A. Unit Overview:
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:
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.
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
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:
Unit Enduring Understandings:
Source 7/20/11: http://www.ehow.com/facts_5705713_do-diabetics-use-math_.html
C. Evidence of Learning:
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.
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):
Kahn Academy: 9 worked examples on functions start here (press “next video” to advance):
Math Is Fun: lesson on functions: http://www.mathsisfun.com/sets/function.html
Math Is Fun Function Grapher and Calculator:
Shodor Interactive: interactive Function Machine activity: http://www.shodor.org/interactivate/activities/FunctionMachine/
Shodor Interactive: Function Flyer activity
Positive Linear Function Machine:
Linear Function Machine:
National Library of Virtual Manipulatives, Function Machine:
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
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):
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.
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:
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.
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:
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?
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.
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.
H. Online Resources :
The following websites were used in preparing additional strategies, accommodations, and modifications in the above lessons: