Excel, pathways course

1. Using Excel
to Build Understanding
in a Pathways Course
Kathleen Almy & Heather Foes
Rock Valley College
Rockford, Illinois

2. Keurig vs. Starbucks
Suppose you buy a venti latte every day at Starbucks. Is it worth
it to buy a Keurig and make your own coffee at home?
Estimate how many days you would need to make your own
coffee at home to recover the cost of the Keurig, assuming you
use one K-cup per day. How much will you spend on coffee for
that number of days?

3. Keurig vs. Starbucks: Algebraic approach
Assumptions:
Each latte is $3.57 plus 8% sales tax.
The Keurig you are interested in costs $99 and the coffee K-cups
are about $16.49 for a 24-pack.
Cost at Starbucks = 3.57(1.08)n n = number of days
Cost with Keurig = 99 + (16.49/24)n
3.57(1.08)n = 99 + 0.69n
n ≈ 32 days

4. Keurig vs. Starbucks: Numeric Approach
Suppose you buy a venti latte every day at Starbucks for $3.57 plus
8% sales tax. Is it worth it to buy a Keurig and make your own
coffee at home? Suppose the Keurig you are interested in costs
$99 and the coffee K-cups are about $16.49 for a 24-pack.
Cost at Starbucks: 3.57(1.08)= $3.86/day
Cost with Keurig: Start at $99 and add $0.69/day
Compare costs with graphs and tables in Excel.

5. Keurig vs. Starbucks
$-
$20.00
$40.00
$60.00
$80.00
$100.00
$120.00
$140.00
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Cost
Days
Keurig vs. Starbucks
Keurig
Starbucks

6. Keurig vs. Starbucks
Takeaways:
Students can approach, and solve, this problem without algebra.
Students can be led from the numeric solution to an algebraic
approach.
Using Excel shows the full story, not just the break-even point.

7. Why use a spreadsheet?
• Numeric approaches take advantage of how students naturally
solve problems.
• Using algebra requires a level of generalization and abstraction
many students are still developing at this level.
• Numeric and graphic approaches can be more accessible and
illuminating. They are not necessarily simpler or inferior.
• Using a spreadsheet minimizes the number crunching and
allows students to focus on making sense of the numbers.

8. Is a Hybrid Car Worth It?
You’re considering buying a 2015 Toyota Prius which starts
around $25,000 and gets 50 mpg. A similarly equipped Honda
Accord will run closer to $22,000 but will get 31 mpg. Can the
Prius make up the price difference with its lower fuel costs? How
long would it take?
$ 𝑠𝑎𝑣𝑒 = $ 𝑔𝑎𝑠 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙
1
𝐺𝑀 𝑛𝑜𝑤
−
1
𝐺𝑀𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑑

9. Hybrid Car: Algebraic Approach
Assumptions: Current gas price is $2.25/gallon
$ 𝑠𝑎𝑣𝑒 = $ 𝑔𝑎𝑠 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙
1
𝐺𝑀 𝑛𝑜𝑤
−
1
𝐺𝑀𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑑
$3,000 = $2.25 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙
1
31
−
1
50
3,000 = 0.02758 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ≈108,772 miles
If you drive 1,500 miles per month, this would take over 6 years.

10. Hybrid Car: Numeric Approach
You’re considering buying a 2015 Toyota Prius which starts around
$25,000 and gets 50 mpg. A similarly equipped Honda Accord will
run closer to $22,000 but will get 31 mpg. Can the Prius make up
the price difference with its lower fuel costs? How long would it
take?
Assumptions: Current gas price is $2.25/gallon
Prius: Start at $25,000 and add cost of gas times number of miles
divided by mileage→ $25,000 + $2.25 ∗
𝑚𝑖𝑙𝑒𝑠
50 𝑚𝑝𝑔
Accord: Start at $22,000 and add cost of gas times number of miles
divided by mileage → $22,000 + $2.25 ∗
𝑚𝑖𝑙𝑒𝑠
31 𝑚𝑝𝑔

11. Hybrid Car

12. Hybrid Car
Takeaways:
Students have some real-world experience and intuition they can
bring to this problem if you let them.
You can bridge from the numeric approach to the algebraic
approach.
The numeric approach can be used to generate the formula and
discuss the factored form as well.

13. Hybrid Car: A related problem
You are considering buying a new hybrid vehicle but are
concerned that the benefits might not outweigh the cost. How
efficient would the hybrid need to be in order to make up a
$5,000 price difference if the standard vehicle gets 25 mpg?
Assume gas costs $2.25/gallon.
$5,000 = $2.25 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙
1
25
−
1
𝐺𝑀𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑑
Can beginning algebra students solve this algebraically?
Can beginning algebra students solve this numerically with a
spreadsheet?

14. Hybrid Car $5,000 = $2.25 ∙ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙
1
25
−
1
𝐺𝑀𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑑
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
20 40 60 80Years
New MPG
Years needed to reach $5000
5,000 miles/year
7,500 miles/year
10,000 miles/year

15. Sierpinski Triangles
Assume the first triangle has an area of 1 square unit.
What happens to the area of the green portion as the number of green
triangles increase?

16. Sierpinski Triangles: Algebraic approach
Find the area of the first four triangles. Generalize the area of the nth triangle.
In general:
9
square units
16
3
square units
4
27
square units
64
1 square unit
1
3
square units
4
n
 
 
 

17. Sierpinski Triangles: Numeric approach
Find the area of the first four triangles. Use Excel to compute successive areas.
Observe the pattern as more green triangles are created.
9
square units
16
3
square units
4
27
square units
64
1 square unit

18. Sierpinski Triangles

19. Sierpinski Triangles
Assume the first triangle has a perimeter of 3 units.
What happens to the total perimeters of the green portions as the number of
green triangles increases?

20. Sierpinski Triangles: Algebraic approach
Find the perimeters of the green portions of the first four triangles.
Generalize the perimeter of the green portion of the nth triangle.
In general:
27
units
4
9
units
2
81
units
8
3 units
1
3
units
2
n
n

21. Sierpinski Triangles: Numeric approach
Find the perimeters of the green portions of the first four triangles. Use Excel
to compute successive perimeters. Observe the pattern as more green
triangles are created.
27
units
4
9
units
2
81
units
8
3 units

22. Sierpinski Triangles

23. Sierpinski Triangles
Takeaways:
The problem of analyzing the patterns of the areas and
perimeters is more accessible with a spreadsheet than with
algebra.
Students can gain insight into geometric sequences with the
numeric approach. This knowledge helps later with exponential
functions.

24. Projecting Ebola in September 2014
From the NPR article: A Frightening Curve: How Fast Is The Ebola Outbreak Growing?
"It's spreading and growing exponentially," President
Obama said Tuesday.
Right now we've had more than 5,000 cases of Ebola, and at least
2,600 people have died.
Five thousand cases of Ebola is bad; 10,000 to 25,000 is
unbelievable.
And with this outbreak, cases are doubling every three to four
weeks.
So if help doesn't arrive in time — and the growth rate stays the
same — then 15,000 Ebola cases in mid-October could turn into
30,000 cases by mid-November, and 60,000 cases by mid-
December.

25. Projecting Ebola

26. Projecting Ebola: Algebraic approach
When will the number of cases reach 60,000? Is the December
estimate accurate?
Start by making some assumptions:
Starting value on Sept 7 is 4,366 cases.
Number of cases doubles every 3.5 weeks.
Number of cases = 4,366 ∙ 2 𝑡
where t = # of 3.5-week periods
60,000 = 4,366 ∙ 2 𝑡
Since developmental students do not yet know logarithms, use
Excel to project the number of cases using the given function.

27. Projecting
Ebola:
Numeric
Approach
t = 3.78 time periods of
3.5 weeks each or
approximately 13 weeks
The December estimate
is accurate.

28. Projecting Ebola
Takeaway:
Using Excel gives developmental students a way to solve
the problem and improve their understanding of
exponential modeling.

29. Projecting Ebola
NOTES:
The CDC predicted 1.4 million cases by January 20, 2015. According to the World
Health Organization, the number of cases in January was actually 21,000.
The CDC used very intricate models incorporating many variables. They
projected the worst-case scenario, which increased awareness and resources.
From The Economist: “For a start, the models relied on old and partial figures.
These were plugged into equations whose key variable was the rate at which
each case gave rise to others. But this “reproduction number” changed as
outside help arrived and those at risk went out less, avoided physical contact and
took precautions around the sick and dead. So difficult are such factors to predict
that epidemiologists modelling a disease often assume that they do not change.
By presenting such grim projections, the experts arguably made it less likely that
they would come to pass. One of their purposes, says Neil Ferguson, a member
of the WHO’s Ebola response team, was “to wake up the world and say that this
could be really bad if we don’t do anything”. They succeeded, and resources
poured in. ”
http://www.economist.com/news/international/21642242-why-projections-
ebola-west-africa-turned-out-wrong-predictions-purpose at all.

30. Using Excel in traditional
algebra exercises
Excel can illuminate the solution to a
linear equation.
Solve the equation 2x – 8 = 5x – 17.

31. Using Excel in traditional
algebra exercises
Excel can help students understand
identities.
Solve the equation 2x – 8 = 2(x – 4).

32. Using Excel in traditional
algebra exercises
Excel can help students use the slope to find
more points on a line, including the y-intercept.
Problem:
Find the equation of line passing through the
points (3, 6) and (8, 31).
Find the slope and use it to fill up and down.
𝑚 =
25
5
=
5
1
Identify the y-intercept.
Write the equation: y = 5x – 9.
x y
3 6
8 31
-5
-4
-3
-2
-1
0
1
2
4
5
6
7
9
10
11
12
13
14
15
-1
+1
-34
-29
-24
-19
-14
-9
-4
1
-5
+5
11
16
21
26
36
41
46
51
56
61
66

33. Using Excel in traditional
algebra exercises
Takeaway:
Using Excel can help students develop a deeper
understanding of algebraic exercises. It encourages
students to move beyond mindless manipulations and
instead make sense of the concepts involved.

34. Questions

35. kathleenalmy@gmail.com
heather.foes@gmail.com
http://almydoesmath.blogspot.com
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