Mtht 400 The geometry of inequality
Homework due Oct. 26, 2005
Rubric and answer sheet.
Scores were generally good; most of the difficulty was in clarifying the mathematical versus sociological issues. If you got less than or equal 11 and want to make-up, read one of the other papers in the Renewing Mathmematics volume and write a brief (1/2 page typed) description laying out the mathematical core and reflecting on its possible effectiveness in a classroom. For your amusement, I am posting an excel spread sheet
Showing the wide variance of answers you provided for the answer to question 2. I guess this validates Scott’s view that these are estimates.
1. How many `square blocks’ are there in a circle of radius 3 miles?
(A `block’ in
Everyone calculated this correctly for 2 points. Some got an extra point for a more refined estimate, using the integer part of the arc sin to count the number of square blocks. There was about a 5% difference.
2. Estimate the number of liquor stores, community centers, theaters and churches in a circle of radius 3 miles centered at Racine and Harrison.
I expected you to count the number of each category by hand
in a small area and multiply. Doing this
myself, in the 24 square blocks bounded
by Ashland, Morgan, Roosevelt, and Harrison I found 7 churches, 2 community
centers, 4 liquor stores, and 1 theater.
Taking someone’s careful calculation that there were 1715 square blocks withing 3 miles of Harrison and Racine (not counting
No one else did this. Most people counted the entire area (via an internet site) or from phonebook. A couple of people included the ratio aspect of the problem by using the internet counts on larger areas. I gave 2/3 for an internet count. An additional point if you preserved the ratio aspect of Brantlinger’s approach.
4. What are the mathematical issues addressed in this lesson? From your experience are there mathematical problems here that 9th and 10th grade students will have trouble with?
Here is a list of mathematical issues that arose:
1) area: Linear measure vrs square measure
2) Covering a circle with squares
3) Estimation – sampling – how big a sample should one choose
4) Basic notions of statistics
5) problems with division
6) The use of arc sin by several of you in answering question 1. shows how this problem could be adapted to an 11th grade class with
7) the notion of function
8) area
9) map reading
5. What are some problems in `scientific sociology’ involved in making the estimates suggested in this article and my problem 2?
Here are some of the issues that arose:
1) experiment versus surveying
2) difficulties of defining categories – is a bar a liquor store? I called a boxing club a community center in my survey – was that appropriate?
3) assumptions involved in sampling. Essentially one is either assuming the population is homogeneous or taking a sufficiently random sample that the inhomogeneiety is accounted for.
4) possible bias in sampling (Some confused this with bias in interpreting the results.)
6. What are some of the advantages and disadvantages of this lesson?
Here are some of the issues that arose:
Advantages:
1) attracts students
2) authentic problem – shows connections of math to real world
3) opportunity for group work
4) critical thinking
5) socially relevant
Disadvantages:
No one in this group mentioned the first disadvantage I heard from teachers: the danger of offending someone: students, administrators or parents.
1) too time consuming
2) not enough math content for the time; does it actually teach geometry?
3) not enough time for exploring the social science
4) Is this an injection of politics into the classroom
5) danger of reinforcing negative stereotypes
6) And as one of you pointed out, students would use the internet and miss the math. (This could be fixed by specifying that the students have to count.)