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LINEAR_EQUATIONS

UNIT 4: SYSTEMS OF LINEAR EQUATIONS

Action Item 4.1: What is a Solution to a System of Linear Equations ?

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Activity 1:
Introduction to
Systems of Linear
Equations
In this activity, you will investigate a situation involving
two linear functions by looking at a graph of the two
in order to develop an understanding of the solution
to a system of equations. Use this graph to review
the concepts you studied earlier: Function, domain,
range, independent variable , dependent variable, linear
function, and rate of change ( slope ). If you need a
review of these concepts, refer to the resource center.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)
Activity 2:
Solving a System
of Equations
Using Tables
In this activity, you will examine a situation involving
two functional relationships by creating a table to analyze
the situation and find a solution.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)

Action Item 4.2: Solving Systems Using Graphs and Tables

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Activity 3:
Solving a System
with Graphs
In Activity 2 you solved a problem by examining a table.
In this activity you solve similar problems by graphing.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)
Activity 4:
A System with
No Solution
In the first three activities you found the solution to a
system of equations. In this activity you will learn that
there are other possible outcomes when solving a system.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)

Action Item 4.3: Solving Systems by Symbolic Methods

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Activity 5:
Substitution
Method
In previous activities you looked at a problem involving
tickets and solved it by examining a table and then again
by graphing. In this activity you will solve this problem
using symbolic methods.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)
Activity 6:
Linear
Combination
Previously, you solved a system of equations using the
substitution method. Another method of symbolic
solution is linear combination.
In the following tutorial we are going to practice this
method on the system:
2x + y = 7
x – y = –1
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)

 

Activity 7:
Mixture Problems
Systems of equations can be used to solve many different
kinds of problems. We will try some of these methods
on another kind of problem.
111.32(c)(4)(A)
111.32(c)(4)(B)
A(c)(4)(A)
A(c)(4)(B)
Activity 8:
Which Method
Should I Use?
You have learned four methods for solving systems of
equations.
Graph
Table
Substitution
Linear combination
Part of understanding the methods for solving systems of
equations is learning to recognize which solution is most
appropriate for a given problem. In this activity you will
practice solving systems in order to develop a sense for
which method to choose.
111.32(a)(6)
111.32(c)(4)(B)
A(c)(4)(B)

UNIT 5: QUADRATIC FUNCTIONSUNIT

Action Item 5.1: Getting Your Money’s Worth

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Activity 1:
Area and
Perimeter
Functions
The creation of the raised beds will require money to
purchase lumber or other equipment. The student group
did some research to determine the cost.
111.32(b)(1)(C)
111.32(b)(2)(B)
111.32(d)(1)(A)
A(b)(1)(C)
A(b)(2)(B)
Activity 2:
Finding Values
of Quadratic
Functions
In the last activity you saw two functions related to
a square bed . In this activity you will examine the
function y = x2 more closely and define other properties
of the function.
111.32(b)(2)(A) A(b)(2)(A)

Action Item 5.2: Donated Materials

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Activity 3:
The Parent
Function
Multiplied by a
Constant
You have been introduced to a new function called the
quadratic function. In this activity you will look at the
effects of multiplying the function by a constant.
111.32(b)(1)(C)
111.32(d)(1)(A)
111.32(d)(1)(B)
111.32(d)(2)(A)
A(b)(1)(C)
A(d)(1)(A)
A(d)(1)(B)
A(d)(2)(A)
Activity 4:
Area of a Circle
CS Ranch Supply has offered the circular raised beds
made from galvanized metal water tanks. In this activity
you will investigate this kind of bed using your graphing
calculator.
111.32(b)(1)(C)
111.32(d)(1)(B)
A(b)(1)(C)
A(d)(1)(B)

 

Action Item 5.3: A New Condition

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Activity 5:
Adding and
Subtracting a
Constant
You have investigated the effect of multiplying the
x2 term in the quadratic parent function y = x2 by a
constant. In this activity you will look at the effect of
adding and subtracting a number from the x2 term. Let’s
look at another situation related to the garden problem.
111.32(d)(1)(C) A(d)(1)(C)
Activity 6:
Multiple Changes
to the Parent
Function
You have investigated the effect of multiplying the x2
term by a constant and adding or subtracting a constant
from the x2 term. In this activity you will investigate
what happens when both changes are made. In other
words, you’ll see how y = ax2 + c compares to y = x2.
111.32(d)(1)(B)
111.32(d)(1)(C)
A(d)(1)(B)
A(d)(1)(C)

Action Item 5.4: Adding a Walkway

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Activity 7:
Shifting Left
to Right
You have investigated the effect of multiplying the
x2 term by a constant and adding or subtracting a
constant to the quadratic parent function y = x2. In this
activity you will be looking at the effect of adding and
subtracting a number from x before it is squared.
111.32(d)(1)  

UNIT 6: SOLVING QUADRATIC EQUATIONS

Action Item 6.1: Binomial Operations

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Activity 1:
Binomial
Multiplication
In this activity you will investigate the product of two
algebraic
expressions called binomials. You will use your
ability to multiply two binomials to answer questions
about the ground level gardens and walkways.
The Cedar Springs garden club would like to have a plot
set aside in the community garden. The diagram shows
the club’s plan for a square area (gray region) in which
to place picnic tables and an area for shrubs and flowers
(green region).
111.32(b)(4)(A)
111.32(b)(4)(B)
A(b)(4)(A)
A(b)(4)(B)
Activity 2:
Factoring
Trinomials
Multiplying two binomials often gives an answer that
is a trinomial. In this activity, you will study the reverse
process. Given a trinomial, express it as a product of
two binomials.
111.32(b)(4)(A)
111.32(b)(4)(B)
A(b)(4)(A)
A(b)(4)(B)

 

Activity 3:
More Factoring
Trinomials
Recall that when you multiplied binomials you
sometimes got answers with a coefficient on the x -term.
In the activity you will practice factoring trinomials of
this type
: ax2 + bx + c. It is important to remember that
not all trinomials can be factored but you will learn how
to deal with those trinomials in later units.
111.32(b)(4)(A)
111.32(b)(4)(B)
A(b)(4)(A)
A(b)(4)(B)
Activity 4:
Solving
Equations by
Factoring
Now that you know how to multiply binomials and
factor trinomials, you will apply these skills to solving
equations.
There is a group that wants an area available for student
groups to do class projects. One of the students has
submitted a plan for a square garden with a 2-foot
walkway around the region.
111.32(b)(4)(A)
111.32(b)(4)(B)
111.32(d)(2)(A)
A(b)(4)(A)
A(b)(4)(B)
A(d)(2)(A)

Action Item 6.2: Modeling With Quadratic Functions

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Activity 5:
Connections
Between Factors
and Roots
In the previous activity, you learned how to solve
quadratic equations by factoring. This involved
factoring the trinomials and setting each factor equal
to zero . Because the trinomials were factored into
2 binomials, there were 2 solutions to the quadratic
equations you solved. However, this is not always the
case. In this activity you will study the various types of
solutions to quadratic equations.
111.32(d)(2)(A)
111.32(d)(2)(B)
A(d)(2)(A)
A(d)(2)(B)
Activity 6:
Quadratic
Formula
You have solved quadratic equations using tables, graphs,
and factoring, and learned that sometimes an equation
has a solution that cannot be found be factoring. In
this activity you will practice using a formula to solve
quadratic equations that cannot be factored.
111.32(d)(2)(A) A(d)(2)(A)
A(d)(3)(B)

Action Item 6.3: Solving Quadratic Equations

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Activity 7:
Projectile Motion
You have solved quadratic equations by using tables,
graphs, and factoring. You have used quadratic functions
to model situations involving area. In this activity
you will consider a quadratic function that models the
motion of a projectile.
111.32(d)(1)(D)
111.32(d)(2)(A)
111.32(b)(2)(B)
A(d)(3)(A)
A(d)(3)(C)
Activity 8:
Mathematical
Models
We have modeled problem situations using quadratic
functions. In this activity you will practice analyzing
problem situations using not only the roots of equations
but also the domain and range of the function that
models the situation.
111.32(b)(2)(B)
111.32(d)(2)(A)
A(b)(2)(B)
A(d)(2)(A)
A(d)(3)(A)

 

UNIT 7: NON-LINEAR FUNCTIONS

Action Item 7.1: How Fast Do Rumors Spread?

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Activity 1:
Changes in the
Exponential
Model
In this activity, you will learn about exponential growth.
You will analyze a variety of situations that can be
modeled with exponential functions.
111.32(d)(3)(C)  
Activity 2:
Graphs of
Exponential
Functions
Although it does not make sense in terms of the problem
of spreading rumors, we might ask the question, “What
happens when x < 0?” If you only look at the graph of
y = 2x in the first quadrant, you may confuse it with the
graph of half of a parabola (a quadratic function). In
this activity you will compare graphs of quadratic and
exponential functions. You will also compare graphs of
different exponential functions.
111.32(d)(3)(C)  
Activity 3:
Understanding
the Rules of
Exponents
In order to use exponential functions to model problems,
we need to take some time to understand how to work
with exponential expressions. In general, an exponential
expression can be written as am, where a is called the
base and m is called the exponent. The tutorial below
is designed to help you develop the four basic rules of
exponents. Make sure to work through all four rules in
the tutorial.
111.32(d)(3)(A) A(d)(3)(A
Activity 4:
Population Data
You have gathered the following data about the growth
of your city over the last ten years. In this activity
you will use this data to make predictions about the
population of your city in the future.
111.32(d)(3)(C)  

Action Item 7.2: Modeling Inverse Variation Data

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Activity 5:
Solving
Equations of the
form D = RT
Throughout this course, you have studied many different
kinds of relationships. You found that linear functions
have a constant rate of change. You found that quadratic
functions have constant second differences. Also, you
have found that exponential functions have a constant
ratio between consecutive y-values. In this activity, you
will learn about yet another type of function, called
inverse functions.
111.32(d)(3)(B)  
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