Group Tests - A summary

In my last post (http://robebymathdude.blogspot.com/2014/02/v-behaviorurldefaultvmlo.html), I mentioned at the end that I would be trying group tests this semester in some of my classes.  I want to write a little bit about that here.  First a few related details: 

•  I am scheduled to present on this topic at the AMATYC national conference in November in Nashville, TN.  (http://www.amatyc.org/?page=2014ConfHome)
•  I was motivated to do this by a post at Casting Out Nines by Robert Talbert.  (http://chronicle.com/blognetwork/castingoutnines/2012/05/14/turn-to-your-neighbor-and-take-a-test/) though in his case it was for a summer class and it was for Calculus I students, so a slightly different set of students.  
•   At my school, we have a departmental final exam (with versions) that each student has to take, so I also have to ‘train’ my students toward that also.

Since I was doing this over the long semester, and as I mentioned each student must take an individual final exam at the end of the course, I chose to set up my exams as two parts.  There would be a take home part that the group would work on, and then an in class part that was individual.  However, each student was warned that some of the in class part would include some ‘extension’ type questions and some ‘traditional’ exam questions.

For example, for the “Math for Liberal Arts” class I had these two question as part of the take home part of exam one: (both are based upon questions in the text A Survey of Mathematics by Angel, Abbott, Runde).  The students were given the take home part a week before the in class exam, and the take home part was due at the end of class 2 days before the exam.  The class day it was due was a “review day” where I went around the room to answer any questions the students had over the material, including anything from the take home part.  After the take home parts were turned in, I would post the take home part with answers for the students to review for the in class portion of the exam. Anyway, here are two of the questions and what I did on the in class exam. 

TAKE HOME QUESTION 1:      A yardstick measures 1/4 by 3 by 36 inches. How many yard sticks will fit in a box: 

•  3 inches wide and 36 inches high, if the girth of the box is 30 inches? 
•  6 inches wide and 36 inches high, if the girth of the box is 30 inches?
•  3 inches wide and 72 inches high, if the girth of the box is 30 inches?
•  3 inches wide and 36 inches high, if the girth of the box is 60 inches?
•   6 inches wide and 72 inches high, if the girth of the box is 30 inches?
•  6 inches wide and 36 inches high, if the girth of the box is 60 inches? 
•  3 inches wide and 72 inches high, if the girth of the box is 60 inches?

TAKE HOME QUESTION 2:     An average newspaper contains at least 16 pages and at most 87 pages. How many newspapers must be collected to be certain: that at least two newspapers have the same number of pages?

I.            That at least two newspapers have the same number of pages?

II.            That at least three newspapers have the same number of pages?

III.            That at least four newspapers have the same number of pages?

The students had a few days to write up their solutions with explanation as a group.  Then the individual exam in class had these two follow up questions as part of the exam:

EXAM QUESTION 1:   On the take home part you were given the following problem:

*** A yardstick measures 1/4 by 3 by 36 inches. How many yard sticks will fit in a box:

• 3 inches wide and 36 inches high, if the girth of the box is 30 inches?
• 6 inches wide and 36 inches high, if the girth of the box is 30 inches?
• 3 inches wide and 72 inches high, if the girth of the box is 30 inches?
• 3 inches wide and 36 inches high, if the girth of the box is 60 inches?
• 6 inches wide and 72 inches high, if the girth of the box is 30 inches?
• 6 inches wide and 36 inches high, if the girth of the box is 60 inches?
• 3 inches wide and 72 inches high, if the girth of the box is 60 inches?

***

Your answers should have been 120, 240, 240, 240, 480, 480, and 480 respectively.

So how many of these yardsticks will fit into a box that is 6 inches wide and 72 inches high, if the girth of the box is 60 inches?

How many will fit into a box that is 3*R inches wide and 36*R inches high, if the girth of the box is 30*R inches, where R is some positive number greater than one?  Explain how you know!

EXAM QUESTION 2:   On the take home part you were given the following problem:

***        An average newspaper contains at least 16 pages and at most 87 pages. How many newspapers must be collected to be certain: that at least two newspapers have the same number of pages?

•  That at least two newspapers have the same number of pages?
•  That at least three newspapers have the same number of pages?
•  That at least four newspapers have the same number of pages?

***

Your answers should have been 73, 145, and 217 respectively. 

So how many should we take to make sure that at least 42 papers have the same number of pages? 

What is the formula to use to make sure that there are at least R papers with the same number of pages, where R could be any positive integer?  (You should write the formula in both mathematical symbols AND then in words)

Overall the students seemed to enjoy the format.  I informally interviewed most of the students, and all of the ones I interviewed liked the format of the exams, and the idea of the extension questions.  Several of them commented that it really helped them to understand the idea of mathematics as the process of finding patterns.  It worked so well that my Calculus I students get to experience these this semester.

Thanks for reading, and as always, comments are welcome.