Monthly Archives: October 2014

A few years ago, one of our local elementary schools had a 50th anniversary celebration. The principal of the school contacted me regarding an anniversary photo the school wished to take with students and staff. The principal asked our Math Theory & Problem Solving class to come up with the "most aesthetically" pleasing dimensions for the photo. Our class was tasked with using mathematical methods to systematically design the dimensions of both digits.

Here's a quote from the newspaper article about the photo before it was taken:

Browning, who has taught at the school since 1976, starting as a music instructor and then principal in 1998, said Friday would begin with the weekly assembly in the gym.

“Each class will perform a song from each decade, starting with the 1960s,” she said. Browning added the day-long celebration for students would also include special drawings for prizes, carnival games and the group photograph – if it’s not raining. The students and staff will form a “5-0” and have their picture taken by Downey’s Photography from the Scottsbluff Fire Department’s aerial ladder truck.

The principal emailed some information about how many students and staff would be involved in the photo, the approximate height of the aerial ladder, and that the photo would be taken in a field adjacent to the elementary school.

MonaLisaGoldenRatioOur class used mathematics to figure out the size of the viewing field of the camera (based on conservative estimates regarding the camera lens viewing angle). The students did some research and found information on the Golden Ratio, a number that appears over and over again in artwork.

The students wrote instructions for the staff members to utilize when organizing the photograph. We discussed the challenge of keeping that many students organized and engaged for a sufficient period of time in which the photographer could take the photo.

Below is a screenshot of the photo taken to celebrate the 50th anniversary of Westmoor Elementary.

(photo by Downey Photography)

Westmoor50

 

After the photo was taken, I asked my students, "If we compute the ratio of the width of each number to the height of each number, how close is the actual value to our recommendation of the golden ratio (approximately 1.618)?"

Below are screenshots from the Geogebra worksheet examining how close the dimensions are to the target value of 1.618.

Westmoor-50-Question

Screenshot with the initial question. Students can use measurement tools to judge whether or not the 5 and 0 in the photo meet the desired dimensions.

Westmoor-50-Solution

Clicking on the checkbox in the Geogebra sheet (Show / Hide Measurements and Ratios) reveals the details in evaluating how close the ratio of height to width for each digit is to the Golden Ratio.

Westmoor-50-Solution-With-Details

Here are the details to the Geogebra sheet. If you would like to download the Geogebra sheet and mess with the values to see what happens, you can find the Geogebra sheet here.

I was working on preparing problems for precalculus class on inverse functions. I started to type the following run-of-the-mill problem type:

If f(x) = blah and g(x) = bleh, show f(x) and g(x) are inverse functions.

And the work usually goes something like this:

f(g(x)) = f(bleh) = algebra kung fu happens here = x
g(f(x)) = g(blah) = some more algebra kung fu = x
Then conclude f(x) and g(x) are inverses.

Without thinking about it, I typed f(x) = -x - 6. Then I wrote the statement on my pad of paper, started working... exchanged the y and x, solved for y.... and got the exact same function.

y = -x - 6. Hmmm.

A part of me wondered if I had made a careless error. Double checked. Nope. No error. I wondered if the graph of the function was its own reflection across the y = x line.

InverseFunction

Sure enough... Since y = -x - 6 is perpendicular to the line y=x, it will be its own reflection across the y=x line and consequently its own inverse. In fact, this made me think of an interesting question to pose to my students...

"Can you define a class of linear functions that are all their own inverses?"

In hindsight, perhaps I should be more mindful when constructing tasks for my students. But then again, this would be a great discussion to have with my students. Why does the function end up being its own inverse? Can we think of non-linear functions that are their own inverse? Can we define these classes of functions carefully?

2014-10-02_1234This post contains resources for the talk "Using Statistics in Mathematics Classes" given by Jason Vitosh (Falls City High School, Falls City, NE) and myself at the Midwest Regional Noyce Conference on Thursday, October 2 from 2:15 pm - 3:00 pm.

SlideImage

 

Click on the link below to access the presentation file containing resources, images, and links.

10-2-14 Stats is not Math

 

 

 

 

 

 

 

 

 

 

lkajsdfl