Activity 4.1c Mathematical Modeling
Introduction
In this activity you will collect and analyze data in order to make predictions based
on that data. You will use both manual and computer methods to record, manipulate,
and analyze the data in order to determine mathematical relationships between
quantities. These mathematical relationships can be represented graphically and by
equations, also known as mathematical models. You will then use the mathematical
models to make predictions related to the quantities.
Equipment



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Engineering notebook
Pencil
Computer with Excel
Wooden puzzle pieces from Project 4.1 Puzzle Design Challenge
Triple beam balance or scale
Procedure
Part 1. Determine a mathematical model for the mass of puzzle pieces as a function
of the number of wooden cubes in the piece. Then use the mathematical model to
make predictions.
1. Record the number of cubes and mass of each of your puzzle pieces.
Color of Piece
Number of
Cubes
Mass (g)
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2. Graph your data points on the grid below such that the number of cubes in the
puzzle piece is represented on the x-axis and the mass of the piece is indicated
on the y-axis. Label the axes and indicate units.
3. Consider the data that you have graphed and answer the following.
a. Would you expect that this data is linear; that is, if you were to measure
the mass of other pieces with more than six cubes or fewer than four
cubes, would the points fall on a straight line on the graph? Explain your
answer.
b. If you were to sketch a line-of-best fit, what would be a reasonable yintercept. That is, where would the line-of-best fit cross the vertical axis?
Explain your answer.
c. Based on your data, what would you predict for the mass of a single
wooden cube? Explain your answer or show your work.
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4. Using your predicted y-intercept, sketch a line-of-best-fit for your data on the grid
above.
5. Using your line-of-best-fit, complete the following.
a. Estimate the slope of your line-of-best-fit (include the appropriate units).
Explain the interpretation of the slope in words.
b. Write an equation for your line of best fit.
c. Rewrite your equation for the line-of-best-fit in function notation where
M(n) = mass of the puzzle piece and n = the number of wooden cubes.
d. Estimate the mass of a puzzle piece that includes two wooden cubes.
Show your work.
e. If a puzzle piece has a mass of 31.5 grams, how many wooden cubes
would you predict were used to create the puzzle piece? Show your work.
6. Use Excel to create a scatter plot of your data and find a trend line.
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Input the data in tabular form. Be sure to include column headings. You do
not need to include the piece color column.
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Create a scatterplot of the data. Format the axes, label the axes, and title the
chart as shown below.
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Add a linear trend line. Set the intercept to the value you suggested in
number 3 and format the trend line to forecast backward to zero cubes and
forward to 12 cubes. Display the equation of the trend line on the chart.
7. Use the equation of the trend line to answer the following.
a. Rewrite the equation of the trend line using function notation where M(n)
represents mass and n represents the number of cubes.
b. What is the domain of the function? That is, what values of n make sense?
c. What is the range of the function?
d. What is the slope of your trend line? Explain the interpretation of the slope
in words.
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e. How does the slope of your line-of-best-fit compare to the slope of the
trend line? Why is there a difference?
f. Predict the mass of a puzzle piece that is comprised of 9 wooden cubes.
Show your work.
g. If a puzzle piece had a mass of 49 grams, how many wooden cubes would
it contain? Show your work.
h. Is the trend line a good representation of the relationship between the
number of wooden cubes and the weight of the puzzle? Justify your
answer.
Part 2. Identify a mathematical model to represent the mass of puzzle pieces if
larger wooden blocks are used. Then use the mathematical model to make
predictions.
8. Assume that a puzzle cube was rebuilt using 1 in. cubes rather than ¾ in. cubes
and the following masses were recorded for the pieces.
Number of
cubes
Mass
(g)
4
39
5
47
6
57
6
58
6
57
Complete each of the following.
a. Create a scatterplot and find a trend line for the data using Excel. Print
a copy of your worksheet that includes
 Table of data
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
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Scatterplot with properly formatted axes, axes labels and units, and
an appropriate chart title
Trend line and its equation displayed on the scatterplot
b. Write the equation relating number of cubes to mass in function
notation. Be sure to define your variable.
c. What is the slope of the line (include units)? Explain the interpretation
of the slope in words.
d. How does the slope of this function (relating the number of 1 in. cubes
to mass) compare to the slope of the function you found in number 5
(relating the number of ¾ in. cubes to mass)? Explain the difference.
e. Use the function to predict the mass of a puzzle piece (using 1 in.
cubes) if the piece includes 8 cubes. Show your work.
f. If a puzzle piece 95 grams, how many 1 in. cubes are most likely
included in the piece? Show your work.
Part 3. Find a mathematical model to represent the minimum jump height of a BMX
bike as a function of the bike mass. Then use the mathematical model to make
predictions.
9. An engineer is redesigning a BMX bike. He is interested in how the mass of the
bike affects the height that the bike reaches when the rider “gets air” or jumps the
bike off of a ramp. He asked an experienced rider to test bikes of various masses
and recorded the following minimum jump heights.
Bike
Mass
(lbm)
19
19.5
20
Minimum
Jump
Height
(in.)
83.5
82.0
79.2
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20.5
21
22
22.5
23
23.5
24
77.1
74.9
73.3
71.0
68.1
65.8
64.2
Use this data to complete each of the following.
a. Create a scatterplot and find a trend line for the data using Excel. Print
a copy of your worksheet that includes the following:
 Table of data
 Scatterplot with properly formatted axes, axes labels and units, and
an appropriate chart title
 Trend line and its equation displayed on the scatterplot
b. Write the equation relating Bike Mass to Minimum Jump Height in
function notation. Be sure to define your variable.
c. What is the domain of the function? Explain.
d. What is the range of the function?
e. What is the slope of the line (include units). Is the slope positive or
negative? Explain the interpretation of the slope in words.
f. If the engineer designed a bike that weighs 18 pounds, predict the
minimum jump height. Give your answer in inches (to the nearest
hundredth of an inch) and feet and inches (to the nearest inch). Show
your work.
g. If the engineer designed a bike that weighs1 pound, predict the
minimum jump height. Give your answer in inches to the nearest
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hundredth of an inch and feet and inches to the nearest inch. Show
your work.
h. Does the predicted height for a one pound bike make sense? Is this
function a good predictor for minimum jump heights at all bike
masses? Explain.
i.
If the minimum jump height of 89.7 inches is recorded, predict the
estimated mass of the bike. Show your work.
Extend Your Learning
10. Assume that you will build your puzzle cube from 2 cm cubes of solid gold and
each cube had a mass of 153 g. Address each of the following.
a. Give a mathematical model (in function notation) that would represent the
mass of a puzzle piece depending on the number of gold cubes used in
the piece. Define your variables.
b. What would be the mass of a puzzle piece that is comprised of four gold
cubes?
c. If gold sells for $60/g, what is the four-cube gold puzzle piece worth?
d. How many gold cubes would you expect to be included in a puzzle piece
that weighs 1071 g?
Conclusion
1. What is the advantage of using Excel for data analysis?
2. What precautions should you take to make accurate predictions?
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3. What is a function? Explain why the mathematical models that you found in this
activity are functions.
4. Are all lines functions? Explain.
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