1. Module 2A Modeling Tool: Excel
Deriving a Mathematical Model: An Experimental and Virtual Approach via Spreadsheets
Materials needed
--Excel
--Internet access
--Three Excelets: Stacking Styrofoam Cups, Linear Regression, Interpolation & Extrapolation
--Hawaiian Island distance as a function of age data in a spreadsheet (from QELP)
--Styrofoam cups – at least 25 cups of all the same volume
--30 cm metric rulers (marked to the nearest millimeters)
Module 2A
Modeling Tool: Excel

2. Agenda Examine how modeling and simulation fit into the classroom
Collect some data and build a mathematical model
Compare physical model with mathematical model
Explore some basics of mathematical modeling
Analyze a data set and build model

3. The big picture…
In this module, we collect data, the spreadsheets produces a mathematical model with no prior modeling skills required. Then we use a simulation to develop an understanding of the model.
Step 1 of the process - DATA
From experimental data, we plot it to examine the data for trends. How does the y-variable change as a function of the x-variable? Is it linear? Think about any measurement error and what is does to the data values
Step 2 of the process - MODEL
Can we fit the data with an appropriate regression (trendline in Excel))? If so, we have a mathematical model (equation) to describe the data. We can make predictions. How good does the model fit the data?
Step 3 of the process - SIMULATE
With the addition of more variables which are usually controlled in the experiment (held constant), we can develop a simulation of our model. Now we can bring the mathematics alive by adjusting variables
Rule of 4 – multiple representations

4. Goals Build a mathematical model from data
Judge the goodness-of-fit of model to data
Experience the process: Data > Model > Simulation
Graphical interpretation – What does the graph tell me?
Here are the goals for the module.

5. Part I: Measuring the Stack Height of Nested Styrofoam Cups
Going to model the stack height of nested Styrofoam cup. Going to use a completely pre-built Excel file with built-in functions. The term “excelets” was coined by Lanny Arvin, an economics professor at the University of Illinois.
Here we use a pre-built “just add data” Excelet

6. Objectives With this Excelet, we will:
collect and plot experimental data
develop a mathematical model and examine its goodness of fit
make predictions with the model
physically interpret the model by simulation
examine the inverse function of the model and its use
simulate possible experimental errors in the model
assess your understanding of the model and error
This is what we are going to do with the model.
With the “simulate the model” tab of the Excelet we will examine the connection between the mathematical model and the physical model of stacking the nested cups (not usually possible in many cases).

7. Collecting the data
In small groups, participants can be given a variety of stacked styrofoam cups
Using a metric ruler, they need to measure the stack height to the nearest millimeter or 0.1cm
Groups can then pool the data to populate the spreadsheet
In front of you, have a stack of cups and a ruler. Each group has a different set.
Would you expect there to be a relationship between the stack height and the number of cups? Explain
Measure that and then come and put data in spreadsheet at front of room. Order of entry does not matter. Participants can also enter data in spreadsheet on their computer.
The spreadsheet is designed to “just add data” for the various stack heights. A regression line will appear with the equation for the model.
Careful measurement of stack heights will lead to an r-squared value very near one!
Refer to Building a Model Handout p. 2
You have created a model! The computer has plotted the best-fit straight line. Spreadsheet also asks students to read the equation and enter the slope and y-intercept into separate yellow cells (this is needed for subsequent tabs of the spreadsheet).

8. The mathematical model
On the “generate a model” tab of the just add data Excelet, as the data is added the points plot and a linear regression fit is done. Participants do not need to know how linear regression works, only that it gives the line of best fit through the data. A mathematical relationship is given along with r-squared, a measure of the goodness-of-fit (see comment cell on spreadsheet).
What is the slope and intercept of the model telling you? Can you interpret their meaning?
How would you interpret the graph/equation?
Possible answers:
Height is increasing by a little over a cm per cup
y-intercept =
Cup without a rim is a little over 8 cm tall
Slope is the rim height
What is R-squared? The higher the value – the better the fit.
Predictions: What is the height of 150 cups? Talk through possibilities - use equations, stack cups, multiply a known value (10 cups x 15 – intercept?)
What if have a height? How many cups? How would I do that?
Enter slope and intercept in the yellow boxes on the spreadsheet.
Now have a model and have a little understanding of it. Pretty sure model is OK as have a good R-squared. Go to simulation (next slide).
Handout p. 3

9. From the equation…
What is the equation or model tell you about the stacking of the cups?
Chart the responses.

10. The “Simulate the Model” tab will cleanly point out the meaning of the slope and intercept.
Participants should discover:
Height of the rim = slope Height of the base = y-intercept
When the scroll bars or sliders are adjusted the line responds as well as the cup dimensions to the right also change.
Can use the sliders (scroll-bars) to change values and check the model.
If you were unsure of the model, would this help?
What would you say about the uniformity of Styrofoam cups? Does your data support this?
When doing this experiment with students, graph will usually be “noisier” because students are not as careful measuring. Can also introduce noise by having them use rulers with larger space before 0.
Handout p. 4

11. From the equation…
Why does the equation or model have a positive y-intercept?
Chart the responses.

12. Go back to stacking cups page, copy and paste data from that worksheet to this one. Follow instructions from handout.
What can the inverse function tell you? Generate the inverse function by reserving the x and y data.
Ask this practical question:
Suppose we had shipping boxes of 61 cm and 100 cm, how any cups can we stack?
Inverse will allow us to answer this question.
Where would you use this in math class? Algebra I
In science class? Measurement.
Handout p. 5

13. What about errors in the model?
Where can errors originate?
Measurement
Manufacturing
Chart the responses.
What type of errors can arise from the ruler measurements? Did the zero mark on the ruler occur at the end of the ruler?
How uniform are the cups manufactured? Since the r-squared is close to one, manufacturing of the cups must be very uniform.
Now let’s simulate some “what if” questions to explore some possible errors.

14. Here a variety of errors are introduced
Here a variety of errors are introduced. The explanation of the errors is given on the spreadsheet – see comment cells. READ the comment cells!!!!!!
We built the model with “ideal” cups. We have a perfect model here. How do we know?
What happens if we “mess it up”?
Talk through random error (would go either direction) vs. systematic error (always goes one way – non-zero position on ruler is an example)
Also talk through the meaning of the 3 physical variables.
Here a variety of errors are introduced. PLAY WITH ONE ERROR AT A TIME. Be sure all other errors are set back to zero. Here we want to see how the model is influenced – Does slope change? Does the y-intercept change? Does the value of r-squared decrease?
Note red triangles (comment boxes) – can click on these to get definitions of variables.
Handout p Just point out: p. 7 has further assessment questions
p. 8 Notes to teachers on activity – answers!

15. Outcome
Participants will have derived a mathematical model for stacking nested Styrofoam cups: H = a*n + b where “H” is the stack height, “n” the number of cups, and “a” and “b” the parameters that depend on the volume (size) of cup used
Be sure participant write a mathematical equation in terms of the variables studied (not x and y) and place the numerical values of “a” and “b” in their equation!!!!!

16. What have we accomplished?
Make a list -
We have gone through the process:
data > model > simulation
We also developed an understanding of the mathematical model and examined goodness of fit and how errors influence the model.
Error analysis is a vital part of the scientific method.

17. Part II: Elements of Mathematical Modeling
Address the following three questions using the pre-built spreadsheets:
1. How does linear regression work?
2. How do we judge the goodness-of-fit of the regression line to the data?
3. What is the difference between interpolation and extrapolation?
Want to get into some basics of mathematical modeling. What you need to know if working with data.
Participants will use two pre-built spreadsheets to develop an understanding of linear regression, goodness-of-fit, and the difference between interpolation and extrapolation.
How does linear regression work? Minimizes the sum of the squared deviations - This is shown graphically (two-ways) and numerically.
Goodness-of-fit is judged by r-squared, the closer to one the better the fit (scatter in data points compared to linear regression line)

18. How linear regression works
Check the turn the eyeball line on and see how well you can do!
Open the Regression spreadsheet
With open spreadsheet – see a connect the dots line. Turn on “eye-ball line of fit”. Linear regression says, minimize the sum of the deviations. Then play with slope and y-intercept to minimize and try to get the smallest number.
Eye ball your best fit of a line to the data points. The vertical green lines on the graph are the deviations. Linear regression works by minimizing the sum of the squared deviations. The sum of the squared deviations is shown numerically (left side) and as a bar graph to the right. An actual linear regression can be added to see how well your eye-ball fit compares.
Try to see if you can minimize the sum of the squared deviations.
Compare with a fellow teacher.
How does linear regression work? It derives the best fit line by minimizing the sum of the squared deviations.
Is this the best way to do things? Flip to deviations tab. Now call them “residuals” – they will show a trend if data has a curve, which R-squared may not reveal.
Does this allow students access to linear regression without too much work?
Handout p. 9

19. Interpolation vs. Extrapolation
What is the difference?
Interpolation – estimating within data or between data points
Extrapolation – estimating beyond the range of the data at either end (Danger – Does the model hold beyond the limits of the data?)
Use the slider to move the tracer point. This spreadsheet helps students see the definition
What is interpolation of data?
What is extrapolation of data? Can you see any dangers to extrapolation? Look at next tab – which shows the mass of Aluminum cans over time. Looks like they have reached the smallest mass.
Handout p. 10

20. TASK: Distance – Age Data
Using data for the Hawaiian Islands, plot distance from big island as a function of age
Develop a mathematical model for the data
Judge the goodness-of-fit
Did the bend in the direction of the island chain made a difference?
Give instructions on how to locate and download file. This data is from QELP
Participants work handout pp with instructor circulating to help and answer questions!
Once participants have the file downloaded and have created a graph: (halfway through p 12)
Is there a trend?
Is it linear?
What would you predict the R-squared to be?
Then walk through the addition of a trend-line (see handout) and statistics.
Talk about how students approach apparently linear data.
May need to show the “forecast” for 100 periods to show why the polynomial is a “dangerous” model.

21. Discussion of Hawaiian Islands data:
The linear model is a pretty good fit (r-squared is high). There appears to be no change in rate due to the bend (change in direction of movement) in the data.
Did the bend (change in direction of movement of the Pacific Plate) influence the model? The bend is at about 42 km.
What does the slope mean in this model? The slope is the rate of movement of the Pacific Plate. Here it is 76 km/million years or with some unit conversions – 7.6 cm/yr

22. Outcome
Participants will derive a mathematical model using real-world data that shows how the age of the islands is a function of distance, age = m*distance + b, by plotting the data, graphing, and performing the linear regression
The slope (m) is the rate of movement – 76 km/million yrs or 7.6 cm/yr
Participant will have produced a labeled graph with data and a linear regression line. The equation and r-squared value should be displayed on the graph as well.

23. What other applications can you see in the classroom?
List any suggested.
You could suggest stacking other objects such as cookies or coins.

24. Classroom Applications
Probeware experiments
Finding data sets – see QELP website
List any suggested.

25. Summary Developed a mathematical model from experimental data
Examined factors that influence the model
Developed a simple understanding of linear regression and goodness-of-fit
Discovered the difference between interpolation and extrapolation
Now, go model some data!
You have seen--
How to build a model in Excel
A few things you need to build a model – linear regression
Can develop a model judging goodness of fit
So now you can get a set of data, build a model and see how good it is.