# 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 y-intercept. 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 piecewise-defined

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

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 best-fit 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 y-axis 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 follow-up 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 TI-83 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 follow-up 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

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. Half-life 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
half-life 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 half-life 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 9-volt 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

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 TI-83 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 follow-up 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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