Math 130
Spring 04
Name _______________________________
Practice Test Problems Taken from Old Tests
For the real test, you will choose
two of the three problems to solve. You
may use a calculator, manipulatives, scrap paper, and a sheet of notes. If you wish to use a spreadsheet, you will
sign in at the front of the room if many people wish to use the computers, you
may be asked to relinquish yours after ten minutes.
If you don’t understand a question, after reading it carefully, feel free to ask for help.
1. Rectangle Trains:
Below are two “rectangle trains” of length 3 and 5, respectively:
Rectangle Train Length 3 Rectangle
Train Length 5
Note that the Rectangle Train of Length 3 contains 6 rectangles of various sizes that follow the original outlines (where the shaded rectangles are the ones being counted):
a. How many rectangles does the Rectangle Train of Length 4 contain? Convince a skeptic that your answer is correct.
b. How many rectangles does the Rectangle Train of Length 5 contain? Convince a skeptic that your answer is correct.
c. Make a table that records the length of the rectangle trains and the number of rectangles each train contains. Make your table go from 1 to at least 8.
d. What patterns do you notice in your table? List as many as possible.
e. How could you convince a skeptic that your patterns will continue?
f. How many rectangles are there in a train of length 50? Explain.
g. How many rectangles are there in a train of length n? Explain.
Scoring Guide for Problem 1
5 -- Solves all parts of the problem correctly and includes good justifications for all parts of the problem.
4 Excellent work on parts a-f (or equivalent
combination).
3 Excellent work on a-e and one of f-g (or equivalent combination).
2 Excellent
work on a-c.
1 Solves a and b.
0 -- Not up to a 1.
2. Speed Dating
In heterosexual speed dating, men and women come to an event, and have seven-minute “speed dates,” with each date involving one man and one woman.
i. Suppose ten men and ten women come to a speed-dating event and that every possible male-female pair has a “date.” How many “dates” take place?
ii. How would you convince a skeptic that your answer to part i. is correct?
iii. Suppose N men and N women come to a heterosexual speed-dating event. What is the maximum number of different dates that could take place?
iv. How could you convince a skeptic that your answer to part iii. is correct?
B) Bisexual Speed-Dating
In bisexual speed dating, “dates” can take place between a man and a woman, between two men, or between two women, i.e., between any two people at the event.
i. Suppose ten people come to a bisexual speed-dating event, and all possible “dates” take place. How many dates take place?
ii. How could you convince a skeptic that your answer in part i. is correct?
iii. What is the minimum number of people that could attend a bisexual speed-dating event in which over 200 (different) “dates” take place? Explain.
iv. Find an equation for the maximum number of dates that can take place if n people attend a bisexual speed-dating event.
v. Explain how you could convince a skeptic that your answer in part iv. is correct.
C) Intergalactic Speed Dating
On another planet, the population consists of “life-forms,” which are not distinguished by any kind of gender, and all “dates” involve three life-forms.
i. Suppose six intergalactic life-forms, A, B, C, D, E, and F show up for speed dating. Suppose that every possible triple of dates takes place, where, for example, ABC and ABD represent different triples. How many dates take place?
ii. How could you convince a skeptic that your answer in part i. is correct?
Scoring Guide for Problem 2
5 -- Solves all parts of the problem, with good justifications.
4 -- Solves most of A, B, and C correctly, with a few mistakes.
3 Solves two of A, B, and C correctly or
equivalent.
2 Solves A, B, or C correctly or equivalent.
1 Solves Ai or Bi correctly.
0 -- Not up to a 1.
Score for Problem 2: ___________
3. Toothed Squares
The motifs below are common motifs
in the art of women in
Note that the number of small squares includes both black and white squares, that the width is the width of the figure at its widest point, and the perimeter refers to the perimeter of the outside of the figure.
a. Assuming that the pattern of motifs continues, make a table of the figure number, number of small squares, width and perimeter for each figure.
Figure Number # Small squares Width Perimeter
0 1 1 4
1 5 3 12
2 13 5 20
3 25 7 28
4
5
6
7
b. What patterns do you notice in your table? Find at least four, including at least one in the each of the last three columns.
c. How could you convince a skeptic that your patterns will continue? Be sure to refer to the toothed square design to link the original pattern to the numbers in your table.
d. What would be the number of small squares, width, and perimeter for figure 100? How could you convince a skeptic that you are right?
# of small squares ______________
width _______________________
perimeter _____________________
e. What would be the number of small squares, width, and perimeter for figure n, where n is a natural number? Give formulas for each and explain how you know your formulas are correct.
Scoring Guide for Problem 3
5 -- Solves parts a-e of the problem correctly (may include minor mistakes) with good justifications.
4 -- Solves parts a-d correctly OR finds at least two formulas in part e, but leaves out justifications in part c or d.
3 -- Finds good patterns and either justifies at least one pattern correctly and makes progress on finding the numbers for figure 100 and at least one equation or finds correct numbers and at least one equation for figure 100 and doesn’t justify patterns.
2 -- Finds good patterns and makes a good start on either justifying or solving the problem for figure 100.
1 -- Makes a correct table and finds patterns.
0 -- Not up to a 1.
Score for Problem 3: ______________
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
To Commentary To Instructions Doc File PDF File