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Mathematics can be used to model physical relationships to help us understand
them better. In this lesson, you will analyze a geometric relationship and look
for connections among its multiple representations. You will be given a
geometric situation to explore and analyze by gathering and interpreting data.
Then you will generalize your findings by creating a mathematical model so
that you can make predictions.
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1-73. ANALYZING DATA FROM A GEOMETRIC RELATIONSHIP
Your team will make several paper boxes using the instructions given
below. Based on the physical models, you will then represent the relationship
between the height of each box and its volume in multiple ways.
Cut a sheet of centimeter grid paper to match the dimensions that your
team has been assigned. Then, cut the same size square out of each corner and
fold the sides up to form a shallow box (with no lid) as shown below.
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Your Task: As a team you will investigate the relationship
between the height of a box (the input) and its volume (theoutput). Do not use
a graphing calculator today. You may use a scientific calculator.
To analyze this relationship, your team will make at least six differentsized boxes by varying the size of the corners. Begin with six equally-sized
pieces of grid paper cut to your assigned dimensions. Record your information
using multiple representations—including diagrams, a table, and a complete
graph.
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How can we collect data for this relationship?
How much data is enough?
What are the appropriate inputs for this relationship?
How are the different representations related?
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1-74. Begin your investigation by building several
boxes, taking measurements, and collecting data.
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a. As a team, choose a starting input value. Note that this
value is the same as the length of the side of one of the
squares cut from the corner of your grid paper, which becomes the
height of your box. Make the first box and determine its volume. Label
the box with its important information. Work in the middle of your
workspace so that everyone understands what is being measured or
calculated, and be sure everyone agrees on the results before recording
the information in an input → output table on your own paper.
b. Each team member should now choose a different input value and build
a new box using this value. Calculate the volumes of the new boxes.
Record everyone’s data in your table.
c. Use the data in your table to create a complete graph to represent the
situation.
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1-75. MATHEMATICAL MODELING
Now you will generalize your results by creating a mathematical model.
Modeling is an important mathematical process. A common way to model is to
write an equation using algebra.
a. Draw a diagram of one of your boxes. Since the box in this diagram is
being used to generalize your results, you want it to represent a
relationship between any possible input and its output. How can you
label the height of the box to represent all possible heights?
Determine the length and width of this box in terms of the height and
label your diagram.
b.
Work with your team to write an equation for the volume (output)
using the generalized height (input) you chose in part (a). It may help
you to remember how you calculated the volume when the height was a
number and use the same strategy.
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1-76. LOOKING FOR CONNECTIONS
Put your table, graph, and equation in the middle of your workspace.
With your team, complete the parts below.
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As you address each question, justify your statements thoroughly. Also,
if you make an observation, discuss how that observation relates your table,
graph, and equation.
a. Is the domain of the relationship limited? That is, are there some input
values that do not make sense? Why or why not? How can you tell
using the graph? The table? The equation? Using the boxes themselves
(or diagrams of the boxes)?
b. Is the range of the relationship limited? That is, what are all of the
possible volumes (outputs)? Are there any outputs that do not make
sense? Why or why not?
c. Should you connect the points on your graph with a smooth curve? That
is, should your graph be continuous or discrete? Explain.
d. Fully describe the graph.
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1-77. What graph do you get when you use a graphing
calculator to draw the graph of your function? Explain the relationship
between this graph and the graph you made on your own paper. How can you
tell from the equation that the graph will not be a parabola?
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1-78. Organize your findings into a stand-alone poster that shows
everything you have learned about all of the representations of your function as
well as the connections among the representations. Use color, arrows, words,
and any other useful tools you can think of to make sure that someone reading
your poster will understand all of your thinking.
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1-79. LEARNING LOG
Your algebraic equation is a mathematical model of the
geometric situation in this lesson. In your Learning Log, explain the purpose
for creating a mathematical model for a situation. That is, give examples of
ways in which your model would be useful. Title this entry “Mathematical
Models” and include today’s date.
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Describing Spread
A distribution of single-variable data can be summarized by describing its
center, shape, spread, and outliers. Three ways to describe spread are explained
below.
Range
The range is the maximum minus the minimum. It is usually not a good way to
describe the spread because it considers only the extreme values in the data,
rather than how the bulk of the data is spread. For the data set below, the range
is 30 – 11 = 19.
Interquartile Range (IQR)
The variability, or spread, in the distribution can be numerically summarized
with the interquartile range (IQR). The IQR is found by subtracting the first
quartile (Q1) from the third quartile (Q3). The IQR is the range of the middle
half of the data. IQR can represent the spread of any data distribution, even if
the distribution is not symmetric or has outliers. In the data set above, the IQR
is Q3 – Q1 = 28.5 – 17 = 11.5.
Standard Deviation
Either the interquartile range or standard deviation can be used to represent the
spread if the data is symmetric and has no outliers. The population standard
deviation is the square root of the average of the distances to the mean, after
the distances have been made positive by squaring. For the data set above:
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There are 13 values in the population and the mean value is about
22.77.
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The sum of the squares of the distances from the mean is
(–11.77)2 + (–9.77)2 + (–7.77)2 + (–3.77)2 + (–0.77)2+ (–0.77)2 +
2.232 + 3.232 + 4.232 + 5.232 + 6.232 ≈ 500.3
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The population standard deviation is:
Because gathering data for entire populations is often impractical, most of the
data sets we analyze are samples. To calculate the sample standard deviation,
divide by one less than the number of values in the data set. If the data set
above is a sample from a larger population, then the sample standard deviation
is
.