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Problem type 1)
Given: You drive from Albuquerque, NM, to Rock Springs, WY, at an average speed
of 55 miles per hour. Your route is along Interstate 25 to Cheyenne, WY, then
Interstate 80 to Rock Springs.
1a) What would be your total driving time (to the nearest half hour)?
Approximately 14.5 hours.
Use an atlas or road map to solve the problem.
1b) Write a brief description of the problem-solving process including steps,
calculations, and tools used. Use actual road distances in miles.
Distance = Rate X Time
On Interstate 25, the distance is estimated to be 500 miles on a map
500 = 55T
T = 9.09 hours (approx)
On Interstate 80, the distance is estimated to be 300 miles on a map
300 = 55T
T = 5.45 hours (approx)
The sum of the times is 14.54 hours (approximately 14.5 hours)
Problem type 2)
2a) Write a problem using the following elements.
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Travel by air from Memphis, TN, to Las Vegas, NV, with stops in Oklahoma
City, OK, and El Paso, TX, en route.
Students have a map of the United States that shows cities and a distance
scale in kilometers.
Require use of straight-line distance in kilometers for each leg of the trip.
Average aircraft groundspeed is 380 km/hour from Memphis to Oklahoma City
and 310 km/hour from Oklahoma City to Las Vegas.
You fly from Memphis, TN to Las Vegas, NV, stopping in Oklahoma City, OK.
Using your map of the United States and the fact that average aircraft groundspeed
is 380 km/hour from Memphis to Oklahoma City and 310 km/hour from Oklahoma
City to Las Vegas, how long will the entire trip take if there an hour layover in
Oklahoma?
2b) Write a brief description of the process you used to construct the problem.
The problem needed to contain all of the information given and give students
an idea of where to find there data. The El Paso leg was omitted since to aircraft
speed was given. There also needed to be some way to account for the layover
time, since this is a different situation than the car problem above.
2c) Write the steps you expect students to follow to solve the problem.
Students will find/estimate the distances from Memphis to Oklahoma City and
from Oklahoma City to Las Vegas. They will then use D=RT and the given rates to
find the time for each length of the journey. Finally, they will add those times to the
one-hour layover to get the final answer.
2d) Write a scoring key with directions for its use and application.
The problem will be based on a five point scale:
Reasonable estimate of the distance from Memphis to Oklahoma City
Reasonable estimate of the distance from Oklahoma City to Vegas
Correct application of D = RT
Final answer (with hour layover included)
You may have to tweak this to make it fit the format of your class.
Problem type 3)
1pt
1pt
2pt
1pt
Figure A)
Use the given information and the diagram in Figure A to complete Task A and Task
B.
Keep in mind:
1. The diagram is NOT drawn to SCALE.
2. There may be multiple ways to show your solution.
Task A: For each statement listed in Table A, provide a reason to justify why each
statement is true.
TASK B: Prove that triangle AGC is isosceles. Insert answers in table contained in
Table B.
Table A)
STATEMENTS
1. GB  GD
2. BGE  DGE
3. GE  GE
4. BGE  DGE
5. GBE  GDE .
Table B)
REASONS
1. given
2. given
3. reflexive property
4. side-angle-side postulate
5. corresponding parts of congruent triangles
are congruent
STATEMENTS
1. GB  GD
2. BGE  DGE
3. GE  GE
4. BGE  DGE
 GBE  GDE
 GED is a right angle
BD || AC
 GBE  A
 GDE  C
 A  C
 AGC is isosceles
REASONS
1. given
2. given
3. reflexive property
4. side-angle-side postulate
5. corresponding parts of congruent triangles
are congruent
6. if two angles are congruent and
supplementary, they are right angles
7. if corresponding angles are congruent, the
lines are parallel
8. if lines are parallel, corresponding angles
are congruent
9. if lines are parallel, corresponding angles
are congruent
10.transitive
11. if the base angles of a triangle are
congruent, then the triangle is isosceles.
Problem type 4)
Prove: If the base angles of a triangle are congruent, then the triangle is
isosceles.
4a) Draw and label a diagram that includes:
1. A triangle with each vertex and the given information labeled.
2. All other information needed to present the proof.
C
o
p
j
B
E
D
4b)Construct a formal proof of the theorem including:
1.
2.
3.
4.
Given statement(s)
Other statements
A reason for each step.
A conclusion that restates the theorem.
STATEMENTS
 B  D
 CEBis a right angle
 CEDis a right angle
 CEB  CED
CE   CE
 BEC   DEC
CB  CD
 BCD is isosceles
REASONS
1. Given
2. Definition of an altitude
3. definition of an altitude
4. all right angles are congruent
5. reflexive property
6. angle-angle-side postulate
7. corresponding parts of congruent triangles
are congruent
8. definition of isosceles
Therefore, if the base angles of a triangle are congruent, the triangle is isosceles.
Problem type 5)
A person is planning to buy some storage units. The storage units are in the shape of
a rectangular solid (See above).
The person needs to compare the dimensions of the storage units before choosing
which storage units to buy.
The dimensions are given in the table in below.
Use the table to investigate how changes in dimensions affect the perimeter, area,
and volume of the rectangular solid.
A. Determine the perimeters of the base of the rectangular solids. Record your
answers in the table in the column labeled "Perimeter."
B. Determine the volumes of the rectangular solids. Record your answers in the
table in the column labeled "Volume."
C. Determine the surface areas of the rectangular solids. Record your answers in
the table in the column labeled "Surface Area."
D. Explain the relationship between changes in dimensions and perimeter of the
base of rectangular solids. Use ratios in your explanation.
Rectangular Solid A has dimensions that are half the size of Solid B. The
perimeter of Solid A is also half the size of Solid B.
E. Explain the relationship between changes in dimension and volume of
rectangular solids. Use ratios in your explanation.
Although the dimensions are half the size, the volume is one-eighth of the
size. This happens because volume is a cubic unit, and so the one-half
relationship is cubed to give one-eighth.
Explain the relationship between changes in length and width and the areas of any
two corresponding sides of the rectangular solids. Use ratios in your explanation.
Length
Width
Height
Perimeter
Volume
Surface
Area
Since the dimensions are half the size, the areas are one-fourth the
size. This happens because surface area is a square unit, and so the one-half
relationship is squared to give one-fourth.
Rectangular
Solid A
4
10
18
28
720
584
Rectangular Solid
B
8
20
36
56
5760 2336