Saturday, January 17, 2015

7 out of 8 Ain't Bad

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!) 

Sunday, September 7, 2014

How We Let Students Wrestle with Definitions

My colleague extraordinare over at The Mathy Murk, asked our Geometry team how we wanted to introduce the notion of midpoint. She was inspired by Dan Meyer's 3 Acts. I wasn't sure what the parameters were and was a little stuck by own notion of midpoint having to be something collinear. And thus the "notion of where is the best midpoint" was born. Please, oh, please comment on how you would make this lesson BETTER!

Students walk in and are handed a blank half sheet of paper:

Teacher: Draw two houses. (Check some of these beauties out)

 After 90 seconds: Teacher again: Put a dot on the midpoint between the houses.

Teacher: Come tape yours up on the whiteboard.
Teacher: Discuss with your table, which midpoint is the best.

 The odd thing was, that 99% of the students floated the midpoint.


Would you call this midpoint a floated midpoint or a foundational midpoint?





Only a very small handful of students put the midpoint on the ground lined up with foundation of the houses.
(One of the team came in at break, "Ack! no student (out of 30) put the point on the ground! But at least someone suggested it, thank goodness.")


Next we made a list of criteria and had a debate about the assertion:

The best midpoint is inline with ground and foundation.

Lots of pro and cons.

Phew. We talked about congruent segments, definition of between, what does middle mean, and finally, definition of midpoint. It was all a worthy discussion, AND, how could have it been better? Oh, and the best house of them all:

The Smurf house!





Thursday, June 12, 2014

Why I Listen--WCYDWT?

Why is it that Regular M and M's are 1.69 ounces per package, but Peanut Butter M and M's are 1.63 ounces per package?
(The M and M Anomaly, Planet Money, June 6, 2014)

How do All You Can Eat restaurants make money?  (How Do Restaurants Set Their Buffet Prices, Marketplace, June 9, 2014)

What does one do when his child has a rare disease and you find out that no pharmaceutical company wants to invest in research because it isn't profitable? (ie, virility drugs sell more than FIVE Billion dollars annually) "For Sufferers of Rare Diseases, Options are Rare Too, Marketplace, June 9, 2014"

Why will it take at least 10 years for the number of US women CEOs to be on parity with the number of male CEOs when women make up nearly 60% of the work force? (Women make up about 3-4% of CEOs in the US currently) (Will Women CEOs Still Standout in 2024? Marketplace, May 21, 2014)

I have been collecting these juicy morsels of audio files for months and days, knowing how they inspire me, and wondering how I can use them in my classroom to inspire my students. I was listening to this line from Marketplace when my students were just finishing exponential growth and parent graphs:

“[This] kind of social change isn’t a line. It's a curve. It's slow to begin with, like the adoption of a new technology, and then it ratchets up. And it has all these spillover effects. Talented women mentor other women. They mentor other women. The curve gets very steep very quickly.”

I was so excited because I knew my students could visualize and draw out reasonable graphs to describe what this professor from Harvard was saying.

These stories light me up. They make me curious. I know somewhere in these stories there are opportunities for low entry, high ceiling questions that can lead to meaningful mathematics:

Compare and Contrast
Academic Vocabulary
Writing
Crafting Meaningful Arguments
Modeling Mathematics

WCYDWT?

Monday, May 19, 2014

Where Have All the Playing Cards Gone?

      As far back as I can remember, my family has played cards. Random memories include:
  • Playing Black Jack with my dad at age 4 or 5 in my grandmother's apartment in San Francisco (1967 or 1968)
  • Mother's bridge parties (1969 to 1975)
  • Playing Casino with my Nana while Merv Griffin hummed in the background
  • Winking, biting my lip, and raising my eyebrows to signal my Mus partner that I had something good
  • Spite and Malice with my sister anytime, anywhere. 
  • Flying and being given a deck of cards
  • Again at 4 or 5 crawling over to my Aunt Polly's sleeping bag to see if she was awake enough to play Cribbage at the first sign of summer light
We played other games too, Parcheesi, Chinese Checkers, Dominoes. My dad taught me Pinocle and Poker too. Don't you think this is where I began my love affair with math? And my parents were always in disbelief...I am not.

During our Probability unit in Algebra 2 I am amazed at how few students have experience with a standard deck of playing cards. They don't know about the four suits, the number of cards in the deck, how many of each suit there are, or even how to multiply by 13's. It seems such a part of my cultural literacy.

How have you translated "a standard deck of playing cards" with today's students? Is there some hip video game I am clueless about? Is there some new board game that is all the rage?

Sunday, May 11, 2014

James Altucher's Podcast Hit Home

James Altucher is a quirky business guru, podcaster, diy learner. If he were a math teacher, he definitely would be in on the MTBoS. He wants you to be better. (Yeah, he wants you to buy his books, but he also wants you to learn from his mistakes and he wants your path to be easier because he learned a lot of it the hard way.)

Why am I bringing up a business dude? BECAUSE, he recently had Austin Kleon on his show. Their conversation was powerful and resonates strongly with the MTBoS philosophy. Some of what they discussed was about how to have creative success. You have to put your work out there. Even if it isn't perfect, those you trust will help you make it better. (So MTBoS!) He talks about how most of us have been influenced by others. Cool. Talk about who inspires you, what articles, blog posts, activities are piquing your interest and making you feel empowered. Acknowledge your heroes and whose floating your boat at the moment and how they have inspired your work.

We have all known the teacher who thinks they are the expert. They must be the keepers of the knowledge and want their students and colleagues to hang on their strategically parceled wisdom. What James points out is that this kind of teacher/speaker/boss/worker/human misses out on a valuable part of the equation: LEARNING. Wow! My self-esteem regarding my quirky, risk-taking, growth mind-set teaching style got a little less feeble, I think I grew an inch. What a relief to know that even though I have been teaching a long, long time, it is not only okay, but actually enlightened to acknowledge that I am NOT the solitary expert on ALL of Geometry!

Perfect timing to meet the challenge of the last two teaching weeks of the year.

I hope you all have a wonderful year end and an invigorating summer.


Friday, April 4, 2014

Check out this Amazing Blog

It has been about 100,000 years since I have posted. Call it survival. Call it a CCSS learning curve.

I did want to mention my amazing colleague's blog, Mathy Murk, especially if you care about mathematical modeling, alternative assessment, and constructing viable arguments. Jessica Murk is a constant inspiration to me.



Mrs. Murk always says to her students, "I want you to be critical consumers of information." And if you are one of Murk's Geometry students, you are getting plenty of practice!

Go check it out!

Tuesday, November 12, 2013

A Post in Honor of the MTBoS and A Case For Dilberate Pseudo-Text

This post doesn't really fit into any of the MTBoS explorations this go round, maybe what makes my classroom uniquely mine round 2.

First off, I made the best smoothie of my green smoothie life, cures whatever ails you and gives you a mid morning nutrition blast that has become quite the habit. Pack it in a mason jar for that needed pick me up: Into a strong blender goes: 3 pitted dates, 1 pear, 1.5 carrots, rib of celery,  1/2 inch cube of ginger, two big handfuls of spinach, and .75 cup of water. Start blending on low, turn off, pack with a spoon, then blend on high until smooth. Heaven!

I have been working to pull my Advanced Algebra students out of the Algebra/Geometry For All model and into the "let's get ready for college thinking" mode. Instead of "here are three ways to know and graph quadratic functions", I let the graphs drive the equations, forcing the students to  come up with silly scenarios to write model equations. They were a lot more fun to grade and a lot of pride went into them. Honestly, I am not certain yet that deeper learning happened, as the kids stressed about getting them done. I did do one CCSS tweak to the whole thing. The first round was so awful and so lacked understanding, that I made comments on sticky notes, and had the students revise them. I realize I could have had the students do this step, and I needed to get the project going in the right direction before the students would be able to comment.

I asked the students to come up with three equations to model quadratic functions, one that utilized vertex form, one that utilized factored form and one that required students to use a system of linear equations to find a, b, and c. A lot of good discussion came up around where to place the axes and what does it mean when a isn't negative, even when we know our graph should be upside down. The graphics, of course, were the best part. I thinking mapping the path of Skittles into the mouth of the a person across the room, frogs lapping up flies mid jump, and swan divers make a case for a little psuedo-text license. Just saying.