How do you make the surface area of cylinders captivating? "Do more of what you love," rings in my ears. (thanks Christopher ) I had to stop looking for approval and go with what is important to me. I had just been through two social justice minded nights of Seder, so I had a lot on my mind. As I was looking around my classroom, I noticed how many students had cylindrical shaped vessels with them. I gathered them all up and put them on display. I had an agenda that was both mathematical and much broader. My underlying goal was to expand student thinking, get some critical evaluation going, and apply a simple notion to having a voice (and maybe sneak in some talk about branding, economics, and consumerism)
The first day I lead a conversation, it looked like this:

The conversation was loud, and active, and thoughtful. The students came up with cylinder-shaped right away. I loved getting thick into their vocabulary, "girth, coney, purpose, mass, volume, recyclable, durable." I got to give them a new word, "ergonomics," and we talked about product placement in movies and tv, why some were "coney" and some straight up and down (right), and I even got to sneak in some SJ with branding which led to tribes, acceptable tribes and not acceptable tribes. (reusable, that is my tribe, tree hugging dirt worshipers, Starbucks, that is my tribe, etc...(we are in the San Francisco Bay Area, so our sports are very tribal) 
I realized that I could include more students if the next day, (we are on a block schedule so I hadn't seen 2 of my 4 sets of Geometry kids yet) I started smaller. I was thinking YOU, WE, ME. 

The next day, I gathered up all the cylindrical vessels, and handed groups of 4 some large paper. I had them do a brain dump.  When the students figured out what we doing, MORE bottles came out of backpacks and book bags, including a glittery one! Their's looked like this:

I had them then get up, and move around to read what others had written. Back at their seats I gave them a few more minutes to be inspired. to write down anything they wanted to remember or add. 

This time, when I called them together, I asked each person to share something that their team wrote, something they heard, or something they saw. More voices, and a much more delighted me. It now felt more okay for me to ask the following questions:
1. What is the net of a right cylinder? (I love the surprise when students recognize the length of the rectangle part of the next is actually the circumference of the circle)
2. What is its surface area--I found a student's bottle, asked for the measurements of the diameter and height. 
3. What is the volume?
4. and most importantly, can you find another set of dimensions for the cylinder that has the same volume? 

We will see what I get on Tuesday and Wednesday. I am curious to know if the different set of classes will have more enthusiasm, more of their assignments completed, and/or more investment in the task. 
Left this activity for my Geometry students with a sub the other day: Please be patient, there is a WOW at the very end, really.

Find the area of the triangle in two different ways. 
Screen Shot 2017-03-05 at 7.47.51 PM.png
Too many students wrote 1/2(6x5) = 15 sq units

Upon my return, and I inquired how did you know that AB was 6 units and and BC was 5 units. 

"I counted them." Oh, really? I mused. How can you be sure. Saddy faces, big bottom lips. "Yeah, I guess I needed to be more accurate. I asked what would you get if you counted by square units instead of linear units? Heads went down, tongues out instead of big bottom lips and pencils scratched across paper. "12!" shouts an otherwise unengaged kid. (yay for counting!)

Which is more accurate I ask, and why? A fun banter moves along, and finally the the gal who is ALWAYS on it, but doesn't want to stick out as a know-it-all, can't take it any longer: This is what I did: Pythagorean Theorem and some subtraction:
Boy, did kids like that. We all were still in the first way because the students who tried 1/2bxh quickly abandoned it when h was really hard to figure out. Now I know you must be thinking, really, this is all you got?

Brandon, my love bug, decides that the triangle is a right triangle and he likes the Pythag stuff, so for his second attempt he writes: A =1/2(sqrt 29)(sqrt 20) = 12.04. Nice I say. How do you know the triangle is right triangle? "Because it looks..." Whoa Stop! Paaaleeesseee. How can we make sure? So we check out the slopes, Ah...not perpendicular. But close right? It is a good approximation, and we High Five.

Meanwhile, across the room the same notion is being utilized, but with this sweet, sweet, amazing and fabulous student work:

He deserves extra large, right? I could weep. And we get the opportunity for a conversation about rounding, how big of a purchase do you need to make when ordering materials to off by .15 of unit per unit, a case for simple radical form (even though it is about slope). What a day!

Something to make you smile:
After doing a lot of searching for how to make square roots meaningful and I decided to give good ol' Theodorus another whirl.

Day 1: Ladies and Gentleman, this is the bar: (As I am cutting and pasting and resizing, I realize this student's name is Estrella!)

(yeah ignore that her right angles aren't right angles, she was inspired and did all that math!)

The next day, I get Jonathan's and I am sobbing. (he told me he thinks he did it incorrectly, and he is correct in his thinking, which is very cool, so he fixed it with pencil underneath) I am so lucky. Kids are just amazing!

Love the lime and cilantro.

What other questions would you ask? What Standards of Math Practice did I hit? Did I miss any opportunities? Please let me know!

Call me slow. Go ahead, you wouldn't be the first.

Someone out there was calling for the worst topics we have to introduce. I would put rationalizing radicals right up there.  Then I had an epiphany. (A little late to the game in the scope of teaching, not proud, but happy that it came nonetheless)

Who cares about rationalizing radicals (and in particular, square roots)? I know from teaching both Precalculus and Geometry, why I want my students to be efficient at it, and I always struggled with; why not just use decimals?

Deep in a quandary about why my students were struggling with the Pythagorean Theorem and in particular, why is was hard for them to wrap their heads around why they needed to take the square root to find the length of the missing side, I realized I wanted my students to use simple radical form. I spent the first few days allowing the students to leave the radical unsimplified. Then I upped the anti and asked that they put all radicals in simplified radical form. (I tried starting with warm-ups of simplifying radicals, and found they were getting too hung up there when also trying to figure out the PT--yeah, some of the "getters" shake their heads and look with great sympathy (actually) at the kids who aren't there yet).

It is difficult to see the value of sqrt 192 When you rewrite it as 8 sqrt 3, and we know sqrt 3 is between 1 and 2, it gets easier to see its value when write an inequality and then build the value.

It is a start for them to place the value, and recall that the hypotenuse is the longest side.

Now we are preparing in Geometry for Special Right Triangles and Right Triangle Trig and it is time for the students to either learn or recall how to simplify radicals in the denominator. Again, why? Why can't the sqrt 2 stay in the freaking denominator.

Easy comparisons, that is why! What is
the value of

Is it more than one? Less than one?   When you start here
And move to here                      
The students give you a little, "oh." The strugglers get a number sense boost and the "Can we please just move ons" get to appreciate a little deeper why we torture need to rationalize a denominator. 

Not a big move, but forward I say!

Are your students struggling or acting needy these days? It sort has been the "water cooler" talk among the teachers. Especially in Science and Math. We are a fairly progressive school and most of the teachers are dedicated to not wasting student's time and giving them tasks that are relevant, and yet, we notice quite a few students falling apart and just asking us to help them every.step.of.the.way. Is this right? I am so confused? I don't know what to do. Students are really struggling with separating similar right triangles from one right triangle. Using Geogebra, cutting them out, using transformations all help, and yet students are reluctant to commit their findings to paper.

I can wrap my head around rationalizing radicals as developmentally a mathematically mature concept to wrap one's head around, but the Pythagorean Theorem?  What do you think for high school Sophomores and Juniors?    

In these crazy times, I wanted to share a positive lesson that actually had the impact I was looking for! Imagine!
I wrote last year about how I was on a proportional reasoning kick. The lesson didn't get the "wow," I was looking for, so I switched it up this year.
Both started with an old project I used one year teaching 7th grade Humanities in an International School in Ghana:
The 100people site translates the world population into a village of 100 people. I asked the students to to do 3 things: Translate our school population to a country of their choice, present their findings and create a poster with at least 4 info graphics. The students had to determine how many people each student at WHS represented in their country and had to translate 10 aspects of their lives to those of WHS students. We used the above website, that was updated for 2017(!), the CIA World Factbook and the Worldbank Data page to translate a country of their choice to the population of Windsor High School--our suburban high school of 1712 students. It was the start of the new semester and a way to front load proportional reasoning in Geometry.

The presentations were so-so. A few got how to move from "proficient" to "exemplar," (Rubric)with some examples that I provided from my chosen country, Cuba. (Imagine if you were a Freshman an d you weren't allowed to use the bathroom at school. That is how many people in Cuba have no access to indoor plumbing.) I haven't graded the posters yet, but they look amazing, pictures to come. What has got my attention at the moment is the reflection papers. I have been making phone calls home to express to parents my appreciation for what their students have taken away from this project.

Angela on Brazil," We are extremely fortunate for all we have here in the United States and this project opened my eyes to that."
Brandon on his project about Switzerland," These types of projects make proportions fun to learn...This project should continue to be done every year because it will help people understand proportions a bit better."
Colton on his project on Italy, "The main thing I have taken from this project is how lucky I am to have grown up in Windsor with the luxuries I've had. So many times people take electricity, sanitation and drinking water for granted."
Tatiana on Greece, "I found this project to be a cool, eye-opening project and I definitely enjoyed doing it."
(Please note, the first names here sound very Anglo, however, the majority of them have Hispanic surnames.)

The students are now going around doing their presentations in their science classes that are studying food, food access, and contemplating how we will feed our 7.5 billion inhabitants.

Here is the project hand-out and Rubric:If Windsor High School Were...

Please let me know how it goes for you if you try it, or if you have any suggestions. Check back for pictures!

There are so many great posts! So many great bloggers.

#MTBoSBlogsplosion #2 asked us to blog about soft skills. This is my arena! I blogged about my High 5 version here: High - 5 Amy Style. My students wrote on their finals that what they liked best about the class is my energy and stories. (My latest Exit High 5 was "What are you happy about leaving behind in 2016" Kids are really remarkable, one dude said, "my girlfriend!"

So to play it forward I wanted to share two bloggers who are not only amazing math teachers, but who also inspire me to be a reflective educator by their deep and honest blogging.

miss calculate--Elissa
Elissa explores deeply and is committed (okay, obsessed) with improving the art of teaching.

An Old Math Dog Learning New Tricks--Lisa Henry
Any person who starts a blog by saying "I have taught for 20 years and still have a lot to learn," will not only have me as a reader, but also has my "mama bear" support.

Who are your favorite soft skill math bloggers?