on the session:

Warm-Up Routines: Developing Mindset While Enhancing Math Understanding

Warm-up activities can maximize class time, set class culture, develop growth mindset in students and fill gaps, or extend student understanding. Participants will engage in a collection of high leverage warm-up routines, learn about research supporting warm-ups, and learn how to use them to grow their students and their teaching practice.

Here are my slides from the session:

Here is the (editable) warm up sheet that I used as a sample sheet to give participants a starting point to adapt this to their own classes and style. I also created and linked to a sheet where I have been collecting awesome warm up routines and tried to align each to the math practice standards.

It was super scary! The room filled to capacity (over 120 participants) at an 8 am session. It was so nice to have the familiar faces of Andrew Browning CouchMegan Schmidt, Stephanie Bowyer, Daniel Schneider and Justin Aion, and Katherine Bryant in the room. I am honored that they came to my session considering they already know the routines I was sharing.

I am curious why this session was so high interest. Was it because the session title included the super-hot-buzz-word: mindset? Was it because the session included specific and awesome, ready to use the next day, teaching strategies?

I learned and grew as a teacher by signing up to facilitate a session at a NCTM conference and I hope to do it again. Proposals for 2017 are due December 1st, and at this point, I am not sure what to present. I want it to be a valuable addition to the overall conference, and be considered useful by the participants.

I would like these conferences to have more sessions facilitated by teachers and researchers, and less by consultants and for-profit businesses promoting their products (I’m looking at you, TI). As teachers, we all are researching effective teaching daily through direct application. We all have something useful that we discovered works for us in our classrooms and I encourage you to apply to present a session too!

# A low-tech unit studying quadrilaterals

In an effort to be better at sharing quality basic stuff that works, here’s how I teach quadrilaterals:

I am hesitant to share the files that I use because I’ve borrowed them from all over the interwebs and at this point I don’t even know where to give credit, but here they are!

[10/11/16 update: Most of this came from Elissa Miller. You can find even more awesome geometry resources on her blog.]

I usually start by having students deduce the properties of parallelograms using an old fashioned ruler and protractor. This is because I’ve noticed that students could use the practice. I know there are a lot of quadrilateral discovery activities on Geogebra, but I still think it is occasionally important to practice using different tools.

After this task, I usually have students complete this checklist measuring, discussing and comparing properties of rhombi, squares, rectangles and  parallelograms.

After these investigations and discussions, students work in pairs to complete an Always Sometimes & Never discussion of quadrilaterals, described here.

Students also complete the task Complete the quadrilateral developed by Don Steward as described by Fawn Nguyen. This year, I adapted this task into a Desmos activity.

I also include some basic quadrilateral practice, like this assignment. It is just a good, basic, practice that helps expose student conceptions & understanding.

# Geometry Planning Guide

Units:

1. Constructions
2. Congruence
3. Transformations & Similarity
4. Right triangles & Coordinate Proof
5. Applied Trigonometry & Solids
6. Circles

In the spirit of Geoff Krall’s Problem Based Curriculum Maps, I attempted to organize my geometry curriculum and learning targets along with associated activities, tasks & lessons. In order to keep this as a useful document, I tried to only include the tasks that I have actually used in my geometry classes. I am interested in adding & deleting from this document regularly to keep if useful for me (and hopefully others). I plan to have the second semester completed this summer, as I am trying to develop this as I go this school year.

[update 7/25/16: It is finally completed!]

I have shamelessly stolen from all over the MTBoS, Math Vision Project, Engage NY & the Unit Blueprints Project and tried to give credit as much as possible.

Please share any criticisms, activities that I should add, activities that are misaligned, etc… in the comments.

# Twitter Math Camp 2016: Get Uncomfortable

Debate – Chris Luzniak & Mattie Baker

### “My claim is _____, my warrant is _____ .”

Structures:

1. Chalk Talk: Posters with questions on them, students respond by writing, no talking allowed.
2. Talking points: read more here
3. Debate! Argument = claim + warrant
1. Soapbox debate: provide class with a debatable prompt, and a minute to think. Then student must stand and state their claim and warrant to the prompt. It’s more fun if you randomly call on students.
2. Always, Sometimes, Never statements: students summarize previous idea, then state their argument.
4. Point-Counterpoint: use would you rather questions. Students must alternate arguments, so they have to disagree with previous person.
5. Table debate: assign student to teams to develop arguments, and then have the teams debate.

How to encourage debate:

• add debatable terms to questions – best, worst, most efficient, should, biggest, smallest, most, weirdest, coolest, always, sometimes, never
• Change boring math into a debate – Given an equation, ask,
• What is the best way to graph this?
• Which number would you change to change the graph the most?
• This graph will never go below the x axis

Full Scale Debate:

Divide the class into 4 teams. Provide students with a carefully constructed scenario and 4 different stance’s to argue (example provided with musician recording contract).

• have a rubric.
• assign students roles (opening argument, , questioner, attacker, defender, closing argument)
• takes about 3 class periods:
1. understand the problem and develop a plan
2. day research & begin calculations
3. finalize arguments

Socratic Seminars:

Students read a variety of texts or resources on a topic, then consider questions in a large group discussion.

examples of questions to consider:

1. What are some strengths and weaknesses of each presentation?
2. When would each text be appropriate to use?
3. What difficulties may students have?

This sounds like a really interesting thing to do in math class. I need to learn more and see it in action so that I can implement this effectively.

Critical thoughts to creating a successful debate culture:

• the accumulation of many intentional, small teacher moves over time sets the culture of student talk
• When you want students to talk to each other, the teacher must SIT DOWN. make yourself small, and not the center of attention. Encourage students to talk to each other. If it is a whole class debate, have the student talking STAND UP. Slowly back out of the center, have students call on their peers.
• Start early
• Keep it simple. Use basic soapbox debate for the first month or two.
• explicitly talk about what active listening looks like – be very specific (not writing, looking at the speaker, knees pointing toward the person speaking…)
• ideally, dedicate about 5 minutes per class 1-2 times per week
• provide structure (argument = claim + warrant ) and verbal cues
• Occasionally, have students do a quick write providing an image and a word bank. This will help students to practice communicating mathematically.

Keynote: Jose Luis Vilson

We need to talk about race with our students and give them a safe space to grapple with their thoughts. In math instruction, the goal is to teach students to grapple with tough problems for which the solution is not already know and work towards a logical and reasonable resolution. This same principal can be applied to social justice issues.

Some questions/statements for students:

• I just want to hear what you have to say
• Why do you feel this way?

We need to become comfortable getting uncomfortable and evangelizing for our truths. Avoiding confrontation and being polite can be destructive in the end.

Getting Triggy With It – Kristin Fouss

This session made me think of this Kate Nowak blog post.

She shared a very complete and organized collection of quality, basic stuff. Progressions, lessons, strategies. I can’t wait to use and adapt it for my first year of teaching pre-calc in a while.

Experience Connecting Representations – David Weiss

This structure connects a visual model to more abstract expressions. This could be graphs & equations, trinomials and algebra tiles, quadratic expressions and their factored forms…

Structure:

1. post more equations than corresponding visuals (task is to match the visual to the equation/expression)
2. provide individual think time – What do you notice?
3. Time to discuss with a partner (teacher circulates, listens & asks a pair if they would be willing to present their thoughts to the class)
4. display verbal cues:

Presenter

We saw ___ so we connected ____.

_____ matches ______ because ______.

Audience

They noticed ____ so they _____.

Their connection works because ______.

5. Get presenter’s to the front. One can only speak and the other can only point. They explain their thinking for one pair. Keep this light, safe & fun. If a student does not explain clearly enough or missing key elements, just let it go, they will most likely come out in later explanations.

6. Ask a student in the class to re-explain the presenter’s thinking

7. Teacher record thinking while a new students explains.

Repeat from step 5 with a new pair of students.

Once all problems have been paired and described by the class, have the pairs try to create a visual mode for the remaining equation that was not paired to a model

Close by having students complete a written reflection.

Explore Math – Sam Shah

Sam talked about a low stakes high reward assignment that he gives his students. They have to complete 4 or 5 mini explorations on any math topics of interest to them (with incremental due dates) and complete a brief written description or some evidence of what they did.

Site of suggestions

Johnathan Claydon – Varsity Math

He turned the advanced math classes into a ‘club’ called varsity math and created t-shirts, stickers, party’s and a summer camp to go with it. He also made recruiting posters and placed them at the middle schools in order to motivate students and create a buzz around taking more advanced math classes.

This is a great idea! I recently convinced 10 students at my school to take a more advanced math class and I think I will have to figure out how to adapt this concept to fit my tiny group in an effort to get this group to grow in future years.

Tracy Johnston Zagar’s Keynote – Link to slides

She opened my eyes to recognizing the different skill sets that elementary and secondary teachers have and the importance of valuing these skill sets and why we should try to break down our comfort barriers to get over ourselves and learn from each other.

I think I need to write a whole additional blog post on how individuals’ comfort seeking needs really limit our happiness, growth, empathy and success. (an ongoing theme this conference)

Variable analysis game – Joe Bezaire

The math game with the lame name

The basics of how it works:

1. Students guess the rule then they add a line of values that matches the rule.
2. Then these students become judges and let their peer know if they got it too.
3. Write it as an expression. Make connections between the various student expressions.

This may be a good warm up activity, so I want to be sure to link to it here so that I can find it in the future.

Six Steps to Modeling – Brian Miller & Alex Wilson

1. Define the question
2. Identify Variables & assumptions
3. Develop Model
4. Test Model
6. Report out