If you're a calculus teacher, you know this struggle...

My non-AP calc students have truly blown me away with the ease at which they've taken to derivative rules this unit. I went into this unit with a relatively open idea of pacing- what my AP kids can do in a day might take 3-4 with these kiddos. But, they're killing it....when isolated, their calculus is beautiful.

Unfortunately, the calculus isn't going to be "isolated" as we move on in the course and the struggle became very, very real as we started implicit differentiation. With that in mind, I write this sorting activity to help scaffold students in their algebraic skills.

I have students identify the initial problem, then work on whiteboards to determine the next steps. They had great conversations and asked even better questions. Once this was finished, I saw a lot more independence as we solved implicit differentiation problems without the scaffolding....success!

Here's the original document if you want it!

## Thursday, December 13, 2018

## Thursday, November 29, 2018

### Did I Finally Figure Out How to Teach Recursion?

Recursion has always been a weird topic for me- one where I try all sorts of different things to try to get kids to tie the relatively basic conceptual idea to the funky notation that is sometimes associated with it. I've taught it at the middle school level (next=now blah blah blah), in Algebra 1, in Algebra 2, and in Pre-Calculus and I inevitably always have some kids who struggle make the transition from the idea to the notation and requirements. So, yet again, I tried tweaking this year. Here's what I did.

I started by putting this slide on the board:

Students had 30 seconds in groups to tell me the exact sequence about which I was thinking....this obviously caused some unrest. Some groups were prepared with random guesses, some got indignant that they didn't have enough information, some argued among themselves about which side to take. I let groups guess and every time I said "WRONG!" until someone finally pointed out that this was rigged.

I let them know that I'd allow the class to ask 2 questions to try to get it right. They needed to prioritize those questions in their groups and then we continued the discussion. The 2 questions they asked:

1) What's the pattern?

We discussed- was this strong enough to give me the EXACT sequence I was thinking about? Nope!

2) What's the first term?

That's it! We need to know where to start and how to proceed from there! Those 2 magical ingredients of a recursive formula were just generated by the class- score.

Next, I challenged groups to complete this table together. They had great discussions, challenged each other, and the wait time to try to get them to ask questions at the end was

*excruciating*because they really felt confident about it.
Welp, guess they don't need me now. They just discovered how to write these without me ever teaching it. I'll be in the Math Office drinking my latte with my feet up on my desk.

After some wrap up notes, we played a round of Quiz Quiz Trade- one of my favorite Kagan Strategies. Below are my rules:

And here's the actual document:

These were small tweaks to how I normally teach recursion, but they made a huge difference for my kiddos this year. The extra work we did with notation previously made a big difference as well (see my last blog post). I loved seeing the kids excited about a topic that has sometimes caused anguish for others in the past!

## Friday, November 16, 2018

### Building Understanding of Sequence Notation

Sequences and series are never a particularly hard topic for my Algebra 2 kids- conceptually, at least. They get it. They've been looking for patterns since they were tiny little humans and it's a fun puzzle for many of them to do so in class. What they struggle with consistently is the notation we demand of them at this level of mathematics. Some mastered it in Algebra 1, but many just show up and pray they do it right this time or memorize what their teacher told them to do instead of trying to battle the sense-making involved to understand and be able to apply this knowledge. This activity is one I use to combat that and it builds to have students generate their own equation for recursive sequences.

__Phase I:__We start by doing a sorting activity, where students are given the cards at right and asked to order them. This is an easy enough task- definitely low floor for some of my weaker students. This is a springboard for discussing the "why" and asking questions like:

- Which comes directly after n? How do you know?
- Which comes directly before n? How do you know?
- If you have n-3, which would come 2 after that?

You can really extend it as far as you want to go here.

__Phase II:__From there, we look at the actual expressions we use for terms.

__Phase III:__
Here is where the wrap up discussion as a class occurs and we begin to test our understanding of the notation. In small groups, we first just examine the differences between position in the sequence and actual value of the term:

Then, we start translating from words into notation.

I've always heard it said that students are more receptive to an idea if they think it came from themselves or another student....this activity has been a huge help in clarifying misconceptions with the help of other students, not just a teacher re-explaining it the same way for the 400th time.

If you have anything else that you love for teaching recursion, sequences, or series, please share!!

## Monday, November 5, 2018

### Limit Definition of the Derivative Carousel Activity

I've been working on building up the limit definition of the derivative conceptually with my non-AP Calc class for the last week (see my recent post on what I've been doing so far ) and today we started putting it all together!

To into this activity, I used an awesome warm up from Math Teacher Mambo (Thanks, Shireen, for sharing on the AP Calc Facebook group). As soon as she shares it publicly, I'll link it here. It's so, so good!

Future Home of Warm Up Activity Link

After completing the warm up, here's how the carousel worked:

1) Put students in to groups and have them start at a blank poster or piece of VNPS

2) Each group completes one step at their poster, then rotates. They will then check the next group's work, correct it, and then add the next step to the poster. Here were the directions:

- Draw a blank axis- I told them just first quadrant- and a function of their choice
- Sketch a secant line
- Label x, x+h, and h
- Label f(x) and f(x+h)
- Write an expression for slope of the secant line
- Transform expression into slope of the tangent line

Here were some of our results:

Awesome conversations ensued, including whether it would make sense for f(x) to equal f(x+h) and what that would mean for the secant line (Helllllloooooo, Rolle's Theorem!). Kids were explaining to each other, critiquing each other's work, and doing a lot of sense making among themselves!

We'll see how this translates to retention beyond today on their next quiz, but I loved seeing the progress that they're making. I think many could explain it better than a few of my AP kids right now- a good "challenge accepted" moment for me to amp this conceptual understanding up more in AP, too.

### Inverses of Exponential and Logarithmic Functions Circuit

Quick post to share a new circuit I just finished.

My kids needed some more practice and I wanted them to begin reflecting on everyone we'd done in our logs unit before we get into more difficult applications and review for our Quarter 1 exam....this took care of 2 birds with one stone!

This is just the basics and doesn't use any laws of logs, so it could be amped up if you wanted to do so.

My kids needed some more practice and I wanted them to begin reflecting on everyone we'd done in our logs unit before we get into more difficult applications and review for our Quarter 1 exam....this took care of 2 birds with one stone!

This is just the basics and doesn't use any laws of logs, so it could be amped up if you wanted to do so.

Steal away!

## Thursday, November 1, 2018

### Discovering the Derivative!

I knew when I took on non-AP Calculus this year that I was going to have to up my game. Last time I taught a non-AP Calculus course I was a first year Calc teacher and I was truly staying only a few days ahead of the kids. I didn't have the perspective of deeper conceptual understanding that added years of teaching a course can bring. I was on a mission this year to build strong conceptual understanding for these students so as they moved on to Calculus in their college career, they would have a strong enough background to face any challenge.

The whole concept behind the limit definition of the derivative can really elude kids. They get that slope is rate of change and they get the mechanics of the derivative rules, but they can be very unsure as to why we're taking a limit and what that "h" means anyway. I decided to tackle this by building up the concept, slowly and intentionally.

Phase I: Have students experiment using what they know about average rate of change and see how they relate it to instantaneous rate of change (on their own, with no intervention from me)

Phase II: Make sure they understand average rate of change and the structure behind the notation

Phase III: Move students slowly from their understanding of average rate of change to instantaneous, moving notationally from middle school to college level

I was really happy with how it went and truly impressed with the ingenuity of some of my students. While there are things I will definitely tweak for next year, I felt like my kids walked out with a firm grasp on what the hot mess of notation known as the limit definition of the derivative means. We did no evaluating, no calculating. We just worked on concept and structure.

There are profound benefits to having a work wife who is a physics teacher and that is being able to raid her lab room for supplies when you're feeling inspired in math class. I just wrapped up a graduate course about utilizing engineering and engineering practices in the STEM classroom and this phase was very much inspired by that. It also helped serve as my final exam project, so should out to my students for writing at least half my paper for me.

I told the students that their objective was to find the instantaneous velocity of a marble on a track using only items found in the classroom. No downloading a radar gun app on their phones. No running to the physics lab to grab a photogate.

Groups took all different approaches, some of which suprised me! I knew many would time the whole length of the track and soon realize that the smaller their interval, the more accurate their approximation should be. Some tried to control for the initial velocity by creating ramps, using the acceleration due to gravity to help them calculate. They really got creative with their reasoning, worked well together, and gave each other great feedback.

I used a BeeSpi to determine the actual velocity of the marble and we found a percent error to determine how accurate groups were in their measured value. And basically, the coolest thing happened. Since we used the engineering design process- which uses iteration of trial, redesign, feedback, etc- the students continually tweaked their process. And each and every group independently determined that you wanted an interval as tight to the actual BeeSpi as possible. We wanted the size of this interval to approach 0. Without that, we'll never have the most accurate prediction. Couldn't have lobbed it in for me any better the next day to build the idea of the limit definition.

If you're interested in the activity and actual documentation of my kid's work, check out the whole document here:

The whole concept behind the limit definition of the derivative can really elude kids. They get that slope is rate of change and they get the mechanics of the derivative rules, but they can be very unsure as to why we're taking a limit and what that "h" means anyway. I decided to tackle this by building up the concept, slowly and intentionally.

Phase I: Have students experiment using what they know about average rate of change and see how they relate it to instantaneous rate of change (on their own, with no intervention from me)

Phase II: Make sure they understand average rate of change and the structure behind the notation

Phase III: Move students slowly from their understanding of average rate of change to instantaneous, moving notationally from middle school to college level

I was really happy with how it went and truly impressed with the ingenuity of some of my students. While there are things I will definitely tweak for next year, I felt like my kids walked out with a firm grasp on what the hot mess of notation known as the limit definition of the derivative means. We did no evaluating, no calculating. We just worked on concept and structure.

__Phase I: Engineering Design Challenge__There are profound benefits to having a work wife who is a physics teacher and that is being able to raid her lab room for supplies when you're feeling inspired in math class. I just wrapped up a graduate course about utilizing engineering and engineering practices in the STEM classroom and this phase was very much inspired by that. It also helped serve as my final exam project, so should out to my students for writing at least half my paper for me.

I told the students that their objective was to find the instantaneous velocity of a marble on a track using only items found in the classroom. No downloading a radar gun app on their phones. No running to the physics lab to grab a photogate.

Groups took all different approaches, some of which suprised me! I knew many would time the whole length of the track and soon realize that the smaller their interval, the more accurate their approximation should be. Some tried to control for the initial velocity by creating ramps, using the acceleration due to gravity to help them calculate. They really got creative with their reasoning, worked well together, and gave each other great feedback.

I used a BeeSpi to determine the actual velocity of the marble and we found a percent error to determine how accurate groups were in their measured value. And basically, the coolest thing happened. Since we used the engineering design process- which uses iteration of trial, redesign, feedback, etc- the students continually tweaked their process. And each and every group independently determined that you wanted an interval as tight to the actual BeeSpi as possible. We wanted the size of this interval to approach 0. Without that, we'll never have the most accurate prediction. Couldn't have lobbed it in for me any better the next day to build the idea of the limit definition.

If you're interested in the activity and actual documentation of my kid's work, check out the whole document here:

**Phase II: Emphasize Average Rate of Change**

Here's the document I used for notes and partner tasks:

I wanted kids to understand not only how to calculate slope and it's meaning (duh, they've been doing that forever), but I wanted them to really examine the structure. Here's what the filled in notes looked like:

I also wrote a matching activity here where students would match the structure to the interval and function to the solution, but I wrote it at 11 pm and when I opened it the next morning I realized it made no sense so we will be revising that for next year :)

**Phase III: Moving from Average to Instantaneous**

These notes and activities are found in the notes above.

We talked about how the slopes of these secant lines (average rates of change) approached the slope of tangent line (instantaneous rate of change) and the groups chatted about how this all related to what we did the day before.

**Smaller interval, more accurate prediction.**
Partners then worked through this exploration (adapted from Calculus God Mr. Korpi ), showing the impact on the slope of the secant line as the size of the interval approaches 0. Students were able to articulate this easily and knew the slope was approaching 1. (They also had a good debate over what the word astute meant, so add that to the list of things they learned today). The very last question took some prompting and we never got to a formula, but the idea you see written here is a good summary of the conversation we had. They made sense of this themselves, I just wrote it down.

Then, we went to the big momma. I made her a few years ago and I love her. She builds from the middle school understanding of slope to the limit definition of the derivative.

The kids walked out today and could explain this in their own words, could write the limit definition for a given function, and could identify the function and x coordinate from a "disguised derivative" given to them. All in all, I'd call that a big win.

## Monday, October 1, 2018

### Google Forms Pre-Assessment

My non-AP Calculus class is about to start our limits unit and I wanted a better picture of what they remember from last year to inform how I lay out the unit. A little background: my kids are coming from a variety of places including Pre-Calc Honors, Pre-Calc Advanced Topics (an advanced functions class- the "non-honors" version for our school), crash course community college Pre-Calc over the summer, and some even from Algebra II. I wanted to try something new to get feedback from kids that would also mean I didn't have to wait until I saw them next, so I threw together a Google Form using my answer key.

For each question, I posted my full solution to the problem, then asked students to tell me if they got it correct or not, how confident they felt on the skill, and then an extra space for comments. Comments were optional, but kids have been sharing some interesting feedback on there. Here are some screenshots of the form:

It's been most interesting to see the correlation (or lack there of) between kids getting a question correct and their confidence. It's helpful to see that kids got something correct, but still didn't feel confident in it....a nice insight outside of just what percent of kids "knew it."

I shared a bit.ly link on the handout and told kids they could only receive credit if they filled out the form, so there's incentive to actually do it. I'm using it to plan out where students will need support and where we can move faster than I'd expected.

Definitely a trial run, but so far so good!

## Tuesday, September 25, 2018

### Unit Circle & Exact Value of Trig Functions Review Activities

The Unit Circle is my jam. Last year, I even went to a tattoo parlor and asked them to slap one on my forearm (to which they replied that I'd need to do it much bigger if I wanted the detail I requested...so that saga continues). When I taught Pre-Calc, it was truly my favorite week of the entire year. But now that I'm teaching all things Calculus, there's less time for the beauty and elegance and just a short window for rapid fire review. My AP students are assumed to know it backwards and forwards. My school level Calculus students needed a bit more work with it based on my pre-assessments, so I did a few new things to practice with them!

I gave each group a hula hoop and had them align it with the floor tiles to form a set of axis. From there, each group was given cards with all radian measures and all coordinates of the unit circle.

You could go even further with this and include degrees, but I wanted to make sure the students were starting to think in radians from very early on in the year. I figured I would give them 2 minutes to try to label everything, but this turned itself into a 15 to 20 minute activity with an

**Unit Circle Hula Hoop Puzzle**I gave each group a hula hoop and had them align it with the floor tiles to form a set of axis. From there, each group was given cards with all radian measures and all coordinates of the unit circle.

You could go even further with this and include degrees, but I wanted to make sure the students were starting to think in radians from very early on in the year. I figured I would give them 2 minutes to try to label everything, but this turned itself into a 15 to 20 minute activity with an

*amazing*debrief.
As I walked around the room, I noticed some great strategies. I took note of them for next year and I want to create some kind of guiding questions displayed to lead our conversation. Here's where the convo wound up going today:

**Where did your group start?**

- Students started with quadrantal angles (which was a word they couldn't tell me before this convo, so glad it came up), then divided from there

- They were able to check themselves by thinking about the order the radians should go around the circle. I saw one group specifically calling each other out for putting ⅚Ï€ in the 3rd quadrant because it clearly had to be less than Ï€. The idea made sense and I saw a few kiddos who'd clearly just tried to memorize their way through this in Pre-Calc have the logic behind it click. This took a huge working knowledge of fractions which is somehow still a struggle for kids who've successfully made it to Calculus.

**What did you do when you got stuck?**

- For my group, this seemed to be on the coordinates. The angles took some discussion, but they were able to reason through it together. The coordinates were a different ballgame.

- Students were at least able to sort the coordinate into quadrants. Many were able to think about the reflections that take place to make angles with the same reference angles have related coordinates. All of these ideas were integral to where we'd go after this- reviewing the circle and how to use it.

**Exact Trig Values Speed Dating**

After a brief review and a few practice problems, we were ready to practice! Instead of just a worksheet or a Kahoot and trying to get my first block to wake up already, I decided to make them start wandering. No pre-planning required...this one was easy to wing!

1) Get a whiteboard & marker

2) Find a random partner

3) Answer the random exact value question Mrs. G puts on the board with your partner

4) Boards up!!

5) Class Discussion (if necessary)

6) Find a new partner and repeat

This not only gave me the chance to get kids working and talking, but I liked that I could go over each question and check in with kids I saw struggling.

I have a whole library of other activities I've done with Pre-Calc classes in the past, but I really liked these for a group that only has 1 day to review all of this!

## Sunday, September 23, 2018

### Methods of Finding Limits Error Analysis

My AP Calculus students just took their big Limits quiz and before we dive into Continuity and IVT, I wanted to make sure we solidified some sloppy issues I saw pop up. I designed this activity based off actual errors from student work, hoping for a little personal reflection before they ever get their quizzes handed back to them. I'm planning to use this as an intervention tool with my struggling AP kids, but then a teaching tool with my school level Calculus kids.

I started this by simply having students choose a technique to evaluate each limit, without actually evaluating. I hope this will be a catalyst for conversations within the group and a baseline for students as they move into the next part of the activity.

The 2nd part of the activity shows examples of mistakes from the quiz that I rewrote in my own handwriting to avoid any embarrassment. Student will need to find the mistake and then fix it. One of my favorite parts of these mistakes is that often they do not lead to an incorrect answer. My Calculus students need to get very used to attending to notational precision in their work and there's no better time than the present for that!

I started this by simply having students choose a technique to evaluate each limit, without actually evaluating. I hope this will be a catalyst for conversations within the group and a baseline for students as they move into the next part of the activity.

The 2nd part of the activity shows examples of mistakes from the quiz that I rewrote in my own handwriting to avoid any embarrassment. Student will need to find the mistake and then fix it. One of my favorite parts of these mistakes is that often they do not lead to an incorrect answer. My Calculus students need to get very used to attending to notational precision in their work and there's no better time than the present for that!

There's a good mix of Algebra and Calculus mistakes throughout the activity, but they definitely lean heavier on notational fluency than anything else.

Feel free to use as is or modify as you see fit! Also, if you have any common issues that you see with your students and limits, feel free to send them to me so I can add them to this!

## Monday, September 17, 2018

### Rational Functions Who Am I? Activity

Pre-Calculus has always been my first baby. As a new teacher, I was thrown into Pre-Calculus and it quickly became my pride and joy...I experimented, learned, and fell in love with teaching it. Not only was it the best preparation I could have had for when I finally got to teach my real, true love (Calculus), but it also helped me connect my knowledge of algebra, geometry, and more in a way I'd never done as a student.

This year I've started teaching a school level Calculus class and it's been so much fun to delve back into some Pre-Calculus topics that I don't normally get to review with my AP kids. This activity was a great review for my students to get them talking, factoring, and thinking. It didn't take long, but generated some good conversations. It would also be a fun extension to have students write one of these "Who Am I?" activities, which could be used for any topic in any course!

Feel free to steal or adapt! Let me know if you make any meaningful changes! Here is the file:

Here's some more of my thoughts about rationals from past posts, if your in the market!

## Tuesday, August 14, 2018

### #MTBoSBlaugust Day 8: Oh The Places You'll Go!

I have been lucky in all my years of teaching to have a "senior" level class each year. For a few years, it was 8th graders- top dogs going off to the wide world of high school. For the majority, it's been Pre-Calculus or AP Calculus with some of the most wonderful 18 year old humans the world has ever seen.

One of the things I try to do with my seniors is give them a space in the classroom that is all their own. I bought this poster as a heading for the corner:

Some students have upwards of 10-15 pennants on the wall by March, posting every acceptance they want to share there. This board has seen big names: Yale, Cornell, Duke, Syracuse, UNC Chapel Hill, University of Texas, and many more. But I wanted this board to be a place to celebrate

If you teach seniors, I really recommend trying this. I don't have pictures of last year's wall since my room is being used for summer school, but I'll take some and post them in the future!

One of the things I try to do with my seniors is give them a space in the classroom that is all their own. I bought this poster as a heading for the corner:

I chose it for a few very specific reasons:

- Dr. Seuss is a boss
- It was available as a Prime item (an instant sales tactic)
- I like that it doesn't specifically mention "college" or anything else like that- just "the places"

I keep a set of blank "pennants" in a folder near the wall and I encourage kids to post anything they're excited about there. They can decorate to their hearts content or they can just fill it in. They can fill out as many as they want. For some, that looks like this.

Always an added bonus when you get to celebrate an acceptance to your alma mater (Go Bearcats!) |

Some students have upwards of 10-15 pennants on the wall by March, posting every acceptance they want to share there. This board has seen big names: Yale, Cornell, Duke, Syracuse, UNC Chapel Hill, University of Texas, and many more. But I wanted this board to be a place to celebrate

*anything*that my seniors are celebrating about their life outside high school. For some of those kids, that doesn't mean a 4 year school. I've had kids post military commitments, jobs, community college, missionary work, and gap years. Not only is this good for my seniors- giving them a sense of pride and ownership- but it's also a place where my underclassmen can get inspired by the things their older counterparts are accomplishing. Most fun for me, it brings kids rushing into my room first thing in the morning- sometimes long before their class- to share happy news with me or confide in me if they're disappointed. They ask their friends to "wait up" after class so they can stay and share something with me. It's such a simple thing, but it's been a powerful tool in helping me connect to my kiddos at a scary and exciting time in their lives.

If you teach seniors, I really recommend trying this. I don't have pictures of last year's wall since my room is being used for summer school, but I'll take some and post them in the future!

## Wednesday, August 8, 2018

### #MTBoSBlaugust Day 7: Using Google Drawings to Encourage Student Independence with Technology

One of the things that was most daunting about beginning to integrate technology into my every day instruction was the thought of having to do any instruction in the actual use of the tech. As much as we think our kids are digital natives, there's a cruel reality we must face. For many our my kiddos, this was the extent of what they could do with technology:

If it wasn't social media or a video game, my kids weren't buying it. And trying to explain the things that need to be clicked in the correct order to a class of 30 kids who may or may not be listening was daunting. I can vividly remember saying the same thing over...and over...and over...and over the first few times I tried a new tech tool. And while some of the kids caught on quickly, I found it hard to balance the kids who were moving ahead of me and the kids who were lagging behind.

Looking for a tool to create more student independence, I started playing with Google Drawings and it's been my go to tool ever since! If you haven't tried it before, start in your Google Drive:

See how you just did that....on your own?

That's because you just used a Google Drawing I made to show you

*exactly*where to look!
It's very easy to insert images (often screen captures of the tool you're using) and then edit then by adding text, shapes, arrows, callouts, and more. You can also add multiple pictures to one drawing, add charts and diagrams, and add word art.

You can download the image as many different things, but I usually use JPEG since it's easily inserted into a Word document or uploaded to my website.

Here are some examples some I've made for my kids in the past:

These can be easily copied, pasted, uploaded, and shared! I put them into student worksheets so kids can work at their own pace and I'm free to move about the cabin and help as needed! It's been a life saver for me (and my own sanity) when teaching with tech!

## Tuesday, August 7, 2018

### #MTBoSBlaugust Day 6: Continuity, Limits, & L'Hopitals, Oh My!

PLT PICs at NASA |

*out of this world*(get it, get it???) PD experience this summer- going to Houston to attend the Advanced Placement Annual Conference. On top of that, I got to attend with my Calculus PLT PIC (yup....professional learning team partner in crime- because there are never enough acronyms in a school setting). She just finished her first year of teaching AP and I finally have enough under my belt to apply to be a reader, so we were coming from slightly different perspectives. Both of us knew one thing for sure, though....we

*could not*stand to hear another word about L'Hopital's Rule by the time we left.

Anti-L'H's sentiment on conference notes |

The big issue was this- there's been a lot of debate online all year about the way L'Hopital's rule is justified. Understandably, the College Board has asserted that it is mathematically incorrect to say that something equals 0/0 since 0/0 is indeterminate. This caused some unrest in the Calculus community, since many teacher have allowed students to write this for years. Easy enough adjustment for my PLT to make...make sure students evaluate each limit separately. But we got a new curve ball this year that brought some amazing scoring statistics with it. Question #5 from the AB exam was a relatively straight forward question- average rate of change, evaluating derivative, candidate test for absolute extrema, L'Hopital's Rule.

So why oh why did only 0.013% of students (under 30 worldwide) get full credit on this question?

The answer came from the scoring guidelines, which allocated 3 points to part d instead of a more typical 2 points:

If a student didn't state that g was continuous, they missed a point. Any while logically we know that this must be true, I think very few of us as Calculus teachers would have expected our students to state this explicitly. This was discussed ad nauseam in many sessions and left me thinking how I could better prepare my students for this type of more rigorous justification. I knew it needed to start during my limits unit and then continue throughout the year. Most importantly, it needed to have my students analyzing

__why__a limit can be evaluated and when it does not exist vs cannot be evaluated because we are missing necessary information.
Here's the activity I designed to start my students thinking about this during the first unit.

I've attempted to use multiple representations, lots of notation to build fluency, and to scaffold up to 2 questions more like the part d on the AP. I also am trying to help kids distinguish between when we need more information, when we have enough information, and how to justify all of that. I will be emphasizing that it doesn't just saw "evaluate," but it also says justify! Please feel free to send feedback! This is still rough and a work in progress!

## Monday, August 6, 2018

### #MTBoSBlaugust Day 5: Be Prepared! (For Parent Teacher Conferences)

My husband and I went to visit Cornell this weekend, where one of our friends in finishing his PhD. While looking out at the view beneath the clock tower, we were stopped by an older couple admiring our dog.

Them: "Your dog is beautiful"

Me: Thinks "Duhhhhh. Most beautiful animal in the world. Glad everyone else sees it too."

Them: (Petting her) You're a sweet old girl!

Me: Thinks "Wait did you just call my dog old?" Fiery rage starts to burn at the proposal that my dog is anything other than a young whipper snapper who will live forever.

Anyone who knows me could guess this would be my reaction...

To all the parents who trust me with their kids everyday: thank you. I just cried leaving a dog boarding "resort." God help me as a parent ðŸ™„— Caitlyn Gironda (@Caitlyn_Gironda) August 30, 2016

I can't help but equate this scenario to the way some parents must feel when they walk into a parent teacher conference. They are being told the way other people view their child- a true extension of themselves. I don't have children so I cannot even fathom what they must really be experiencing. I only know that no one calls my sweet puppy old!

No one has ever accused me of being underprepared, especially when it comes to dealing with parents. My first years as a private school teacher in "the crucible that is our very scrutinizing...parent clientele" (my former principal's word....in my actual letter of recommendation) trained me well to come prepared and leave with parents feeling over-informed. It also taught me a lot about being able to give constructive feedback without someone feeling attacked or criticized. Since those days, I've always seen parents as an ally- someone I want on my side. Some may not help their children's attitude towards my class, but they can certainly hurt it. If I get them on my team from the start, it's never bad news for the student.

This relationship is more difficult when a student struggles throughout the year, as many parents are at a loss about how to help their students. They want tangible things they can do to help their child or that their child can do to improve, as well as real feedback on specific struggles. With that in mind, I started using this form at each parent teacher conference:

The form leads with the positive: naming students' strengths.

I try to not only address my concerns, but also list some ideas of things students could do to make improvements. From there, I include extra help opportunities, missing assignments, and upcoming assignments. Here are 2 examples from previous meetings:

I then attach a multitude of items to this- a copy of the printout from my online grade book, a copy of the schedule for our Math Learning Center, and any other information that I want the parents to get directly from me. One of the things that I like most about this packet is that it frees parents to actually have a conversation with me, rather than frantically jot down notes. One of the other things that I love is that it emphasizes just how many opportunities for extra help the students have, which has effectively eliminated the "what are you doing to help my student" accusation that can rear it's ugly head sometimes.

A simple form, but one that's been really effective for me! New teachers, don't fear parent conferences! Just make sure you're prepared, informed, and speaking with love and respect about someone's baby!

## Sunday, August 5, 2018

### #MTBoSBlaugust Day 4: Graphing Calculator Bootcamp & DIY Breakout Boxes

"No Calculator"

We all know that these 2 words, in combination, can cause pure terror in some students. Our kiddos have developed a dependency on technology in many arenas of life and math class is definitely one of them. Graphing calculators are a powerful tool, but I find when most students walk through my door in September they are using them for one thing and one thing alone- arithmetic.

One of my first tasks with my students in upper level classes (Pre-Calc, Calculus, AP Calculus) is to test their fluency in using their graphing calculator for more than just arithmetic. While other tools like Desmos provide

I decided to take it to a new level this year, so I am turning it into a "Breakout" activity a la Breakout EDU or an "Escape the Room" at your local mall.

I started by building "Break Out" kits. Each one contains:

That makes 4 different types of locks and 4 different things that can be locked. Total, these cost me $42 at my local Dollar Tree, but they can reused for any course and topic and my husband can steal them for his Chemistry classroom too. I thought it was worth the investment and still way cheaper than the name brand one that Breakout EDU sells.

My plan is to adapt the "Bootcamp" to be a set of clues and problems whose answers make up the numbers in the 2 number-dependent locks. Then, the padlock keys will need to be accessed by solving other problems. It won't add a ton more time to the Bootcamp activity, but should considerably up the engagement and competitive nature. I'm still in the process of writing it, since each lock has a different code and each box needs to be personalized, but this is a start! Everything is organized in a spreadsheet & color coded with colored stickers so it can be tracked and put back easily. I'll share more when I have the final activity finished and when it's implemented, but I'm excited to try this out, especially with my non-AP Calc kids!

We all know that these 2 words, in combination, can cause pure terror in some students. Our kiddos have developed a dependency on technology in many arenas of life and math class is definitely one of them. Graphing calculators are a powerful tool, but I find when most students walk through my door in September they are using them for one thing and one thing alone- arithmetic.

One of my first tasks with my students in upper level classes (Pre-Calc, Calculus, AP Calculus) is to test their fluency in using their graphing calculator for more than just arithmetic. While other tools like Desmos provide

*much*better worlds to explore mathematics, the reality is they will need to use their handy dandy TI's when we get to the exam. I want to be able to both pre-assess what they already know and help remind them of skills on which they might be rusty. In the past, I've used this:I decided to take it to a new level this year, so I am turning it into a "Breakout" activity a la Breakout EDU or an "Escape the Room" at your local mall.

I started by building "Break Out" kits. Each one contains:

- A Toolbox
- A Bike Lock (Numerals)
- A Combo Lock
- 2 Padlocks
- A small plastic container (tupperware-ish)
- A fabric sunglass case
- A laundry bag

Here are the containers that can be locked:

And here are the locks:

That makes 4 different types of locks and 4 different things that can be locked. Total, these cost me $42 at my local Dollar Tree, but they can reused for any course and topic and my husband can steal them for his Chemistry classroom too. I thought it was worth the investment and still way cheaper than the name brand one that Breakout EDU sells.

My plan is to adapt the "Bootcamp" to be a set of clues and problems whose answers make up the numbers in the 2 number-dependent locks. Then, the padlock keys will need to be accessed by solving other problems. It won't add a ton more time to the Bootcamp activity, but should considerably up the engagement and competitive nature. I'm still in the process of writing it, since each lock has a different code and each box needs to be personalized, but this is a start! Everything is organized in a spreadsheet & color coded with colored stickers so it can be tracked and put back easily. I'll share more when I have the final activity finished and when it's implemented, but I'm excited to try this out, especially with my non-AP Calc kids!

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