More Barbies–in Calc

I start my (non AP) Calculus class every year with two days reviewing lines, slopes, functions, etc…and we spice it up by (quickly) doing the Barbie Bungee Jump activity! There’s lot of places to read more about that. So I won’t go into that any further.

However, I really wanted to add both more fun activities to the rest of my units AND have more of a continuous through-line. So, I tried to include a Barbie-centered activity in my introduction to derivatives, including average velocity vs. instantaneous velocity, etc. I came up with Barbie walking the runway!

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Page 2:

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It worked really well! They had fun, and we talked a lot about slope=velocity. The last page had some questions more textbook-y, asking them to apply the ideas in new problems:

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Email me if you want the full doc!


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Rearranging Limits

I struggled a bit with teaching limits the past few years. Not only are they difficult for students to grasp, but the texts I have used as guides have been a little bit jumpy. Add that to the fact that I teach a non-AP Calculus class. So, this year I really wanted to make a clear exploration of limits, building a strong intuitive understanding of what a limit was asking for. The result: instead of making daily objectives about limit concepts (what is a limit, one-sided limits and infinite limits), I changed my objectives to limit strategies. In other words, our objectives/topics for each day were:

  • Day 1 & 2: Finding limits by tables
  • Day 3: Finding limits on a graph
  • Day 4: Finding limits using algebra

The first two days, we just spent finding limits using tables (on Desmos and in the graphing calculator). I started with the “buffering” idea I mentioned here. Then, we just made lots of tables, explicitly talking about what numbers you would plug in to get “really close” to a certain x-value. Below is a screen shot of the worksheets I used. Notice that we got into infinite limits without talking about them as anything different.

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The next day, we talked only about finding limits on graphs. I gave lots of graphs, and we found the limit by reading what the graph is approaching on each side.

The last day we spent some time using “algebra tricks” to also find limits.

My goal was that by the test, students could find limits using whatever method(s) they prefer, having a deep intuitive understanding of what a limit is. And the scores were so much better this year!

As an additional test of how it worked, the day before the test, I gave the following as a warm up problem. I’ve done it every year, and this is the first year students could do it on their own, without any help! They did a great job!

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RSBCMTA Conference

Yesterday, I crashed the Riverside San Bernadino Counties Math Teachers’ Association first annual conference, and I am so glad I did! Just the day of inspiration and insight I needed.

The day started with a great call to action from Chris Shore, focusing us on the Common Core Standards for Math Practice.

He also demonstrated the difference between traditional quizzing and some variation, but more on that later…

Chris’s key note focused our sessions around the SMPs in a good way. I gave a session on debate in the math class. Then, inspired by Tracy Zager’s TMC16 talk, I went to an elementary school teacher’s session on classroom discourse. It was a great session led by Mary Vongsavanh, and it got me thinking more about classroom discussion structures. Something she said that really struck me was “if want want them to do x, give them that experience”…as well as “The more you talk about something, the deeper your understanding.”

One take away is an extension of a Falsification activity I already do. Her example is below. Based on the picture, what is the rule for what fractions go inside the circle? You can “test out” the rule when you think you know it by naming a fraction that you believe goes in (or out of) the circle, and the teacher will confirm.

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I have lots more to say about discussion and assessments and all that, but I’ll get more into that later. Thanks RSBCMTA for a great conference!

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Start of Year: Introducing Vocab

In the past two years, I’ve been convinced to introduce definitions always using the Frayer Model. If you don’t know of it, there’s a video I just found online here. This is what a blank one looks like:

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Now, let me say that I avoided it for many years because it felt to me to be a bit too contrived or too geared towards younger students. However, I’ve come around on my thinking, and I use it in all my classes, 7th grade through Calculus, and I LOVE it.

Part of the reason I love it is that it forces me to thoroughly flesh out the new vocab word, for the benefit of my students.

Another reason I like it is that I’ve found two ways of structuring how we fill in the boxes (see below).

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Let me just add that I hand out to students a page where this box covers half a page (pictured to the right). I usually have two boxes on the front and one or two on the back, depending on how many important definitions I plan to go over in the span of a class or two.

I then also display one box on the board when we are filling it out in class.

Now, on to the two ways I work through the boxes:


Style 1:

A nice, straightforward way to use the model is to follow it in a U-shape. It flows like this:

  • Start in the top left corner: I give a formal definition.
  • Move to the bottom left corner: as a class, let’s come up with some examples.
  • Move to the bottom right corner: with a partner, come up with some non-examples (which I may have some students put on the board).
  • End in the top right corner: As a class, we add any notes that came up during our examples of things we want to remember. I may also come back to this box in the following day or two to add more notes if necessary.

Here’s an example we did in my Integrated Math 3 (Algebra 2) class.

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We filled in the examples with tables and sequences that had a quadratic pattern. The “Essential Characteristics” popped up as an animation after we filled in the bottom boxes. We later went back to fill in “Essential Characteristics” with notes like forms a parabola when graphed.


Style 2:

My newer use of the box (inspired by one of my awesome colleagues at TMC17) forces students to create good definitions based on examples. I fill out the bottom boxes and they come up with a definition that is “unbreakable.” It flows like this:

  • Start in the bottom left and bottom right: Prepared examples or on the spot examples. Have students look at those, consider.
  • Move to the top left box: have students pair share a good definition (that is unbreakable) and then discuss as a class. The teacher tries to “break” the definitions students come up with using examples and non-examples (possibly the ones already there), until the class comes to a solid definition.
  • End in the top right box: fill in notes about anything important that came up in the “trying to make a definition” process. I may also come back to this box in the following day or two to add more notes if necessary.

This time, the thing I start with looks like this:

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Here, students started to say things like “it goes through the vertex.” However, my horizontal and diagonal lines go through the vertex, too. So I was able to “break” that definition. Students love playing this “game” of making a solid definition, and in this class they eventually honed in on the point that it had to vertical and cut the parabola “in half.” The fact that it went through the vertex was a “cool side effect” as one student said, and that went in our Essential Characteristics box.






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Start of Year: Day 2 (Introducing Debate)

After my first day activities, which (I hope) build some community and positive culture in my classroom, I like to begin Day 2 by introducing my debate format. I have a whole blog post sharing more of the details of how I go about it here. I just wanted to show the two start of debate activities I used in my classes. The earlier blog post describes all the steps I follow.

Note: The debate activity only lasts 5-10mins. If I have a small enough class, I try to get everyone to stand and say one claim/warrant…but I also feel out the room and stop if we’re getting repetitive or I’m losing engagement.

In Calculus class, I started with this slide:

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In my Integrated Math 3 (Alg2-ish) class, I started with a Which One Doesn’t Belong. The first slide was:

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I then dive right into our first Unit. During this time, I have students go to the board to do problems, introducing another routine of my class. Using Vertical Non-Permanent Surfaces (VNPSs) is something I blogged about here. I have 6 different dry erase boards around my room for students to work on in pairs or groups of three.



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Start of Year: Day 1

I read this article a few days before school started, and it really got me thinking. The gist is that I want to start with my actually curriculum, rather than “cool/fun math” activities because I don’t want to send any subtle messages that the curriculum is too boring to start on day one…or the curriculum can’t build community. The quote that stuck with me is:

if you’re teaching content well, the class culture stuff will fall into place.”

So here is a run down of how my classes tackled Day 1:

  • When students enter, they fill out a notecard with name, best person to contact at home, favorite number, and if you were a creature, what would you be and why. I found this summer that changing the word animal to creature allowed for a wider umbrella, including fictional creatures, plants, and Cinderella.
  • When students enter, they also fold a large notecard in half and make a nameplate. I have them use markers (I provide) and write their name (just first name, what they want to be called) on both sides of the nameplate, in case it gets turned around.
  • While students are filling out the notecard and nameplate, I quickly go around the room taking a picture of each one with my phone. I make sure to get the nameplate in the picture. I now have a photo directory to learn students names at home.

Here’s what the first slide of the day looks like for all my classes:

Day1 Screen

  • After quick introductions (tell me one thing about you I can’t tell just by looking), I pass out the syllabus. On Day 1, I only have students look at the supply list. I don’t go over anything else. I will go over other parts (homework policy, grading, etc) when those things actually arise the first time (i.e. when I give the first homework, when the first quiz grade is put in the grade book, etc).
  • Next, I do want to get students talking more about classroom rules and culture, but I do it in the form of Talking Points (a la Elizabeth Statmore). Here is my list of statements. I leave some blank space at the end for students to come up with their own controversial statements. They have a lot of fun with those!

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  • Now that we are about half-way through the class period, in the spirit of that article I mentioned above, I have students work on content. In Calculus, it was the first part of Barfing Monsters (a la Sam Shah). In my Integrated 3 (think Algebra II) class, it was this great problem that I got from Bill Thill and Peg Cagle about patterns (see the picture below). It’s a great open ended problem, that eventually lets students pull sooooo much math from a simple pattern. Students create linear and quadratic equations based on what they use to fill in the blank.

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  • Lastly, inspired by Sara Van Der Werf, I have students write about their thoughts on the first day of class inside their nameplate they created. Just a sentence or two. That evening, I write back to each student, and we continue to have a brief written, private conversation in this nameplate throughout the first week. It does wonders for making connections and building relationships!
  • The homework for the first week is to email me two paragraph about themselves in general and as a math student. It’s called a Mathography, and in the first weeks of school, I respond to each email.

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#talklessAM (or the nitty gritty on starting debate)

I just got back from TMC16, where I was lucky to get to co-facilitate a three-part morning session called “Talk Less, Smile More: Getting Students to Discuss and Debate Math” for a big group of awesome teachers! My co-facilitator Mattie was awesome! The group was awesome! What a great time!

Gloating aside, one question that always comes up when I talk about introducing debate is How to Start? In the session, we talk about sentence starters, writing good questions, and give participants time to experience the activities, but there’s still that question on how to explicitly begin. On top of this, I just had a phone conversation with a great public school teacher in LA that had been to a workshop of mine a month earlier. He is going into his third year, but his commitment to try debate this coming year and his deep questioning of the how to get started was inspiring. Through all these conversations, I’ve tried to nail down all the important details of how I begin, and I thought I’d share the nitty gritty on how I get these routines started in my classroom.

(Disclaimer: this is just a detailed account of how I do it. There is no one way, but hearing this may help you figure out your way.)

Day 1: First day of school is usually a little chaotic: students need to find the room, find a seat, fill out and decorate a nameplate, get a syllabus or supply list, etc. Sometimes the classes are also short. So I usually don’t debate on Day 1. However, I do get the community of discussion going by forming a circle and having students  introduce themselves, usually with their name, where they are from, and then something goofy like “my one superpower would be” or “if I could be any cartoon character.” I follow that up by doing some math. It is my philosophy that in a math class, we should do some math every day. There are a lot of great open ended, all level problems to start with like the Locker Problem, The Camel Problem or the card challenge “Skip Flip.” I will have to share these in another post.

Day 2: I want to introduce debating as soon as possible. So Day 2 is usually the day. When students enter, they see directions on the board to create an argument for three different topics. As they file into their assigned seats, they silently take out their notebooks and attempt to write an argument for each prompt. The slide looks like this:

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After some writing time, I have all the students stop and tell them that before we begin, I need to teach them HOW to make an argument/how to debate. (This used to make total sense at my old school when I was the debate coach, but students at my new school don’t seem to mind.) I show the following slides:

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I briefly give the definition of an argument, but I try to minimize that. The key is the way we breakdown an argument into Claim and Warrant. So I define those two words for the students and then say that the mathematical formula for an argument is


I stress that the only thing they have to takeaway from this slide is the sentence starter:

 “My claim is…   and my warrant is…”

I also have this hung on all four walls of my classroom. It is important to have it on multiple walls so that students can see it no matter what direction they are facing during a discussion.

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The next slide takes us back to the original three questions that I want students to make arguments about. This is when I give them the directions for how to do a “Soapbox Debate.” (This phrase is a bit outdated to our students, but I love explaining it to them.) Directions are:

  1. When you are called on, you must stand.
  2. You must use the words “my clam is…   my warrant is…”
  3. If you are not speaking, your shoulders and knees must turn toward the speaker.
  4. Our eyes should be on the speaker (who should be standing)
  5. Only the person standing is speaking. The rest of us are listening.

When I give the directions, it is done orally and they are not numbered. I simply explain them. I just listed them out for ease of reading.

Then (and this is where it takes a moment of bravery on the teacher) I jump in with the first speaker! Some notes on this:

  • I usually ask for volunteers in the first days of school because I don’t want to make a shier student too uncomfortable (and I will work out ideas with them later)
  • That said, sometimes students can all be quiet. So I may have to cold-call on students. It can be startling, but I try to do it with as much positivity and as non-pushy as possible.
  • I make the speaker stand. Even if they don’t want to. I just smile/laugh, say it’s ok, and tell them to stand up.
  • I correct the speaker on the spot if they don’t use the words Claim and Warrant.
  • I sit down before I call on the first speaker. I want the speaker to be the only one in the room standing.
  • I am hyper-vigilant that every student is turned toward the speaker and looking at the speaker. Though I have passed over the discussion control to a student, I am working even harder to make sure everyone is following along (while trying to keep all messaging positive).
  • After the first student, I (still sitting) call on another.
  • Depending on the question and responses, I may only have 3, 4 or 5 speakers for each question. That means 3 students share their favorite movie, then I move onto the second topic and have a few students share out. Sometimes it’s only 1 or 2 for whatever reason. I try to read the room.

And that’s it. Overall, it may take 5-10mins only. I am very actively watching students, correcting body language and sentence starters.

The rest of class may not involve explicit debating again. Early in the year, I tend to just spend roughly 5 minutes per day doing debate activities as the warm up. Then, even though I do not explicitly do a formalized soapbox debate, the culture of student discussion often spills over into rest of class.

Final Thoughts:

  • Standing is important. Students may be a little uncomfortable but it is important in establishing the culture and getting students a little out of their comfort zone/passive learning.
  • Using the sentence structure “my claim is…my warrant is…” is something I stick to in every response. In later weeks, I may not bother to correct students who don’t use this exact structure, but stressing it in the beginning sets the pattern that answers always have two parts in my class. Students just get in the habit of always explaining their work.
  • When I introduce all the structures, I always start with some kind of “fun” topic, like the best movie or best musician. After that we only debate math. It helps to lower the stress of learning a new structure if the content is something students find interesting and are easily opinionated about.

And that’s really it. Everything else I do throughout the year is just variations on soap box debating, a mix of oral (standing) debates and written work that includes warrants (explaining).

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