Created on: June 1, 2012

Website Address: https://library.curriki.org/oer/Unit-2-Linear-and-Exponential-Relationships

TABLE OF CONTENTS

- Lesson 2.1: Solving Systems of Linear Equations
- Lesson 2.2: Exponential Relationships with Equations and Inequalities
- Lesson 2.3: Understanding Negative Exponents and the Zero Exponents
- Lesson 2.4: Graphs of Exponential Functions
- Lesson 2.5: Division Properties of Exponents
- Lesson 2.6: Understanding Scientific Notation
- Lesson 2.7: Exponential Growth Functions
- Lesson 2.8: Exponential Decay
- Lesson 2.9: Building New Functions
- Unit 2 Project: Baseball Mania

IN COLLECTION

In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students explore systems of equations and inequalities, and they find and interpret their solutions. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

In this lesson we build upon student experience with graphing and solving systems of linear equations from middle school to focus on justification of the methods used. We rove that given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

In this lesson, students will learn to simplify expressions involving exponents and inequalities, use scientific notation, graph exponential functions, and model real-life situations using exponentials.

In this lesson we will distinguish between situations that can be modeled with linear functions and with exponential functions. We will recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

In this lesson we represent and solve equations and inequalities graphically. We focus on linear and exponential equations. Also we develop the understanding that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, which could be a line.

Both multiplication and division properties of exponents have the basic rules of operation for exponents. The associative and communicative laws of multiplication underlie the multiplication properties of exponents. We use the associative law to remove parentheses and the commutative law to rearrange factors within a product. In this lesson, students will learn to relate these considerations to the Division Properties of Exponents. Through using these division properties people can find useful information such as ratios.

Students will be able to: • write very large and very small numbers in scientific notation. • translate numbers from scientific notation to numerical form. • add, subtract, multiply and divide numbers in scientific notation.

In this lesson, students will learn to simplify expressions involving exponents, use scientific notation, graph exponential functions, and model real-life situations using exponentials. Lesson 2.4 was about graphing exponential functions. Lesson 2.7 is about one use of exponential functions, which is to model exponential growth.

In this lesson we will distinguish between situations that can be modeled with linear functions and with exponential functions by proving that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals. Also we focus on constructing linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Further, we interpret the parameters in a linear or exponential function in terms of a context.

In this lesson we focus on vertical translations of graphs of linear and exponential functions. We relate the vertical translation of a linear function to its y-intercept. This lesson is based upon the common core standard, F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. We experiment with cases and illustrate an explanation of the effects on the graph using technology.

In this project, we address many of the skills and standards taught in the nine lessons in this unit. We ask students to work in small groups to create a baseball-related business. Students will reason quantitatively and use units to determine costs of creating a part of a proposal for a new baseball organization; this includes costs of building, team selection and salary, marketing and sales, building maintenance and management requirements and salaries and stadium concessions.