Pick's Theorem

In 1899, a man named  George Alexander Pick discovered a new and very different method of finding the area of a polygon.  This could be ANY polygon, as long as it can be placed on the Cartesian Plane and each of the vertices has coordinates that are integers.

His formula?  Let "i" be the number of lattice points (where the grid lines intersect on the coordinate plane) on the interior of the shape; let "b" be the number of boundary points (where the grid lines fall on the sides of the polygon), then Area = i + (b/2) -1.

For example, the green triangle has 4 interior points and 4 boundary points (vertices are included as boundary points).  The area is 4+(4/2)-1=5,  Another way to calculate area would help us verify this answer for area. We could verify this area found by Pick's Theorem by taking the 3x4 rectangle that completely encompasses the green triangle and subtract the three right triangle areas.
(12)  -  (1/2) 2x2 - (1/2)2x3 - (1/2)1x4 = 5. (Note: using our area estimation algorithm from class only gives an estimate, not necessarily an exact value, so don't use that method here.)

 Find the areas in TWO ways (Pick's thm and one other -- either rectifying or subtraction from a larger rectangle) shown in the previous paragraph of the remaining colored polygons.

More for practice and deeper understanding: 

Now use the illustration below to find the areas of regions A, Q and R. Verify that A = Q + R.  

Now, for your blog, find the areas outlined above using Pick's Theorem (yes, all of them) and verify them using techniques you have learned in class.  THEN, go to this website to see what happens that's interesting when you use Pick's Theorem to find areas of regions with "holes" in them.  Feel free to comment on the similarity between the proof of Pick's Theorem and how we found the areas of triangles placed on the coordinate plane in class. 

Remember your blog must be STAND ALONE. Do not require your reader to read Jammnpeaches to know what's going on in your blog. 

G. A. Pick died in a concentration camp in about 1943.  

Sources: 

http://www.cut-the-knot.org/ctk/Pick.shtml

https://en.wikipedia.org/wiki/Pick%27s_theorem

http://jwilson.coe.uga.edu/emat6680fa05/schultz/6690/pick/pick_main.htm

https://nrich.maths.org/1867

Proving Trig Identites

You have in your possession an amazing tool called your graphing calculator.  While the TI 84 can't do algebra for you (but there are "CAS" algebra systems on calculators....just not allowed on your standardized tests), the TI 84 can help you simplify or verify identities using the graphing application of the calculator.

Let's start with those exercises where you need to "simplify" a trig statement.

The idea: if you are looking at one of those trig statements that looks crazy busy and are not sure where to go, sometimes it's easier to work through finding the answer (that is, writing the *solution*) if you know the answer. But darned it if the teachers want you to "show your work" and not just guess, so you still need to do that algebra to the trig functions.  So how can you find the answer so you can be helped to find the solution?

Let's say you are asked to simplify: (sinx+cosx)^2-2secxcscx and you don't know where to start.
Type the expression into Y1, then graph.   You'll want to be conscious of what mode you want to consider; but you'll find that the graph of this expression is simply a horizontal line at y = 1.  That information suggests to you that the expression simplifies to 1.

Now how about proving trig identities?

To review the definition of identity: An equation that is true for ALL values of the variables; not just one or two, but for all values of "x."

Perhaps you've been trying to prove an identity and have found yourself in algebra hell for a while and you're not sure if the equation is an identity after all.  How might your calculator help determine if your teacher or the text made a mistake?

This is your question to answer for your blog.  Start with the following link:
http://mathbits.com/MathBits/TISection/Trig/trigidentity.htm

In your blog, find an equation that is NOT an identity. It could be an equation with one or more solutions or it could be an equation with no solutions.   Show how your calculator would disprove the equation as an identity. Be creative. Have fun.

Then find your own identity to use. Use the basic identities or the Pythagorean identities to help you find one.  Again, be creative. Show how you would use your calculator to strongly suggest that the statement IS an identity.   Notice my change in wording...  Can't you use your calculator's graphing (or tables) tools to prove that an equation is an identity?  You'll need to address that also.

In sum:
a. Find an equation that is not an identity.  Use your calculator to demonstrate that it's not an identity. Be creative. Take screen shots, pictures, images, etc.
b. Find an identity.  Use your calculator to show what an identity looks like on the screen. Be creative. Take screen shots, pictures, images,etc.
c. Why can you not prove an identity using the graphing or tables tools? Explain.

Fourth Dimension

In class, we began a brief conversation about the tesseract or the hypercube.  It's amazing how simple transformations can create such a powerful concept.  In this blog, you have an opportunity to think about this more deeply or see some beautiful images.

Here are a couple links for fourth dimension discussion:

High School Student Discussion. (This is an academic discussion by a bright high school student, dry in imagry, but with some great ideas.

Forget about your familiar world. (This could simply blow your mind with the simplex, It's 14 minutes that just keeps getting more and more interesting. Watch the first 8 minutes then I challenge you to just try to stop the video. At 9 minutes you have the hypercube. )

And here's someone's story of building a visual or a double-rotation of a tesseract. It has a great set of visuals.

For those of you who say, "Hey, this could be a process for finding more than the 4th dimension..."  Here's an image:

To blow your 1-dimensional mind further: here's a 2-dimensional image of a 4-dimensional shape represented in 3-dimensions casting a 2-dimensional shadow.

OR, find your own resource for thinking about a tesseract or hypercube or the 4th dimension (Please, no Dr. Who links....keep this mathematical.)

I hope this is fun for you; please keep asking yourself the "Why?" or "How?" questions so you can delve a little more deeply into your blog.  Find something fun or interesting and write about it.  Let is be recreational!

Geo/Algebra II,Due 10/20/2016: Fourth Dimension

In class, we began a brief conversation about the tesseract or the hypercube.  It's amazing how simple transformations can create such a powerful concept.  In this blog, you have an opportunity to think about this more deeply or see some beautiful images.

Here are a couple links for fourth dimension discussion:

High School Student Discussion. (This is an academic discussion by a bright high school student, dry in imagry, but with some great ideas.

Forget about your familiar world. (This could simply blow your mind with the simplex, It's 14 minutes that just keeps getting more and more interesting. Watch the first 8 minutes then I challenge you to just try to stop the video. At 9 minutes you have the hypercube. )

OR, find your own resource for thinking about a tesseract or hypercube or the 4th dimension (Please, no Dr. Who links....keep this mathematical.)

I hope this is fun for you; please keep asking yourself the "Why?" or "How?" questions so you can delve a little more deeply into your blog.  Find something fun or interesting and write about it.  Let is be recreational!

Without Trig, your House Would be Lopsided.

This blog post follows the "Grade of the Road" post; you'll want to read that one first.  Some of you have commented on this blog last year already (hopefully, you will remember that blog.) If you have not, take a minute to evaluate what trig function (using the angle of elevation) is the slope (or grade) of the road.  Professionals who work on roads use the word "grade" to refer to the steepness of the slope of a road.

Professionals constructing roofs use the word "pitch" instead of "grade."  Clearly to understand grade, pitch, and their relationship to each other, one must have more than a simple understanding of the basic trigonometric function called "tangent" and the algebraic concept of "slope" that you've learned in previous classes.

The pitch of a roof is described the following way. If a plan calls for a 11/18 pitch roof, then the roof rises 11 inches for every 18 inches of horizontal.  (Horizontal is "run" in the image shown.)

Find the angle of elevation in the 11/18 pitch.  Then discuss the relationship between "pitch," "grade" of a road and "slope."  Furthermore, what does "tangent" have to do with "pitch," "grade" and "slope"?

Then find the full length of the total rise and the "rafter line" (refer to the illustration above) if the full length of the "run" of the rafter is 22 feet.  You'll need to think about similar triangles.

Your blog needs to (a) stand alone, that is, not require your reader to read JammnPeaches. (b) be written in complete sentences  (c) complete the math with explanations in English.

There's more cool math and geometry in these constructions, particularly when there's gables or two different pitches involved in a single house.  If you'd like to explore this OYO or in a blog post, feel free to do so. The reference below is excellent for this blog and for your future studies.  If you have Geometer's Sketchpad, then you can see the sketchpad illustration of the "valley" between two roofs of different pitches.

reference:
http://jwilson.coe.uga.edu/EMAT6680Su09/King/Roofing/Application%20of%20%20Mathematics%20in%20Construction.htm

Grade of a Road.

Driving East on I-70 into Denver, there's signs posted about the road being a "6% grade."  What's that mean?  Why is this posted on the highway?   A 9% grade is steeper.  Rock climbers think nothing of a 100% grade; many even climb an infinite grade.  mmm.  What's that mean? We've explored this briefly in class, in particular in regard to the Pike's Peak Cog Railway that sports a 25% grade in one section.

Do a short quick google search on "Grade of Road" or "Incline."

I've never seen or driven on an 18% grade or a 25% grade on a road; the 9% grade I drove on was steep enough.  What grades have you seen? I think there's a 12% grade into Jackson Hole, Wyoming.  Anyone drive on that grade?








Your blog: How is the grade of a road established? What's it mean to have an x% grade?  Explain.  Is this helpful information?

The grade of a road has EVERYTHING to do with the slope of a line.  What is the slope of a line?  How is this related?

Here's an image to help in your description:

In your blog, to earn full credit, you will need the following: (a) to connect trigonometry, slope, and grade together; (b) to include an application -- maybe something you saw on a road or read about.  You  may NOT write about the Manitou (Pike's Peak) Incline (that was already done in class); (c) to write complete ideas in full sentences. Your blog needs to stand alone -- that is, not require your reader to also read my blog. 

Images are great, but this is a blog with an implication that there's an essay there.  If you have any questions, please ask! 

The 5 WHY's or HOW's: Writing an Effective Math Blog

So you have a blog; TUMBLR works great for you or blogger is the best thing since a spiral notebook.  So, how do you proceed writing a math blog?  How do you get those coveted perfect scores on your math blog?

When the year starts, your blog is a response to something posted by the teacher. You might have to look up something specific online or watch a video. Your comments need to be something about the mathematics involved and not just, "I liked the sound of Steve Inskeep's voice," or "google had 9,832,568 different search results, so it must be important."  So how do you think about the mathematics being presented here or explored?  This blog is to help you ask yourself questions so you can delve more deeply and more meaningfully into the mathematics of the blog.  You have thirty minutes to work on the blog -- you have time to develop something thoughtful here.

"The 5 Why's" might help you keep digging to a depth that's 5 times as deep. (You can also substitute "how" if that seems to fit more into what you are digging into.)  So if you feel like writing, "It was interesting," then ask yourself, "WHY?"  You might then say, "Because it applied to the real world."  Then ask yourself, "HOW?"  and so forth, until you've asked and answered 5 questions. Your goal is to keep digging until you get some real meat in your essay and not just "It was nice."

Another method is to ask yourself how this relates to the current or recent topic in math.  How is this blog relevant?  More often than not, the blog that I write for you to respond to is somehow linked to the topic in class.  For example, you will be asked to write about the grade of a road and its relationship to a trig function, or for a different topic you'll be asked to watch a video about something in the real world.  You could ask yourself, "Why would Ms Peach think this is relevant to what we've been studying?"  Try to second guess me -- I know you try to do that all the time anyway, just take the same idea into the math realm.

As the year progresses, you'll likely be developing your own blog ideas.  In January, previous students have said and asked, "Hey, Ms Peach, I found this really cool thing called Taxi-Cab Geometry, Can I do a blog on that?"  or "Hey Ms M, What if you put that triangle on a sphere, what would happen?"  or "My uncle is an engineer and he says he uses trig. Can I find out what he does and write about that?"  I've said YES to each of those, even though they were very different from the prompt that I wrote.  While I'll always be giving you some kind of prompts in case you have writer's block and need a jump start, you DO have freedom to choose your own topics.  They do, however, need to have a connection to mathematics.

I hope this helps some of you.  If you read this and have comments or suggestions for your classmates, please add them here on my blog on the comment page.

Err in the Direction of Optimism (aka Perseverence)

If you've been in my class before, you've already read the "Err in the Direction of Kindness" blog. And further, you've even written a blog about it.

You're free to read it again and write a NEW comment on that blog, but here's a second option.

Kindness to others is of the utmost importance.  The other characteristic that will be important to you in your life is PERSISTENCE.  So your second option is to watch (or read) something meaningful about persistence and write about it in your blog.

Persistence.  It goes hand in hand with optimism.  If you are optimistic, then you'll likely have greater persistence.

Calvin Coolidge spoke beautifully about persistence:

“Nothing in the world can take the place of persistence. Talent will not; nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not; the world is full of educated derelicts. Persistence and determination alone are omnipotent. The slogan ‘press on’ has solved and always will solve the problems of the human race.” ~Calvin Coolidge

Hear Calvin Coolidge speak this paragraph HERE. 

Here's a couple Ted Talks that could be interesting for you -- choose one or find your own to write about. 

Ted Talk: Dennis E. Murphree: “The Importance of Persistence in Your Life.”

“Err on the side of Optimism; … optimism is so much more fun to live.”

https://www.youtube.com/watch?v=Z3TJCLWTgt8

Ted Talk: Derek Clark: “Power of Determination.”

“Every Child is Worthy”

https://www.youtube.com/watch?v=trcYKxj48HA


Olympians, people who persevered. Stories of Olympians.
http://www.upworthy.com/when-life-hit-them-hard-these-4-olympians-persevered-and-hit-back-even-harder?c=sr1&sr_source=lift_facebook 

The 5 WHY's or HOW's: Writing an Effective Math Blog

So you have a blog; TUMBLR works great for you or blogger is the best thing since a spiral notebook.  So, how do you proceed writing a math blog?  How do you get those coveted perfect scores on your math blog?

When the year starts, your blog is a response to something posted by the teacher. You might have to look up something specific online or watch a video. Your comments need to be something about the mathematics involved and not just, "I liked the sound of Steve Inskeep's voice," or "google had 32,568 different search results, so it must be important."  So how do you think about the mathematics being presented here or explored?  This blog is to help you ask yourself questions so you can delve more deeply and more meaningfully into the mathematics of the blog.  You have thirty minutes to work on the blog -- you have time to develop something thoughtful here.

"The 5 Why's" might help you keep digging to a depth that's 5 times as deep. (You can also substitute "how" if that seems to fit more into what you are digging into.)  So if you feel like writing, "It was interesting," then ask yourself, "WHY?"  You might then say, "Because it applied to the real world."  Then ask yourself, "HOW?"  and so forth, until you've asked and answered 5 questions. Your goal is to keep digging until you get some real meat in your essay and not just "It was nice."

Another method is to ask yourself how this relates to the current or recent topic in math.  How is this blog relevant?  More often than not, the blog that I write for you to respond to is somehow linked to the topic in class.  For example, you will be asked to write about the grade of a road and its relationship to a trig function, or for a different topic you'll be asked to watch a video about something in the real world.  You could ask yourself, "Why would Ms Peach think this is relevant to what we've been studying?"  Try to second guess me -- I know you try to do that all the time anyway, just take the same idea into the math realm.

As the year progresses, you'll likely be developing your own blog ideas.  In January, previous students have said and asked, "Hey, Ms Peach, I found this really cool thing called Taxi-Cab Geometry, Can I do a blog on that?"  or "Hey Ms M, What if you put that triangle on a sphere, what would happen?"  or "My uncle is an engineer and he says he uses trig. Can I find out what he does and write about that?"  I've said YES to each of those, even though they were very different from the prompt that I wrote.  While I'll always be giving you some kind of prompts in case you have writer's block and need a jump start, you DO have freedom to choose your own topics.  They do, however, need to have a connection to mathematics.

I hope this helps some of you.  If you read this and have comments or suggestions for your classmates, please add them here on my blog on the comment page.

Piecewise Function: Dog Years.

How many of you have heard that one year of a dog's life is equal to seven years of a person's life?  

This would imply the following equation: y = 7x where x is the dog's age (in years) and y is the human age (in years). According to this site, "One explanation for how this formula got started is that the 7:1 ratio seems to have been based on the statistic that people lived to about 70, and dogs to about 10."

Using graph paper with your x scaled from 0-12 and y scaled from 0-84 (think about why these values for domain and range make sense), draw this line ( y = 7x) to represent a MODEL of the relationship between dog-years and human-years. 

(An option other than graph paper is to use Desmos.  Should you do this, you may want to plot human years on x axis and dog years on y-axis -- this is the inverse of the functions given here.  Desmos can have you switch the x's and y's quite nicely should you want to do this. ) 

However, there's more to it than that.  Dog development is different from human development.  Even different breeds of dog develop at different rates. Below are two different additional models that compare a dog's age to a human's age.

This site has the following conversion: "For the first two years, a dog year is equal to 10.5 human years. After that, each dog year equals 4 human years. This calculation is based on studies that indicate dogs, except maybe larger breeds, develop more quickly in the first two years of life."

This implies a piecewise function to MODEL the relationship between the age of a dog and the age of a person: one function for 0<x<2 and a second function for 2<x, where x is the age of the dog. In the language and symbology of math:

 

In your blog, explain how these "pieces" were found.  In particular, where did the "13" come from? 

Using the graph paper from the first model (with your x scaled from 0-12 and y scaled from 0-84 and containing the linear function already graphed), graph this piecewise function. This is a second, likely improved, model of the relationship between dog-years and human-years.

Yet the  American Veterinary Medical Association has yet another MODEL of the relationship between a dog's age and a human's age:

• 15 human years equals the first year of a medium-sized dog’s life.
• Year two for a dog equals about nine years for a human.
• And after that, each human year would be approximately five years for a dog.

Your task for this blog is three-fold.  

1. You need to identify the piecewise function that models the American Veterinary Medical Associations description of the relationship between a dog's age and a human's age (as described above in green with bullets). 
2.  You need to add this function to the graph paper that contains the other two models. Then choose three different dog ages and find how many "human years" is the age of the dog for each of the three models.  (You'll be turning in this graph on paper on the due date for this blog.)  
3. Find another piecewise function in the Real World.  It can be a different age conversion for dogs (or for cats).  It can be how the power company bills for services.  It can be a discussion of the examples used by Khan Academy or another site (DO credit your source.) Describe and explain the piecewise function and how it models what you've chosen  Graph it, if that's necessary in your explanation,  As always, find something that's fun for you. 
OF COURSE, you'll need to (a) add this to your current blogsite and not create a new one.  (b) turn in the blog accurately and carefully through canvas. Be sure I do NOT need a password to access the URL you give me.  (c) If I send you a message through canvas that I can't read your blog, you MUST respond within 3 days or your grade will remain a 0. 

Iowa Caucus Statistics in Albuquerque Journal

I am always looking for good examples of statistics or geometry to use in class with an eye for misrepresentations of the math that might lead people to make incorrect assumptions. Tuesday's paper (Feb. 2, 2016) offered coverage of the results of the Iowa caucus and also provided an excellent opportunity for us to explore how the results were displayed to the readers of the Journal. Indeed, election year media coverage seems to always provide good material for me to use in class.

This is not meant to be a partisan blog; both sides of the aisle are guilty of misrepresenting the truth.  News media, however, does tend to be particularly biased in their reporting.  HOWEVER, if the readership is savvy and informed, the presentation of "facts" backfires. People tend to get irritated when they think they are not being told the truth.

So the above image appeared on the Front Page (FRONT page, mind you) of our own Albuquerque Journal. Based on the image, you'd think that Rafael "Ted" Cruz had won the primary in Iowa by a landslide -- after all, look at the size of his circle compared to those of  DonaldTrump and Marco Rubio.  I'm glad they printed the percentage of votes right there so I don't have to look them up.  Let's do some math.

I pulled the image into Geometer's Sketchpad and measured the radii of the three circles representing the votes for each of these three republican candidates.

We can then find the approximate areas of the three circles.  Cruz: 87.9 square cm; Trump: 17.5 square cm; Rubio: 16.6 square cm.  Senator Cruz appears to be cruising, with his image covering an area just over 5 times that of Trump's image and nearly 5 and 1/3 times that of Rubio's image.

But reality?   Look at the percentages of the votes received.

First of all, notice that 28%, 24% and 23% do not add up to 100%.  We are missing 25%.  How did that 25% vote (likely for other candidates)?  We don't know from this front page image. Also notice that 28%, 24% and 23% are not that far apart. Furthermore, Cruz did not win a simple majority (that is, more than 50% of the vote.) Numerically, this does not particularly look like a landslide.

Of the 75% of the votes represented by these three circles, Cruz earned 37% of those votes.  Trump and Rubio earned 32% and 31%, respectively (notice my 37, 32, and 31 sum to 100% of the 75 percentage points). Cruz earned nowhere near 5 times the votes of Trump.   No landslide here, either.

So the newspaper was interested in having people believe, on some level (perhaps even unconsciously), that Cruz won by a wide margin.  Might that bias affect what the readers think or or what they might perceive their peers in Iowa think?  We can debate that until the cows come home.  But this visual image seems to misrepresent the success of Cruz: it was actually a pretty close election.

Your blog can take one of several routes.

1. Find something else that's mathematically misrepresented in the media.  Anything.  And explain why the data is misrepresented.  You'll need to show an image and complete an interpretation of what's incorrect and perhaps how the information should have been represented.

2.  Explain my geometry.  How did I construct the exact center of each circle?  (I did NOT just guess!)  Then how did I find the area of the circles?  Why did I use ratios of area and not ratios of radii?  How might the numbers have turned out it I had compared radii? Find those values and interpret.   Is the ratio of the radii the same as the ratio of the areas?

Why Mathematicians find math thrilling.

Why mathematicians find math thrilling. 

great article/ted talk.