Differentiating the Holidays

I am shamelessly stealing this from a coworker. It is a horrible way to come back after a long hiatus of no blog posts. I apologize. I have blog posts in my head, but am not motivated enough to write.

I also hate the idea of FWD FWD FWD e-mails, but I did think this one was fun enough to share beyond our school:

The Top Ten Ways to Differentiate Thanksgiving Dinner

10. Serve all of your guests on different-sized plates
9. Make each of your guests focus on just one or two foods instead of the whole buffet
8. Eat in shifts in different rooms
7. Let your visual guests just enjoy looking at the food while your kinesthetic guests get to eat it
6. Allow the turkey to have a say as to whether he should be oven-roasted or deep-fried
5. Serve dessert first, then the hot dogs
4. Pair everyone up with an eating buddy
3. Serve the simpler foods first like mashed potatoes and work your way into the more complex foods after your guests have showed mastery of eating
2. Allow your quick eaters to put their food in a blender for faster consumption
And the number one way to differentiate Thanksgiving Dinner…
1. Serve Thanksgiving, but call it a Fourth of July celebration

"Real World" Problem Solving: I Give Up!

I have a SMART board in my room. One of maybe 3 in our school. I also share my room for one period a day with a teacher who doesn't like to use it. So, the board is on a wheeled mount so I can move it out of the way for the 45 mins a day he is in my same classroom. This means I have to orient it at least twice a day and don't really get to ever lock it into place.

We take attendance online in our gradebook program and it is the expectation that attendance be done in the first 10 minutes of class (to catch skipping students). Since doing this on my own daily leaves the first 5 minutes of class where the students goof around instead of working, I decided to use the SMART board to my advantage.

This is what our attendance looks like (names cropped to protect the young):

I set everyone to the far right column (absent) and when students arrive, they are to hit the leftmost bubble next to their name to mark themselves present (the middle option is for tardies). Generally, they like being able to come up to the SMART board and having it work for them. Win-win.

Here's the issue: Since it's never perfectly calibrated (especially with students going to the board and slightly moving the mount to one side or the other), hitting the board in the right spot is often difficult. When you hit the board, a time diamond-shaped cursor shows up and blinks where it thought you touched, but even moments after I orient/calibrate it, some sections of the board will be off by a centimeter or two.

Many of my students have the hardest time getting it to work. I understand that it's somewhat unpredictable on your first touch, but they will continue to hit the same spot (usually harder and harder) and continue to get more and more frustrated.

Part of me wants to laugh and part wants to cry. Is it really that hard to compensate for the issue? You hit it on the button exactly the first time and you see the cursor a bit below your name so it doesn't work. Try hitting it a centimeter higher, right? They will try four or five times and then either try to get a friend to do it or give up and walk away mad and throw their hands up in the air saying, "I can't do it. You just do it for me."

Then I try to teach them how to problem solve math problems and they react the same way. Is it surprising?

I'm all for trying to tap into student intuition and their own internal motivation (a la Shawn Cornally, but what can I do with students like this? Am I being overly dramatic in this observation of student behavior?

(NB: Not all are like this. About half of them figure it out and there are no problems, but about half have the issue as described.)

Pretests (2011-12)

Presented without real comments, here are my pretests for Algebra 2 and Precalculus this year:

Alg2Pretest

Pre Cal Pretest

Transparency With Students

One thing I'm wondering about for the coming school year is how much to talk with my students about the background of the logistics of teaching. Certainly many of you have said that it is important to regularly talk about the meaning of SBG to get students on board with the system. Providing outlines for units is also important for connecting the concepts to one another.

I wonder, though, how much of it just becomes "execu-speak" as in the clip above. Of course I get excited about the whys and whats of not only the content, but the teaching methods, but how much do they care about it? How much SHOULD they care about it?

How much detail should I go into when explaining why I don't take homework grades? Should I even bring up the issue that I don't think I should take off for late work, but the school put a policy in place to make me do it? Do they care WHY I'm just repeating their questions back to them and never actually giving them the "real answer" (as seen in Rhett's awesome post)? Is it worth mentioning the reason I have the classroom set up in the way I do, the reason I go out of order from the book, the reason I grade on a 10 point scale, the reason I spend hours each night after school thinking about all the little details to give them the best possible learning experience that I can?

Real-world Math: Lampshades

My wife wants to make something like this:

The issue is that she wants the shadow part to be words, so getting the angles right is essential.

So, how should she design it in photoshop so that we can print it out and glue it on? Enter math.

She measured the top and found that it has a circumference of 26 inches. The bottom of the shade has a radius of 32 inches. The "slant height" is 7 inches. The "true" height is harder to measure because you have to keep it level and ours is already on the lamp, so it wasn't measured, but could be "if you REALLY need it".

So, what to do? Well, here's what I knew:

1. This shape is called a "frustrum." In particular, this is a conical frustrum.
2. You can create a cone by cutting a sector from a circle and gluing the ends together.
3. Since this frustrum shape is a large cone with a top cone cut off, it could be created by taking a sector from an annulus (a "ring" made by taking a large circle and cutting our a concentric, smaller circle).
4. Formulas I knew and may or may not need here:
   
   • For a sector created from a circle of radius R and sweeping out angle T (in radians), the arc length of the sector (S) is S = R*T
   • Circumference = 2 * pi * radius
   • Pythagorean theorem (in particular, using the "true" height, slant height, and radius of a cone)

So, here's what I tried first (aka The Hard Way):

(Background info: I've never taught geometry and am more of a functional analysis person, so that's why this stuff came to me first.)

Consider the sector that would form the large cone (from which we cut the top off to make the lampshade): I knew the arc length is 32, but that's about it. The angle I called T and the radius of this large circle I called x.

Things I learned about this cone by playing around with the formulas above:

• If we call the radius of this large cone R, then 32 = 2 *pi * R, so R = 16/pi.
• 32 = x * T
• If we call the "true" height of the cone H, then H^2 + R^2 = x^2 (and could use R as above).

Consider the sector that would form the cone we cut off the other one: I know the arc length is 26 and that's it for that one. This angle is also T, but the smaller circle would have a radius of y (so that y < x above). Similar to the above things, but with a couple added to relate the two cones I now know:

• If we call the radius of this smaller cone r, then 26 = 2 * pi * r, so r = 13/pi.



• 26 = y * T



• If we call the "true" height of this cone h, then h^2 + r^2 = y^2 (and can use r from above).



• x = y + 7 (the 7 was measured as the slant height of the shade, see above)



• H = h + m, where m is the measured "true" height of the shade.

Putting these pieces together, I ended up with these two equations:

• H^2 + (16/pi)^2 = (y+7)^2
• (H-m)^2 + (13/pi)^2 = y^2

Two equations with two unknowns may be solvable, so I tried to actually measure m and got about 55/8 inches. I could solve these for y, then find the angle T = 26/y and have the sector for the small circle. Adjusting a bit to get x would give the large circle.

This was messy and gross. In fact, I gave up from here and decided to find a more "elegant" solution.

Second attempt (The Easy Way):

This time I tried to go back to basics of what I remembered about geometry (bear in mind I've not had anything really geometry related since 8th grade which was almost 20 years ago now).

I know there is such a thing as "similar triangles," so maybe there's such a thing as "similar cones" (seeing as how they are just rotated triangles). So, I went with that. Variables are already defined as above, so here are the ratios I used:

x/y = R/r = (2*pi*R)/(2*pi*r) = 32/26 = 16/13

So, clearly y = (13/16)*x. We also know that y + 7 = x and using substitution, (13/16)x + 7 = x, so (3/16) x = 7, so x = 112/3. Now that I know x, I can get T = 32/x.

Whoa! That's easy, but were my assumptions ok? Well, I'm not really doing a general proof here, so I'll just check it with these numbers: To the Bat Geogebra!


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


I'm still getting to know this application, so I'll explain how I made this:
First I put A at the origin and defined a to be 112/3 as discovered above. Create a circle with center A and radius a (the program called this c as you can see in the variables to the left). I needed another point on this circle, so I just used the one on the x-axis by creating line b (y = 0) and then intersecting the circle with the line; this created point B.

So, create a circle centered at B and radius a; this is circle labeled as f. (I now realize this is somewhat redundant since I could just switch the roles of A and B without this extra step, but I went into it thinking A would be a point on the circle's edge and would create the center of the circle elsewhere.) Anyhow, I then figured out the angle (which I called T above, but is labeled e in this applet). 0.86 is in radians, though, and since my wife will do the measuring, I figured degrees would be better, so I converted it which is labeled as d.

Use GGB to find a point which is d degrees away from A which it labels as A'. Just to make sure, create the arc between A and A' and check the length: 32. Perfect!

Now to finish, I made a circle centered at A and radius 7 (since I knew that was the slant height of the shade). Where that intersects the x-axis is the point C which will be the beginning of the smaller arc. Make a circle centered at B and radius y (which you may remember is equal to x - 7 as above) and find where that intersects with the line between A' and B. Double check by finding the measure of arc CD (26 yay!) AND the angle CBD (49.11 degrees yay!).

So now we have it. Measurements can now be passed back to the wife who is the real Illustrator/Photoshop guru and she can bend the text she wants for the lampshade to fit.

Pshew!

I'm still not sure what she has in mind and maybe this project won't even get finished by her, but I'll post a picture if it ever gets there!

Secondary Skills and Opportunities

Sometimes I wonder how things would be if we didn't need the secondary skills to go with certain things or how the world would be if everyone had the same opportunities.

Consider this: Something within Sam would make him the best violin player ever to walk the earth.

Unfortunately, Sam has grown up in the slums of Chad and never even hears a violin, much less gets the chance to practice or own one. OR perhaps Sam has all the skills and brains to be a great violinist, but he has stage fright or he loses a hand in an accident or some other issue that keeps him from actually showing this talent to the world.

Then I wonder how that plays out in my classroom. Bad test takers? Less-than-ideal backgrounds? Other handicaps physical or mental that might keep a kid from doing her best?

Cheap Gas

How far is it worth driving for "cheap gas?"

Let's say I have a gas station near me (or on my route or whatever), but that I know another gas station elsewhere is selling gas for ten cents less. Is it worth it for me to go out of the way to buy it cheaper?

When I fill up, I usually get about 12 gallons. So, the trip must be worth $1.20 for me to make it. Since gas is about $3.60 right now, I am saving about 1/3 of a gallon of gas to get to this other station.

My car gets about 27 miles to the gallon, so to be worth the 1/3 of a gallon, it must be less than 9 miles (round trip) for me to go out of the way to get it and come back.

This seems like a decent open-ended question for students to fill in with their own numbers for their cars.