**A. Unit Overview:**

**Content Area:**Math**Unit Title:**Define, Evaluate, Compare and Use Functions**Target Course/Grade Level:**Resource Room Math/Grade 8**Instructor’s Name:**Suzanne Brummitt**School:**Lakeside Middle School, Millville, NJ**Date:**July 26, 2011

** ****Unit Summary: **

This unit demonstrates what functions are and how they are different from other types of relations among numbers. It develops fluency in how to interpret and represent functions in four ways: algebraic expressions, tables, graphs and words. Throughout the unit, students are exposed to data, statistics, and problems that we encounter all of the time, in the news, in school, at work, and in our private lives.

**Primary interdisciplinary connections:
**

- Language Arts
- Science
- Health
- Social Studies

**21st century themes:**

** **All themes will be incorporated through the specific selection and/or creation of real-life projects and problems involving the interpretation or creation of mathematical functions.

- Global Awareness
- Financial, Economic, Business and Entrepreneurial Literacy
- Civic Literacy
- Health Literacy
- Environmental Literacy

Source, July 17, 2011: http://www.p21.org/route21/index.php?option=com_content&view=article&id=6&Itemid=3

**21st century skills:**

** **Learning and Innovation Skills

- Critical Thinking and Problem Solving
- Communication and Collaboration
- Information, Media and Technology Skills
- Information Literacy
- Life and Career Skills
- Initiative and Self-Direction
- Social and Cross Cultural Skills

**Unit Rationale**:

This unit builds on an understanding of relations or rules, content which should already have been mastered in an 8th grade unit on expressions and equations. Altogether, the 8th grade pre-Algebra curriculum provides the foundation for students to be successful in high school Algebra. It is also critical to the 8th grade geometry content, which involves significant use of formulas. Students need to learn how to manipulate functions in order to take advantage of their predictive power, which allows us to calculate the impact of change in many real-world correlations, for example, the impact of food consumed on our health and the impact of our actions on the environment.

**Learning Targets : **

**Common Core Standards :**

1 Common Core Standards :

Content Standards: Define, evaluate, and compare functions.

8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Note: Function notation is not required in Grade 8.

8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Content Standards: Use functions to model relationships between quantities.

8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Source, July 17,2011: http://www.corestandards.org/the-standards/mathematics/grade-8/functions

Mathematical Practice Standards:

a. Make sense of problems and persevere in solving them.

b. Reason abstractly and quantitatively.

c. Construct viable arguments and critique the reasoning of others.

d. Model with mathematics.

e. Use appropriate tools strategically.

f. Attend to precision.

g. Look for and make use of structure.

h. Look for and express regularity in repeated reasoning.

**Unit Essential Questions:**

- What is a function?
- How is a function different from other types of relationships?
- How are functions used to represent relationships between real-world data sets?
- How can functions be expressed?
- What are the parts of the algebraic representation of a function?
- What do linear and non-linear equations look like?

** ****Unit Enduring Understandings:**

- The amount of insulin that a diabetic needs to take is a function of his personal insulin ratio and the amount of carbohydrates he has consumed. In other words, if you know a person’s insulation ratio and total carbohydrates consumed, you can calculate the exact insulin dose that he will need to adjust his blood sugar after a meal. Miscalculation can be fatal. (Note: Students in Lakeside Middle School, of the Millville Public School district, are able to understand and make strong emotional connections to the importance of this function because of the annual fundraising that we do for the Juvenile Diabetes Research Fund. Part of the drive is an assembly during which students view a video about how a person their own age manages her diabetes on a daily basis.)

Source 7/20/11: http://www.ehow.com/facts_5705713_do-diabetics-use-math_.html

- Functions are different from other mathematical relations.
- Functions have predictive uses that other relations do not have.
- The variables in an algebraic expression are arbitrary. Changing variable names does not change the function.
- Not all functions can be represented by straight lines.

**C. Evidence of Learning:**

**Summative Assessments **

- Weekly Quiz
- Unit Quiz
- Final Exam
- State Tests

** ****Formative Assessments **

- Pre-Quiz
- Exit Tickets
- Classwork
- Homework
- Jigsaw presentation

** D. Equipment / Resources**

**Equipment / Technology needed throughout the Unit:**

SMART Board, laptop, document camera, flip video, graphing calculators (one per student), graphing calculator software for SMART Board display, classroom computers for student access to Internet activities, Geometer Sketchpad.

**Teacher Resources:**

*Activities for Algebra with the TI-73*. Texas Instruments, 2002.

18 investigations from Illuminations: http://www.thinkfinity.org/partner-search?start=0&partner=6&partner_value=no&from_links=&txtKeyWord=functions&txtKeyWord2=functions&narrow=1&chkGrade%5B%5D=grades%3A6%7Cgrades%3A7%7Cgrades%3A8&chkPartner%5B%5D=Illuminations

Kahn Academy: 5-part lesson on functions starts here (press “next video” to advance):

http://www.khanacademy.org/video/introduction-to-functions?playlist=Algebra

Kahn Academy: 9 worked examples on functions start here (press “next video” to advance):

http://www.khanacademy.org/video/functional-relationships-1?playlist=Algebra%20I%20Worked%20Examples

Math Is Fun: lesson on functions: http://www.mathsisfun.com/sets/function.html

Math Is Fun Function Grapher and Calculator:

http://www.mathsisfun.com/data/function-grapher.php

Shodor Interactive: interactive Function Machine activity: http://www.shodor.org/interactivate/activities/FunctionMachine/

Shodor Interactive: Function Flyer activity

http://www.shodor.org/interactivate/activities/FunctionFlyer/

Positive Linear Function Machine:

http://www.shodor.org/interactivate/activities/PositiveLinearFunct/

Linear Function Machine:

http://www.shodor.org/interactivate/activities/LinearFunctMachine/

National Library of Virtual Manipulatives, Function Machine:

http://nlvm.usu.edu/en/nav/frames_asid_191_g_3_t_1.html?from=grade_g_3.html

Function Grapher

http://nlvm.usu.edu/en/nav/frames_asid_109_g_3_t_1.html?open=activities&from=grade_g_3.html

Function Transformations

http://nlvm.usu.edu/en/nav/frames_asid_329_g_3_t_2.html?open=activities&from=grade_g_3.html

Gizmos activities at http://www.explorelearning.com/

** ****E. Lesson Plan Topics / Titles (One lesson per day):**

**Lesson 1: **Formative Assessment ** **

**Lesson 2:** Defining Terms and Basic Concepts

**Lesson 3**: Compare functions in verbal descriptions and graph form

**Lesson 4:** Translate between functions in verbal descriptions and graph form.

**Lesson 5:** Weekly quiz. Compare functions in table form and graph form

**Lesson 6:** Translate functions between table form and graph form

**Lesson 7: **Compare functions in table form and algebraic expression

**Lesson 8:** Translate functions between table form and algebraic expression

**Lesson 9: **Translate functions expressed in words to table, graphic, and algebraic form.

**Lesson 10:** Review and Unit Quiz.

**F. Teacher Notes about Lesson Plans**

** Each lesson: **

- Begins with an activity to connect to
**Previous Knowledge**and to**Student Interest / Preferences.** **High Quality Curriculum**- Prentice Hall Mathematics: Pre-Algebra 2004.- Includes
**Flexible Grouping**via – Whole group, independent, pair/share, Jigsaw activities (students in groups work on different problems, according to their skill level, and present to the class). Different problems address various real-world domains (science, social studies, finance, etc.) **Respectful Tasks**within a**Supportive Learning Environment**via**Multiple Means.**(see next page)

**G. Universal Design for Learning Options**

All sites referenced in this section were obtained through links on the following web site, which provides the UDL Guidelines, Version 2 (as of July 26, 2011):

http://www.cast.org/udl/index.html

**Multiple Means of Representation**

**Guideline 1: Provide options for perception**

Study of functions involves a great deal of visual information. Functions must be represented in tables, algebraic expressions, and graph form. It is essential that these representations be made accessible to everyone. One important feature of the presentation is the use of color for information and emphasis and the contrast between background and image. The following site describes effective contrast.

http://www.lighthouse.org/accessibility/design/accessible-print-design/effective-color-contrast

** ****Guideline 2: Provide options for language, mathematical expressions, and symbols**

The study of functions includes very specific, mathematical use of words that have different meanings in different contexts (e.g., “function,” “slope,” “vertical line,” “variable.” Understanding this vocabulary is essential. The following site helps students to learn vocabulary through a visual map of the word’s various meanings in different contexts:

**Guideline 3: Provide options for comprehension**

**Students are more likely remember the essential nature of functions if they understand the connections between the components of a function, the between functions and other mathematical relations, and among the vocabulary . An important tool for graphically organizing these connections is Webspiration, an Internet tool for which Millville Public Schools has licenses:**

http://www.mywebspiration.com/

**Multiple Means of Action and Expression**

**Guideline 4: Provide options for physical action**

The game, “Hidden Secrets of Al-Jabr,” combined with a computer touch screen or a SMART Board, provides options to students who have difficulty using a mouse, thereby varying the methods for response and navigation.

http://techmatrix.org/ViewProduct?itemId=548

**Guideline 5: Provide options for expression and communication**

This unit depends on extensive use of graphing calculators. If individual, student graphing calculators are not available, web-based calculators are also helpful:

http://www.math.com/students/graphing.html

**Guideline 6: Provide options for executive functions**

The study of functions in entirely new to 8th graders, and it is both complex and abstract. Therefore, it is important to break the concepts and skills down into components that can be tracked by the student and the teacher, so that both are aware of the progress that the student is making. A great way is for the student to maintain a graph of his own achievement. What could be better in a unit whose content has so much to do with graphing?

http://nces.ed.gov/nceskids/createAgraph/default.aspx

**Multiple Means of Engagement**

**Guideline 7: Provide options for recruiting interest**

During the jigsaw problem-solving activities, students’ interest will be stimulated by options in the tools used for production of their reports and the design of their reports. One exciting option will be to publish a newspaper online with their investigations and results. This site offers a special tool for this:

** ****Guideline 8: Provide options for sustaining effort and persistence**

The jigsaw activities are specifically designed to allow students to collaborate and learn through social interaction. The following site provides valuable classroom guidelines for social learning. It is particularly apt because it uses solution of a simple algebra problem as an example.

http://www.edutopia.org/math-social-activity-cooperative-learning-video

**Guideline 9: Provide options for self-regulation**** **

This particular subject matter can be very difficult to access. Students are likely to become frustrated unless they are able to have success and build confidence. Knowing their learning styles and choosing activities that fit their learning styles is an important component. This video, while it is for the teacher, could also be useful for students to see, especially since it shows a student with dyslexia juggling and shows the related math: graphs of functions showing the parabolas of the items juggled.

http://www.edutopia.org/gateway-brain-learning-styles-video** **

__H. Online Resources : __

**The following websites were used in preparing additional strategies, accommodations, and modifications in the above lessons:**

** **

http://daretodifferentiate.wikispaces.com/