8.3.3Ma


 * Grade: 8 Unit: 3 Week:3 Dates: 12/10-12/14 **
 * Content: Rate of Change and Slope **


 * Theme Essential Question: **
 * How are functions used to describe relationships that happen in our lives? **


 * Essential Questions: **
 * ** How do you find a rate of change or slope? **
 * ** How does rate of change or slope relate to real world problems that are associated with a constant change? **
 * ** How can you show that the slope of a line is the same between any two points on the line? **


 * Standards **
 * Understand the connections between proportional relationships, lines, and linear **
 * equations **
 * ** 8. EE.5 ** Graph proportional relationships, interpreting the unit rate as the slope of the graph . Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
 * ** 8. EE.6 ** Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


 * Objectives **
 * The student will define rate of change.
 * The student will investigate rate of change using tables.
 * The student will investigate rate of change using graphs.
 * The student will define slope.
 * The student will make the connection between rate of change and slope with/without a proportional relationship.
 * Slope
 * The student will compute the constant rate of change (slope) given a table or a graph of a linear function.
 * The student will relate the point (1,r) of a linear function and the unit rate.
 * The student will construct a viable argument extending their understanding of slope with respect to similar triangles.
 * The student will apply the concept of similar triangles in finding slope of a linear function.


 * Reflection and/or Comments from your PCSSD 8th Grade Curriculum Team **

As noted in the objectives, the understanding of rate of change and slope needs to be very extensive. This should include investigation of slope with technology. Allow students the opportunity to explore steepness of slope, zero slope, undefined slope, negative slope.

Reviewing the focus discussion in the Common Core State Standards for Mathematics, p. 52, students need to recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b). They need to understand that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. This association has not been a major issue in our previous framework, but it is now.

Earlier in the OnCore book, section 5.4 had a connection between similar triangles and slope. After reviewing the information, we felt it needed to be introduced in Unit 3 with Functions. This will be a good time to review similar triangles as they relate to slope.

[|**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**]
 * Background Information **
 * Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. **
 * (Taken from [|Ohio Dept of Education Mathematics Model Curriculum 6-28-2011] ) **

Provide multiple opportunities to examine the graphs of linear functions and use graphing calculators or computer software to analyze or compare at least two functions at the same time. Illustrate with a slope triangle where the run is "1" that slope is the "unit rate of change." Compare two different situations and identify which is increasing/decreasing at a faster rate.


 * Assessment **
 * Product **
 * Students will continue creating a portfolio detailing their growth pertaining to the topics addressed throughout this unit on functions. The Entry provided follows the theme of construction repair and remodel.
 * Entry #3 (a) **
 * This product should be a re-creation of their study of slope.
 * Using graph paper, start with the equation having a slope of zero (y = 0x) and then select other lines with “steeper” slopes (y = mx).
 * Students are to summarize their finding for slope in the equation y = mx.
 * Entry #3 (b) **
 * Students are to design a word problem that uses the concept of proportional relationships in the theme of construction repair and remodel.
 * Create a table.
 * Graph the proportional relationship.
 * Find the unit rate (slope).
 * Use the unit rate (slope) to write the equation y = mx.
 * What does the slope mean in your problem?
 * Design a question and show how it uses your table and equation.


 * Key Questions **
 * What are key concepts that must be considered when creating a table from a word problem, a graph, and/or an equation that has a proportional relationship?
 * How does the slope of any two values from a table or graph correspond to the unit rate//?//
 * What geometric figure is formed when the rate of change is a constant//?//
 * How do you interpret the unit rate as the slope of the graph?
 * What is the connection between the concept of slope and similar triangles?


 * Observable Student Behaviors **
 * Students can determine slopes given tables, ordered pairs, or graphs.
 * Student can associate the unit rate to the slope.
 * Student can interpret the meaning of a unit rate or slope.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||
 * ** Mathematical Practices **

Rate of change, unit rate, constant rate of change, variable rate of change, slope, linear equation, linear function, slope-intercept form, proportional relationship
 * Vocabulary **

Adjust the constant of variation and explore how the graph of the direct variation function changes in response. Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake. Highly Recommended: @http://illustrativemathematics.org/illustrations/129 (EE.5) @http://illustrativemathematics.org/illustrations/57 (EE.5) @http://illustrativemathematics.org/illustrations/55 (EE.5) @http://illustrativemathematics.org/illustrations/86 (EE.5) @http://illustrativemathematics.org/illustrations/184 (EE.5) Nothing Available at this Time (EE.6) The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
 * Suggested Activities ** [see Legend to highlight MCO and HYS]
 * Houghton Mifflin OnCore Mathematics Unit 2-3 p. 43-46
 * Houghton Mifflin OnCore Mathematics Unit 5-4 p. 125-126
 * ABC Mastering the Common Core in Mathematics Chapter 5-4 to 5-6, p. 51-55
 * Gizmo Lessons
 * Direct Variation
 * Estimating Population Size
 * __ Teaching the Common Core Math Standards with Hands-On Activities __ by Judith Muschla
 * 8.EE.5 p. 169
 * 8.EE.6 Activity 1 p. 171, Activity 2 p. 172
 * JBHM Grade 8 GP2, Unit 4, p. 265-296
 * Glencoe Pre Algebra Chapter 8, Lesson 5, p. 393-397
 * Page 392: Algebra activity (slope and rate of change)
 * Glencoe Algebra I, Chapter 5, Lesson 1-2, p. 256-270

Odyssey (teacher discretion) Skills Tutor (teacher discretion) Algebra’scool: Unit C Module 9
 * Diverse Learners **

Suggested:
 * Homework **
 * ** @http://www.kutasoftware.com/free.html ** to print assignments on variety of topics
 * See appropriate Glencoe, OnCore, JBHM, and ABC materials


 * Terminology for Teachers **

** E ** thnicity/**C**ulture | **I**mmigration/**M**igration | **I**ntercultural **C**ompetence | **S**ocialization | **R**acism/**D**iscrimination ** High Yield Strategies ** ** S ** imilarities/**D**ifferences | **S**ummarizing/**N**otetaking | **R**einforcing/**R**ecognition | **H**omework/**P**ractice | ** N ** on-**L**inguistic representation | **C**ooperative **L**earning | **O**bjectives/**F**eedback | ** G ** enerating-**T**esting **H**ypothesis | **C**ues, **Q**uestions, **O**rganizers ||   ||
 * || ** Multicultural Concepts **
 * Lesson Plan in Word Format (Click Cancel if asked to Log In) **


 * Resources **


 * Professional Texts **


 * Literary Texts **


 * Informational Texts **
 * See New York Common Core Aligned Task (other resources)
 * @http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htm


 * Art, Music, and Media **


 * Manipulatives **
 * ** @http://nlvm.usu.edu/ ** National Library of Virtual Manipulatives
 * Algebra: Function Machine, Function Transformations
 * Graphing calculators
 * TI CBR (Data Collection Device used with TI 84) to explore slope


 * Games **


 * Videos **
 * Discovery Learning @http://www.discoveryeducation.com/


 * SMART Board Lessons, Promethean Lessons **
 * Smartboard Resource Website Smartboard lesson search engine


 * Websites **
 * @http://illustrativemathematics.org/standards/k8 PARCC Online (released items)
 * http://www.khanacademy.org Online tutorials
 * @http://www.tli.net/ TLI Quiz Builder


 * Other Activities, etc. **

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