Introduction to Functions 9th Grade Algebra Unit by Rachel ...

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Introduction to Functions 9th Grade Algebra Unit by Rachel McGuire

Table of Contents

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Context Analysis ................................................................................................................................. 3 Content Analysis ................................................................................................................................. 5 Content Outline ................................................................................................................................... 7 Concept Map.......................................................................................................................................... 9 Common Core State Standards ................................................................................................ 10 Pre-Test.................................................................................................................................................... 13 Pre Assessment................................................................................................................................. 17 Lesson Plans............................................................................................................................23

Lesson Plan 3.1 Relations and Functions, Domain and Range ....................................... 23 Lesson Plan 3.2 Linear vs Nonlinear Functions .................................................................... 34 Lesson Plan 3.3 Function Notation ............................................................................................ 39 Lesson Plan 3.4 Graphing Standard Form .............................................................................. 47 Lesson Plan 3.5 Graphing Slope-Intercept Form ................................................................. 55 Lesson Plan 3.6 Transformations ............................................................................................... 67 Unit Overview..................................................................................................................................... 84 Post Test........................... .................................................................................................................. 85 Post Assessment............................................................................................................................... 90 Reflection and Self Evaluation ................................................................................................. 96 Teaching Materials .......................................................................................................................... 98 Resources ............................................................................................................................................. 98

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Context Analysis

Community Factors

Milan High School is located in Milan, Michigan and part of the Milan Area Schools district. The building houses ninth through twelfth grade students. Milan is a small community with a population of about 14,253 people and is located about 15 miles south of the Ypsilanti/Ann Arbor area. Most of the area consists of farms and many students work on their parents land. Only 14.6% of adult men over 25 and 18.5% of women went on to college after high school. Of those percentages, only about half of them graduated with a bachelor's degree. The median household income is about $82,000 and the average family size is 3. A majority of students come from two parent households and 92% of the student body (out of about 600 students) is Caucasian.

Classroom Factors

In my classroom we have 36 individual student desks. Most of the classes are relatively full, with only 1 or 2 desks left empty. In my class I like to promote cooperative learning. Most days the desks are arranged into groups of 4 in order to do so. Unfortunately we have very limited technology available to us. We are limited to a classroom set of TI-84 calculators. Every day in class the agenda is written on a side whiteboard and materials for the day are on the front board typically accompanied by a "do-now" startup problem. The students know to come in, pick up any handouts, a calculator, and their binder that they are allowed to keep in the classroom.

Student Characteristics

A majority of my algebra classes are freshman students. Many of them are still getting used to high school and can tend to be lacking in maturity levels. This being the case, I have to keep the class as structured as possible and I can't put too much responsibility on the students. I also have a high number of students on IEP plans and another portion of students are retaking the class. For many students, algebra is a new concept and students do not have very much prior knowledge relevant to the class material. A lot of these characteristics will play a significant role when it comes to planning my unit.

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Implications for Instruction Due to the high level of IEPs and students retaking algebra, my unit will need to move at a slower pace. Most of this material is brand new and I will have to demonstrate a lot of the concepts. Unfortunately this will limit the opportunities for inductive and cooperative lessons. Also due to the characteristics of the class, I believe cooperative group work may be difficult for some students. Many students tend to have issues focusing and staying on task when they are permitted to work in a group. I will need to make sure I give very clear directions and explain what my expectations are for the class periods when we are working in groups.

Rationale

Students: Have you ever wondered how much it would cost you for every minute you talk on the phone? How about how long it will take to fill up that pool you've been thinking about all summer? In this unit we will discover how we can model real life problems with linear relationships. We will also learn how to graph these functions in order to visualize the relationship. Several of the problems we encounter on a daily basis can be modeled using a linear function and we don't even realize we are doing it.

Teachers: This unit covers all of the material pertaining to the introduction of linear functions including domain and range, function notation, graphing, and transformations. The unit directly reflects several of the High School Functions Common Core State Standards and aligns with the school district's benchmarks. The material is presented in a variety of ways including group work, hands-on activities, and individual work to monitor personal progress and to prepare students for the summative assessment.

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Content Analysis

Common Core State Standards

HSF-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

HSF-IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSF-IF-LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

HSF-IF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).

HSF-BF.1a Write a function that describes a relationship between two quantities.

HSF-BF.B.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. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Generalizations

We can model real world situations with linear functions. Focus Question: Can you think of any problems that have a linear relationship?

A linear function has two variables and has a constant rate of change. Focus Question: How can I determine if two objects will have a linear relationship?

Function Notation can be used as another way to write a function to easily represent the input and output.

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Focus Question: How can I represent an application problem using function notation?

Graphs help to visualize information and allows us to easily read facts from the graph Focus Question: In what type of problems will it help me to visualize the information?

Concepts

Functions Domain and Range Linear functions Standard form of equations Slope-Intercept form Graphing Slope Linear transformations Application

Facts

Ordered pairs, graphs, mapping diagrams, and tables are representations of functions.

The domain is a list of the inputs of a function and the range is a list of the outputs.

A linear function has a constant rate of change. A linear function has two variables. The standard form of a linear equation is Ax + By = C. The y-intercept of a function is where it crosses the y-axis on a graph and

when x = 0. The x-intercept of a function is where it crosses the x-axis and when y = 0. The slope-intercept form of a linear equation is y = mx + b. In the slope-intercept form, m represents the slope and b is the y-intercept. The slope of a line is rise over run and represents the rate of change. The linear equation y = x (or f(x) = x) is the parent function for all linear

equations. A series of transformations can be performed on y = x to obtain every other

linear function. Many real world problems can be modeled by linear functions.

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Content Outline

I. II. III. IV.

Relations a. Pairs inputs with outputs i. A function pairs an input with only one output 1. Vertical line test ii. The domain is the set of the input values iii. The range is the set of the output values b. Can be represented multiple ways i. Ordered Pairs ii. Input/Output Tables iii. Coordinate Planes (Graphs) iv. Mapping Diagrams

Linear Functions a. Two variables and a constant rate of change i. X and Y ii. Slope b. Tables i. Follow an addition or subtraction pattern c. Graphs i. Graph is a straight line d. Equations i. Can be written in the form y = mx + b.

Function Notation a. Another way to name an equation i. Replaces y with f(x) in an equation ii. Can use any letter, most commonly f. b. Can easily visualize the input and output i. Example: f(2) = 1 1. 2 represents the input, 1 is the output

Graphing a. Standard Form ? Ax + By = C i. Find the x-intercept by plugging 0 in for y. (x, 0) ii. Find the y-intercept by plugging 0 in for x. (0, y) iii. Graph both intercepts on the same coordinate plane iv. Connect the points with a line b. Slope-Intercept Form ? y = mx + b i. M is the slope typically represented as a fraction (rise / run) 1. Rise ? how many units to go up or down 2. Run ? how many units to go left or right ii. B is the y-intercept 1. A point on the y-axis iii. Graph the y-intercept (0, b) iv. "Do" the slope starting at the y-intercept in both directions

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1. If positive, go up and right, and go down and left 2. If negative, go up and left, and go down and right v. Connect the three points with a line V. Transformations a. The parent function of all linear functions is f(x) = x or y = x b. Perform transformations on the parent function to obtain new lines i. Translations 1. Move up or down: g(x) = f(x) + k 2. Move left or right: g(x) = f(x+k) ii. Shrinks and Stretches 1. A shrink makes the slope of a line smaller or shallower.

a. g(x) = f(ax) b. 0 < a < 1 2. A stretch makes the slope larger or steeper a. g(x) = f(ax) b. a > 1 iii. Reflections 1. Flips the graph over an axis a. Y-axis: g(x) = f(-x) b. X-axis: g(x) = -f(x) VI. Applications a. Model real-world problems with linear functions. b. Interpret what a linear equation in a contextual problem means.

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