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?