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Math 302A Review for Final
The final is a 2 hour exam that will have a slightly different arrangement
from the in
class midterms. There will be three different parts to the final part 1 will be
short answer
computation problems, part 2 will be similar to the midterms in format, and part
3 will be
essay.
• Representations, Connections, and Problem Solving
1. In a barnyard there are chickens and rabbits. If there are 50 heads and
120 feet,
how many of each animal are there?
2. Caroline’s name and address starts getting distributed to advertisement
companies
that start sending her catalogs in the mail. On day one Caroline receives 1
catalog, day two she receives 3 catalogs, day three she receives 5 catalogs, day
four she receives 7 catalogs, and so on. If she does not get rid of any
catalogs,
how many would she have on day 200? How do you know this? (once you see the
pattern it may help to justify it by representing the number of catlogs Caroline
has on two consecutive days in a clever way and shade in the portion of the
second
model that represent the new catalog).
3. Let’s say you have a 4 gallon pail and a 9 gallon pail.
(a) How can you measure exactly 1 gallon of water?
(b) How can you measure exactly 3 gallons of water?
(c) How can you measure exactly 6 gallons of water?
1. What comes after each of the following numbers in the given base:
(a) 124_{5}
(b) 99_{12}.
(c) AAA_{11}, where the symbols (from smallest to largest) are 0, 1, 2, 3, 4, 5, 6,
7,
8, 9, A.
(d) F9_{16}, where the symbols (from smallest to largest) are 0, 1, 2, 3, 4, 5, 6,
7,
8, 9, A, B, C, D, E, F.
(e) DEF_{16}, where the symbols (from smallest to largest) are 0, 1, 2, 3, 4, 5, 6,
7,
8, 9, A, B, C, D, E, F.
2. Convert the given number in one base to the given base:
(a) 124_{5} to bas 10.
(b) 329_{10} to base 8.
(c) 329_{10} to base 12.
(d) 213_{5} to base 2.
3. Recall the Roman number system: I = 1, V = 5, X = 10, L = 50, C = 100,
D = 500, and M = 1000. Position of the symbols matters in the sense that
IX = 9 and XI = 11.
(a) Write 444 using Roman numerals.
(b) Convert XIX, CMIV , and MCMXCV I to our base 10 system.
(c) Solve: XXI + XC.
(d) Solve: CXC − CLX.
4. Find all bases a such that 21_{a} describes a quantity greater than or equal
to the
quantity represented by 8_{10} and less than or equal to the quantity represented
by
24_{10}.
5. Which quantity is larger: 743_{8} or 743_{9}? Justify.
6. List any similarities between the process you use when comparing the
quantities
14_{5} and 14_{6} and the process you use when comparing 6/13 and 11/25 .
1. Describe a connection between student’s addition
strategies and a student’s understanding
of the base 10 number system.
2. Identify and explain the error in each of the following
cases:
(a) 70 − 48 = 28
(b) 76 − 48 = 32
(c) 700 − 482 = 228
3. Explain why 26 + 47 = 27 + 46 = 46 + 27, but not equal to 62 + 74.
1. Your friend teaches 7th grade and wants to revisit
division strategies in her classroom.
Help your friend by modifying one of the division strategies covered in
class so that it incorporates the use of negative numbers
2. Compute 132_{4} × 23_{4}.
3. Write a word problem in the partitive context for 46 ÷ 6.
4. Modeling 37 as 3 longs and 7 units and 52 as 5 longs and 2 units, diagram
37×52
relying on the long and unit representations of the numbers.
• Number Theory
1. Compute the following values : GCF(1,3), GCF(2,4),
GCF(3,5), GCF(4,6), GCF(5,7),
GCF(6,8), GCF(7,9), GCF(8,10), GCF(36,38), and GCF(65,67).
2. Basing your answer on what you answered in the previous
example , what could
you say about GCF(a, a+2)? As best as you can explain why your answer is
correct.
3. Do you think a similar statement is true for
GCF(a,a+3)? If so, what would it
be?
4. If a set of bathroom tiles measure 15cm by 20cm, what
is the smalles square
section that can be covered by these tiles?
5. Find all pairs of numbers which have a GCF of 1 and an
LCM of 15.
6. Find all pairs of numbers which have a GCF of 2 and an LCM of 16.
7. Find all pairs of numbers which have a GCF of 15 and an LCM 315.
• Fractions
1. 3/4 of the people that favor Amazon.com over BN.com are
women, and 3/7 of people
that favor BN.com over Amazon.com are women. Which online bookstore is
prefered by more people if the number of men that prefer BN.com more is the
same as the number of men that prefer Amazon.com?
2. A big game between Stanford and UA is attended only by
fans of one team or
the other (no neutral people in sight). If 3/5 of the UA fans in attendence are
UA
alums, and 3/4 of the people in attendence are alums of one of the two schools
what
fraction of the Stanford fans are Stanford alums if the number of UA fans is the
same as the number of Stanford fans? (Assume that no alum is a fan for the other
team).
3. Suppose 8 equal sized squares represented 3/4 , what
would 5 of the same squares
represent?
4. Solve times using the area model.
1. Please use the sheet from Monday as a study guide.
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