Ideas floating around my head these days:

Estimation:

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?

Statistics:

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

Parenting:

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?

# Ms. Z Teaches in Mathland

A High School Math Teacher/Poet/Crafter/Athlete/Mom

## Sunday, July 19, 2015

## Tuesday, May 19, 2015

### Attending to Precision, Excel, Number Sense, Sticky Problems, and Rolling Thoughts

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!

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:

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!

## 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)

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):

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.

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:

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!

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

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.")

*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?

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?

(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:

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?

- 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

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.

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.

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