Math 151

Fall 99

Midterm Pyramid Exam

 

The Rules:

            * The three parts of the exam count equally.  They will be scored and recorded separately.

 

First part of the exam, general directions:

            * Work is to be done individually. 

            * If you don't understand a question, feel free to ask.

            * You may use a calculator and any manipulatives you would like.  Feel free to go to the cabinet to get manipulatives, or ask for help.

            * If you wish to use a computer, feel free to get up and do so.  Since we have more students than computers, access will be rationed if there is a problem. 

            * The parts of questions that are marked with one asterisk comprise part one of the exam.  Work on these parts first.  If you have extra time, begin working on other parts.

            * Look at the scoring guide: remember, you need to explain how you did a problem and how you know your answers are correct.

 

Second part of the exam:

            * For this part of the exam, you will work in groups of five.

            * If your group wishes, you may split into one group of two and one of three.

            * The only people you are allowed to work with for this part of the exam are people in your group.

            * For the second part of the exam, you are responsible for all problems with one or two asterisks beside them.

            * Turn in one paper per group.  Papers are due Friday October 22 at the beginning of class.

            * Questions will be scored using the scoring guidelines, however, each question now contains more parts.

           

Third part of the exam

            * The class will work together on this part of the exam. 

            * You will have the entire class period on Friday October 22 to work together.

            * For this part of the exam, you are responsible for problems with one, two, or three asterisks beside them.

            * You will submit one paper for the entire class; this paper is due Monday October 25.  You may include photocopies of portions of your group papers as part of this paper.

            * On Monday October 25, there will be an oral portion of the exam.  You are all equally responsible for all of the material.

            * The scoring scale is basically the same here, however, it will also be based on how well the entire class understands the answers. 

 

 

Scoring Guidelines:

            * Questions will be scored separately and scores will be recorded separately.

            * The various parts of each question will be scored together.  These parts change depending on the phase of the exam.

            * The three questions will be scored according to the following general rubric:

                        Level 5 -- Answers the question accurately and completely, with a very                                                             convincing justification.

                        Level 4 -- Answers the question accurately and completely, with the                                                      exception of possibly minor mistakes, with a reasonably                                                             convincing justification.

                        Level 3 -- Answers the question accurately and completely, with the                                                      exception of possibly minor mistakes without a good                                                     justification OR does some strong work on the problem                                                      with good justification.

                        Level 2 -- Does some strong work on the problem without good                                                                       justification or does fair work on the problem                                                                with good justification.

                        Level 1 -- Does some work on the problem, gets started.

                        Level 0 -- Doesn't really get started on the problem.

 

 

1.  Cuisinaire Rod Trains

 

            The manipulatives on your tables are called Cuisinaire Rods.  If the white rod is a unit, then there are Cuisinaire Rods measuring from one to ten units.  A "rod train" is a set of rods placed end-to-end, where order matters.  For example, the trains RED -WHITE and WHITE-RED are distinct trains, both equal in length to the light green rod.

 

For part 1 of the exam, please choose one of parts A or B to complete.

 

* A) How many different rod trains consisting solely of red and/or white rods are there equal in length to the orange rod?

 

* B) How many different rod trains consisting of exactly three rods are there equal in length to the orange rod?

 

** C) How many different rod trains, with no restrictions on the types of rods, are there equal in length to the orange rod?

 

*** D) Generalize parts A, B, and C to any length rod.  What can you say about other similar problems (e.g. use red, white, light green only or use 4 rods, etc.)?

 

2. Trio Schedules

 

            This problem is a generalization of the appointment problem we did in class.  Now we would still like to schedule separate appointments at 1:00, 2:00, etc., however, each appointment will be for three people.  The goal in scheduling is that every pair of people meets exactly once.   Thus, if Heather, Lisa, and Emma meet at 1:00, then Heather will not meet Lisa or Emma at any of the other times.  We will call such a schedule a trio schedule.

 

*A) Is it possible to design a trio schedule for 6 people?  If it is possible, do it.  If it isn't possible, explain why not.

 

*B) Design a trio schedule for 9 people.  Explain your strategy. 

 

**C) For which numbers of people is it theoretically possible to design a trio schedule?  (For example, in the problem where meetings had two people, it was clearly impossible to make a schedule with an odd number of people.  In this problem, clearly we need to start with a multiple of three, but the number of pairs of people who need to meet might also impose additional constraints on making a schedule.)

 

****D) (optional -- this is hard!)  Design a trio schedule for 15 people.

 

3.  Maximum Products

 

            Let n be a positive integer (i.e. a whole number).  Your goal in this problem is to find a set of positive integers that adds to n, with the product of these integers as large as possible.  We will call such a set the maximal set and we will call the product of its elements the maximal product.  Repetitions are allowed.  For example, for n=10, two possibilities are S = {1,2,3,4} and T={1,1,1,2,5}.  The elements in S and T each add up to 10.  The product of the elements in S is 1x2x3x4=24 and the product of the elements in T is 10.  Neither of these products is maximal however. 

 

* A)  For each integer from 1 to 10, find a maximal set and the maximal product.  

 

* B) What patterns have you observed so far in finding maximal sets?  What rules might you use in constructing them for bigger numbers?

 

** C) Find the maximal set and the maximal product for integers from 11 to 20.

 

** D) Find the maximal products for 100, 101, 102, and 103.

 

*** E) Explain how to find the maximal product for any integer.

 

Copyright 2005, Debra K. Borkovitz.  You may copy or edit this material for non-profit, educational use only.

 

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