I had a little cry today. I knew I needed to go home all day, these were appreciation tears. Thanks to Sarah over at Math = Love, for the great idea. This submission is a keeper! 

ABC Advice to a new class for fall: College Readiness (Math for Seniors not on the STEM track, but with an interest in keeping up their math chops--Math for Humanities Majors???)

Don't want to bore anyone to tears, AND, if you had some trials trying to differentiate some of the SMP's, here is how I see it:

SMP 7-Look for and make use of structure vs. SMP 8-Look for and express regularity in repeated reasoning. 

With some help from a very LOGICAL colleague, I have wrapped my head around "Look for and Make Use of Structure" as being about taking a big chunk and breaking it down into pieces. (This is one of my favorite SMPs--what a geek!) With Pre-Calc we are working with Polar Graphing. Looking at each piece of a polar equation and determining where to begin, what part drives the petals, the loop, and the dimple has been a lot more enjoyable and pragmatic.
We started with this hand-out: file here
after a some explores with this Desmos Activity Builder: Polar Equation Investigation
Notice and Wonder. Can you make sense of the structure of the equation? Can you break it down by family? Are you curious what wonderment it will draw and why? If you want to alter it, can you? How much do you need to see before you can fill in the rest by yourself?
In an earlier post, I commented on how I want my students to view equations as stories. They (the equations) have families and quirks, and regularities, and some stuff that isn't so regular. It is kind of like taking apart a machine and putting it back together so that you understand how it works. 

What story do these polar equations tell?
Objective: Demonstrate Mathematical Curiosity
What: Geometry Projects
Help Wanted: 25% of Final or Extra Credit?

Extra Credit means less to grade and higher quality work. 25% of Final means students have more control over their semester final and broader view for me for summative assessment. 

Objective: Demonstrate mathematical curiosity

To Do: You will choose one of 5 projects to complete

Here is the beautiful file for Rolling Cups that my TA made. The rest of the files will get added, as I make them! Download all files here

I worry about being in control of all the technology in my classroom. I worry about asking students to all take out a device, because there will be a few who don't have one or don't have data to jump online. So what do you do to preserve the teachable moment when you haven't booked the the Lab or Chromebooks weeks in advanced? I take control (unfortunately) of the technology. Am I not helping when I assume that at least I am capturing the moment? My husband just opined from the other room that if I use a formative assessment to measure if they have learned what I intended then it is not a problem.

I thought I was being clever with this hands-on investigation of the Intersecting Chord Theorem.

It was a fail. There wasn't enough curiousity, the measurements were so all over the place that no quantitative regularity could be found to make a conjecture.

I had the students but a giant X through the front and and turn the paper over.

I asked them to make a grid with six columns and 5 rows.

Then I turned on the projector and trotted over to Geogebra.

I asked the students to make a column for BF, FD, CF, FE. Ss "Ms. Z there are 6 columns." Me: "I know, try your best to ignore them for the moment."

Nothing much interesting happened until I used the arrow tool to pull one of the points around. The students liked that. So I pulled Pt. E a bit away from Pt. D and we again wrote down the values. We did it twice more. And then I gave the students the remaining 2 columns: BFxFD and CF x FE. I then asked the students to notice and wonder. Now cool stuff was happening. I asked the STUDENTS how we could summarize this cool property. What could we call everything to make the property clear.

AND here is the kicker! I was too tired to make up my own example so I opened the textbook (can you believe it, I haven't opened that thing for months!) and just at that moment, Kelly, one of my most enthusiastic Geometry students,  asked, "Really Ms. Z, I don't mean to be rude, but WHY are we learning this?" And there in front of my eyes the answer:

So I drew the shard, but before we got into the mathematics, we had such a fabulous conversation about the archaeology of the shard.

  • Where was it found? 
  • What else was it found with?
  • Was it made of material from the area?
  • Was it a gift?
  • What kind of plate was it?
  • Did it have any markings on it?
I went back to the question, what kind of plate was it? I asked if there was more than one kind of plate. This usually quiet male student pipes up, "we have salad plates." (That comment made me happy!) We talked about the significance of the size of the plate, was it a platter? An offering plate? 
About this point the students knew what I was on about and were begging me to draw in the chord. They had to fuss a little to remember that if the diameter it perpendicular to the chord, it bisects the chord. Was it possible to find the perpendicular bisector of the chord, sure! And every thing fit together nicely. Thanks Textbook. 

We had just learned the construction for finding the center of a circle too. So I had to ask was there another way...does it matter the size of the shard?

So back to the question, did I own too much of the technology? And of course next year, I will schedule the computer lab way in advanced!

Also, my imagination went on such overdrive with the shard, that I am going to incorporate archaeology into one of the circle projects the students get to choose. When I get those more together, I will post them. What are your favorite Circle Projects?

It's easy to post when your articulate, gracious, loving daughter does it for you! So surprised and honored! Get the tissue!


Growing up with teachers for parents, I encountered (all too often) students who assumed teachers evaporated into thin air as soon as the classroom door shut. Only to be bewildered and horrified when they ran into us at the grocery store, proclaiming, "Ms. Zimmer, you eat Annie's microwavable pizza too?!?" While every week should be teacher appreciation week, I am glad this week is dedicated to honoring the work of our educators both in the classroom and behind the scenes.

I was fortunate enough to not realize office jobs exist until I had one sophomore year of college. I was under the impression all professionals got summers off and could be home every night to cook family dinner and help with homework. While this description makes teaching seem like a walk in the park, it is anything but easy. I am humbled every day by the patience and empathy teachers exhibit accommodating student's weaknesses and cultivating their strengths. (Patience applies to working with parents too!)

If your parent is a teacher like mine, or you have a teacher who has significantly impacted your academic experience, please take this week to APPRECIATE them and let them know that (while compensation for teachers in most countries is grossly incomparable to the amount of work they do and the positive impact they have on the community) we are grateful for their devotion to our development and contribution to our accomplishments.
 — with Amy Ellen Zimmer andJeffrey Diamond.
Look for and Make Use of Structure. Play. Investigate.
We started a polar unit with this lab I made with Desmos Activity Builder.

Polar Graphing Investigation

Here are some of my student's pearls of noticing:

Just wanted to share something that pushed me outside of my comfort zone and that the kids liked too.

Here is their learnings and questions

So many things to talk about now! And because the STUDENTS want to know.

Here is a follow up I am thinking of doing.

Comparing Polar Graphs

Pretty Trippy MAN

PS The students loved the sliders! Thanks, Desmos. Now would you please get me into the kids dialogue boxes in real time?

I have been gathering evidence for my fantasy "Don't go it alone," series. We all work so hard, it is crazy that we feel so compelled to do so much ON OUR OWN. I subbed for a colleague teaching Algebra 1. She is so dang creative! Her class was working on graphing quadratics in factored form.
Unfortunately, working solo, she missed an entire page of graphing written as quadratic  = 0 (instead of y = or f(x) =)

I have done this too many times myself. Would this teacher have caught her mistake if I did not happen to cover for her? Sometimes we find 'em, sometimes we don't. With 15 sections of Algebra taught at our HS, the going it alone is too common place. I believe this because I am guilty of going it alone and I know the benefits of collaboration.

Mr. H. My BTSA Participating 1st year teacher is a systematic guy. He cares so much that the students get what they need step by step. He is careful not to overwhelm them with too much information.
Here is his pool problem for our unit on solids:

My Brain exploded with extensions: Will there be 1/2 the volume if the diameter is cut in half? Are there other dimensions that use the same amount of water? What is the largest pool you can have that amount of water? 

Yeah, I know that might not have been the intention in this lesson...but I loved the premise and wanted Mr. H. to give the students something to juicy to discuss and share. 

I had all these cool ideas for exploring circles in Geometry. Circles are a great bridge topic between Geometry and Algebra.
I have lots of good ideas, I am just not the most organized person in the world. Lucky for me, I have colleagues that are. Here was my lesson:
Me: Multiply  (x+1)^2
                      (x+12)^2      Stop when you find the pattern and just write the product.

Me: What did you notice?
       How would it change if all the sums were differences?
Me: Can we work backwards? Can we find a magic number that makes a perfect square trinomial, etc...
Me: How are these equations related to the circles we have been graphing?
Me: Can you find the center and radius of this circle in standard form? x^2 + 8x +y^2 = 9?

My students were happy enough and there were "oh, I get its." We got into fractions, and even radii whose lengths were square roots.

My lovely colleague asked me how I was approaching complete the square for circles. And I shared the above. Here is how she interpreted my lesson:

To see her entire development get the file here: file

I just love how she saw the whole thing. How she was able to put the lesson into a form I could not.

I have many things I want to do with my students that I am not sure how to structure, so I shy away. The more I share with colleagues, the more we create rich and inviting experiences for the students.