A Recursive Process

Altitude Sickness

by Dan Posted on July 29, 2011

Act 1

View from Pikes Peak:
On a trip out west with my wife, we took a trip up to the top of Pikes Peak by Colorado Springs. I’m calling this post Altitude Sickness because despite having this post in mind when we went up, I didn’t take the one picture that I wanted to take. Idiot. Nincompoop. Sigh. Lets move on. Here is a very similar picture (from wikipedia) to the one I wanted to take:

At the top of Pikes Peak I sealed a very similar water bottle. Pikes Peak is 14,110 ft tall. I took the water bottle back home and here is a picture of the water bottle.

So. What altitude am I at right now?

Act 2

You’ll need the following bits of information to solve.

$P_{1}V_{1}=P_{2}V_{2}$
Where $P$ is pressure, and $V$ is volume.
$P=P_{0}(1-2.25577*10^{-5}*h)^{5.25588}$
$P$ is pressure, $P_{0}$ is the pressure at sea level (101,325 pascals), and $h$ is the altitude in meters.
The volume of the crushed (by air) bottle is 395 mL (using Archimedes’s water displacement method and the fact that 1mL of water = 1g of water. Video of measurement found here.)
The volume of the bottle is 555 mL.

Act 3

The pressure-altitude formula is used twice. Once to get the air pressure at the top of Pikes Peak. This information can then be used to find the air pressure for my current location. Then you need to buy tadalafil in UK solve for h in the pressure-altitude formula to find the current altitude. The answer ended up being 162m, while the internets puts me at 91.4m of altitude. Not too shabby! Pretty damned good if you ask me.

Tweaks

Take the right picture at the top of Pikes Peak.
Keep the bottle crushed (I had to open it up back at home to get the volume of an uncrushed bottle).
Get an identical bottle for comparision.
… (any more)

Questions

Is the question: “What altitude am I at right now” legit? Do you think this question is natural to the students? I’m sure there’d have to be some explanation given to why the bottle gets crushed. But will this question come up? Give me your best read.
Should I keep the height of the mountain hidden so that what the student solves for? This seems better, but please you tell me what you think.
Is it helpful to bring in both bottles to class, the crushed one and the non-crushed one?

Posted in Full Posts | Tagged #wcydwt, calculate this | 9 Comments

SBG: Please Help – A Question

by Dan Posted on June 30, 2011

“Math class is tough.”
Barbie (1992)

How can I use SBG to challenge the top learners?

So after weeping quietly into my cereal while looking at the Geometry Regents results at the end of this year (just kidding, I don’t eat cereal, it was my toast), I noticed that I didn’t have as many “mastery” scores as I had in the previous years. Now for those of you blissfully unaware of the Regents tests, mastery is an 85 or higher on the year end exam. One of our school board’s goal is 90% passing and 40% mastery on all Regents exams (those of you who know of the Algebra II exam, you may return reading when you can manage to stop laughing). In reality I want the students who struggle to pass or better and the rest of the kids I want at mastery.

Before I continue, I want to list the faults with trying to answer this question:

Sample SBG size of one year.
Regents exam content varies each and every year, and the curve is set based on how “hard” they want the test to be.
Sample size of only 40 something students in my classes taking the Geometry Regents. Maybe earlier years were “stacked” in comparison to this year, a feeling that I’ve had since the start of this year.

That Said.

Compared to the last couple of years, I had far fewer students reach mastery this year. Why?
How do I challenge the students who relatively breeze through the material?

There were maybe 10% of typically high achieving students who did ok on the first assessment and then knocked the second assessment out of the park. These students where to buy cialis in USA didn’t have to stay after-school very often to reassess, and they probably spent less time working on math under SBG then they would have in a traditional classroom.
Should I make the quizzes more difficult? Should I cap grades at 85% and to get higher I could design a project for them to get more in depth on a topic? (but if its useful for them, isn’t it useful for everyone?)

To quote Leeloo, Please Help.

(the fifth element is … love, ha. haha. hahahahahaha.)

Posted in Full Posts | Tagged sbg | 8 Comments

Math Notebook Questions

by Dan Posted on June 22, 2011

End of the year reflection time.

In my 7 years of teaching, I’ve never required my students to have a math notebook. The sum total of my speaking on the subject is probably a couple of minutes at the beginning of year, “You need a math notebook or binder to write notes in and keep handouts for class.” Then throughout the year some minor reminders to take out your notebook and we can write some stuff down. With the SBG hoopla this year, I gave out manila folders for the students to keep their old concept quizzes and their grade sheet in. That’s about it.

I’ve never:

had notebook checks
given/graded notebooks
required dividers and pencil pouches
math design on front
et cetera

Have I made an error? Is it worth my valuable time and effort to get the students to construct and maintain a math notebook? Now that I have a 1:1 student:computer ratio should I move in the electronic notes direction? What advice do you have for me? No judgement on my behalf; I’m just wondering if I should change the way I do things.

Posted in Full Posts | Tagged teaching methods | 6 Comments

Dartboard Problem

by Dan Posted on June 20, 2011

From Infinigons, etc: Putting myself in kids’ shoes

You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?

Superb problem.

I took the lazy route. I made a python program to run 100,000 trials, and graph those trials on a coordinate plane.

Visual result:

And the numerical result:

closer to center 21939
closer to edge 78061
ratio is 0.21939

After 1,000,000 trials, the ratio is:

closer to center 219148
closer to edge 780852
ratio is 0.219148

The code for the vpython simulation is found here (and pasted below).

from math import sqrt
import random
from visual import *

#Distance Formula
def distance(xa,ya,xb,yb):
    return sqrt((xa-xb)**2+(ya-yb)**2)

totalthrows = 0
closertoedge = 0
closertocenter = 0

#Creates the White points at the boundries of the screen
points(pos=[(-500,-500,0),(500,-500,0),(500,500,0),(-500,500,0)],size=5, color=color.white)

while (totalthrows < 100000):

    #All the random point coordinates are in the range:
    # 0 <= (x or y coordinate) < 1
    #They are multipled later for display purposes
    xcoord = random.random()
    ycoord = random.random()

    #The min distance to edge is simply the coordinate closest 0 or to 1
    mindistancetoedge = min(xcoord,ycoord,(1-xcoord),(1-ycoord))

    if (mindistancetoedge > distance(xcoord,ycoord,0.5,0.5)):
        #Random point is closest to center
        closertocenter += 1
        points(pos=[(int(1000*xcoord)-500,int(1000*ycoord)-500,0)], size=1, color=color.red)
    else:
        #Random point is closest to edge
        closertoedge += 1
        points(pos=[(int(1000*xcoord)-500,int(1000*ycoord)-500,0)], size=1, color=color.yellow)
    totalthrows+=1

print "closer to center " + str(closertocenter)
print "closer to edge " + str(closertoedge)
print "ratio is " + str(float(closertocenter)/(float(closertoedge+closertocenter)))

Update: Btw, several people have REAL solutions (bravo): @infinignons and Ms. Cookie

Posted in interesting stuff | Tagged calculate this, puzzler, python | 2 Comments

SBG: A Small Tweak and a Feedback Inequality

by Dan Posted on June 16, 2011

The grading system in Geometry this year was described to the students with the following description:

The final concept grade will be out of a grade of 8. The best you can do on the first quiz is a 4 out of 8 (50%). Every concept will quizzed at least twice. If the second grade is higher than the previous grade, then you double the new grade. Otherwise you add the two grades together. Here are some examples:

3 on the first, 4 on the second, means a 8 / 8 grade

2 on the first, 2 on the second, means a 4 / 8 grade

4 on the first, 3 on the second, means a 7 / 8 grade

2 on the first, 2 on the second, 4 on the optional (after school) quiz, means a 8 / 8 grade

That is how the grading system ran for three quarters. For the fourth quarter, I made one small change: The first assessment of a concept was not graded and just feedback was given. In reality this meant that their grade for an assessment was based solely on one assessment, and this was usually a week behind when we actually worked on that concept in class.
At first they didn’t like the change because I think they were relying on the grade to tell them all the information. But after a week they had gotten used to it and I heard no complains.

Pros To Feedback Only On The First Assessment

Students actually read the feedback and they asked more questions about stuff they didn’t get.
Students moved one step up a cognitive scale. They took the feedback and tried to put themselves in my shoes and give themselves a grade.
“Oh, I’d only get a 3.5 because I screwed up solving the equation.”
“This is only a 1 or 2 because I have no idea what I’m doing on this concept.”
Despite my initial fears, the students still worked on a concept quiz that they knew wasn’t going to be graded. I was worried that they’d leave it blank, or give it a cursory try, but there was no drop in effort on the first assessment.
My grade book became a lot easier to manage and explain. The grade on the second look was their grade and any reassessments after school simply replaced the old grade.

Cons To Feedback Only On The First Assessment

Many students still walked into the second reassessment over confident despite the feedback on their first assessment. They knew that they could retake if they needed and buy cialis online legally so not enough effort was being made to knock the assessment out of the park on the first pitch that counted.
Sometimes a kid’s grade on a concept may be a bit lower. If they would have gotten a 4 then a 3.5 on the second assessment then their old grade would be a 7.5/8 and their new grade would be a 7/8. Not a biggie, but it lead to some unnecessary reassessments for the few grade grubbers that I have.

Feedback Thoughts

Overall I’ve increased the length and breadth of my feedback to the students. After being goaded into writing more feedback, I’ve written more on students quizzes than ever before.

I started the year off by giving a grade and some feedback.
To me it looks like this:

But to most students it looks like this:

So I’ve come up with the following inequality (nerd!) to describe the situation:

Feedback > Feedback + Grade > Grade

Do you have any feedback? If you do give me feedback with a grade, be careful, I may not read the feedback carefully.

Update (6/17/2011): Just found this post from (6/6/2011) that talks about very similar things in a great way: http://symmetricblog.wordpress.com/2011/06/06/sbf/

Posted in Full Posts | Tagged sbg | 10 Comments

Geometry SBG Concept list

by Dan Posted on June 13, 2011

To copy Ms. Nowizee from last year, I want to share my Geometry Concept List. This was the first year of Standards Based Grading in Geometry, and before seeing the end of the year results, I’d give it a B (more to follow).

The hard part was designing the list so that each concept was not too big and not too small. If the concept covered too much stuff, then it was difficult for the students to nail down what they knew, and what they didn’t. If the concept was too small then it could make it too obvious to the student how to solve a problem. For example take the Circle topics — please. A concept that is too small is Exterior Angles in Circles, which only essentially has one formula to apply. A concept that is too large would be Circles (circle angles, circle segments, circle proofs). The student may know the circle angles cold, but be totally lost on the proofs.

As the title of this blog would suggest, I’ll keep changing this list as I teach Geometry. Sadly (or not), for the first time in 6 years, I won’t be teaching Geometry next year. I have another section of Computer Programming, and for the first time a section of Pre-Calc Honors. So the list will stay as it is until I teach it again.

No more delay, here you go: link.

Update:
Here is the wall in class where we kept track of the concepts. Was really nice having them up at all times.

Posted in Full Posts | Tagged Geometry, sbg | 3 Comments

Car Talk Puzzler (The Colored Caps)

by Dan Posted on June 12, 2011

The newest car talk puzzler. Draw a diagram, it may help.

Three men, Mr. White, Mr. Brown and Mr. Green, were in the habit of meeting in a local doughnut shop every morning for coffee and doughnuts.

One morning as they were sitting at their usual table, Mr. White remarked, “Hey, will you look at that. We’re each wearing a colored baseball cap today.” One white cap, one brown cap, and one green cap. But interestingly, no one was wearing a cap of the color that matches his name.

At this, the guy wearing the green cap says, “Oh you stup’, that doesn’t mean anything, it’s just coincidence. Shut up and eat your doughnut.”

The question is what color cap is each man wearing?

Answer.

Posted in interesting stuff | Tagged car talk, puzzler | Leave a comment

Why Math? Neil deGrasse Tyson Responds

by Dan Posted on June 11, 2011

Must watch:

Math needs better marketing.

There are people who say “I’ll never need this math, these trig identities”, from 10th grade, or 11th grade, or maybe you never learned them. Here’s the catch. Whether or not you ever again use the math you learned in school; the act of having learned the math established a wiring in your brain that didn’t exist before. And it’s the wiring in your brain that makes you a problem solver.

-Neil deGrasse Tyson

I wish some of today’s leadership would see things from his point of view.

Posted in interesting stuff | Tagged math, technology | 2 Comments

Wonderlic Test

by Dan Posted on June 10, 2011

Wonderlic Sample Test

If you’ve never heard of the Wonderlic, it’s an exam given to players entering the NFL draft every year.

The Wonderlic is an exam comprised of 50 questions. Players have 12 minutes to complete the test. The average score (out of 50) is a 21.

Vince Young made headlines last year when it was rumored that he scored a 6. As we all saw in 2006, the Wonderlic doesn’t mean much. St. Louis’ Ryan Fitzpatrick scored a 50, and he’s nothing more than a backup quarterback.

Some sample scores of other players include: Drew Bledsoe – 37; Steve Young – 33; John Elway – 30; Dan Marino – 16; Deltha O’Neal – 15; Donovan McNabb -14.

Source: http://walterfootball.com/draftwonderlic.php

On that website, they provide 20 sample questions that should be completed in 5 minutes. Of the 20 questions, probably 15 are math related. Some straightforward questions. Some tricky questions. Your students may be interested in the draft and how their intellect compares to a college quarterback. So I put the questions (and the answers) in a google doc for easy printing. Enjoy. (I got 2 wrong, sigh.)

Posted in interesting stuff | Tagged fun stuff, interesting stuff | Leave a comment

Scavenger Hunt: Geometry Regents Review

by Dan Posted on June 9, 2011

Quick post:

In NY our end of the year exam is the Regents exam (no comment). The schedule of the exams this year means that I won’t see my students for 13 days. Great.

I made up a Geometry Regents Scavenger Hunt found here: http://bit.ly/qbyregentsreview
The students solve one set of problems (all the problems are old geometry regents questions), and combine the answers in some way to find the link to the next problem (hence the “scavenger hunt”). All you have to do is print the first page, and hand that to the students. Two previous scavenger hunts are found here and here. I’ve made the last page editable by anyone, so students can tag their name on the document.

Posted in interesting stuff | Tagged Geometry, regents, scavenger hunt | 4 Comments