Attachment A
GLE: NO.1.B.10
Assessment Questions
1). At a school basketball game, the box office sold 400 tickets, which totaled revenue of $1750. Tickets cost
$6 for adults and $4 for students. In the rush of selling tickets, the box office did not keep track of how many
adult and student tickets were sold. The school would like this information for future planning.
a). Let "a" represent the number of adult tickets sold and "s" represent the number of student tickets sold.
*Write an equation showing the relationship among a, s, and the number of tickets sold.
Answer: a + s = 400
*Write an equation showing the relationship among a, s, and the total revenue from tickets sold.
Answer: $6(a)+$4(s)=$1750
b). Solve the System of equations. How many adult tickets were sold and how many student tickets were
sold?
Solution: Solve for either "a" or "s".
6(a) + 4(s) = 1750 a + s = 400
a = (400 ñ s)
Now substitute.
6(400 ñ s) + 4(s) = 1750 Now that we have found s
2400 ñ 6(s) + 4(s) = 1750 we can substitute that
2400 ñ 2(s) = 1750 value into the above
- 2400 -2400 equation.
-2(s) = -650
s = -650 a = (400-325)
-2 a = 75
s = 325
Answer: a=75 and s=325
The box office sold 75 adult tickets and 325 student tickets.
c). Check your answers by substituting your values for a and s back into the original equation dealing with
total revenue from the tickets sold.
Answer: 6(a)+4(s)=1750
6(75)+4(325)=1750
1750=1750
This is a true statement. Therefore, I have solved the problem correctly.
415
Attachment A
GLE: NO.1.B.10
Assessment Questions (continued)
2). Jeff is planning to buy a DVD player. The price is $72 more than twice the amount he saved last month.
How much did Jeff save last month? The Cost of the DVD player is $650.
Solution: Let x = the amount of money that Jeff saved last month.
a). From the information gathered in the above problem we
know that:
2x + 72 = 650
Solving this 2x = 650-72
equation 2x = 578
x = 578
2
x = 289
Thus, we know that Jeff saved $289 last month.
b). Now check your answer. Does $72 more than twice 289 equal $650.
Solution: 2(289) + 72 = 578 + 72 = 650
Therefore, our answer is correct. Jeff had saved $289 last month.
3). Nate goes shopping with his buddies and sees a pair of basketball shoes on sale at 25% off the original
price. If the sale price is $45, what was the original price?
Solution: Let p = the original price of the shoes
Since the pair of shoes is on sale at 25% off the original price, the sale price = the original price ñ 25% of the
original price. Therefore, 45 = p - .25(p)
a). Solve this equation.
45 = p - .25(p)
45 = .75(p)
45 = p
.75
60 = p
b). Check your answer. Is $60 minus 25% of 60 equal to
45?
60 - .25(60) = 60 ñ 15 = 45
Our answer satisfies the specific solution, and therefore, the original price of the pair of shoes was $60.
416
Attachment B
GLE: NO.1.C.10
Large Numbers
Day 1: Have the question "how big is a trillion?" on the board. This day is designed to get the students
interested in math. Discuss with the students the importance of math in everyday life and give examples of
how they will use math after they graduate. Quickly explain and draw the place value chart (to one million)
on the board and ask, "Where is a trillion?" After having the chart to a trillion, ask "How long would it take
you to make one trillion dollars, if you made one dollar every second 24 hours a day, 365 days a year?" Take
guesses and then tell them it would take about 30,000 years. Prove it by showing them the process (i.e. 60
times 60 would be $3,600 every hour, 3,600 times 24 is $86,400 a day, 86,400 times 365 days is
$31,536,000 a year, 1,000,000,000,000 divided by 31,536,000 will round to 31,710 years.) Discuss this and
other interesting math facts.
Day 2: Why do I need to know the place value chart? Discuss the place value chart and hand out a place
value graphic organizer. Explain the base ten concept of math and that every time you move to the left of the
chart you multiply by ten, to the right divide by ten. Explain the number line and how to compare numbers
using the place value chart and a number line.
417
Attachment C
GLE: NO.3.E.9-12
Performance Task
Summary:
Begin by discussing the vastness of the universe. For example, tell students that light travels at the
unimaginably fast speed of 300 million meters per second, and yet light takes years to travel to us from the
stars and takes thousands or even millions of years to travel the depths of space between galaxies. When
we’re dealing with those kinds of distances, it’s no wonder that we often think of them as being beyond our
grasp. One way to put these distances into perspective is to think of them as multiples of smaller-scale
distances. By putting these quantities in the context of a well-understood frame of reference, they begin to
have more meaning.
Internet Resource Links:
Link 1: http://www.newton.dep.anl.gov/newton/askasci/1993/astron/AST005.HTM
Link 2: http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/971124x.html
Link 3: http://www.abarnett.demon.co.uk/atheism/universe.html
Link 4: http://www.aspsky.org/mercury/mercury/30_01/universe.html
Link 5: http://mathforum.org/library/drmath/view/56366.html
Student Directions:
1. Parallax is the apparent change in position of an object when it's viewed from two different places. Astronomers use
this phenomenon to measure the distances to some stars. They
assume that the stars are fixed, and as the Earth moves in orbit they take measurements of the apparent shift in
position. Then they calculate the distance based on a trigonometric relationship between the parallax angle and the
baseline (the radius of Earth's orbit).
Considering that the more distant an object is, the smaller the angle it will make, why would parallax measurements be
better suited for stars than for galaxies?
2. What is the value of using exponents? Give some examples of when they are commonly used. (Exponents are used
to express and calculate large numbers. For example, if you
needed to multiply 1,000,000,000,000,000 by 1,000,000,000, instead of dealing with all those zeros, you could write
the equation as 1015 X 109 and add the exponents to get the answer, 1024. Exponents are also used in business to
express large sums of money and in science to express pH levels, the magnitude of earthquakes, and the brightness of
stars.)
3. Vast distances in space are often measured in light-years. A light-year is the distance that light travels in one year, or
about 6 trillion miles. Altair, a star in the constellation Aquila, is 16.6 light-years away, which means that the light we
see now from that star left its surface 16
years and 219 days ago. Describe what was happening in the world when the light we are seeing from Altair first left
that star. How far away is Altair in miles?
4. Explain why it would be impossible for scientists to measure stellar distances that are accurate to within a few feet.
Why is it not critical to attain such accuracy when dealing with astronomical distances?
5. Does knowing how to use a scale on a map help you understand how to use scale to measure distances in the
universe? How are they similar? How are they different?
6. Describe how you could measure the height of a mountain without having to climb it. (Hint: Imagine that you're
standing 10 miles from the base of the mountain.)
418
Attachment C
GLE: NO.3.E.9-12
SCORING GUIDE
Summary:
You can evaluate students using the following three-point rubric:
Three points: active participation in classroom discussions; cooperative work within groups to complete the
Classroom Activity Sheet; ability to answer more than three questions correctly
Two points: some degree of participation in classroom discussions; somewhat cooperative work within
groups to complete the Classroom Activity Sheet; ability to answer three questions correctly
One point: small amount of participation in classroom discussions; attempt to work cooperatively to
complete the Classroom Activity Sheet; ability to solve one problem correctly
419
Attachment D
GLE: GSR.1.A.9
Student Directions:
Your group is going to be creating a skit about how the Pythagorean Theorem was
developed and used in a culture. After you sign up for a culture (one group per
culture), you must research their proof of the Pythagorean Theorem and various
aspects of the culture. Summarize your findings as well as how the theorem was
developed or proved and why it was used. Create a skit which portrays this
information. Develop a worksheet of 10 real world Pythagorean Theorem problems
that your culture may have used. Also, provide detailed solutions for the worksheet.
SCORING GUIDE
Level 1: A
To receive an 'A' a group must do the following:
1. Turn in a well-written summary of their culture and its use of the Pythagorean Theorem.
A bibliography must be included. 15 points for the summary 5 points for the bibliography.
2. Perform a skit (without notes) with an accurate proof/use of the Pythagorean Theorem
and portrayal of the culture - 40 points.
3. Turn in a worksheet with ten well-developed scenarios using the Pythagorean Theorem
and correct solutions - 4 points each problem for a total of 40 points.
420
Attachment E
GLE: GSR.1.A.10
Fibonacci Questions for Assessment
1. Imagine that scientists in the rain forest have discovered a new species of plant life.
Where might they look for the Fibonacci sequence?
2. Suppose that you're shooting baskets with a friend. After a few practice shots, you decide that you want to
keep score. The first basket either of you makes is worth one point. Just to make things interesting, you
suggest that every time either of you makes another basket, you add your previous two scores to get a new
total. To make the game even more appealing, you offer to start from zero, while your friend can start from
one. What sequence of numbers would emerge after shooting eight baskets?
What is the difference in points between you and your friend? What pattern has emerged from the point
difference?
3. The Fibonacci sequence continues indefinitely. If all its terms were added together, it would be called a
series and the result would be infinite. But not all infinite series add up to infinity. For example, adding all
the terms of 1/n (where n is 1, 2, 3, etc) does not result in a large sum at all, even though the series could go
on forever. What would the sum of five numbers in the series 1/n be? Explain why it wouldn't be infinitely
large.
4. Explain that numbers missing from the Fibonacci sequence can be obtained by combining numbers in the
sequence, assuming that you're allowed to use each number more than once. For example, how could the
number 4 be obtained from the sequence? How about 11? 56? Think of a number not in the sequence and try
to figure out what numbers to combine to get it.
5. At first glance, the natural world may appear to be a random mixture of shapes and numbers. On closer
inspection, however, we can spot repeating patterns like the Fibonacci numbers. Are humans more apt to
perceive some patterns than others? What makes certain patterns more appealing than others?
6. Try to solve this problem: Female honeybees have two parents, a male and a female, but male honeybees
have just one parent, a female. Can you draw a family tree for a male and a female honeybee? What pattern
emerges? Are they Fibonacci numbers?
(The male bee has 1 parent, and the female bee has 2 parents. The male bee has 2 grandparents, and the
female bee has 3 grandparents. The male bee has 3 great-grandparents, and the female bee has 5 greatgrandparents. The male bee has 5 great-greatgrandparents, and the female bee has 8 great-great-grandparents. The male bee
5 has 8 great-great-great-grandparents, and the female bee has 13 great-great-great-grandparents.)
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Attachment F
GLE: GSR.1.B.9
Creating Congruent
Triangles
Under what conditions are
triangles congruent?
Materials: rulers and protractors
(students may work alone or in groups)
1.
2.
Draw a triangle of any size and shape and label it ABC.
Using the ruler and protractor, measure and record the angles and sides of
the triangle. Remember that there are 180 degrees in the sum of the angles
of a triangle.
3. Draw triangle DOG where DO = AB, angle D = angle A,
and angle O = angle B.
4. Measure and record the remaining angles of triangle DOG.
5. Are triangles ABC and DOG congruent?
6. Is it possible to draw triangle DOG so that it is NOT congruent to triangle
ABC?
7. Draw triangle KIT where KI = AB, angle K = angle A, and
angle T = angle C.
8. Measure and record the remaining angles of triangle KIT.
9. Are triangles ABC and KIT congruent?
10. Is it possible to draw triangle KIT so that it is NOT congruent to triangle
ABC?
11. Discuss your finding from this activity.
422
Name __________________________________
Attachment G
GLE: GSR.1.B.9
Performance Event
A fast food restaurant recently gave toys away in its children’s meals. The toys came inside
a cardboard pyramid. Unfolded, the pyramid had this design:
3
2.5 inches
A.
Calculate the area of wasted (scrap) material if each pyramid is stamped out of a
square piece of cardboard that is 7 inches on each side. Provide the work that shows how
you arrived at your answer and write your answer on the line.
Area: ____________ sq. inches
423
Name __________________________________
Attachment G
GLE: GSR.1.B.9
B.
Can the amount of scrap material be reduced? If so, write a letter to the president
of the fast food restaurant company explaining how it can save money on the production
of these cardboard pyramids by determining the amount of scrap material saved. Include
drawings if you wish. If the amount of scrap material cannot be reduced, explain why.
Write your letter to the president of the fast food restaurant here.
424
Attachment G
GLE: GSR.1.B.9
Answer Key
Exemplary response –
Part A – the piece of cardboard 7 inches on a side has 49 square inches; using the Pythagorean
Theorem, the box is about 21 square inches, so 28 square inches is wasted.
Part B – a piece of cardboard about 5 inches on a side (4.95 inches) could be used to make the box,
resulting in a savings of 24 square inches of cardboard for each box.
Many valid processes could produce these results.
Many valid letters could be written to the CEO, but it should contain the above info and point out the roughly
50% savings in materials used to make the toy boxes by using the smaller piece of cardboard.
Scoring Guide:
4 points: The student’s response fully addresses the performance event by:
 demonstrating knowledge of mathematical principles/concepts needed to complete the event, such as
accurately computing the areas of the toy boxes
communicating all process components that lead to an appropriate and systematic solution, such as
illustrating the savings to the company by using the smaller piece of cardboard to make the toy boxes
having only minor flaws with no effect on the reasonableness of the solution
3 points: The student’s response substantially addresses the performance event by:
demonstrating knowledge of mathematical principles/concepts needed to complete the event, such as a
generally accurate computation of areas of toy boxes
communicating most process components that lead to an appropriate and systematic solution
having minor flaws with minimal effect on the reasonableness of the solution
2 points: The student’s response partially addresses the performance event by:
 demonstrating a limited knowledge of mathematical principles/concepts needed to complete the event, such
as computations of areas of toy boxes with errors
 communicating some process components
 having flaws or extraneous information
1 point – The student’s response minimally addresses the performance event by:
 demonstrating a limited knowledge of mathematical principles/concepts needed to complete the event, such
as an inaccurate calculations
 communicating few or no process components
having flaws or extraneous information that indicates a lack of understanding or confusion
0 points – Other; such as merely copying prompt information.
425
Student _________________________________
Attachment H
GLE: GSR.1.B.9
Building codes for Smallville state that no building may be more than 35 feet tall. A carpenter constructing
the building in the figure would like to build the roof so that the vertical rise of its two inclines will be 10
inches for every foot of horizontal distance as they extend toward the peak of the house.
20 feet
30 feet
Given the other dimensions shown, will the carpenter be able to build the roof the way he wants to and still
meet the code? Show work that supports your conclusion.
426
Attachment I
GLE: GSR.1.B.9-12
Experiment with Volume
The Problem Take a sheet of paper, and roll it up to form a baseless cylinder. Now take another sheet, rotate the paper, and form
another baseless cylinder. Think about the volume of each cylinder and make a prediction.
A Prediction



Would the two volumes be equal?
Would the short cylinder have greater volume?
Would the tall cylinder have greater volume?
Explanation Why did you predict as you did?
A Demonstration Tape two sheets of paper to form the two cylinders, one short and one tall (Stiff paper is helpful. I use
transparency sheets). Hold the tall cylinder upright in a shallow box and fill with rice. Now fill the shorter cylinder, and compare
the two amounts of rice. Was your prediction correct?
The Calculation Calculate both volumes. (You may need these formulas)


Circumference = 2(pi)(r)
Volume = (pi)(r^2)h
V=
V=
427
</UL< b>
Attachment I
GLE: GSR.1.B.9-12
Volume Functions
The Problem Consider the perimeter of the two rectangles above - 39 ins. Think of making baseless
cylinders from all the rectangles with a 39 inch perimeter. What are the dimensions of the rectangle that
would produce the cylinder with the greatest volume?
Explore You may want to cut some rectangles with various lengths and widths with the correct perimeter
and roll them to form the cylinders. It's fun to predict which of them you think will have the greatest volume.
Formulas you may need:


Circumference = 2(pi)(r)
Volume = (pi)(r^2)h
428
Attachment I
GLE: GSR.1.B.9-12
Solution Choose which method you want to use to solve the problem:



Tabular
Graphical
Calculus
Tabular Method
Complete the table below.
Height of Rectangle Width of Rectangle Radius of Cylinder Volume of Cylinder
1
3
5
7
9
11
13
15
17
19
Where is the volume greatest? Try a value on each side of the radius with the greatest volume. Did one of
those give you a greater volume? Keep trying to get closer and closer to get the best answer that you can.
Graphical Method
Write an expression for the volume of the cylinder as a function of its radius. (Hint: Use the two formulas
provided.) Graph the expression, preferably using a graphing utility. Where does the function seem to
achieve its maximum value. Can you zoom in to get a more exact answer? Continue to zoom in until you get
the best answer that you can.
Using Calculus
Using calculus you can find an exact solution. Write an expression for the volume of the cylinder as a
function of its radius. (Hint: Use the two formulas provided.) Find the dimensions of the rectangle that give
the volume function's maximum value.
Extension: Solve the problem using any P for the perimeter.
429
Attachment J
GLE: GSR.1.B.11
430
Name _______________________
Attachment K
GLE: GSR.2.A.9
A(1,5)
D(-1,-3)
B(4,5)
C(2,-3)
1. Given the vertices of parallelogram ABCD in this standard (x, y) coordinate plane, what
is the area of ΔABC in square units?
A.
B.
C.
D.
10
12
15
16
2. Find the straight-line distance from the store to the school. Round your answer to the
nearest hundredth of a mile. Provide the work that shows how you arrived at your answer
and write your answer on the line.
Each block grid = 0.75 mile
________________ miles
431
Attachment K
GLE: GSR.2.A.9
Answer Key
1. B
3 13
; on the grid, the school is 4 over and 6 down; using the Pythagorean
2
3 13
Theorem, 42 + 62 = x2; x =
or approximately 5.41; or another valid process.
2
2. Distance = 7.21 miles or
432
Attachment L
GLE: GSR.2.A.10
Coordinate Geometry
Choose 5 exercises to complete by graphing the points on the graph paper given.
1. Given points:
(-11, -5), (-15, -3), (-15, -13), and (-19, -11).
Is this quadrilateral a rectangle?
2. Given points:
(-13, -12), (-4, -3), and (5, -18).
Is this a right triangle?
3. Given points:
(2, -10), (-4, -19), (4, -4), and (-2, -13).
Is this quadrilateral a parallelogram?
4. Given points:
(3, -6), (5, -11), (-7, -10), and (-5, -15).
Is this quadrilateral a rectangle?
5. Given points:
(-11, -15), (4, -3), and (-27, 5).
Is this a right triangle?
6. Given points:
(-17, 56), (-21, 55), (-10, 65), and (-14, 64).
Is this quadrilateral a parallelogram?
7. Given points:
(71, -69), (66, -65), (69, -74), and (64, -72).
Is this quadrilateral a square?
8. Given points:
(22, 14), (-18, 30), (30, 34), and (-10, 50).
Is this quadrilateral a rectangle?
433
Attachment M
GLE: GSR.3.A.9
Household Decoration
Topics: Reflections, Translations, Rotations, and Dilations
Many of the geometric figures that we see daily in our real world can change in size, in
shape and can move from side to side or forward and backward. These changes are
descriptions of transformational geometry that have applications found in the areas from
science and architecture to music and history.
The student Goal is to inform the customers of a shop about the beauty of geometric
designs that are used and have been used through-out our world of shapes in motion.
The student Role is to be an engineer of geometric designs that will be "hot-selling"
decorative creations.
The Audience is composed of fellow students and the teacher who wants to buy unusual
items that appeal to the eye to display in their homes.
The Situation for each student is to research, propose, design and create geometric
designs that illustrate transformational geometry.
The Product/Performance and Purpose is to design a proposal, a plan and a project.
The goal is to convince the shop's proprietor of the plan (orally) and to inform the
customers about the appeal of the designed works that may be displayed throughout their
homes.
The project should include 3 representations of transformational geometry with a written
explanation of these types of transformational geometry illustrated in the design and why
they were chosen.
The project should be one that can be used in everyday life in customers' homes.
Customers may ask questions about the design of the project. Also, a self-assessment will
be performed at the time of the presentation.
434
Attachment M
GLE: GSR.3.A.9
Scoring Guide
Transformations
Student knowledge
Written
Explanation
3 points
Includes 3 different
types of
transformations
Clear understanding ,
able to identify the
transformations orally
Clear written
explanation of design
and the transformations
used
2 points
Includes 2 different
types of
transformations
Some understanding of
transformations
Written explanation of
design and the
transformations used
435
1 point
Includes 1 type of
transformation
Very little
understanding of
transformations
Explanation is unclear
or not provided
Attachment N
GLE: GSR.3.B.9
Student Directions:
On a recent test students were asked how many hours they studied. The following list shows the
results with X representing the number hours spent studying and Y representing the number of errors
(wrong answers).
(1,5) (2,4) (2,3) (3,1) (1,7)
Graph this data. Be sure to label all axes as well as correctly scale the axes. Identify the function rule.
Create graphs that show the translation and reflection of the function. Explain what the data indicates
about the relationship between hours studied and performance on tests.
Rubric(s)
Rubric: What's the trend scoring
Trait: What's the trend scoring
Performance Type:
4
3
Response shows
a complete and
thorough
understanding of
the mathematical
concepts. All
graphs correct
and complete.
There may be
minor errors in
scaling that do
not effect
conclusions.
Function is
identified
correctly
Explanation of
data is clear and
concise
Response shows
a complete
understanding of
concepts.
Graphing is
correct, some
minor errors.
Function is
identified
Explanation of
data is generally
correct
2
Response shows
some
understanding of
concepts.
Two of the three
graphs are
correct.
Function has
minor flaw
Explanation of
data has flaws.
1
Some effort was
made that
demonstrates
some minimum of
mathematically
correct
representation
On of the three
graphs is correct.
Function has
major flaws.
Explanation of
data has major
flaws
436
0
Response shows
no understanding
of mathematical
thought or no
response was
given.
Attachment O
GLE: GSR.4.B.9-12
Hidden Irrationals
On the dot paper below, the horizontal and vertical distance from one dot to the other is 1 unit. Draw
line segments with the following lengths and label with their lengths.
1. square root 2
2. square root 5
3. square root 8
4. square root 18
5. square root 20
6. square root 26
7. square root 32
8. square root 34
9. square root 40
10. Choose 3 other lengths of irrational numbers that you can draw by connecting dots. Draw and label.
11. Give 3 other irrational-number lengths that you can not draw connecting dot to dot.
12. Explain what has to be true for you to be able to draw the number lengths in this way.
13. Are all the diagonals irrational? Convince me with an argument or a counter-example.
437
Attachment P
GLE: M.2.C.9-10
Student Directions:
The Giant Tetrahedron Project
We will make a giant tetrahedron that is created out of each student's individually created tetrahedron.
Accuracy is important for all tetrahedrons to fit together.
Part One- Construction of the Tetrahedral Net
1. Draw a line segment of exactly 3 inches in the middle of the paper 3" from the bottom.
2. Using a compass construct an equilateral triangle.
3. Construct another equilateral triangle off each side of your original triangle.
4. Check to see that each line measures exactly 3 inches.
5. Cut out your net on the outside lines. You may want to leave tabs to glue it together.
6. Carefully fold along each of the lines of your original (base) triangle and glue together .
Part Two- Planning for Construction of a Giant Tetrahedron
We need to find out how many layers high we can make our tetrahedron with about 180 students each
creating a tetrahedron. To do this we need to create a table and look for a pattern. We need to know how
many tetrahedron will be in each layer and how many total tetrahedrons there will be. Write a formula for
extending the pattern.
Part 3 - Total Surface Area
We will figure the total surface area of our tetrahedron but to do that we will begin with the surface area of
each small tetrahedron.
1. To find the surface area of each individual tetrahedron you need to sketch your tetrahedron. How many
faces can you see? How many are hidden?
2. Draw one face and label the length of the base and sketch in the altitude.
3. We now have a special right triangle. Label the angles and find the height.
4. Calculate the area of each face and multiply by the number of faces that make a tetrahedron.
5. Calculate the total surface area of our giant tetrahedron.
Part 4- Pascal’s Triangle
The face of our Tetrahedron has the shape of Pascal's triangle. The number on the face of each small
tetrahedron in the structure will be the sum of the 2 small tetrahedrons above it, unless it is a corner in which
case it has only one tetrahedron above it and the sum is only 1.
Find the first nine rows of Pascal’s Triangle so that we will know how to label each layer.
438