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.


You can download a editable (word) copy of this sheet here.

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

You can download these files: images & checklist here.


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 as described by Fawn Nguyen.


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



Geometry Planning Guide

Click here to access and comment on the Geometry Planning guide


  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 _____ .”


  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?
  • Where is your compassion/empathy?

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…


  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:


We saw ___ so we connected ____.

_____ matches ______ because ______.


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.

A blog post about it

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


Image from this Dan Meyer blog post


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

Moody’s Math Modeling Guide – Free Download

In this session we progressed through these steps to develop a model for ranking roller coasters, but the big idea here is more about how to facilitate this process. It would apply well to geometry tasks including 3 act’s such as best square or Mathalicious’ Face Value (my post on this task).

More than Resources – Dylan Kane’s Keynote

Clever Ideas ≠ Coherent Curriculum

We need to be thoughtful and intentional, not just resource collectors.

This resonated with me as I am an avid idea collector, but I struggle with how to make a curriculum coherent. I want to work on criteria for coherence and re-evaluate the content of my current classes.


Analyzing Triangle Congruence with AngLegs

I’ve tried to explain why AngLegs are a must have for high school geometry and should not only be considered a tool for younger students. Here is an example of how I find them indispensable in teaching triangle congruence. This lesson is adapted from MARS: Evaluating Conditions for Congruency.

“Ok class, you are sitting in pairs and at each table is a bag of AngLegs. On the board I have written the question we are trying to answer for each of the scenarios I will present.”

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“I made a triangle that includes a blue AngLeg. Can you make a different triangle that also has a blue AngLeg?”

“Make one. Hold it up.”

“How do you know that these triangles are different?”

“So, you are saying that keeping one side the same does not mean that the triangles must be congruent.”

Next, lets look at Card 3:



“I made a triangle out of a blue, a purple and a yellow AngLeg. Can you make a different triangle using the same three AngLegs?”

“What if you put them in a different order? …or move the purple between the blue and the yellow? Are you sure they have to be the same? How can you tell?”

Student:  “The triangles still fit perfectly on top of each other”.

Student: “If the three sides on one triangle are the same as three sides of another triangle, then the triangles must be the same.”

Look at Card 7:

Capture“How can you tell if an angle is the same in two different triangles?”

Student: “They fit perfectly on top of each other!”

“Is there a way to make these triangles so that they are not congruent?”

Student: “No way. These have to be the same”

Student: “Wait! I made two different triangles with all three angles the same and one side the same.” Does this count? Look!”

Student:”If two angles are the same, then the third angle always has to be the same because they add up to 180 degrees!”

“So what is the conclusion for this one?”

Student:”The triangles can still be different sizes, but their angles are all the same.”

“For the remainder of this class period, individually analyze card 5 and any other card, so 2 additional cards. Take good notes and write down your conclusions for tomorrow, where you will be randomly assigned a partner to complete the triangle activity. ”

From here the lesson continues as described in the SHELL center teacher guide linked above and described further in a blog post here.

Removing the hurdle of constructions allows students to focus on the learning goal for this activity: determining the minimum information required to guarantee triangle congruence, and what congruence means. It also connects nicely to congruence proofs through transformations as students are physically checking of the triangles fit on top of each other.

Puma Period: Supporting Executive Function Development

We have a homeroom period for 40 minutes after lunch every day, we call it Puma period (our mascot). Students of all grades are assigned randomly to a teacher when they come to the school. The goal is to have very mixed groups of students from a variety of backgrounds, achievements and grades in each class. The students stay with the same Puma teacher until they graduate, so each year a few leave and we get a few new freshman, but over half of the students remain the same. As their Puma teacher, I make contact with their parents and am a point of contact when parents have a question or concern about their student. I also keep track of each of my Puma students’ grades and missing work and stay alert if something seems off or if they need some motivation or a high five.

Here is our current weekly Puma routine:


A quick explanation of Planner Points: Students currently have teachers sign their planners in each class and they get 5 out of 5 points if they are doing everything well. They lose points at teacher discretion for behavior, tardiness, missing work, etc. This is used as a daily tool to communicate with parents. Students calculate their average for the previous week on Mondays. The goal is to separate grades from behavior. Most of our staff is frustrated with this aspect of our system and we are considering eliminating it next school year.

On Mondays and Fridays we stay together as a small group. On Mondays, students look up and list any missing assignments and their grades and write them in their planners. I meet briefly with each student to help them plan to be successful for the week. On Fridays we have a open whole group heart to heart where we focus on the things in their life that are causing stress and then discuss strategies to manage those things. Since we meet every day and hear about each other’s lives, we become a pretty close group, making the school culture more empathetic to their peers. This leads to students developing a goal for the next week.

Some recent goals I have seen are:

  • I will go to the gym for 1 hour three times.
  • I will take a few deep breaths to calm myself down when I notice myself getting mad.
  • I will go to bed by 10 pm every night this week, with my phone in another room.
  • I will complete all of my missing assignments this week.

On Tuesdays and Thursdays, after checking in with their Puma teacher, students can go to classes where they need help to complete their assignments. Any student who claims to have nothing to do can go to our gym. On Tuesdays, students with no missing work and acceptable grades get “free Puma” which means their lunch is extended, this is a major incentive for students to remain caught up.

On Wednesdays we have a whole school assembly where we occasionally have guest speakers, awards, plays by the theater class,  updates on upcoming school events. Sometimes we sing happy birthday to individuals who are having birthdays that month. If time is available we have question and answer time where students are invited to suggest ideas on a school policy (usually cell phone issues) or general questions and they are discussed and considered with the principal and their peers.

The Puma period affects the culture of the school. Students feel included and valued, they learn to reflect and take responsibility over what they can control and they define what that is (and what it isn’t). They also look out for each other more than students do at other schools where I have taught.

2016-02-23 16.40.01

Students learn executive functioning skills through reflection and discussion. This poster is in every classroom and has been adopted as a common vocabulary among the staff. Being able to define a deficit instead of feeling inferior helps students to have something actionable to develop. I am in the process of creating a document listing each skill and what it looks like in students as well as strategies to manage students with a deficit in that area.



A Day in the life of a Math Teacher


Tuesday, January 12th 2016

4:30 am: Wake up before my alarm and turn it off.

5:00 am: grade in my pj’s

6:00 am: Still dark out,  10 degrees F, Run 6 miles with my 2 standard poodles.

7:00 am: Quick! Shower, dressed, feed the dogs, get to school!

8:00 am: Staff meeting (Kid Talk)

8:45 am: Programming class students setting up their accounts with Globaloria. The introduction part is a little tedious – setting up a blog, creating an about me page…students are eager to start programming.

10:00 am: Geometry class – Taco cart project!  Students did taco cart on the first day of the semester. It is a great pre-assessment of their problem solving skills & experience applying the Pythagorean Theorem. The project part is for them to find the fastest path to the taco cart and then write a paragraph on what they did, how they did it, how they decided they were correct, etc. Similar to the Math Forum POW’s.

11:45 am: Lunch time – Google Hangout with Globaloria in order to complete the PD portion of the programming software we will be using.

12:30 pm: Puma period – On Tuesdays, students can go to any class to get help. I had about 10 students who missed a class at some point last week, they sat in groups by missed assignment, and I ran between groups answering questions. There were other stragglers in my room using the laptops. One student completing a math online credit recovery and she asks for help with each question. I try to do a quick mini lesson on solving equations with her.

1:15 pm: Planning period – Now I have a minute to eat. Finalize my plan for Intro to programming next period. Upload assignments onto Google Classroom. Put together plans for my classes tomorrow. Remember I haven’t graded programs from last class, log in to CodeHS and see that I have about 30 programs to grade and provide feedback. I also notice that one student was programming at midnight last night for fun. Awesome!

2:45 pm: Intro to Programming – I lean heavily on CodeHS for this class. Since we are still in the initial part of this class, students are having fun completing the introductory programs and they are all moving at their own pace. I want them to feel free to progress quickly, but I am not sure what I will do when some are very far ahead of others. It’s a good problem. I’ll figure something out.

4:05 pm: School ends. I clean up, make copies for tomorrow, respond to a few emails, chat with some students and leave within an hour.

5:00 pm: Home. Walk the dogs. Make Dinner. Clean up.

8:00 pm: sleep


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