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Curriculum of the Algebra 2 Collaboration
Linear Data Analysis
1.Guess the Age
Students are asked to guess the ages of a group of famous people. The actual age
is paired with the
student’s guess to begin discussion of information in a scatterplot, a linear
model (y=x), and the
accuracy of the model. In an attempt to identify the best guesser in the class,
techniques are used that
anticipate residuals.
Technology: PowerPoint slideshow to show faces, Graphing calculator
2. Introduction to Linear Data Analysis
Using a collection of eight scatterplots, students determine graphs of lines or
curves that might be
appropriate models to describe bivariate data. Data that associates the length
of a spring with the
weight hung from the spring is used to find the linear regression line.
Interpret the slope and the yintercept
of the line, and forecast other ordered pairs from the linear model.
Technology: PowerPoint slideshow of scatter plots, Graphing calculator
3. Price of Apples
More practice with scatterplots, linear regression line, interpreting slope and
y intercept , and
prediction. Students will shift data left to give meaning to the yintercept.
Students discuss possible
criteria of linear regression line. (Taken from the Algebra II Indicators from
NCDPI.)
Technology: Graphing calculator
4. Hurricane Fran
The criteria of the linear regression line are defined. The Geometer’s Sketchpad
geometric
illustration is useful to showing the sum of the squares of the residuals. The
Fran data is definitely
not linear. Fitting this data with a line will show the usefulness of the
residual plot. (Taken from the
Algebra II Indicators from NCDPI.)
Technology: Geometer’s Sketchpad web site, The WRAL website, Graphing
calculator.
5. Piecewise Defined Functions as Models
The data for the Olympic swimming records for the 400 Freestyle found in the
Algebra 2 Indicators
show data with two definite trends over time. In this lesson students develop a
piecewisedefined
linear function using domain restrictions and the linear regression line. This
model provides specific
information in the slopes to compare the data of the two trends. Since both
men’s and women’s data
is given, one data set can be discussed in class and students can follow up the
lesson with the other
data set.
Technology: Graphing Calculator
The Linear Function
6. Linear Inequalities with a Parallelogram
Students are given the coordinate of a point in the second quadrant. Based on
this point they develop
a group of four linear inequalities whose solution forms a parallelogram that
falls in the second
quadrant of the coordinate system. The task asks students to find equations of
lines and look at points
of intersection. They learn to graph inequalities on the graphing calculator.
(Taken from the Algebra
II Indicators from NCDPI.)
Technology: Geometer’s Sketchpad and Graphing calculator.
7. Linear Programming
Using a problem setting of varying hours of summer work with two possible jobs,
students explore
the restrictions of domain and range and linear inequalities to set boundaries
on a region of the graph
where solutions may lie. Within this region, they must determine the solution
that gives the
maximum weekly income. (This problem is based on a problem from the NC Algebra
II Indicators.)
Once a solution is reached, generalizations about the method are made, and
another problem tackled.
Technology: Graphing calculator, PowerPoint slideshow.
The Quadratic Function
8. Wile E. Coyote
Wile E. Coyote creates a catapult to catch Road Runner. Using a quadratic
function that describes the
trajectory of the Wile E. as he is shot from the catapult. Students find maximum
values, zeros, and
domain to answer questions about the antics of Wile E. (This problem is based on
a talk by Wally
Dodge.) A second problem from the NC Algebra II Indicators leads students
through a similar
procedure in tracking the path of a space shuttle.
Technology: Graphing calculator, Animation (from studio video) showing Wile E.
and Road Runner.
9. Pig Problem: Writing and Solving Quadratic Equations
Given several problem settings, students develop quadratic functions for which
they investigate
maximum values, zeros, and specific values to answer specific questions about
the settings.
Technology: Flash animations to illustrate problem settings, Graphing
calculator.
10. Football and Braking Distance: Model Data with Quadratic Functions
Students are given data to describe the trajectory of a football tossed from the
tallest bleachers of a
stadium. The data is fit with a quadratic function using least squares criteria.
Given data extracted
from page 288 of Glencoe’s Algebra II book, students investigate braking
distance versus speed of a
car. Using quadratic least squares, the student finds a bestfit function for
the data. Data is given on
reaction distance versus speed of the car. When reaction distance is added to
braking distance to find
total stopping distance, students fit another quadratic function. A Follow Up
Problem relates number
of sides of a polygon with the number of vertices to create a quadratic
function.
Technology: Flash animation to illustrate problem setting, Graphing calculator.
11. Questions about Quadratics
Using a group of questions from the Algebra II Indicators from DPI, students use
both the calculator
and paper and pencil to answer questions about characteristics of quadratic
functions.
Technology: Graphing calculator.
12. Collecting and Fitting Quadratic Data with the CBL
Using the CBL and the graphing calculator, students work in groups to collect
data describing the
freefall of an object over time. The data collected includes data not relevant
and that must be
eliminated , and data is shifted near the yaxis to make the intercept
meaningful. The students
describe the meaning of the coefficients. The experiment is run again with an
object that has drag
(like a hat) and a model is found. The followup problem works with the football
data from the
lesson: Football and Braking Distance: Model Data with Quadratic Functions.
Technology: CBL, Graphing calculator, Balldrop and Hiker programs for the TI83
plus.
Other Functions
13. Distance Formula
Using rulers, students measure distances on a diagram to find a shortest path.
They create ordered
pairs and a scatterplot. With the motivation that the scatterplot has a clear
message, the students
develop a function that measures the distances using the distance formula. Based
on the function, the
shortest distance can be estimated and then considered on the diagram. A
followup problem
involving determining the best place to put a new Post Office is included.
Technology: Ruler, Graphing Calculator
14. Equations with Radical Expressions
Data representing the period of a swinging pendulum versus the length of the
pendulum can be best
modeled by a square root function. Data and an appropriate model are both given
to the students.
Questions from the NC Algebra II Indicators require students to solve equations
involving radical
expressions. Solutions are also investigated from both a graphical and an
analytical point of view.
Technology: Graphing Calculator
15. Applications of Rational Functions
By developing a function to describe the annual cost of a refrigerator and given
a function describing
concentration of drug in the body, students relate the behavior of the graph of
a rational function with
the phenomenon it describes. Asymptotes and particular points become important
information about
the application.
Technology: Graphing calculator.
16. Composition and Inverses of Functions
Concepts of composition are used to develop functions that describe volumes of
pyramids with
specific bases and combinations of special discounts when purchasing a car. The
connection between
study time and number of courses leads to a function using inverse function that
can help students
determine the number of courses to take for available weekly study time.
Technology: Graphing calculator.
17. Polynomials as Models
A data set of the average price of gasoline for each year from 1993 to 2001
shows data with many
changes. Using all the different regression curves and the regression line from
the calculator, the
students investigate the best model of the data and discuss its ability to
predict.
Technology: Graphing calculator
The Exponential Function
18. The Drug Problem
Using ideas presented in Jim Sandefur’s article from the February 1992
Mathematics Teacher we
model the amount of cough syrup in the body over time with water and food
coloring. Next, students
calculate the amounts of medicine in the body every four hours using an informal
iterative process.
From these ordered pairs of time and amount, we fit the data with an exponential
function found
using the exponential regression fit on the calculator. This function is then
interpreted within the
context of the amount of cough syrup in the body.
Technology: Measuring cups, food coloring and spoons, Graphing calculator.
19. Halflife and Doubling Time
Skittles or M&M’s are randomly thrown onto a paper plate. The candies that fall
with a letter face up
are removed. Students document throw number and number of pieces remaining.
Using the
exponential regression fit, we find a decreasing exponential function with a
halflife of one. A similar
data collection that leads to an increasing exponential function with a doubling
time of one results
from cutting a sheet of paper, stacking the resulting pieces and cutting again.
With these definitions,
the Hurricane Fran data from the Algebra 2 indicators is fit with an exponential
function. Students
then determine if this data has a halflife or doubling time.
Technology: M&M’s, scissors, Graphing calculator.
20. Money and the Exponential Function
Using the ideas of compounding, students use shorter and shorter compounding
periods that lead to
the definition and meaning of e.
Technology: Graphing calculator.
21. Voltage Data Collection for Exponentials
Using the CBL with voltage probe, a 9volt battery, resistor, and capacitor,
students collect data
describing how the voltage drains from a capacitor when it is disconnected from
the battery. A
comparison of the ratios of the voltage reading at one second with the voltage
reading of the next
second reveals that the voltage is falling by a consistent percentage.
Therefore, the data is described
by a decaying exponential function as a model.
Technology: Graphing calculator, CBL with voltage probe, battery, resistor, and
capacitor for each
group of students.
Mathematical Modeling
22. The Box Problem
Students build open top rectangular boxes from a standard sheet of paper by
cutting congruent
squares from each corner. Data is collected that pairs the length of the side of
the cut out square with
the volume of the resulting box. To describe a clear pattern shown in the
scatter plot, students
develop a function through analysis of the box design. Based on this function,
the length of the side
of the square is determined that will create a box of maximum volume, and two
squares that will
produce a box of equal volume.
Technology: Graphing calculator, Geometer’s Sketchpad sketch, Animation (from
studio video)
showing 3 different versions of how to make a box.
23. Relationships in Rectangles
Using random integers between 0 and 30, students create lengths and widths of
rectangles. In the list
facility of the graphing calculator, these lengths and widths can be used to
calculate perimeters and
areas of the rectangles. Students investigate several relationships using
scatter plots—the most
exciting is area versus perimeter. Using the ideas of a function forming a
boundary on the scatter
plot, students discover information about the perimeter and area of a rectangle.
This lesson is based
on an article “Connecting Data and Geometry” by Tim Erickson found in the
November 2001
Mathematics Teacher.
Technology: Graphing calculator.
Miscellaneous Topics
24. Univariate Data Analysis
Using the techniques of line plots and stem and leaf plots, but focused on box
and whisker plots,
students investigate which baseball player they would most like to have on their
team: Barry Bonds,
Mark McGwire, or Sammy Sosa.
Technology: Graphing calculator, data program for TI83 plus.
25. Matrix Operations
Using three settings students apply matrix addition and multiplication .
Statistics of recent NFL
quarterbacks from several years allow students to see the definition of matrix
addition, matrix
subtraction, and scalar multiplication. Using a problem setting from
Contemporary Precalculus
through Applications students investigate orders of students at a lunch counter
using a probability, the
transition matrix, and matrix multiplication. A similar technique is used to
investigate the location of
a mouse in a maze as a followup activity
Technology: PowerPoint slideshow for NFL problem, Flash animation for mouse in
the maze
problem, graphing calculator.
26. Complex Numbers
Using the definition of complex numbers and operations with complex numbers,
students add,
multiply, and graph with complex numbers using some sample items from the NC
Algebra II
Indicators. Once familiar with the operations and graphing, students iterate
complex numbers in
functions to determine whether the iteration stabilizes. With some
experimentation, rules are
developed that show patterns in stabilization that carry into graphs by special
coloring schemes. The
result is a fractal. Examples from the Julia Set and the Mandelbrot Set are
shown.
Technology: Graphing calculator, Power Point slides.
27. End of Course Test
Using nine sample items from the Algebra 2 End of Course test, students develop
strategies for taking
the end of course test.
Technology: Graphing calculator.
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