EDSEC 437/637 UNIT 2: ELEMENTARY ALGEBRA
You are to submit this file electronically into both BlackBoard and LiveText after you
have completed all work. For the alpha problems, you should first try the problem on
your own and then read through and work through the solution. For the numeric
problems, solve the problems, addressing each of the prompts and justifying your
response. Before submitting the assignment, return to the score sheet and complete the
following.
The solutions for the alpha problems have been completed for you. However, you should
first try the problem on your own and then read through and work through the solution.
On the score sheet, you will be asked to identify at least one Common Core standard (at
the indicator level) for each problem. For example, a sixth grade standards you might
cite, if appropriate, is:
Expressions and Equations 6.EE
Apply and extend previous understandings of arithmetic to algebraic
expressions.
1. Write and evaluate numerical expressions involving whole-number
exponents.
See http://ed.sc.gov/agency/pr/standards-andcurriculum/documents/CCSSI_MathStandards.pdf.
For the numeric problems, indicate which technological tools would be appropriate for
students to use for a particular course you have identified for the problem and justify your
choice.
In addition to the score for each problem, you will be evaluated on the following
objectives:
 Monitors and reflects on the process of mathematical problem solving by
making thoughtful comments on each problem
 Provides sound arguments for problem solutions and recognizes that solutions
are not complete if they are not justified
 Communicates mathematical thinking clearly and with appropriate
mathematical language
 Demonstrates an understanding of the appropriateness of technological tools
based on the problem
 Uses technology efficiently and effectively
 Uses technological tools efficiently and effectively to explore algebraic ideas
and representations of information and in solving problems
For each of the objectives listed immediately above, you will earn either 5 or 0 points.
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ELEMENTARY ALGEBRA UNIT SCORE SHEET
Six-Point Scoring Rubric
6 Points: Work shows solid understanding of the underlying concepts, sufficiently
addresses each component of the problem, and has a correct solution.
4 Points: Work shows good understanding of the problem, but has some minor flaws or
minor omissions.
2 Points: Work shows understanding of the problem and underlying concepts, but has
significant flaws or omissions.
0 Points: Problem is not attempted or shows virtually no understanding.
Person Who Completed This Assignment: _____________________________
People with whom you worked: _____________________________________________
PROB
PTS
EARNED
SC MATH STANDARD (alpha problems) or
COURSE & APPROPRIATE TECH TOOLS (numeric
problems)
2-A
2-B
2-C
2-D
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10
TOTAL
(max: 84)
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
OBJECTIVE
MAX PTS
Problem Set
84
Monitors and reflects on the process of mathematical problem
solving by making thoughtful comments on each problem
5
Provides sound arguments for problem solutions and
recognizes that solutions are not complete if they are not
5
justified
Communicates mathematical thinking clearly and with
appropriate mathematical language
Demonstrates an understanding of the appropriateness of
technological tools based on the problem
Uses technology efficiently and effectively
5
5
5
Uses technological tools efficiently and effectively to explore
algebraic ideas and representations of information and in
5
solving problems
TOTAL
114
3
EARNED PTS
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-A: THE FIELD TRIP
The American History Club is taking a field trip to Philadelphia and Boston. The
club has raised enough money to provide transportation and lodging for everyone.
However, each student must take enough money to cover meals and extras. You plan to
take $425.00 and spend $30.00 each day. Your friend is taking $550.00 and is planning to
spend $45.00 each day.
A.
Write an algebraic expression for the amount of money you will have remaining
at the end of any day. Define any variables you use.
B.
Write an algebraic expression for the amount of money your friend will have
remaining at the end of any day. Define any variables you use.
C.
Use a table to determine the day on which you and your friend will have exactly
the same amount of money remaining.
D.
Explain how you decided on which day you would both have the same amount
of money.
E.
Write an equation to determine when you and your friend would have the same
amount of money. Use the equation solver on your calculator to solve this
equation. How does this answer compare with your answer from problem C?
Explain any differences in the answers.
F.
Exactly how much money would you both have on this day?
G.
At most, how many days would you like the field trip to last? Do you think your
friend would agree with you? Explain your reasoning.
H.
What is the maximum amount your friend could spend each day in order to be
able to take a 21-day trip and only spend the $550.00? Justify your reasoning.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ONE SOLUTION TO PROBLEM 2-A: THE FIELD TRIP
A. Write an algebraic expression for the amount of money you will have remaining
at the end of any day. Define any variables you use.
We’ll start by identifying a variable to help us. Let X = the number of the day
on the trip. Since we start with $425, after one day we’ll have 425 – 30 dollars. After
two days we’ll have 425  30  2 dollars left. After 3 days, we’ll have 425  30  3
dollars left. After X days, we’ll have 425 – 30X dollars left.
B. Write an algebraic expression for the amount of money your friend will have
remaining at the end of any day. Define any variables you use.
Again we’ll let X = the number of the day on the trip. Using logic similar to
that in part A, after X days, our friend will have 550 - 45X dollars left.
C. Use a table to determine the day on which you and your friend will have exactly
the same amount of money remaining.
From the MAIN MENU, call up the “Table” function. Then,
 If necessary, press F3 to choose the “Type,” followed by F1 for “Y=.”
X, ,key
T for X. Press EXE .
 Type in 425 – 30X for Y1, using the
 Type in 550 – 45X for Y2. Press EXE . See below left.
 If the equal sign in Y1 or Y2 is not highlighted, move the cursor to the function
and press F1 to select it.
 Next, we’ll set the range of values for X. Press F5 to access the “Range”
function. Since X represents the number of days on the trip, we might want to start
at day 0 and end by day 15 (we would already be out of money). The “pitch”
refers to the increment the calculator will use for X. Press EXE after each value
you enter. See below right. Press EXIT when finished.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
The table should have the values shown below. You can use the arrow keys to
move through the table. From the data shown, we determine that sometime during the
9th day you will have the same amount of money as your friend.
X
Y1
Y2
1
395
505
2
365
460
3
335
415
4
305
370
5
275
325
6
245
280
7
215
235
8
185
190
9
155
145
10
125
100
11
95
55
12
62
10
13
35
-35
14
5
-80
15
-25
-125
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
D. Explain how you decided on which day you would both have the same amount of
money.
After examining the table of values, the students should recognize that after
the eighth day, the values in the Y2 column become less than the values in the Y1
column. Therefore at some point during the ninth day, the two students will have the
same amount of money. Students may estimate that both students have
approximately $170.00.
E. Write an equation to determine when you and your friend would have the same
amount of money. Use the equation solver on your calculator to solve this
equation. How does this answer compare with your answer from problem C?
Explain any differences in the answers.
From the MAIN MENU, call up the “Equation” function. Then,
 Press F3 for the “Solver.”
 Type in the equation, using SHIFT
.
for the equal sign. See below left.
(NOTE: At this point, you may have a different value for X from the one shown
on the screen.)
 Press F6 to solve the equation. See below right.
1
Our result tells us that after 8 days (part way into the ninth day of the trip),
3
you will have the same amount of money as your friend. Further, the calculator shows
that, for this value for X, both sides of the equation evaluate to 175. Thus, after 8
days, we will both have $175. This is consistent with our estimate from the table.
F. Exactly how much money would you both have on this day?
7
1
3
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
We already answered this in looking at the screen for part E. Both the left and
right sides of our equation evaluate to 175, indicating we both have $175.
G. At most, how many days would you like the field trip to last? Do you think your
friend would agree with you? Explain your reasoning.
The student may examine the table and determine that at the end of the 14th
day $5.00 remains, thereby concluding that the trip should last no longer than 14
days. She/he may also use the equation solver to determine how long the trip should
last if all of the money is to be spent. We can determine that your money will be gone
in a little over 14 days (see below left), but that your friend’s money will be gone in a
little over 12 days (see below right).
Therefore the friend would not want to take a trip that would last 14 days.
H. What is the maximum amount your friend could spend each day in order to be
able to take a 21-day trip and only spend the $550.00? Justify your reasoning.
One way to explore this problem is with a table, looking at various
expressions. From the MAIN MENU, call up the “Table” function. Enter the
following expressions, pressing EXE after each entry. (See below left.)
Y1: 550 - 45X
Y2: 550 - 40X
Y3: 550 - 35X
Y4: 550 - 30X
Y5: 550 - 25X
Y6: 550 - 20X
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
 Next, press F5 to set the “Range.” You may wish to start at 0 and end at a value
like 25, again using a pitch of 1. Press EXE after each value you type in (if the
correct value is there, use the down arrow to move to the next field). See below
right. When all values have been entered, press EXIT .
 Press F6 to see the table.
After examining the table of values created, the student should determine that
if the friend spent $25.00 per day, at the end of 22 days, the friend would have no
money remaining. Therefore, a trip for 21 days would be possible. Since we used
increments of $5 in our functions, we may wish to see if the friend can spend $26
each day.
 Press EXIT to return to the “Table Function” screen.
 Use the down arrow cursor to highlight Y5. Use the right cursor arrow to
highlight the 5 in 25X and change this 5 to a 6. Press EXE and F6 to see the
table.
Notice that the friend can spend $26.00 and still go on a 21-day trip. Similar steps
can be used to determine if the friend can spend $27 each day. This time, -17 is shown in
the table. Consequently, if the friend spends $27.00 per day, there will not be enough
money for a 21-day trip.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
AN ALTERNATE SOLUTION TO PROBLEM 1 USING GRAPHING
There are, of course, other techniques for investigating this problem. One good
way is with a graph. You may wish to come back to this problem and explore it with
graphs after students have been exposed to equations in two variables, but you also may
wish to use this as an introduction to the topic. The solution below addresses the first part
of the FIELD TRIP problem.
From the MAIN MENU, choose “Graph.” Then,
 Delete all the functions that are there by highlighting them and pressing F2
followed by F1 .
 We’ll now set the viewing window. Press SHIFT
F3 to access the window.
Type in appropriate values, pressing EXE after each value you type. So that you
can clearly see the axes, you may wish to extend your domain and range slightly
beyond the values you wish to explore. See below left for a possible window.
Press EXIT when finished.
 Enter the two functions in as Y1 and Y2, pressing EXE after you type in each.
See below right.
 To see the graph, press F6 . See below left.
 To explore the point where the two lines meet, we can TRACE the functions to
see approximately where they meet. To access the TRACE function, press F1 .
Then use the right and left arrow keys to move along a function and the up and
down arrow keys to switch between functions.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
 Once the cursor is close to the intersection, you can use the ZOOM function to get
a closer look. To do this, after you have moved the cursor close to the point of
intersection, press F2 for ZOOM and then F3 to ZOOM IN.
 You can then use the TRACE feature again by pressing SHIFT
F1 .
If you are interested in a more accurate solution, the calculator can also help.
 With the graph displayed but no functions shown, press F5 for the “Graph
Solver.”
 Press F5 for “Intersection” and wait a few seconds. The calculator shows the
point of intersection. See below right.
This tells us, once again, that after a little over eight days into the trip, both will have
$175 remaining.
REFERENCE: Addison-Wesley Secondary Math. An Integrated Approach. Focus on
Algebra, 1996.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-B: GIFT GIVING
Gift-giving season is approaching and you wish to purchase gifts for four relatives
and six friends. The amount you spend on each relative is usually, but not necessarily,
different from the amount you spend on each friend. You have saved your money and can
spend any amount up to $200. How much should you spend, on average, for each
relative, and how much, on average, should you spend on each friend?
EXTENSIONS
1.
Change the amount of money from $200 to $100 and then to $500. What
happens to the graph as you make these changes? Why?
2.
Leave the amount at $200, but change the number of relatives. What happens to
the graph as you make these changes? Why?
3.
Leave the amount at $200, but change the number of friends. What happens to
the graph as you make these changes? Why?
4.
Compose some general rules that show how the graph of ax  by  c is affected
by a, b, and c respectively.
5.
Investigate the problem if you were to spend the money on 4 close relatives, 6
distant relatives, and 8 friends.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ONE SOLUTION TO PROBLEM 2-B: GIFT GIVING
This problem can be addressed any number of ways. Two reasonable approaches
include using a table and using a graph. In either case, however, some algebra simplifies
the problem.
Let x represent the average amount spent on each relative and y represent the
average amount spent on each friend. Because you have four relatives for whom you will
buy something, 4x represents the amount you will spend on relatives. Similarly, 6y
represents the amount you will spend on friends. You can spend any sum up to and
including $200, assuming any tax has already been included in the expressions listed
above. Our relationship can thus be described by the inequality 4 x  6 y  200 .
Let’s begin with a graph. To enter the relation into the calculator, we must first
solve for y. First, subtract 4x from both sides to obtain 6 y  4 x  200 . Then, divide
both sides by 6, obtaining y 
2
100
. (NOTE: If we chose not to simplify, we could
x
3
3
use y  (4 x  200)  6 .)
Now we’ll graph it. From the MAIN MENU, call up the “Graph” menu.
 If relations from previous work are entered, either delete them (highlight them
and press F2 followed by F1 ) or turn them off (highlight them and press F1
so they are no longer selected). The following assumes that Y1 is available for the
problem.
 To enter our relation, we first want to enter the correct type of relation. With Y1
highlighted, press F3 for TYPE, F6 for more options, and F4 for the less than
or equal to relation. This returns you to Y1.
 Simply type in the relation, using the a b/c key for the two fraction bars and
press EXE . (NOTE: If the division symbol is entered instead, either enclose the
–2 3 in parentheses or put a multiplication symbol between the 3 and the x. If
not, the calculator will treat the variable term as
The entry screen is shown below.
13
2
2
x .)
instead of
3x
3
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
After entering, the relation, press F6 to view the graph. Assuming a standard
window, which uses a domain and range from –10 to 10, the entire graph will be shaded
in! This is due to an inappropriate window.
We’ll fix the problem by determining a reasonable domain and range. First,
discuss with students a reasonable domain for x. Keep in mind that x represents the
average price of each gift for relatives. Obviously, it doesn’t make sense to make the
average negative, so a reasonable minimum value for x is 0. Because there are four
family members and a maximum of $200 to spend, if the person decides to spend all of
the money on relatives, the average still cannot exceed $50, so 50 is a reasonable
maximum. Setting the scale at 5 may be an agreeable number of tick marks on the x-axis.
When thinking about the range for y, students should also realize that 0 is a
minimum value to use. Because there are six friends to buy gifts for, the maximum the
average could be is 200/6 or $33.33, rounded to the penny. If students wish, they can use
200/6 as the maximum value for y, or they may wish to round to perhaps 35. A scale of 5
again is appropriate, although students may prefer a different value.
To enter these values, when looking at the graph,
 Press SHIFT F3 to access the viewing window.
 Type in the numbers discussed above.
 Press EXE after each entry. After the last entry, press EXIT to return to the
input screen.
 Pressing F6 will draw the graph with the new window. The window and graph
are shown below.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
An alternate method of setting up an appropriate viewing window is to input the
desired values for x as explained above and drawing the graph. Then,
 Press F2 to obtain the ZOOM window.
 Press F5 for an "AUTOMATIC" window determined by the calculator.
This ZOOM AUTO feature can be very helpful when you have determined a
reasonable domain, but wish help in determining a reasonable range.
Although it does not transfer well to paper as shown below, turning on the grid
lines may make it easier for students to read and interpret the graph. Because we set both
scales at 5, both horizontal and vertical moves of one dot represent a change of $5. To
turn on the gridlines, with the graph displayed,
 Press SHIFT MENU to reach the graph set-up screen.
 Then, use the down arrow until GRID is highlighted.
 Press F1 to turn the grid on and EXIT followed by F6 to return to the graph.
(See below left.)
 The Trace feature, accessed by pressing F1 , can help students recognize the
vast number of possible solutions. See below right.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
Depending on their maturity and background, students may have expected only
one specific answer when you first presented the problem. Obviously, that is not the case;
several values are possible for the mean cost of a relative’s present and the mean cost of a
friend’s present. After they have studied the graph and traced through the boundary line,
ask them to identify points in the shaded region, reminding them that the scale has been
set at five. As they create their list, they should keep in mind what the point means. For
example, the point (20, 15) is in the shaded region. This means that to keep within the
$200 limit, it is okay to average $20 per gift for each relative and $15 per gift for each
friend.
Then ask other questions, such as the following:

What is the significance of moving to the right within the shaded region?
(Because the x-value is increasing, this means they are spending more on
average for each relative gift.)

What is the significance of moving up within the shaded region? (This
represents increasing the average cost of a friend’s gift.)

Are there values that indicate spending exactly $200? (With this scale, four
points, (5, 30), (20, 20), (35, 10), and (50, 0), are on the line that defines the
region. Because they are on this boundary, they indicate that the maximum
amount available, $200, was spent.)

Based on your graph, what would you do? Support your answer. (Answers will
vary.)
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ANOTHER TECHNIQUE USING A TABLE
Another way to explore the problem is with a table, looking for the maximum
values for x and y. Once again the calculator can provide an excellent means for
investigating. From the MAIN MENU, call up the “Table” function. To look at the table,
we need an equation, not an inequality.
 To change the type of relation, press F3 for type and F1 for Y=.
 With Y1 highlighted, type in
2
100
, again using the a b/c to enter the
x
3
3
fraction bar.
 The next step is to set the range for our x-values. Pressing F5 allows us to
choose the values of x we would like to investigate. Because x can vary from 0 to
50, we should set 0 as our Start value and 50 as our End value.
 The pitch is up to us. If we want, we could look at every possible monetary value
for x by entering a pitch of .01. This seems a little extreme, but some students
may wish to find a solution that is this precise. Alternatively, we can check each
dollar by entering a pitch of 1, or perhaps we could check fewer values by
entering a pitch of 2 or even 5.
 Below left shows a pitch of 2 and below right shows the beginning of the table,
obtained by pressing EXIT and F6 after the range was set.
Use the down and up arrows to scroll through the table, noting the possible
maximum values for x and y. These are average values for relatives’ and friends’ gifts in
which all $200 is spent; any values less than these would, of course, spend less than the
allotted $200.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-C: RAISING MONEY
You and your classmates decide to sell sweatshirts and T-shirts to raise money for
a school trip. You decide that you should sell at least thirty items, but do not want to
exceed 120 items. Based on a small survey of students, you also decide that the number
of T-shirts should be at least twice the number of sweatshirts.
A.
Assign variables to the unknown quantities and write a system of inequalities
that model the given restrictions.
B.
Graph the system, indicating an appropriate window and scale and shading the
feasible region.
C.
Determine the vertices of the polygonal feasible region.
D.
Assume the profit on each sweatshirt is $5 and the profit on each T-shirt is $2.
What is the maximum profit you can obtain?
E.
How many sweatshirts and how many T-shirts should you sell to maximize your
profit?
EXTENSIONS
1.
Use inequalities that do not involve “or equal to.”
2.
Change the restrictions to ones determined by the class. Use numbers that do
not result in integral solutions.
3.
Assume different profit margins and determine the maximum profit.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ONE SOLUTION TO PROBLEM 2-C: RAISING MONEY
A. Assign variables to the unknown quantities and write a system of inequalities
that model the given restrictions.
Let x represent the number of sweatshirts and let y represent the number of Tshirts that you intend to sell. Since you want to sell at least thirty items, the sum of x
and y must be 30 or greater. Algebraically, we write x  y  30 .
Similarly, if you do not want to exceed 120 items, the sum of x and y must be
120 or less. Algebraically, this becomes x  y  120 .
Our third restriction indicated that the number of T-shirts (y) must be greater
than or equal to two times the number of sweatshirts. We can write y  2 x .
If we care to, we can include contextual, common sense inequalities stating
that both x and y must be at least (greater than or equal to) 0.
B. Graph the system, indicating an appropriate window and scale and shading the
feasible region.
From the MAIN MENU screen, call up the “Graph” menu.
 Delete any functions by pressing F2 for “Delete” and F1 to confirm the
deletion.
The first inequality we wish to enter is x  y  30 . First, however, we need to
solve for y. Subtracting x from both sides gives us y  30  x .
 To change the TYPE of relation, press F3 , F6 for more options, and then F3
to obtain the “greater than or equal to” relationship we are looking for.
 At this point, just type in 30  x and press EXE .
 To put in the second relation, which is a “less than or equal to” relationship, again
solve for y first. Then press F3 for TYPE, F6 for more options, F4 for the
“less than or equal to” relation, 120 – x, and EXE .
 Again change the type to enter the third relation, y  2 x . After pressing the up
arrow a couple of times so you can see all three relations, your screen should look
like the one shown below on the left.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
If the standard viewing window is used, you will not see much of the graph. What
you do see is shown below right.
The problem, of course, is with the window. The standard window only shows
values between –10 and 10 for both the x and y variables. To begin a search for a
window, we may decide to look at values between 0 and 120 for both x and y. The
logic behind this choice is that we cannot sell a negative number of sweatshirts or Tshirts, and certainly if the total is restricted to 120, neither value alone could exceed
120. A scale of 10 might be appropriate for both axes to avoid an over-abundance of
tick marks.
 When looking at the graph, press SHIFT
F3 for the viewing window. Type in
the desired values, pressing EXE after each entry to obtain the screen shown
below left.
 Pressing EXIT and F6 will redraw the graph with the new viewing window.
 If you wish, the window may be further refined, perhaps by setting the maximum
x-value to 50. To do so, press SHIFT
F3 , make the change and press EXE ,
EXIT , and F6 . The result is shown below right.
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
C. Determine the vertices of the polygonal feasible region.
The vertices of the shaded feasible region can be found numerically (by using
table values on the calculator if desired), algebraically (by solving for the
intersections of pairs of lines), or graphically. An approach that takes advantage of
the calculator’s capabilities is described below.
Assuming we are restricting ourselves to non-negative values (using x  0 ,
the y-axis) as the left boundary, our region is framed by a quadrilateral. We wish to
find the vertices of this figure. Two of the points are on the y-axis; by definition, these
are the y-intercepts of two of our boundary lines. Our three lines are x  y  30 ,
x  y  120 , and y  2 x . Before we find the y-intercepts graphically, let us note that
it can be easily determined that the y-intercepts are 30, 120, and 0 respectively. Since
(0, 0) is not a boundary point, two of the four vertices are (0, 30) and (0, 120).
The calculator also gives us these points quickly. While looking at the graph:
 Press F5 to access the graph solver
 F4 for Y-ICPT (the y-intercept), the up or down arrow key until the desired line
is shown, and EXE .
 Repeat this process for each of these three lines. You should quickly identify the
two desired points. Again, (0, 0) is not desired.
To find the other two vertices:
 Again access the graph solver by pressing F5 , but then press F5 to find the
intersection points.
 Use the up or down arrows to find a desired function and press EXE when a
function you want is listed.
Note that when you try to find the intersection of Y1 and Y2, you receive a
“NOT FOUND” message. This is because the two lines are parallel and have no
intersection. Repeat the process for other pairs of lines. The intersection of Y1 and Y3
is (10, 20) and the intersection of Y2 and Y3 is (40, 80). We now have our four
vertices. Again, they are (0, 30), (0, 120), (10, 20), and (40, 80).
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EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
D. Assume the profit on each sweatshirt is $5 and the profit on each T-shirt is $2.
What is the maximum profit you can obtain?
To maximize (or minimize) a value in linear programming, one need only
check the vertices of the polygonal feasible region. Since x represents the number of
sweatshirts and y the number of T-shirts, we need only check the value of the
expression 5 x  2 y at each of the four points found above. Several techniques can be
used, but for values such as these, simply plugging into the expression may be the
simplest. This is shown below:
(0, 30)
-
5(0) + 2(30) = $60
(0, 120)
-
5(0) + 2(120) = $240
(10, 20)
-
5(10) + 2(20) = $90
(40, 80)
-
5(40) + 2(80) = $360
Clearly, the maximum profit that can be achieved is $360. Students may wish
to check other points shaded in the feasible region to evaluate the profit; this may
help convince them that maximum and minimum values can occur only at vertices.
E. How many sweatshirts and how many T-shirts should you sell to maximize your
profit?
This question has actually been answered within the work for question D
above. The point that produced the maximum profit of $360 was (40, 80).
Consequently, you should sell 40 sweatshirts and 80 T-shirts.
22
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ANOTHER TECHNIQUE WITH LINEAR PROGRAMMING
Another technique that can be effective in investigating linear programming
problems, such as RAISING MONEY, involves the DYNAMIC capabilities of the
calculator. The authors of this material have come to call this technique the “Nina
Technique,” named after a participant in the pilot sessions who shared her ideas with us.
Begin as normal, using the GRAPH menu to establish the feasible region. The
window for this problem has been changed to the one shown below left. Then,
 With the graph displayed, press OPTN , F1 for picture, F1 to Store the
picture, and F1 or any other function key to store the graph as a picture in any
desired memory location.
 Press MENU for the MAIN MENU. Call up the “Dynamic Function” screen.
 Press SHIFT MENU for SET UP, scroll down to “Background,” press F2 for
Picture, and F1 or whatever function key is needed for the location in which you
stored the picture. Then press EXE to return to the “Dynamic Function” screen.
(Remember to set the Background to None later.)
 Highlight each of the functions and delete them by pressing F2 and F1 .
Our profit function can be represented algebraically as P  5 x  2 y . We want to
see what the maximum profit is that remains within our feasible region. Solving this
function for y, we have y 
P  5x
. Enter this function as shown below right.
2
23
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
We now wish to set P up as a dynamic variable, that is one that can vary
according to conditions we set. To do so:
 Press F4 for Variable. P will be automatically listed as the dynamic variable.
 Press F2 for Range. Select values that you think might describe the possible
profits. If you are unhappy with the values you have chosen, you can simply try
again. One possible set is shown below left. Press EXE after typing each value.
 When you are finished, press EXIT .
 To select the speed, press F3 . Highlight the speed you want and press F1 to
select it. “Stop&Go” has been selected, as shown below right.
 Press EXIT to return to the primary “Dynamic” screen and then F6 . This will
take your calculator a few moments to set up. If you get a MEMORY error, try
setting a different range of values for P.
 Press EXE to move the profit function through the feasible region. Note that
when P = $350, the profit line is still within the feasible region (below left), but
when P is $400, the profit line has left the feasible region (below right).
Changing the range so that the pitch is smaller allows you to be more
accurate. This technique can provide an excellent visual for the students.
24
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-D: FOOD WEB OF SELECTED ANIMALS
The line diagram below is called a digraph and represents a small food web. The
directed segment joining cat and rat, for example, indicates that cats eat rats. Numbers 1,
2, and 3 are associated with each animal for later reference.
RAT (2)
CAT (1)
MOUSE (3)
The relationship expressed in this digraph can be represented by a matrix if we let
the numbers 1, 2, and 3 represent the respective rows and columns in a 3-by-3 matrix. In
constructing the matrix to convey the information of the digraph, let the position for each
entry be designated by the ordered pair (i, j), the i indicating the row position and the j
indicating the column position. Put a 1 in that position if i eats j, and put a 0 in that
position if i does not each j. Thus the matrix associated with the above digraph, which we
0 1 1 
will call F, is 0 0 1 . The three 1’s in the matrix represent the three directed


0 0 0
segments of the digraph.
The matrix F2 represents indirect food sources. For example, in row 1 column 3 of
F2, a “1” appears because cats use mice as an indirect food source; that is cats eat rats,
and rats eat mice. Determining why this happens is a worthwhile exploration in itself.
Now suppose that because of damp, rainy weather, the insect population of an
area has increased dramatically. The insects are annoying to people and animals. State
authorities are in favor of using an insecticide that would literally wipe out the entire
insect population. You, as an employee of the Environmental Protection Agency, must
determine whether this action will be detrimental to the environment.
Consider the following possible digraph of a food web for seven animals,
including the insects that are causing the problem.
25
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
BEAR (1)
FIELD MOUSE (2)
INSECT (3)
CRAYFISH (4)
FOX (7)
FROG/TADPOLE (5)
TROUT (6)
PART 1
A.
Construct the associated matrix to represent this web. Call it Matrix A. Notice
that rows 2, 3, and 5 each contain a single 1. What does this indicate?
B.
Column 1 contains all 0's. What does a column of all 0's indicate?
C.
Notice that bears and trout have the most direct sources of food. This can be
determined by finding the sum of the numbers in the rows. Find the sums of the
seven rows.
D.
Column 3 has the most 1's. What does this suggest about the food web?
E.
The matrix A2 denotes indirect (through one intermediary) sources of food. Find
A2. Notice that column 3 contains all nonzero numbers. What does this
indicate?
F.
Find A + A2 and the associated row sums. This matrix denotes the total number
of direct and indirect food sources of food for each animal.
G.
What animal has the most food sources?
26
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PART 2
Now suppose that an insecticide has been introduced and all of the insects have
been killed. Several animals will lose a food source.
H.
Construct a new Matrix B to represent the food web with no insects. What effect
does this have on the overall animal population?
I.
What has happened to the row sums? What has happened to the food source of
the tadpoles and the field mice? What are the implications?
J.
Find B2 and B + B2.
K.
What are the row sums? Compare these answers with those of the original
Matrix A.
L.
Will all the animals be affected by the insecticide? Which animals will be least
affected?
M.
Organize and summarize your findings in a brief report to convince the
authorities that this insecticide is harmful to the total environment.
REFERENCE: Mathematical Modeling in the Secondary School Curriculum, A
Resource Guide of Classroom Exercises, Edited by Frank Swetz and, J.S. Hartzler,
NCTM, 1991.
27
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
ONE SOLUTION TO PROBLEM 2-D: FOOD WEB OF SELECTED ANIMALS
PART 1
A. Construct the associated matrix to represent this web. Notice that rows 2, 3, and
5 each contain a single 1. What does this indicate?
We will use the graphing calculator for our matrix work. From the MAIN
MENU, call up “MATrices.” Then
 Define the dimensions of Matrix A. Type 7, press EXE , again type 7, and again
press EXE .
 Type in each number in the matrix, reading across the row and pressing EXE
after each entry. Press EXIT when finished.
The entire matrix will not show on one screen, but it should be equivalent to
the matrix shown below.
0
0

0

0
0

0
0

1 0 0 0 1 1
0 1 0 0 0 0
0 1 0 0 0 0

0 1 0 1 0 0
0 1 0 0 0 0

0 1 1 1 0 0
1 0 0 1 0 0
On your calculator, the first five rows of Matrix A should look as below.
The single 1 in rows 2, 3, and 5 suggests that, at least for the animals listed in
the food web, field mice, insects, and frogs/tadpoles have only one source of food.
28
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
B. Column 1 contains all 0's. What does a column of all 0's indicate?
A column of 0’s indicates that the animal has no predator. Here it tells us that
none of the listed animals eats bears.
C. Notice that bears and trout have the most direct sources of food. This can be
determined by finding the sum of the numbers in the rows. Find the seven row
sums.
Row sums in order are {3, 1, 1, 2, 1, 3, 2}.
D. Column 3 has the most 1's. What does this suggest about the food web?
More animals rely on insects for food than any other source.
E. The matrix A2 denotes indirect (through one intermediary) sources of food. Find
A2. Notice that column 3 contains all nonzero numbers. What does this indicate?
From the MAIN MENU, select “Run.” Then,
 Press OPTN , F2 for “Matrices,” F1 for matrix names, ALPHA for letters,
the key with the letter A on it (the same key that is used for the x-variable), and
the squaring key. Your screen should look like the one below left.
 Press EXE to obtain the result. See below right.
The fact that column 3 does not contain any 0’s indicates that every animal
eats an animal which feeds on insects. In other words, insects are an indirect source of
food for every animal listed in the web. The entire matrix follows.
29
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
0
0

0

0
0

0
0

1 2 1 2 0 0
0 1 0 0 0 0
0 1 0 0 0 0

0 2 0 0 0 0
0 1 0 0 0 0

0 3 0 1 0 0
0 2 0 0 0 0
F. Find A + A2 and the associated row sums. This matrix denotes the total number
of direct and indirect food sources of food for each animal.
First, we want to store A2 in for Matrix C (we have already identified a
purpose for Matrix B.)
 To do so, after pressing the EXE key above, press AC/ON to clear the screen.
 Now repeat the procedure for squaring Matrix A, then press the store key (the
right arrow), F1 for matrix names, ALPHA and the key with C on it. Before
you press EXE , your screen should like the one below left.
 After you press EXE your screen should look like the one below right.
30
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
Had we chosen to do so, we could have done this when we first squared A.
We now wish to add Matrix A with Matrix C. Simply using F1 to tell the calculator
you are working with matrices, perform the operation as shown below left. After
pressing EXE your screen should look like the one below right.
The entire matrix sum is shown below.
0
0

0

0
0

0
0

2 2 1 2 1 1
0 2 0 0 0 0
0 2 0 0 0 0

0 3 0 1 0 0
0 2 0 0 0 0

0 4 1 2 0 0
1 2 0 1 0 0
G. What animal has the most food sources?
The bears have the most food sources. The sum of the first row is 9, the
highest of all the row totals.
31
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PART 2
H. Construct a new matrix B to represent the food web with no insects. What effect
does this have on the overall animal population?
From the MAIN MENU, choose “Matrix.” Highlight B and again make it a 7
by 7 matrix (so we can keep the same labels), but construct the matrix as if there were
no arrows going into or out of the insects. Obviously, there will be a large impact
upon these animals. Your matrix should be equivalent to the following:
0
0

0

0
0

0
0

1 0 0 0 1 1
0 0 0 0 0 0
0 0 0 0 0 0

0 0 0 1 0 0
0 0 0 0 0 0

0 0 1 1 0 0
1 0 0 1 0 0
I. What has happened to the row sums? What has happened to the food source of
the tadpoles and the field mice? What are the implications?
The row sums of the animals that depend on insects have decreased by 1. The
only food source for the tadpoles and the field mice has disappeared, and they will
soon die.
J. Find B2 and B + B2.
Using the same technique as described earlier, we find that B2 and B + B2 are:
0
0

0

0
0

0
0

1 0 1 2 0 0
0 0 0 0 0 0
0 0 0 0 0 0

0 0 0 0 0 0
0 0 0 0 0 0

0 0 0 1 0 0
0 0 0 0 0 0
0
0

0

0
0

0
0

32
2 0 1 2 1 1
0 0 0 0 0 0
0 0 0 0 0 0

0 0 0 1 0 0
0 0 0 0 0 0

0 0 1 2 0 0
1 0 0 1 0 0
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
K. What are the row sums? Compare these answers with those of the original
matrix A.
The row sums of B + B2 are {7, 0, 0, 1, 0, 3, 2}, considerably smaller than
before the insects were eliminated.
L. Will all the animals be affected by the insecticide? Which animals will be least
affected?
All the animals will be affected by the insecticide. The bear will be affected
the least.
M. Organize and summarize your findings in a brief report to convince the
authorities that this insecticide is harmful to the total environment.
Answers will vary, but all should recognize the potential impact of eliminating
an animal from the food web, especially one low on the food chain.
33
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-1: CLASS PLAY
The freshman class is planning its annual play. Production expenses were high
and you determine that the revenue from ticket sales must be at least $2,000.00. If you
charge $5.00 admission for every adult and $2.50 for every student, determine the
combinations of adult tickets and student tickets that must be sold to produce the
$2,000.00. Show this relationship algebraically, graphically, and with a table. Describe
the value each representation has.
PROBLEM 2-2: THE PIGGY BANK (Reference: Mathematics Teaching in the Middle
School, NCTM, Vol.2 No 6, May 1997, Menu of Problems.)
John has $150.00 in his piggy bank and plans to add $2.00 each week. Parneshia
has $200 in her piggy bank and plans to add $6.00 dollars each week. After how many
weeks will Parneshia have twice as much money as John? Develop your solution first
with lists, then with a graph, and, finally, with an algebraic equation.
PROBLEM 2-3: SWIM RECORDS (Reference: UCSMP, Algebra, 1988.)
In 1988, the women's record for the 100-m freestyle in swimming was 54.73
seconds. It had been decreasing at a rate of 0.33 seconds a year. The men's record was
49.36 seconds and had been decreasing at 0.18 seconds a year.
A. Assuming that these rates continue, after how many years will the records be the
same? What year is this? Justify your solution with a table, with a graph, and
algebraically. Do you think this will happen? Why or why not?
B. Use the provided information to predict the world record times in this event for
the years 1992, 1996, 2000, 2004, and 2008. Compare this with the actual records.
What does this say about the information provided in the problem?
C. Two ways to address the mathematical issues raised in this problem are to limit
the domain of the linear model or to seek a different model. Comment on both of
these strategies.
34
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-4: HEIGHT AND FOOT LENGTH
Someone claims that there is a strong relationship between people’s heights and
the length of their feet. You are to investigate this claim
A. After gathering the data, construct a scatterplot. What domain, range, and scaling
factors work well? Do height and shoe size appear to be related? What type of
relationship seems appropriate?
B. How linear are the data? Perform linear regression and determine how well the
regression line describes the data.
C. Interpret the slope of the regression line. Be sure to include units in your
interpretation. Does this have real-world meaning? If so, what?
D. Interpret the y-intercept of your regression line. Be sure to include units in your
interpretation. Does this have real-world meaning? If so, what?
E. Find five people whose data were not used for the regression model. Determine
how well the model predicts their shoe size when you know their height and how
well the model predicts their height when you know their shoe size.
PROBLEM 2-5: PEANUT BUTTER AND JELLY
Eric loves peanut butter and jelly, sometimes together and sometimes separate.
Peanut butter has 190 calories per serving, and grape jelly has 50 calories per serving.
How many servings of each can Eric have if he keeps his total calorie intake from these
two foods under1000? Use multiple representations to demonstrate your solution.
35
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-6: GEOMETRIC NUMBER PATTERS (Reference: Mathematics for
Elementary Teachers: An Activity Approach, Albert B. Bennett, Jr / L. Ted Nelson. WCB
McGraw - Hill, 1998.)
Consider the following sequence of geometric number patterns and the number
represented by each term.
1
5
13
A. Draw the 4th term in this sequence. How many tiles did you use?
B. How many squares would you need to represent the 5th term?
C. Create a table relating the figure number with the number of tiles.
D. Graph the information from the table. Describe the graph.
E. Find a pattern in the table of values. In order to help discover the pattern,
investigate the first order differences of the function values.
F. Investigate the second order differences.
G. Using the general form for a quadratic function, set up a system of equations from
your table of values.
H. Solve this system of equations to find the quadratic function that describes this
geometric number pattern.
I. Find the number of squares needed to build the tenth term of this sequence.
36
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-7: THE DOG WALKERS
Max and Moritz decide they would like to start their own business to earn some
extra money. They decide to start a business that offers a service.
After several weeks of observing their neighborhood, they decide that a dog
walking service is needed. In addition to establishing their own web page, Max and
Moritz have 200 advertisements for the M & M Dog Walkers to send to families in their
neighborhood. To get the ads out in a timely manner, Max and Moritz hire three people to
fold the ads, stuff and seal the envelopes, and apply the address labels they printed on
their computers. They will pay $0.03 to fold an advertisement, $0.06 to stuff and seal an
envelope, and $0.02 to apply an address label.
Person 1 folds 58 advertisements, stuffs and seals 60 envelopes, and applies 50
address labels.
Person 2 folds 77 advertisements, stuffs and seals 72 envelopes, and applies 78
address labels.
Person 3 folds 65 advertisements, stuffs and seals 68 envelopes, and applies 72
address labels.
A. Create a matrix showing the amount of work each person does. What do your
rows and columns represent?
B. Create a matrix showing how much money is paid for each action. What do your
row(s) and column(s) represent?
C. Determine how much each person will be paid. Explain how you arrived at your
answer.
D. Can matrix multiplication be used to show how much money each person earns?
Explain, using the context of the problem.
E. Can matrix multiplication be used to show how much money is spent on each of
the three actions? Explain, using the context of the problem.
F. Write a matrix representing the number of actions performed by each person.
Label the rows and columns.
37
EDSEC 437/637 Unit 2: Elementary Algebra Problem Set
PROBLEM 2-8: PLANNING A ROUTE
A map shows major routes connecting some of the towns in the Clemson area.
These routes are shown in the graph below; assume each route is two-way.
A. Create a matrix that represents the information contained on the graph. Use the
town names to label the rows and columns. If there is a direct route between two
towns, enter a 1 in the appropriate cell. If not, enter a 0.
B. Use a matrix operation to determine the number of ways you can get from one
town to another by going through exactly one other town. Explain why this
operation works.
C. How many ways are there to go from Simpsonville to Clemson by going through
exactly one other town? Explain your answer both in terms of the graph and in
terms of the matrix operation.
D. In the squared matrix, explain the entry in the Clemson row, Clemson column. Be
specific.
E. Use matrix operations to determine the number of ways to get from Mauldin to
Clemson by going through exactly two other towns. Explain your result both in
terms of the graph and in terms of the matrix operations.
F. Identify two different contexts in which the underlying mathematical ideas for
this problem can be applied?
GREENVILLE
EASLEY
MAULDIN
SIMPSONVILLE
PIEDMONT
CLEMSON
PELZER
38