Quick -- which is bigger: 1/3 or 1/4? Apparently, there is a significantly large proportion of the American Public that believes that 1/3 is less than 1/4. After all, 3 is smaller than 4.
Evidence is cited in the New York Times Magazine, July 27, 2014 in the article (New Math)-(New Teaching)= Failure, by Elizabeth Green. The premise of the article is that "the common core is the best way to teach math, but no one has shown the teachers how to teach it." The example above appears in the article.
Follow this link; it has the same article but with a different title that's a little more provocative. Read enough of the article to find the reference to the McDonald's restaurant competitor that offered the "third" pounder to be in competition with McDonald's "quarter" pounder. In your response to this blog, be sure to mention the more provocative title AND the name of the competitor restaurant, just so I know you read at least a little of the article.
I'd like your response to anything in this article in a meaningful but brief essay. Some ideas follow: (Go ahead, be provocative!)
Do you think it's the education system? Do you think it's the poor education of our math teachers in this country? Many of your teachers (like me) at independent schools (like AA) have never received any formal training in math teaching. What's up with that?
Why do you think we have so much "innumeracy" in our fellow Americans (to use a vocabulary word coined by John Allen Paulos). If we had as much "illiteracy" as "innumeracy," then people would be up-in-arms. Why are folks not so upset about the "innumeracy"? What's up with that?
Do you have examples of innumeracy that you'd like to share? I wrote a post on this last year; I've made it available again if you'd like to see that. (Find it below....)
Due Date: The THIRD day 9 in our semester 2 occurs Feb 17. If you do not have class that day, of course it's due Feb 18.
Monday, January 12, 2015
Wednesday, January 7, 2015
To Infinity and Beyond. Due Feb 2
Infinity. Several things we said in class:
Consider a segment of length 1 mm. Then consider a line of infinite length in both directions. There is a one-to-one correspondence between each point on the segment and each point on the line. Yep, there's the same number of points on the segment (infinite) as there is on the line (infinite).
Let's say you are the proprietor of a hotel with an infinite number of rooms. You have the good fortune of having all rooms rented. Yet when a new customer walks in the door and asks about lodging for the night, you say, "Why yes, of course, I have space for you...." There's a variety of ways that can happen, but the solution requires involvement with infinity.
We can also use infinite sets in calculating probability. The example given in class: If a Natural Number is chosen randomly, what's the probability that the number will be a multiple of 8? (1/8).
Do these examples blow your mind? Find another story about infinity and write about it in your post. Have fun; include jokes if you'd like. But the jokes must have some kind of foundation or understanding or exploration of the infinite.
Consider a segment of length 1 mm. Then consider a line of infinite length in both directions. There is a one-to-one correspondence between each point on the segment and each point on the line. Yep, there's the same number of points on the segment (infinite) as there is on the line (infinite).
Let's say you are the proprietor of a hotel with an infinite number of rooms. You have the good fortune of having all rooms rented. Yet when a new customer walks in the door and asks about lodging for the night, you say, "Why yes, of course, I have space for you...." There's a variety of ways that can happen, but the solution requires involvement with infinity.
We can also use infinite sets in calculating probability. The example given in class: If a Natural Number is chosen randomly, what's the probability that the number will be a multiple of 8? (1/8).
Do these examples blow your mind? Find another story about infinity and write about it in your post. Have fun; include jokes if you'd like. But the jokes must have some kind of foundation or understanding or exploration of the infinite.
PreCalculus. BIG Numbers. Due Feb 2
So it's back in hunter-gatherer times for our species; around the campfire one night, there's a couple of cave people geeking out and talking about numbers.
Thor: "I can think of a number bigger than you can."
Lana: "Oh yeah? Try me."
Thor: "Two."
Lana: "Three."
Thor: "Oh man. Got me."
So how big is big? We can talk in class about a billion seconds as being a lot of seconds -- nearly 32 years worth -- and say WOW that's awesome...what big numbers..... but what about the number of seconds that's in a billion years? That number makes our billion seconds seem pretty trivial. Just as we may chuckle at Lana and Thor, there may be a group of people several thousand years hence (assuming we don't kill ourselves before then) that laughs at our inability to contemplate what they consider to be large numbers. This is almost like my 1986 computer I told you about where 40 megs was SO MUCH SPACE that people thought, "What is she ever going to do with that space?"
So how big is big? Exponential functions (with a base greater than one) are rapidly increasing functions and just beg the question: how big is that big number? Choose your method of describing large numbers; give some kind of numerical idea of how we might contemplate large ideas. Other than Jenny Lee's or Tony Borek's idea of how much time is equal to 1 billion seconds, what are other ways that might allow us to wrap our minds around large numbers? That's your task. Yep, pretty open-ended and this asks you to think a little creatively and outside the box. Google whatever you'd like, but be original and write your own stuff.
Let me give you some starter examples of what you can think about: How many blades of grass are there on AA property? How about how many gallons of water are in Lake Superior? How many pennies would you stack together to reach the moon?
Have fun. If you have questions, let me know.
Thor: "I can think of a number bigger than you can."
Lana: "Oh yeah? Try me."
Thor: "Two."
Lana: "Three."
Thor: "Oh man. Got me."
So how big is big? We can talk in class about a billion seconds as being a lot of seconds -- nearly 32 years worth -- and say WOW that's awesome...what big numbers..... but what about the number of seconds that's in a billion years? That number makes our billion seconds seem pretty trivial. Just as we may chuckle at Lana and Thor, there may be a group of people several thousand years hence (assuming we don't kill ourselves before then) that laughs at our inability to contemplate what they consider to be large numbers. This is almost like my 1986 computer I told you about where 40 megs was SO MUCH SPACE that people thought, "What is she ever going to do with that space?"
So how big is big? Exponential functions (with a base greater than one) are rapidly increasing functions and just beg the question: how big is that big number? Choose your method of describing large numbers; give some kind of numerical idea of how we might contemplate large ideas. Other than Jenny Lee's or Tony Borek's idea of how much time is equal to 1 billion seconds, what are other ways that might allow us to wrap our minds around large numbers? That's your task. Yep, pretty open-ended and this asks you to think a little creatively and outside the box. Google whatever you'd like, but be original and write your own stuff.
Let me give you some starter examples of what you can think about: How many blades of grass are there on AA property? How about how many gallons of water are in Lake Superior? How many pennies would you stack together to reach the moon?
Have fun. If you have questions, let me know.
Saturday, January 3, 2015
Geo/Algebra 2, misspelled: Sets of Numbers Due Jan 15
Watch Sal Khan talk about sets of numbers.
Compare what he talks about to what we do/did in class about sets of numbers. I hope that his lesson will help clarify the sets of numbers discussion from class. Know that you can always use his lectures/site for helping you learn the material from class. I know some of you already do. If you don't know of this resource on line, then this blog is all about having you learn about Khan Academy.
In your blog, please write about how YOU will learn to keep these sets of numbers straight. Find a mnemonic device that will help you remember, for example, the difference between the integers and rational numbers. Go ahead, google to see if there's something that someone else has already come up with that you think will be helpful. Or come up with your own idea.
Before you start on this blog, be sure that you make yourself familiar with the blog grading rubric.
Also, I'm thinking that we'll do a better job with blogs this semester; we'll have a blog due every day 9. I'd like your responses, positive and negative, to this.
Compare what he talks about to what we do/did in class about sets of numbers. I hope that his lesson will help clarify the sets of numbers discussion from class. Know that you can always use his lectures/site for helping you learn the material from class. I know some of you already do. If you don't know of this resource on line, then this blog is all about having you learn about Khan Academy.
In your blog, please write about how YOU will learn to keep these sets of numbers straight. Find a mnemonic device that will help you remember, for example, the difference between the integers and rational numbers. Go ahead, google to see if there's something that someone else has already come up with that you think will be helpful. Or come up with your own idea.
Before you start on this blog, be sure that you make yourself familiar with the blog grading rubric.
Also, I'm thinking that we'll do a better job with blogs this semester; we'll have a blog due every day 9. I'd like your responses, positive and negative, to this.
Precalculus: Welcome Back! Topic: Transcendental Functions. Due Jan 15
What does the word, "Transcendental" mean? In English literature, you may (or may not) have studied the "transcendentalists" like Emerson or Thoreau. Why are these authors called the Transcendentalists?
In mathematics, we discuss "Transcendental Functions." Trigonometric functions are transcendental. We are now exploring exponential and logarithmic functions which are also transcendental. Why? How? What does this have to do with the use of the word, "Transcendental" in English literature?
Before you write this blog, you'll want to read the instructions and rubric as posted on canvas. Be sure you are doing what you need to be doing to earn all the credit you need in this blog.
Also, in your blog, let me know how often you think it's reasonable to write a blog. Is once each cycle good? That's about once every 2 weeks. I'm thinking that we could make the blog due every day 9, unless of course, there's a test that day. I'd like your response to that.
In mathematics, we discuss "Transcendental Functions." Trigonometric functions are transcendental. We are now exploring exponential and logarithmic functions which are also transcendental. Why? How? What does this have to do with the use of the word, "Transcendental" in English literature?
Before you write this blog, you'll want to read the instructions and rubric as posted on canvas. Be sure you are doing what you need to be doing to earn all the credit you need in this blog.
Also, in your blog, let me know how often you think it's reasonable to write a blog. Is once each cycle good? That's about once every 2 weeks. I'm thinking that we could make the blog due every day 9, unless of course, there's a test that day. I'd like your response to that.
Wednesday, April 23, 2014
Quadratic Function. Parabola. Second Degree Polynomial. GA2 and TPC
Quadratic functions. Parabolas. Second Degree Polynomials. There's so many names, you just know this has to be important. For your blog, you'll find one aspect of quadratics to focus on and to develop further. Choose something that's a little outside what we did in class but still includes quadratic functions. My blog, hopefully, will send you to a bunch of different places where you can check out two categories of information about these functions. The first category is the cool mathematics of these things. The second category is the application of quadratics to the "real world". Of course, if you want to expand this and discuss something else relevant, then that's ok, too. If you have questions about whether your idea is ok, then just ask me.
As an intro, here's a site that has a slider (you'll have to scroll down to see what I mean) that shows you how the transformations to the quadratic function are paralleled with an athlete kicking a soccer ball. mathisfun.
Please go to that site now and use the slider. Scroll down. See if there's something else on that page that catches your eye. There's several good ideas for your study there.
How many different mathematical ways can you think about a quadratic function?
1. Geometrically, as a conic section (for GA2 kids, if you don't know what that means, google conic section).
2. Geometrically, as a set of points equidistant from a single point and a line. Here's an example of this. Again, you'll have to scroll down a little to see the circles and line. (Artsy folks: you might like the first image on this page; surely a child of the 1960's came up with this idea.)
3. Using algebra to plot points on a Cartesian plane using a quadratic equation..lo and behold, you get the parabola. (Sorry, no link for this one; you've all done this hundreds of times before.)
4. Use algebra to express the function in a variety of helpful forms: vertex form, descending form, factored form. (Again, we did this in class.)
5. Using creative forms of construction: folding paper or using the circles paper shown in #2 link above. (There's some interesting youtube videos that show paper folding to create other conic sections; that can be an interesting exploration for you.)
So...what about interesting applications?
1. A parabolic reflector. (Someone's trying to make money with this one...)
2. A parabolic dish for a telescope.
3. A football punt. (NFL and NSF got together for this video.) The idea applies, as you'll see in the video, to any projectile.
Find something that strikes your fancy. Choose ONE idea and develop it. Make it more than 8 sentences; it must be a single full idea. Cut and paste. And, of course, submit through canvas.
As an intro, here's a site that has a slider (you'll have to scroll down to see what I mean) that shows you how the transformations to the quadratic function are paralleled with an athlete kicking a soccer ball. mathisfun.
Please go to that site now and use the slider. Scroll down. See if there's something else on that page that catches your eye. There's several good ideas for your study there.
How many different mathematical ways can you think about a quadratic function?
1. Geometrically, as a conic section (for GA2 kids, if you don't know what that means, google conic section).
2. Geometrically, as a set of points equidistant from a single point and a line. Here's an example of this. Again, you'll have to scroll down a little to see the circles and line. (Artsy folks: you might like the first image on this page; surely a child of the 1960's came up with this idea.)
3. Using algebra to plot points on a Cartesian plane using a quadratic equation..lo and behold, you get the parabola. (Sorry, no link for this one; you've all done this hundreds of times before.)
4. Use algebra to express the function in a variety of helpful forms: vertex form, descending form, factored form. (Again, we did this in class.)
5. Using creative forms of construction: folding paper or using the circles paper shown in #2 link above. (There's some interesting youtube videos that show paper folding to create other conic sections; that can be an interesting exploration for you.)
So...what about interesting applications?
1. A parabolic reflector. (Someone's trying to make money with this one...)
2. A parabolic dish for a telescope.
3. A football punt. (NFL and NSF got together for this video.) The idea applies, as you'll see in the video, to any projectile.
Find something that strikes your fancy. Choose ONE idea and develop it. Make it more than 8 sentences; it must be a single full idea. Cut and paste. And, of course, submit through canvas.
Friday, March 14, 2014
A New Pi. For GA2 and TPC
I must credit Trevor Kann and his most recent post with this idea. Click on his blog.
I'll summarize below, but I recommend you see his blog as the "original
source." He also gives, in his blog, a lot of the answers to the
questions I ask in my blog. He explored the ratio of the circumference
to the diameter of a circle (otherwise known as pi) in "Taxicab
Geometry." We had discussed this as a tangent in class, but being a
good thinker and good blogger, he started asking himself, "What if...."
To set this up, in class we talked about a new way to look at "point," "line," and "plane," after all, they are undefined terms. What if we defined these things such that a plane was a grid, much like city streets on a NS EW grid. Points would be lattice points, or intersections of the "streets" on the grid, and we could travel only on the grid its self, much like a taxicab would drive. A circle is still the set of all points equidistant to a single point.
In
the grid to the right, the center point of a "circle" is shown with a
circle of radius 2 (diameter = 4). Remembering that you can only travel
on the lines and points are only places where the lines intersect, find
the circumference. Then take the ratio of the circumference to the
diameter. This ratio is known as pi. (On a Euclidean plane, like we
have been using, the ratio is about 3.14159....). Verify that this
calculation of the new "pi" is indeed 4. Make your own grid; identify a circle with a different radius. Is the ratio still 4?
The above is our "tangent" in class last week.
So what did he do that was so cool? Trevor asked, What if we have the same rules about lattice points, traveling only on the lines, but we set up the plane to be equilateral triangles. What would pi be? (Actually, he said, "This got me thinking....") To the right is a snip of his new grid. One unit is the length of the side of an equilateral triangle; points are only vertices of the triangle. You can see the center and a circle shown. The radius of this circle is 2. Find the circumference. Find the value that is the ratio of the circumference to the diameter ("pi"). It's not 3.14159..... and it's not 4.
Then he thought some more. What if the grid was tessellating
hexagons. In the process of thinking, he figured out that the only
tessellating regular polygons are squares, triangles, and hexagons. To
the right is his image of the tessellating hexagons. Of course, he used
a radius of 2 and found a value for the circumference, diameter and the
ratio of the circumference to the diameter. Go ahead, find that ratio. (Answer is in his blog :) )
So ok. In this next blog, I know Trevor is still thinking about this (DO keep thinking, Trevor!).
For your blog, you can expand on this idea. OR better yet, find something like, "Why are there only three regular polygons that tessellate?" or "What is the relationship between regular tessellations and regular polyhedra?" or "What if we put a circle on a soccer ball?" (Trevor's question, lay off that one, I think he may be writing about that one.....) BTW, a soccer ball is also a "truncated icosahedron." OR even better, think of a tangent we took in class (or maybe your brain just took in class without us being with you on that mental adventure....) As yourself, "What if..." or be like Trevor and just start thinking....
To set this up, in class we talked about a new way to look at "point," "line," and "plane," after all, they are undefined terms. What if we defined these things such that a plane was a grid, much like city streets on a NS EW grid. Points would be lattice points, or intersections of the "streets" on the grid, and we could travel only on the grid its self, much like a taxicab would drive. A circle is still the set of all points equidistant to a single point.
In
the grid to the right, the center point of a "circle" is shown with a
circle of radius 2 (diameter = 4). Remembering that you can only travel
on the lines and points are only places where the lines intersect, find
the circumference. Then take the ratio of the circumference to the
diameter. This ratio is known as pi. (On a Euclidean plane, like we
have been using, the ratio is about 3.14159....). Verify that this
calculation of the new "pi" is indeed 4. Make your own grid; identify a circle with a different radius. Is the ratio still 4? The above is our "tangent" in class last week.
So what did he do that was so cool? Trevor asked, What if we have the same rules about lattice points, traveling only on the lines, but we set up the plane to be equilateral triangles. What would pi be? (Actually, he said, "This got me thinking....") To the right is a snip of his new grid. One unit is the length of the side of an equilateral triangle; points are only vertices of the triangle. You can see the center and a circle shown. The radius of this circle is 2. Find the circumference. Find the value that is the ratio of the circumference to the diameter ("pi"). It's not 3.14159..... and it's not 4.
Then he thought some more. What if the grid was tessellating
hexagons. In the process of thinking, he figured out that the only
tessellating regular polygons are squares, triangles, and hexagons. To
the right is his image of the tessellating hexagons. Of course, he used
a radius of 2 and found a value for the circumference, diameter and the
ratio of the circumference to the diameter. Go ahead, find that ratio. (Answer is in his blog :) )So ok. In this next blog, I know Trevor is still thinking about this (DO keep thinking, Trevor!).
For your blog, you can expand on this idea. OR better yet, find something like, "Why are there only three regular polygons that tessellate?" or "What is the relationship between regular tessellations and regular polyhedra?" or "What if we put a circle on a soccer ball?" (Trevor's question, lay off that one, I think he may be writing about that one.....) BTW, a soccer ball is also a "truncated icosahedron." OR even better, think of a tangent we took in class (or maybe your brain just took in class without us being with you on that mental adventure....) As yourself, "What if..." or be like Trevor and just start thinking....
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