I was so inspired by Ilana Horn's (@tchmathculture) guest post by her own teenager, Faking Excellence: The Art of Milking Mediocrity For All it's Worth, that I have to tell you how.

The writing is my favorite kind of prose: creative non-fiction, self-reflective, witty, smart. Having a Senior in high school, we, (my daughter and I) can totally relate. As I read, my heart went thunk, as I fall into every teacher trap she laid, "Greet your teacher upon entrance, have book and supplies with you at all times..." My daughter carries a certain book with her at ALL times, how long could it possibly take to read one novel? Now I know it is just for show, as is her overflowing pencil case. I must admit, I do love the oversupplied student...

"Live life like an overworked student..." Oh so my daughter...though she ALWAYS has time to respond to friends, go to movie nights, and fit in a fantasy shopping spree online. "Do you want me to be sad and lonely? Aren't you the one who told me to spend more time with people?" Though I must wonder if this makes me a flakey teacher because I feel for students and let them slip things to me late (with explanation.) I so don't want to the one who gives the students busy work. I try to pick and choose assignments wisely, does this make for lazy students?

To linger over the greatness of the writing and try to make peace with my gullibility, I gave my students the opportunity to read the blog as extra credit. I wrote a Google Form,

And feel so excited to make more response sheets like this one to check for not only content understanding and mastery, but to hear my student's voices. We are not a 1-1 school so I am not sure I can REQUIRE a response.

How do you use Google Forms and response sheets to assess students and inform instruction?

It is important for Newbies, Oldbies, and Inbetweenbies to look for resilience in ourselves. How do we handle a dismal result from a well-planned lesson?

Blame and negative talk to ourselves are not useful or productive. So acknowledge it didn't go so well, and tweet out to your PLC, share with a trusted colleague, sleep on it, check the MTBoS Search Engine, take a run or a bath, or both!

I wanted my students to discover which triangle congruence shortcuts worked and which didn't. Last year I had good success with lesson inspired by the original CPM Math 2 curriculum, where the students cut out various triangles, we put them up on the board and decided which were the same and which were not. Problem was, not enough accuracy with protractors and notion that vertices had to be the endpoints of the triangles, so sometimes non-congruent triangles ended up in the SSS category for example.

So this year I started with a lesson inspired by one I found on James Rahn's website: Discovering Ways Triangles are Congruent. I wanted to use Patty Paper to try to get a grip on accuracy. The idea was to create triangles from all the permutations of six parts of a triangle, each of three vertices and each of three lengths. Problem was there were too many triangles (9). The students didn't like making a triangle out of one single angle, or just an angle and a side.

Most students used the arrows at the end of the rays as endpoints and closed the triangle, so many of the triangles looked the same and plus, each student had about a zillion pieces of patty paper to try to keep organized and categorized. There wasn't enough time to debrief and I wasn't prepared to store the bazillions of pieces of patty paper for the students.

After reaching out to MTBoS community and reading a January 2015 post by Kate Nowak on the same topic, I re-organized thanks to Kate's helpful recording sheet and tried again the next day. I focused on what I wanted the students to specifically see, and decided to concentrate on the triangles given three pieces of information.

This time, instead of every student doing every iteration of  a triangle with the six pieces from James' lesson,  I gave the groups of 4 and 5 students the same 6 pieces of information as they saw previously, and using Kate's recording sheet, specifically asked the students to use the patty paper to create one each per group of very specific triangles in specific orders. For example, create a triangle using side AC, then, CB, then, side BA. ENDPOINTS MUST BE VERTICES AND ANGLE VERTICES MUST LINE UP WITH SEGMENT ENDPOINTS!

I told the students they could divide the work up anyway they wanted...each take a triangle to create, work all as a team, as long as when they were finished they had 5 triangles, on 5 separate pieces of patty paper, labelled for the corresponding letter with the directions for the triangle. You have 20 minutes, GO!

I put 5 pieces of colored paper on the board and had a member tape up their triangles with the ones that were built the same way.
My husband teaches Multi-Media studies at the Santa Rosa Junior College. He started as a K-8 music teacher, including band, chorus, classroom music, and drama. He did musicals with all the grades.

What does this have to do with Math, and in particular, graphing? Well my husband can tie his careers together too, because in the end, he is really a storyteller. This is his mantra. Does your photo tell a story? Does your website tell a story? Does your logo, animation, digital portfolio, tell a story? If not, go back and make it tell one.

So when I was about to teach Polynomial Functions, it dawned on me, equations are little stories. They tell us about their shape, their domain, their range, their limits, their relative extrema, their x-intercepts, their y-intercepts, etc... They also tell us what family they belong to and how they are different from their parents. Each is unique and this became really clear when the students got to building a polynomial function.

For this lesson I wanted the students to tell the story of the equation through its graph.
The students had to understand:
→What role the degree played in the graph’s end behavior.
→What role the multiplicity of the zeros played.
→What was the effect of the leading coefficient have on the graph.

First we used Amy’s Polynomial card sort (thanks Amy) to make sense of these equations. I loved Amy’s script and pretty much followed it.

After we got used to the shape of the graphs, we explored methods of finding the roots. We used all the typical methods: factoring, rational root test, quadratic formula, long division and synthetic division. Along the way, we found out that some of the roots were imaginary, that they came in these conjugate pairs (we giggled, bc someone said, “oh like “Orange is the New Black.” Now I can’t think of these pairs of numbers without immediately translating it into “conjugal.”)

It was fun to watch them discuss and grapple with zeros vs. intercepts, (more on that later) and found out that their factoring skills stink. (How do we involve CCSS with the mundane task of factoring, I want my Pre-Calc kids to be able to have TOOLS to factor nearly anything.) Then they got this gift:

Find the equation of a third degree polynomial the following roots such that f(1) = -60.

2,  3 + 4i

Most students got this far: f(x)=x^3-8x^2+37x-50 but couldn’t figure out what to do with the
f(1) = -60.
So we talked about what story does f(x)=x^2-x-6 have? How many different graphs have zeros at 3 and -2? What does f(1) = when a is 2, 3, -5, 6, -1, 1, etc…
The students understood from this exploration that there are infinitely many equations with roots 2 and 3 + 4i. Next time I will use the slider function in Desmos to help the students find “a” before we do it by hand. (Why I didn’t think of it this year is beyond me.)

I wish I had done this first: (I am a tad intimidated by sliders)

In the end, the students did get that the leading coefficient makes f(x) have a unique story.

It was then way more fun to move on to rational expressions. The students were now curious to see what an analysis of f(x)=N(x)/D(x) would produce.
Dear MTBoS,

I stole and embellished, and added (hopefully). And I caught your sense of humor.

Meg at Insert Clever Math Pun Here  so inspired this post. I could not figure out why with her post on Polynomial Functions she had a clip of John Travolta and Saturday Night Fever. What? Then the light bulb went off and I made this:

Here is link. Maybe the students should make their own? I was going to make it matching, but thought the conversation would be richer if they had to make them up. Thoughts?

Happy Blogaugust <3
Dear Math Bloggers,

This one is for you. You have given me so much, I have been racking my brain trying to figure out what worthy goodness I could share with you. Well, nearly every morning, I treat myself and family to a reminder of our two years spent at the American School of India, New Delhi. Chai tea.
Here is a a coveted recipe I hope will sweeten your day and take you on a quick retreat.

Chait tea for 2

For 2 cups of water add:

5-6 pods caradamom smashed
5-6 whole cloves
1 inch fresh ginger smashed
A 2-3 inch stick cinnamon
sprinkle of nutmeg

Boil spices in water until water is color of spices:
Then add 2 cups of milk, at least 2%
and 1 rounded tablesppon of sugar

Bring to a boil and then turn off heat and add three bag of black tea: You can use Lipton's, Lyon's, PG Tips, or any Irish Breakfast or English Breakfast (I use Trader Joe's Irish Breakfast).
Let steep 5 minutes. Pour out through a tea strainer after 5 minutes. Pour into a to-go cup or mason and jar and make your journey to work a mini vacation!
Save it for at work, for a pick up at break.
Be a tea wallah, and double the recipe and deliver it by thermos to unsuspecting teachers and watch them smile!

Thanks to John Mahlstedt @jdmahlstedt, I am motivated to blog about one of the first activities we do in Geometry, school-wide. (Not sure if article came from Danielle Buckman or Jessica Balli, both amazing teachers I am privileged to call colleagues).

We all share and read this article with our students: The Myth of I am Bad at Math, from the Atlantic Magazine. We all teach/share it our own way. 
(One teacher, whose baby boy was due in two weeks, came in the first two days of school, just because she thought the lesson was so important, she didn't want to trust it to a substitute teacher!)

Here is my approach: 

The exit ticket for the day was to agree or disagree with the statement, 
       "Math ability is mostly genetic." 
After the students wrote their reflection, I gave them time to move to an AGREE and DISAGREE side of the room and explain their thinking. 

I sent the article home with each student and gave them the following assignment:
Read the article and highlight 3 new vocabulary words (if you know them all, highlight three interesting words), determine and write the definition from the context, write down 2 big ideas in your own words, and 1 question you have. 

The next day, we discussed the vocabulary, then we broke up into groups of 4 and practiced active listening using "Collaborative Conversation Notes" I learned about at an EduImpact Conference held at our county office of education (SCOE.org) (here is an article about how to use the template). I had the students summarize one or two big ideas they could agree on in the center.  Our group conversation after was touching. All of the students moved to the DISAGREE side. Here is one group's change of heart, "We disagree because if you understand it well, you do good, but if you don't, you have to try harder and get better."

This year, I will ask the students to come up with some "myth" statements. When I have enough, I will give each group one to rewrite as a "growth mindset" statement perhaps taking something from what they learned reading the article. I wrote in Blogaugust #1 that I was inspired by Sarah Hagan and her post on a growth mindset bulletin board that she made from inspiration from @druinok http://statteacher.blogspot.com/. So this is how I will approach crafting the statements to create the bulletin board of Growth Mindset statements.

Moving from I can't to ...yet.

Student size whiteboards. Check. Now we need verticalness, and my DH is outta town. Not that I mind swinging a drill, it is just going through the shower board and not sure what to do if it shatters. (Also, who knows if I am allowed to screw things into the wall, I just did it and figure I will ask forgiveness, but wait, this is why I am so sure I have little to worry about...)
Mirror holders! Yay! I am so happy and the boards are so secure. There is a plastic bit that expands as you screw it in.
(Sorry about the ugly brown walls and the funky picture)

After a full last day before the kiddos also occupied the school, I did collected these in my backyard:
And made a peach, rhubard, raspberry, apple crisp (sorry no pic).

Happy First Days of School ya' all. More on that next post.
So two things are running around my head today.

First, is all the growth mindset stuff inspired by Jo Boaler from Stanford and Carol Dweck who wrote the book Mindset. I love it! I love the message. Productive struggle makes positive change. I was inspired by Sarah Hagan and her post on this bulletin board that she made from inspiration from @druinok http://statteacher.blogspot.com/

Gorgy, right? But then I started thinking about the students owning the space and the words. So now I am thinking, I will ask the students to write their negative thoughts about math and learning and then have them work as a group with one card at a time to change the words to growth mindset statements (we will do first one together), then I will make a bulletin board! I promise I will take pictures and share them. The Dollar Store has gold starts for, yup, $1.00. Hah.

The second thing is Christopher Danielson said at TMC 15, "Find what you love (about math) and do more of that in your classroom," (see my previous post). Well, I do love listening to NPR and I do love Freakonomics, and I just love this very, very special episode:
How to Create Suspense
See the graph paper in the background, math, right?

Everything is in there, a movie producer, sports, a crime novelist and the question, what makes something suspenseful? What is the most suspenseful sport? What could you do to make a sport that is more suspenseful? (silly economists didn't know the rules to Quidditch).

We can appreciate this episode, or we can try to bring into our math lives. #WCYDWT

What/whom do you listen to that makes your math heart go pitterpat?

Got a new Chromebook (Thanks Tech Guy and District) and I am having fun keeping track of all my Mathy Friends from TMC15 and those I have followed for a while now.

I love sayings and being inspired and inspirational. (Do more of what you love, Christopher Danielson) Here are some of my favorites old and new that decorate my room.

New: Hannah, 17, a rising Senior, shared this one with me from her Twitter Feed:
The next one I have on my wall and also on the cover of my daily binder where I can see it.

The next one needs no explanation and has a prominent place in my room:

This one is front and center:

My favorite and well worn poster is this one because I have worked most of 29 years in two high schools in the heart of Sonoma County wine country and it is the heart of the growth mindset.

And this one:

Ummm...this too:

I do have the 8 math practices and "Welcome to Mathland" so you don't think I am a complete hippie. Women, where are you? I want the 4 Claims too.

I would like to add this one but maybe I just need it personal-size:
My teenager and her friend think I should add this one:

What's on your walls?
BTW: If you like Make and Takes, or need an awesome activity for Pre-Calc/Trig, I presented this "My Favorite," (#TMC15) a foldable for finding the formula for the Sum of Angles for Sine and Cosine that SHOWS why cos (x + y) ends up being subraction!

#tmc15 Experiencing Community
     Truly I don't know where to start this post. Partly an ode FOR all the MTBoSers, partly that experience where you just process and download. I did wrtite this all on paper first, just to get the thoughts out, hoping the sensation of pen to paper would help get the flow going.
(Also, I haven't read anyone else's posts yet, because I wanted to share my authentic experience of attending Twitter Math Camp 2015. Fancy that.)

Twitter Math Camp is literaly camp for math educators to gather, be a community, share, encourage, figure out, laugh, sing, and get their Geek on.  It is sponsored by us, paid for by us, and no one is getting any money for participating. The Keynotes are speaking from their hearts, the organizers are donating time and passion. That is what this is all about. And it. is. AMAZING.

     Starting with Gratitudes is always a good strategy. To Lisa Henry, her DH, her folks, and her kids, and all the tweeps who laid the ground for this amazing community, and it is about COMMUNITY.

It was one the few times in my professional life that I felt like I was in the right place. Sure I wish I had more of Fawn's consistency, and Matt's general hipness, and Michael F's mad computer skills, and Rachel's spunk, and Julie's infectious personality, and Sam's "take no prisoners" risk taking, and Chris' synapse stickers, and a thimbleful of the brainpower, energy, and creativity in that auditorium, AND stll, I felt cherished, valued, and believed in because I was there with the communal knowledge that it was our collective endgame to leave becoming better teachers, reaching more students, and grow more learning.

A few gems from the conference for you to use:

Christopher Danielson asked us to find what we love and do more of it in our classrooms. I love curiousity, I love helping students find their capabilities, I love  listening to NPR...

Fawn Nguyen said a lot, and had many of us in tears, and what I needed to hear most was, "Share ALL the lessons." Why? Because it is not about us, it is about growing learning for students. Mind Blown. Fuck the popularity contest.  (I get permission to say that from Fawn). Sure, give credit where credit is due and do please let the author know how you used the material, what tweaks you made, and how it went. And time to bring what works out of hiding.

* Do this cup rolling activity that Lisa and Jim shared with us: http://mathforum.org/pcmi/hstp/resources/papercup/papercup.pdf

Curtesy of PCMI 
Add the extension: what formula could you come up with to generate the circle made by any cup?

* Ask Rachel's  two questions:
--What is the best question you asked a student today?
--What the best question a student asked you?

(Keep track and tweet or post them!)

* Google these for inspiration (thanks Fawn): Estimation180, Global Math, Mathtalks, Visual Patterns, WODB, MTBoS

* If you get stuck looking for the best way to make a lesson student centered look for the backwards version of the question (thanks Brendan and Sam)

* Get the students "Butts up," and use VNPS (Thanks Brian and Alex) to engage students in talking about math.

* So, so, so much more!

This is round one.

If I have left you out, please, like Sam says, don't take it personally. (I can't believe misscalcul8 was there and I didn't even know it. OMG, OMG.) I want to get this out, so you know you can do it too.

Ideas floating around my head these days:


GIVEN a shower puff that is intact, how would you estimate its length unravelled?

GIVEN a plum, who can do the best job peeling it? How would you determine "best job," in a measurable way?



A Danish economist wonders if sleep effects productivity. He has 10,000 surveys of citizens that include sleep habits. What could he do with this data to measure productivity? Click here to find out how he does it!
Freakonomics Podcast 7/16/2015 Sleep Part 2


Just having come back from Europe where EVERY woman goes to the beach in her bikini, young, old, skinny, overweight, glamorous, tall, short, you get the idea...I mentioned this to a friend that has a teenage daughter and a teenage son. She said, "It is so important for our boys especially to see all kinds of woman at the beach, otherwise they get a warped view from TV and magazines about what a woman's body is supposed to look like." Isn't she smart? (I don't have boys, and as everyone knows who works with or has girls, it is so important to remind them to view themselves naturally without some warped Hollywold image about what is attractive.)

Time Off:

I am going to survey my students this fall (ack, I mean in three weeks) about who feels they got a "real" vacation this summer for at least a week. I am going to let them define "real."

Then I am going to track their grades for the first semester and see if there is any correlation. Anyone in? What do you think will show up? Did you know that of the 10 countries with the fewest hours worked weekly, 9 have the highest gross domestice product per capita? (Organization for Economic Co-operation and Development (OECD))

I am super excited to meet some awesome peeps at Twitter Math Camp next week. I am sure I will have much to report. I just hope that I am as groovy and smart as they are.

What are you thinking about this summer?

Let's talk about what we did do shall we, not what we didn't, at least at first.

Attending to Precision and Excel:

I did get my Precalculus students in the computer lab to have them run a spreadsheet for estimated area under a curve. Their homework was to graph and find the following:
The next day we went into the lab and I gave them some basic Excel Spreadsheet how to's and let them try to organize the information and check their answers.

We came up with the following columns:

I find it super good brain food to get the students to recognize and articulate the subtleties between rectangle number, index number, and sub interval. Do you want index 0 to 7 or 1 to 8? Do you want to multiply by cell A2 or by cell A3? Do you want the spreadsheet to calculate the x-value and then the height, or do you want the argument to do it all? The feedback is immediate with an ERROR message or some weirdness like #########. The students appreciated the challenges and intricacies of attending precision and perhaps what coding is like.

One groups final product for problem number 5:

Student K went a bit overboard here, but she caught herself in the LH area and RH area cells.

Not the most exciting thing ever, but really gave me pause for attending to precision and the value of giving students an opportunity to learn a real world skill.

Here is another group's final product for number 3. I like that the area cell for Right hand Area is blank.

Speaking of Attending to Precision:

I am pretty sure I want to try this activity, using Excel in Geometry too. Again, I am just convinced that find the surface area and volume of ANY regular polygonal prism is a good place for students to look for and make use of structure and attend to precision. How can each problem be soooooo different? How can I get the students to slow down, ask themselves, what do I need, what do I have, How can the I make this as simple as possible? (It also occurred to me that the entire final could be one regular pentagonal prism and one regular pentagonal-based pyramid to find the surface area and volume of and the student knowledge of just about all of the second semester would be used (the students are speaking of isosceles triangles as two congruent triangles that are reflected from a vertex perpendicular to a base!).

Which leads me to Number Sense and Spatial Reasoning:

Since it is late and I need to go to bed, I will leave you all with this one last thought, what is the best final product you've asked for to convince you that a student can imagine, picture, draw, internalize the height of a prism? I so badly want them to have an internal vision. We have built, drawn, and broken down many a pyramid, and really, it isn't so much about the "usefulness" of knowing the volume of pyramid, as it is for me at least, that a student can reach into their toolbox, the ones I know they have, and SEE what needs to be done and make something of a problem on their own.

Comments welcome!

Standards for Mathematical Practice (SMP's) in one problem that is! I had so much fun interacting with the students on this one PreCalculus question that I tweaked my neck!

Here it is...Solve this baby over [0, 2pi)

That's it, that little thing. Such beauty and so much discussion. (SMP 1: make sense of problems and perservere in solving them)

We first went after it like this: cos^2(2x) + 2cos2xsin2x + sin^2(2x)=1

The students suspected that cos^2(2x) +sin^2(2x) was a pythagorean identity for one and would leave the equation to be solved 2cos2xsin2x=0. It was pleasant to divide out the 2, and look at the zero product property that was left: cos2x = 0 or sin2x = 0 yaddah, yaddah (SMP 6: attend to precision)...nice discussion about where is cosxsinx = 0 (SMP 8: look for and express regularity in reasoning) and which identity would be nice to use for cos2x.
Eventually they got to : 0, 45, 90, 135, 180, 225, 270, and 315. (SMP 7: look for amd make use of structure)  And the question remained: are these the same results for the original equation? (SMP 3: critique the reasoning of others) How to check without all that work?

DESMOS of course: (SMP 4: Use appropriate tools strategically)

The students noticed immediately that both the original equation and the simplified equation had the exact same x-intercepts. Wow, cool! Yay! Done!

Then Mr. Clever (The only student truly awake at 8 am) said, Zim, can't we just square root both sides? Well, yes we can! Let's do it!

cos2x + sin 2x =1 or cos2x + sin 2x = -1

They wanted to see those graphs in Desmos too. (I made them beg just a little):

They were a little sad when the graphs did not have any common intercepts. Until of course, we remembered that the equation had the word OR right there. (SMP 2: reason abstractly and quantitatively)  So much dancing, so much skipping! So fun! (so much neck tweaking!)