Wednesday, March 4, 2015

Standardized test prep -- A circle and rhombus problem

[This is a repost but it seems to be a good fit with the test series we've been running.]

In an earlier post, we talked about "step-back problems." The idea is that, wherever possible, each problem should be associated with at least one problem that uses similar format and relies on similar concepts but which "steps up" (is more difficult) or "steps down" (is easier).

In that previous post we talked about problems where you had to find the shaded area of a circle. This problem covers similar territory but takes things up a notch.

Circle 1 and Circle 2 both have radius 2. Each passes through the center of the other. Find the area of the rhombus formed by the two points of intersection (A and B) and the centers of each circle (C1 and C2).







Solution after the break.


Monday, March 2, 2015

When discipline crosses the line

I've been meaning to write this up for a while now, but recent news about attrition (see here and here for the conversation up to now) has brought the issue back to the forefront.

When you take a close at the increasingly dominant charter model (the "no-excuses" school) and some of the highly touted success stories (such as the KIPP schools), you will soon notice how extreme some of the discipline can be.
A tiny padded room at KIPP Star Washington Heights Elementary School was a real-life nightmare for two young boys who were repeatedly detained in the tot cells, the Daily News has learned.

The students, who were enrolled in kindergarten and first grade at the highly regarded charter school, were both removed by their parents in the past two weeks after they suffered anxiety attacks as a result of their confinement.

“He was crying hysterically,” said Teneka Hall, 28, a full-time Washington Heights mom whose son, Xavier, was rushed to the hospital after he panicked and wet himself while he was holed up in the padded room. “It’s no way to treat a child.”

The school’s so-called “calm-down” room is small, about the size of a walk-in closet, said Hall, who visited it with her son at the start of the school year. It’s empty, but for a soft mat lining the floor and a single light on the ceiling.

The room’s only window is an approximately 2-foot by 3-foot panel in the single door. It’s partially covered so staffers can look inside, but children cannot. Students were placed in the room, alone, for 15 to 20 minutes at a stretch, their parents said.State law requires that children placed in a time-out setting be in a space where they can be seen and heard “continuously,” but it does not require adults to be in the room where children are stashed.

When 5-year-old Xavier was confined to the room on Dec. 3, he suffered an anxiety attack so severe that staffers called for emergency workers to take him to the hospital.

“I was scared,” said Xavier, who was taken to New York Presbyterian and released to his mom, who pulled him from the charter and enrolled him in another school immediately.
There are two points that cannot be overemphasized here.

The first is how rough, even traumatic, this and other policies of the get-tough, "no excuses" schools can be. There are kids who thrive in highly structured and disciplined environments, but there are many others who respond with varying degrees of anxiety, depression and/or anger. Then, to add injury to injury, this psychological toll is matched with an educational one. These kids are denied instruction through suspensions then forced out and sent to other generally underfunded schools, often in the middle of the year, a practice which maximizes the disruption and minimizes the chance to learn.

The second is the way the incentive system of the reform encourages these often brutal policies. These policies are an extraordinarily effective way of getting rid of kids whom you can't handle or who put a drain on you resources. The result is that the very thing that traumatizes these children produces promotions for administrators and funding increases for their schools.

Monday Video -- Max Maven predicts the future

[Every Monday for the next few months, we'll be posting a short video clip here at You Do the Math. All will have at least a tenuous connection to science, technology, engineering, or mathematics. Teachers can use these as writing prompts or as starting points for larger lesson plans (I'll try to include some hints now and then), but the main purpose is simply to have a little fun.]

In 1987, an otherwise forgettable Dick Clark production featured a number of tricks from mentalist Max Maven. This piece of interactive magic from the show makes for an interesting classroom exercise (though I might skip the first fifteen second -- somethings don't age well). I would print out copies so that the kids could each follow along at their desks. After running the video and having all of the kids play along, I would diagram out all the permutations and show how each path leads to the red circle.








Wednesday, February 25, 2015

Standardized test prep -- Another circle problem

[Another SAT/GRE style geometry problem, slightly more advanced than the last one.]




The circle pictured here has radius 5 with a center indicated by the black dot. What is the area of the shaded region?

How to get started: You may have already thought of this own your own (it’s OK to get ahead of the class here), but if you’re stuck, it’s often a good idea  to ask yourself “is there a simpler related question I can answer?” In this case, how about finding the area of the whole circle?



Now we just need to figure out what part of the circle is shaded. Since the angle corresponding to the shaded region is vertical with the angle that measures x degrees, x degrees is also the measure of that angle. That means that the ratio of x to 360 is the same as the ratio of the area of the shaded region to the area of the circle.

How do we figure out what x is? Look at the two labeled angles. Put together they form a straight line. That means that the two angles are supplementary.

That gives us x equals thirty and since thirty is one twelfth of 360…


The area of the shaded region is one twelfth the area of the circle.



Monday, February 23, 2015

Monday Video -- Möbius Maglev

[Every Monday for the next few months, we'll be posting a short video clip here at You Do the Math. All will have at least a tenuous connection to science, technology, engineering, or mathematics. Teachers can use these as writing prompts or as starting points for larger lesson plans (I'll try to include some hints now and then), but the main purpose is simply to have a little fun.]


Levitating Superconductor on a Möbius strip




Wednesday, February 18, 2015

Standardized test prep -- A simple circle problem

[I'm working on a larger project involving test prep for the math sections of tests such as the SAT and GRE. The exact details are still up in the air. I'm pretty sure that there will be various online components but the details are very much still in flux. In the meantime, I'll be posting some examples to get the ball rolling.]
















The circle pictured here has radius 7. Find the shaded area.


Monday, February 9, 2015

Monday Video -- very slow lightning

[Every Monday for the next few months, we'll be posting a short video clip here at You Do the Math. All will have at least a tenuous connection to science, technology, engineering, or mathematics. Teachers can use these as writing prompts or as starting points for larger lesson plans (I'll try to include some hints now and then), but the main purpose is simply to have a little fun.]


Three Strokes of Upward Lightning