Unit Conversion’s Fifteen Minutes of Fame

Working on plans for a summer program. Thought y’all might like this one.

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In 1968, Andy Warhol said: In the future, everyone will be world-famous for 15 minutes.

In honor of this, Cullen Murphy coined a “warhol”, a unit of fame. For every fifteen minutes you’re famous, you get one warhol.

1. Suppose you’re famous for an hour. How many warhols do you have?
2. Suppose you’re famous for a week. How many warhols do you have?
3. Suppose you’re Britney Spears. How many warhols do you have?
4. As a group, make a list of five famous people and estimate the warhols for each. Is there any way of speeding up your calculations, or do you have to use the 1 warhol / 15 min. ratio to start every time?

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In 1958, the fraternity Lambda Chi Alpha at MIT decided to take one of their members – one Oliver R. Smoot – and use him to measure the length of nearby Harvard Bridge. He laid down at the start, let the members mark his position in chalk, and moved on to the next spot. All told the bridge measured 364.4 smoots, “plus or minus an ear”.

1. Given Oliver Smoot was five feet and seven inches tall, how long is Harvard Bridge?
2. Using the same technique and masking tape, invent your own unit of measurement and figure out how wide this classroom is. Convert your measurement into smoots.

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The ability to use the latter lesson may of course depend on the age of your students, your own chutzpah, and whether or not the floor is carpeted.

Open exposition problems?

Nice quote from a recently revised paper by Timothy Chow, A Beginner’s Guide to Forcing:

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.

Missing a Test Question: A Case Study

I recently gave a test to my Geometry Concepts students which included the following:

Write a counterexample for the following: There is no number between 2 and 3.

First off, the only right answer students put was:
2.5

No 2.1, no 2 1/2, no 2.67. Only 2.5, in decimal. I don’t know if there’s a conclusion to be made here.

Wrong answers the students put were

? (or a blank)

In some cases the students hadn’t been paying attention to the meaning of a counterexample so they forgot it. But in at least one case (based on what the student said) someone wasn’t able to think of a number between 2 and 3; the statement seemed to them to be true. Now, it isn’t that the students didn’t know of the existence of 2.5, it’s just they were mentally “closed” in the natural number system and assumed from context “number” meant something you can count.

250

Some students were getting tripped up by the idea of a counterexample here and instead made an example of a number not between 2 and 3. Other students may have made a double negative; converting “there is no number between 2 and 3″ into “there is a number between 2 and 3″ and forming an counterexample to that statement rather than the original.

4.5

Perhaps most interesting is the students who put a number not between 2 and 3, but still used decimal. It’s like they started with the correct idea of a counterexample and latched on to the part about being between two natural numbers, but made a mental error afterwards when it came to picking a number.

Granularity of Multiple Intelligences

(This is in response to this comment, which I originally wrote as a comment, but then I realized I needed pictures. So.)

In regards to Gardner’s “multiple intelligences” theory, I’ve always found the division lines a little ad hoc and arbitrary. I have my own personal theory that multiple intelligences are to an extent valid, but there’s a great more granularity of categories than in Gardner’s theory.

Let me give an example.

Suppose you’ve got a baseball diamond like the above. What’s the distance between second base and home base?

What pentomino needs to be added to make the figure above a 5×3 rectangle?

Both problems have come up in class. The first problem involves removing visual information; some students are confused by there being two triangles and one student could solve the problem only when I covered the left triangle with my hand. The second problem involves adding visual information. Both problems would be lumped into the “spatial” intelligence, yet I have students who find the first easy but not the second, and vice versa.

I suspect that intelligence could be sorted into 100 or so categories, and then (sometimes overlapping) sets of intelligences could be formed into Gardner’s smaller list.

(Mind you, this is all personal theories, with no hard data whatsoever.)

Counterexamples

In retrospect, this lesson would make a good start-of-year icebreaker; it’d simultaneously check how much math vocabulary the students remember.

This is a bit customized for my own school and classroom.

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Find a counterexample to each statement.
1. All sports have a score that starts at 0 and goes up.
2. No US state has only four letters in its name.
3. Nobody in this classroom has more than 2 brothers or sisters.
4. No poster in this classroom has the word “joy” on it.
5. All the students in this class are 17 years old.
6. None of the food served at lunch tastes good.
7. Intersecting circles must intersect at only one point.
8. No even numbers are prime.
9. Nobody in this classroom owns a dog.
10. All quadrilaterals are squares.
11. No Pueblo students like to draw.
12. All TV shows are set in Manhattan.
13. No movie stars are talented.
14. Nobody has white hair.
15. All mammals have four legs.
16. Lines never intersect.
17. Two triangles cannot be congruent.
18. Two triangles cannot be similar.
19. No teacher at Pueblo has an interesting class.
20. No student in this class has a birthday in October.

Venn Diagrams List Some Of My Favorite Things

Not the most original lesson, but some of my Geometry Concepts students were having trouble with Venn Diagrams.

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music / movies / TV shows / videogames

1. Pick a category.
2. List 20 things.
3. Think of 6 properties some of these have.
Example (videogames): shooting, jumping, old, recent, 2D, 3D

Halo 3 (shooting, recent, 3D)
Super Mario Galaxy (jumping, recent, 3D)
Frogger (old, 2D)
Space Invaders (shooting, old, 2D)

4. Pick 3 of those categories and make intersecting circles. Fill your things appropriately.

5. Describe the spaces that do *not* have things in words.

No videogame is both recent and old.
No shooting games are not recent or old.

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Hopefully when we get back to proving things including logical statements like “nothing small is large” it’ll make more sense.

Incidentally, this was done on the fly when I realized a few days before things weren’t going like they needed to. I just can’t, can’t do a total plan a week in advance; I change things up too much to adapt to the students.

Infinite Geometric Series Through Philosophical Confusion

So, I stole a page from Benjamin Baxter’s playbook and taught a lesson with the aid of comments from the Internet.

Specifically, in Pre-Calculus I was teaching the formula for an infinite geometric sum:

so is the first term and r is the number a term is multiplied by to get the next term.

For example, take .9999…, which is the same thing as

So and . Therefore .999… is

I looked at the posts from the .999…=1 saga at Polymathematics (check the sidebar) and picked choice excerpts from the comments, i.e.

My first reaction is that I am very worried if you are a maths teacher.

We read them over in class and discussed why the reaction to this problem would be so strong, and why so many people would doubt the mathematics.

One conclusion was that there’s an almost philosophical confusion here; an unwillingness to accept the mathematician’s version of infinity. Last semester I had my Pre-Calculus students write an essay on “do you think infinity exists in real life?” because I wanted them to wrestle the strangeness head-on. The abstract leap to the infinite and infinitesimals is really the crux of Calculus.

Junk Statistics Article

Something to look over with students: Recent article on bad statistics

The case echoes a similar statistical legend — the belief that to be healthy we should drink eight glasses of water a day. Last week, researchers at the University of Pennsylvania decided to search for the source of this statistic. Their conclusion: “It is unclear where this recommendation came from.” In other words, they could not find any study to support the “eight glasses” claim.

See this and other fun non-facts at the link.

Forces on a Truss

(Today’s student guest is Erik Lundvall.)

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This problem pertains to construction and the way a given force acts upon a given structure. Trusses are triangular pieces used in the construction of some buildings to support weight of other objects such as the weight of the roof for example. In this particular problem the truss extends from the side of the building and is used to support an object. Say you have a business and over the entrance you have a pretty good sized sign advertising your business. The sign sits on a truss that extends over the sidewalk.

If the downward force of the sign towards the building is 250 pounds, the angle of the truss between lines A and B is 20 degrees, and the angle between line A and the sign is 120 degrees, what is the amount of force in pounds that is pulling down on the truss?

Call of Duty 4, Meet Vectors

Today’s student guest blogger is Emilio Inocencio.

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One sunny afternoon after completing your homework, you decide to play your favorite video game Call of Duty 4. When you enter your room you find that someone has replaced your Xbox 360 with a machined labeled “Mega 64”. The contraption has a note attached that reads, “To Exit-say the magic word: Delaware.” You notice that the device looks like an old fax machine wirelessly connected to a bicycle helmet. Out of pure curiosity you put the helmet on and wonder what will happen. After a few moments your heart is racing as your mind is being sucked through cyber-space and vivid images of the game Call of Duty 4 flash past your eyes. Suddenly, you find yourself inside a building which looks to be abandoned. To your left you see a .50 caliber sniper rifle pointing outside the window. You then immediately notice that you are in the video game Call of Duty 4! You run to the rifle knowing that you must eliminate the target to prevent nuclear war and ultimately save the world. You look down next to where the rifle is set and see a paper with scribbled notes of the velocities and angles of the bullet and the affecting wind.

The bullet is shot heading N30˚W at a speed of 2600 ft/sec and there is a wind velocity of 30ft/sec with a direction of N45˚W. What is the resultant speed and direction of the bullet?

As you continue to look through your scope you can see the bullet flying towards the target. You gain a faint sense of discomfort and slight taste of blood in your mouth. Just before the bullet hits you whisper the word, “Delaware” and are sucked back into reality. The whole experience has left you hungry and even more prepared to fight the good fight in Call of Duty 4.

Special note: The “Mega 64” idea is not mine and I did infact get permission to use this from the people who created the T.V. show. If you are a fan of many old and new video games I highly recommend buying the DVDs from their website www.mega64.com, they are very funny and full of fun times.