This past Friday, my school had an inservice day for the teachers. I was responsible for a small portion of the day. It was my task to provide a demonstration of the Four Corners differentiated instruction strategy to the rest of the faculty. I decided the best way to accomplish that was to actually go through the activity instead of just telling my colleagues about it. So of course, I based the activity on a mathematical statement. Usually, any discussion of mathematics with the faculty in general is met by moans, groans, and eye-rolling. This time, I was surprised. But I’ll get to that in a minute.Four Corners works like this:

1. Teacher writes a debateable or controversial statement on the board.
2. Students move to the corner of the room where the sign is posted that most closely represents their opinion on the statement.
3. Discussion, questions, and debate are then allowed between the four groups.
4. Next, students are allowed to switch to another corner if they changed their opinion.
5. Students are invited to explain what caused any change of opinion.

The statement I posed to the faculty was this:

There are more integers total (… -3, -2, -1, 0, 1, 2, 3, …) than there are decimal numbers just between 0 and 1.

The four statements that I posted in the four corners were:

I was quite surprised by the amount of discussion that my statement generated. It was very interesting to see how “non-math” people think about mathematics in a purely theoretical setting. The faculty really “dug” the discussion and many of them switched corners as thoughts and arguements were shared.

Ultimately, they ended up fairly evenly divided amongst the four corners. By the end of the activity, they were begging me to tell them which response was actually the correct one (I had told them that one and only one of the responses was indeed correct). I eventually did tell them. But in this post, I’ll leave it to one (or more) of my readers to leave a comment as to which response should be chosen to the above statement.

Here is an excerpt from an Associated Press report that was published in my local newspaper this past Sunday (17 Sept. 2006). Some of the information presented in it raises some questions in my mind. I’ve outlined the key statistics in red.

So from the article here’s a summary of the statistics:

In 2004, the percentage of certain borrowers that “paid a higher-than-typical interest rate on their home mortgages” are:

• 32.4% for blacks
• 20.3% for Hispanics
• 8.7% for whites
• 11.5% (all borrowers)

I’m wondering how such a small percentage of all classes are paying “higher-than-typical” rates. Doesn’t logic tell us that exactly 50% of all borrowers will pay a higher-than-typical rate and the other 50% will pay a lower-than-typical rate? Certainly this would be true if the definition of “typical” in this case was “average” or “mean.” So perhaps, “typical” indicates a range of interest rates. Then, the 11.5% indicates the percentage of all borrowers that had interest rates above that range. But if that is the case, it would be very helpful to know the size of that range. Any thoughts?

I tried something new today for a review over the basic probability chapter in my Statistics textbook. I created worksheets that contained approximately 50 exercises from the topics we have been discussing. I then assigned each problem a point value - easier problems were worth one point, harder problems were worth 3 points, and ones in the middle were worth 2 points.

Then I told the students that they had two tasks to complete today. The first was to complete 15 points worth of problems to be turned in and graded. This way, they could choose any combination of problems to solve whose combined point value was at least 15 points. The second task was for them to correctly solve a problem of their choosing and then present that solution to the class. If they chose to solve and present a 3-point problem, then they would earn 3 mythical extra credit points. If they chose to solve and present 1-point problem, they had an easier time at it, but the rewards were less (1 point extra credit).

This seemed to work very well today. The students enjoyed having the ability to choose which problems to solve. They also enjoyed listening to their peers explain solutions and rationale to the exercises.

[The phrase “mythical extra credit points” is one that I saw used by Dan Greene on his blog. As a math teacher, and I’m sure others can attest to this, I know that giving a student 3 or 5 or x extra credit points can mean as much or as little as you want it to. “Mythical” seems so fitting.]

Here’s a quick (maybe) brainteaser from Elias over at Ramblings of an Australian Teacher.

In the multiplication

P Q R
x 3
——
Q Q Q

each of P, Q and R represents a different digit. The sum of P, Q and R is
(A) 16 (B) 14 (C) 13 (D) 12 (E) 10

Think you know the answer and can give a clear explanation? Then email Elias the solution (and post it in a comment here).

The last post contained a fairly easy problem. Now here’s one that is a bit more challenging.

In the figure below, a circle with a center at O and a radius of 4 is inscribed in triangle ABC. Point D is a point of tangency. Segment AD has a length of 8 and segment BD has a length of 6. Find the lengths of the three sides of the triangle — without using trigonometry (sine, cosine, tangent).

[This problem was posed to some mathematics teachers by Victor Katz at the San Antonio national conference of the NCTM.]

Here’s a problem that is a good one to give your Algebra 2 students (or anyone who’s interested).

Find the two distinct numbers whose product, quotient, and difference are all equivalent.

[in layman’s terms] Find two different numbers that, when you multiply them, you get the same thing you do when you subtract them or divide them.

I’ll not give the answer in this post. You give it to me in a comment.

For those of you who teach Statistics… here’s an idea that may help your students get a better grasp on what the mean and standard deviation really are. How about asking them to come up with a set of data that has a given size, mean, and standard deviation? For example: Give me a list of 15 numbers that have a mean of 182 and a standard deviation of 3.5. Using a calculator and a lot of trial and error, most students would be able to get really close.

The learning comes through all the trial and error. They will see how changing the data affects the mean and standard deviation. After a few different problems like that, I think that their understanding of the concepts will be much better than just using formulas and reading a definition out of a book.