Course Title, Assignment #1
Author, Date
Solutions and Hints are shown in red
Problem: Our friend Nathan (who saved his weekly allowance) is going on vacation
with his family for 12 weeks. During this time, his parents will not give him an
allowance.
Nathan decides to take a Savings of $150 with him on this trip. The table below
demonstrates how much money Nathan spends each week for the first five weeks of the
trip:
Please answer the following questions:
(a) How much money is Nathan spending each week of the trip? Justify and explain
how you get your answer.
Nathan is spending $15 each week. You can tell by subtracting the Savings from one
week to the next. For example, 150 – 135 = 15, 135 – 120 = 15 and so on.
(b) Where is the initial condition on this table? What information does the initial
condition provide? Explain.
The initial condition is the starting condition and it’s in the first row. It says that in Week
0, Savings is 150. This means that when Nathan starts this trip, he has $150.
(c) Assuming Nathan continues to spend money as demonstrated in the table, please fill
in Nathan’s Savings for Weeks 6 through 12.
Keep subtracting $15 and here’s what you get:
(d) Assuming Nathan continues to spend money as demonstrated in the table, do
Nathan’s Savings follow a linear pattern? Explain your answer.
Yes, Nathan’s Savings do follow a linear pattern because he is spending the same amount
of money for each passing week. If you spend a constant amount of money for each
passing week, then your growth is linear.
(e) Write an algebraic equation that models Nathan’s Savings per week. Please show
your work and explain how you get this equation.
There are two ways to do this. I will demonstrate both. Remember that the equation for
a line that we work with is y = mx + b where:
m is the slope (or the constant change),
b is the y-intercept (or the initial condition).
Method 1:
To find the slope m: Since we know Nathan is spending $15 each week, then the slope
will involve the number 15. However, we need to come up with the sign (positive or
negative) of the 15: because he is spending instead of saving, do you think the slope will
be positive or negative?
Take an educated guess!
To find the y-intercept b: b is the initial condition, which is the same as the Savings at
Week 0. Since the initial condition is 150, then b = 150.
So the equation of the line is y = –15x + 150.
Method 2:
To get the slope m: just take two points from your table, say (x1, y1) = (0,150) and
(x2, y2) = (1,135). Now use the formula:
y −y
m= 2 1
x 2 − x1
to get the slope. Obviously m come out to be –15.
€ m = –15 into y = mx + b to get y = –15x + b.
To get the y intercept b: Substitute
Then just pick any point from the table, say (x, y) = (10, 0). Plug these numbers in for x
and y and you will get an equation that can be solved for b. Obviously, b should be 150.
(f) Draw a graph that relates the Weeks to Nathan’s Savings on this trip. Please make
sure the graph starts at Week 0 and goes all the way through Week 12. Explain, in
words, how you drew your graph.
Here you can just plug in the points from the table and connect them.
The other thing you can do is use your equation y = –15x + 150, start at (0, 150), and then
use
rise −15
=
m=
.
run
1
Remember that the rise is –15, so which vertical direction will you go in as you move:
up or down?
€
Here is the resulting graph:
(g) If Nathan continues with these weekly expenditures, will his Savings last the full 12
weeks of the trip? Please answer this question using:
• the Table representation
On the Table Representation, you can see Nathan run out of money when the Savings
become 0 or negative, in this case, when they become 0. This happens in Week 10, so
no, this spending habit will not last the 12 weeks of the summer:
• the Algebraic representation
When you solve this, you have to imagine that you do not have information from the
table. All you have is the algebraic equation which is y = –15x + 150 and the question of
what happens in Week 12.
Since we are interested in what happens in Week 12, let’s let x = 12 and do the
calculation:
y = –15(12) + 150 = –180 + 150 = – 30. Same as on the table for Week 12! But clearly,
he is in debt because the amount is negative! So again, his Savings won’t last!
• the Graph representation
(h) Which of the three representations (table, algebraic, or graph) do you think was most
useful in answering part (g). Explain why.
This is a personal opinion for everyone. But please write a few complete sentences
describing which one you like best and why.
(i) Please consider Nathan’s Savings from Module 1 and the use of the Table, Algebraic
and Graphing representations to model his savings when he was receiving an allowance.
In words, compare and contrast the Table, Algebraic and Graphing representations used
in Module 1 to the Table, Algebraic and Graphing Representations generated in this
assignment.
That is, compare the tables to each other, compare the algebraic equations to each other,
and compare the graphs to each other.
To respond to this, look at the patterns in the table and whether numbers are increasing or
decreasing and what this means; look carefully at the algebraic equations and whether
numbers are positive or negative and what this means; look at the graphs and directions
they are slanted and what this means.