LINEAR_EQUATIONS
UNIT 4: SYSTEMS OF LINEAR EQUATIONS |
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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 |
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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 |
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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 |
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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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