1. Mathematical Modeling
Making Predictions with Data

2. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Function
A rule that takes an input, transforms it, and produces a unique output
Can be represented by
a table that maps an input to an output
a graph
an equation involving two variables
Domain – the set of inputs
Range – the set of outputs t d 2 13 3 18 5 28 8 43 10 53
t ≥ 0
Imagine that we want a function to represent the distance that a toy car moves from a line on the floor. [click] We can measure and record the distance of the car from the line at various times. A table is one way to represent the function.
According to our collected data, at t = 2 seconds, the car was 13 ft away from the line. In this instance the input is 2 seconds, and the output is 13 ft.
[click] We can also represent the relationship between time and distance with a graph. The input variable is graphed along the horizontal (or x-) axis. The output value is graphed along the vertical (or y-) axis. Note that the data points from the table fall on this line. [click]
[click] The relationship between time and distance can also be represented with an equation. This function or equation can be referred to as a mathematical model since it provides a representation of the phenomenon.
[click]Domain. The set of input values to a function are referred to as the Domain. Domain values displayed in the table are 2, 3, 5, 8, and 10. However, if we know that the input, t, represents time, we can assume that the input could be one of many, many other values, such as 21.5 seconds. The time could range from the starting time (assume t = 0) to whatever the ending time. We call this a continuous variable since the value can be anything within a range of numbers, not just whole numbers. So assuming time could continue endlessly, the domain could be written as t≥0.[click]
[click] Range. The set of output values from a function are referred to as the Range. Range values displayed in the table include 13, 18, 28, 43 and 53. However, if we look at the graph, the car started at t = 0 at 3 feet away from the line and moved farther away as time increased. So the distance will range from 3 and could potentially keep getting larger and larger as time continues infinitely. So the range can be described as d ≥ 3. [click]
The graph demonstrates the continuous nature of the input and output by showing that any time (greater than 0) will yield an output value or a distance.
y = 5t + 3
d ≥ 3

3. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Linear Function
A function that demonstrates a constant rate of change between two quantities
Can be represented by a line on a coordinate grid
Can be represented by a linear equation involving two variables
Can represent real-life situations
Distance traveled over time
Cost based on number of items purchased
The function we just examined, distance traveled over time, is a linear function. The relationship represented by the graph and equation here is also a linear function.

4. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Linear Equation
A linear function can be expressed by a linear equation
An equation involving two variables
Independent variable, x
Horizontal axis
Dependent variable, y
Vertical axis
[click] In algebra, you often deal with equations that relate the variables x and y. Most often x is considered the independent variable and is graphed on the horizontal axis. Y is considered the dependent variable and is graphed on the vertical axis.
[click] In science and engineering, we often assign letters to variables that are somehow related to the quantity that we are dealing with. For instance distance might be represented with a d and time with a t. [click]
Variables can represent any two related quantities

5. Linear Equation Data is often collected in tables
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Linear Equation
Data is often collected in tables
Data is graphed on a coordinate plane as ordered pairs
(5, 28)
(3, 18)
(2, 13) d t 2 13 3 18 5 28 8 43 10 53 2 13

6. Linear Equation A line can be drawn through data points
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Linear Equation
A line can be drawn through data points
Line-of-best-fit
Trendline
Slope-intercept form
y = mx + b
m = slope = Rise Run =
b = y-intercept
y = 5x + 3 5 3
5 1 = 1
The slope of a line is the vertical rise divided by the horizontal run between any two points on the line. In this case the slope is 5 divided by 1, or 5.
The y-intercept is the y-value of the point at which the line intersects the y-axis. In this case the y-intercept of 3 is also referred to as (0, 3), a coordinate pair. 5 =

7. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Function Notation
Functions are often denoted by letters such as F, f, G, w, V, etc.
G(t) represents the output value of G at the input number t
Garbage production over time
t is a member of the Domain
t ≥ 0
G(t) is a member of the Range
G(t) ≥ tons
slope m = tons/year
Garbage production increases by tons/year
Function notation is a method of denoting a function by a letter and showing the input variable or value. The function shown here represents the garbage production of York County from 1970 through 2005 and is represented by the letter G. This is a mathematical model of the garbage production in York County.
[click] Assuming that the time starts at t = 0, the domain is any t greater than or equal to zero.
[click] For this function, the output is initially tons when t = 0 and continues to increase. So the range G(t) is greater than or equal to tons. Note that the y-intercept is tons.

8. Function Notation Example: d(t) = 5t +3 Slope, m = 5 ft/s
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Function Notation
Example: d(t) = 5t +3
Slope, m = 5 ft/s
The toy car moves 5 feet for every second of time
y-intercept, b = 3 ft
The toy car is initially 3 feet from the line at time t = 0
What is the distance at t = 6 s
d(6) = 5 · (6) + 3 = 33 ft
This is an example of a linear function. The slope is 5.
[When presenting the example, show that the graph gives the same mapping. That is, if you input t = 6, the d-value (y-value) of the point on the line is 33. And if you want the time when the distance is 23 ft, the graph shows a t-value of 4 when d = 23. ]
When will the object be 23 feet from the line?
d(t) = 23 = 5t +3, t = 4 33 4

9. Correlation Coefficient, r
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Correlation Coefficient, r
Measure of strength of a linear relation
-1 ≤ r ≤ 1
r = ±1 is a perfect correlation
r = 0 indicates no correlation
Positive r indicates a direct relationship
As one variable increases, so does the other
Negative r indicates an inverse relationship
As one variable increases, the other decreases
Strength of relationship
r > 0.8 is a strong correlation
r < 0.5 is a weak correlation
How well does the regression line fit the data? There are two measures that are typically used to determine how well the mathematical model fits the data – the correlation coefficient, r, and the coefficient of determination, r^2.
The correlation coefficient indicates how strongly the data follows the trendline.

10. Coefficient of Determination, r2
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Coefficient of Determination, r2
Measure of how well the line represents the data
0 ≤ r2 ≤ 1
Portion of the variance of one variable that is predictable from the other
Example: r2 = 0.65, 65% of variation in y is due to x. The other 35% is due to other variable(s).
Square of the Correlation Coefficient
Excel easily provides the coefficient of determination and can be displayed with the trendline.
The Coefficient of Determination provides a measure of how much of the variation of a quantity is due to another quantity.
For instance, if you plotted data for the weight of a BMX bike against the height of jump and found a trendline that gave a coefficient of variation of 0.62, you could say that 62% of the difference in the height of jump is due to the weight of the bike. The other 38% of the variation is due to other factors such as speed, skill of the rider, surface, etc.
Calculation of the correlation coefficient (and therefore the coefficient of determination since it is simply the square of the correlation coefficient) involves a rather complicated mathematical calculation. Fortunately, we have software tools that will perform this computation for us.

11. Finding Trendlines with Excel
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Finding Trendlines with Excel
Fiscal Year
Year 2003 = 1
Sales (millions $)
2003 1
2.35
2004 2
2.22
2005 3
2.34
2006 4
2.54
2007 5
2.55
2008 6
2.75
2009 7
3.11
2010 8
3.24
2011 9
3.15
Fiscal Year
Sales (millions $) 03
2.35 04
2.22 05
2.34 06
2.54 07
2.55 08
2.75 09
3.11 10
3.24 11
3.15
Create table of data
Common practice to re-label years starting with n = 1
Select data
One software application that can be used to identify a mathematical model to represent data is Excel. Let’s look at an example of how you can use Excel to find the trendline.
Allow students to input the table of data into an Excel worksheet.
To select the data, simply left click and drag the mouse to select both columns of data.

12. Finding Trendlines with Excel
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Finding Trendlines with Excel
Insert Scatterplot
From the insert tab, choose Scatter in the Charts panel. Select the first scatterplot with only markers.
[click] A graph of the data points will be displayed.

13. Finding Trendlines with Excel
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Finding Trendlines with Excel
Format the Scatterplot
Select the scatterplot
Choose the Layout tab
Chart Title
Axis Titles
Gridlines
Legend (delete)
In order to format the scatterplot, select the chart on the worksheet.
Under the Chart Tools Layout tab, you can adjust the chart title, axis titles, gridlines, and delete the legend if desired. [Click, then allow students to adjust the Layout to display the graph as shown here.]

14. Finding Trendlines with Excel
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Finding Trendlines with Excel
Format the Scatterplot
Select the scatterplot
Under Chart Tools
Choose Format tab
Select Horizontal (Value) Axis in drop down menu
Choose Format selection
Adjust the axis options
Select Vertical (Value) Axis
Again select the scatterplot.
Under the Chart Tools Format tab, you can adjust the axis options for both the horizontal and vertical axes. [Click through instructions, then have students make adjustments to display the axis as shown here. Allow students time to adjust settings to obtain the desired graph format.]
Note that the horizontal axis was formatted to show several years in the future.

15. Finding Trendlines with Excel
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Finding Trendlines with Excel
Add Trendline
Select the scatterplot
Under Chart Tools
Choose Layout tab
In the Analysis panel
Choose Linear Trendline
Select Trendline (either within chart or in Current Selection panel)
Forecast
Display Equation
Display R-squared value
Again select the scatterplot.
Under the Chart Tools Format tab, you can adjust the chart and axes titles, the gridlines, and the legend.
[click] A graph of the data points will be displayed.

16. Making Predictions Use the trendline to make predictions
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Making Predictions
Use the trendline to make predictions
Function notation
S(t) = t
where S(t) = projected sales
t = year number (t = calendar year )

17. Making Predictions Use the trendline to make predictions
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Making Predictions
Use the trendline to make predictions
What is the sales projection for 2015?
t = 2015 – 2002 = 13
S(13) = (13) = $3.76 million
You can use either the trendline equation or the graph to make predictions or estimations.
In this case, the trendline can be written in function notation to represent the sales forecast of the product.
It is important to note that r^2 = 0.89, so r = sqrt(0.89) = This indicates that the data displays a strong linear relationship and that the trendline provides a good representation of the data. This doesn’t mean that the predictions it provides will be true, it only indicates that the trendline provides a good representation of the data used.
r2 = 0.89
r =
= 0.94
2003
2015

18. Making Predictions Use the trendline to make predictions
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Making Predictions
Use the trendline to make predictions
When will the sales reach $4 million?
S(t) = t
4 = t
0.1335t = 4 –
t = = 14.8
Say t = 15
t = calendar year – 2002
15 = calendar year – 2002
Calendar year =
Calendar year = 2017
1.9731
In this case, you know the value of the sales to be $4 million, so S(n) = 4. You will solve for the year in which this will occur.
Substitute 4 for S(n)
Then subtract from each side
Then divide each side by
When rounded to the next year, n = 15. Note that since the data is based on fiscal year, annual sales of over $4 million will be recorded in the following year.
Because we chose to let n = 1 in 2003, we can calculate the calendar year by adding 2002 to n.
Using the trendline, sales are predicted to reach 4 million in 2017.
You can obtain similar results using the graph. When sales equal 4 million dollars [click], the year is approximately 15 [click]
2002
2003
2017