Grade 8 - Shelby County Schools Q1 16-17... · Web viewThe Standards for Mathematical Practice describe varieties of expertise, ... arnold/graphics.html. Visual Calculus Tutorials

Grade 8

Curriculum and Instruction Mathematics

1st Quarter

Calculus

Introduction

In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,

80% of our students will graduate from high school college or career ready

90% of students will graduate on time

100% of our students who graduate college or career ready will enroll in a post-secondary opportunity

In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, college and career ready aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and career readiness is rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor.

The Standards for Mathematical Practice describe varieties of expertise, habits of minds and productive dispositions that mathematics educators at all levels should seek to develop in their students. These practices rest on important National Council of Teachers of Mathematics (NCTM) processes and proficiencies with longstanding importance in mathematics education. Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice.

This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that, ultimately our students, can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their practice so that it is in alignment with the three mathematics instructional shifts.

Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around the content standards and mathematical practice standards that teachers should consistently access:

The TN Mathematics Standards

The Tennessee Mathematics Standards:

https://www.tn.gov/education/article/mathematics-standards

Teachers can access the Tennessee State standards, which are featured throughout this curriculum map and represent college and career ready learning at reach respective grade level.

Standards for Mathematical Practice

Mathematical Practice Standards

https://drive.google.com/file/d/0B926oAMrdzI4RUpMd1pGdEJTYkE/view

Teachers can access the Mathematical Practice Standards, which are featured throughout this curriculum map. This link contains more a more detailed explanation of each practice along with implications for instructions.

Purpose of the Mathematics Curriculum Maps

This curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready (CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and contentthe major work of the grade (scope)and provides a suggested sequencing and pacing and time frames, aligned resourcesincluding sample questions, tasks and other planning tools. Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with colleagues to continuously improve practice and best meet the needs of their students.

The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional practice. In fact, our goal is not to merely cover the curriculum, but rather to uncover it by developing students deep understanding of the content and mastery of the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and assessment) of the learners are best-positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected--with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgement aligned to our shared vision of effective instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigorhigh-quality teaching and learning to grade-level specific standards, including purposeful support of literacy and language learning across the content areas.

Additional Instructional Support

Shelby County Schools adopted our current math textbooks for grades 9-12 in 2010-2011. The textbook adoption process at that time followed the requirements set forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. We now have new standards; therefore, the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief State Officers (CCSSO) and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of conceptual knowledge development and application of these concepts), of our current materials.

The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still incorporating the current materials to which schools have access. Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and external/supplemental resources (e.g., EngageNY), have been evaluated by district staff to ensure that they meet the IMET criteria.

How to Use the Mathematics Curriculum Maps

Overview

An overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items.

Tennessee State Standards

The TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards that supports students learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teachers responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard.

Content

Teachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.). Support for the development of these lesson objectives can be found under the column titled Content. The enduring understandings will help clarify the big picture of the standard. The essential questions break that picture down into smaller questions and the objectives provide specific outcomes for that standard(s). Best practices tell us that clearly communicating and making objectives measureable leads to greater student mastery.

Instructional Support and Resources

District and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find instructional/performance tasks, i-Ready lessons and additional resources that align with the standards in that module. The additional resources provided are supplementary and should be used as needed for content support and differentiation.

Topics Addressed in Quarter 1

Preparation for Calculus

Limits and Their Properties

Differentiation

Overview

During the Preparation for Calculus chapter, students will review several concepts that will help prepare them for their study of calculus. These concepts include sketching the graphs of equations and functions, and fitting mathematical models to data. It is important to review these concepts prior to moving forward with calculus. In Chapter 1: Limits and Their Properties, students will become acquainted with the relationship between algebra/geometry and the development of Calculus. Evaluating limits both analytically and graphically is a major area of the unit and will be emphasized. Students should use the graphing calculator to help develop the intuitive feel of limits and graph behavior. This chapter will allow students to have a complete understanding of limits and how they are used. The main topics addressed will be rational exponents, simplifying expressions, writing linear equations, and average rate of change. Chapter 2: Differentiation prepares the students for applications in differential calculus by giving them a firm grasp of methods of differentiation. Emphasis is placed on what a derivative represents (slope of a tangent line to a point on a curve), and the graphical differences between f(x) and f(x). The relationship between differentiability and continuity is also a major point of interest in this chapter.

Fluency

The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebra can help students get past the need to manage computational and algebraic manipulation details so that they can observe structure and patterns in problems. Such fluency can also allow for smooth progress toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to help students gain fluency. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency.

References:

https://www.engageny.org/

http://www.corestandards.org/

http://www.nctm.org/

http://achievethecore.org/

TN STATE STANDARDS

CONTENT

INSTRUCTIONAL SUPPORT & RESOURCES

Chapter P: Preparation for Calculus

(Allow approximately 3 weeks for instruction, review, and assessment)

Preparation:

Algebra I & Algebra II

A-APR.A.3

A-CED.A.2

A-REI.A.1

A-REI.B.4b

A-REI.C.6

A-REI.C.7.

A-REI.D.10

A-REI.D.11

F-IF.B.4

F-IF.C.7a

F-IF.C.7c

F-BF.B.4

Enduring Understanding:

Calculus builds on the foundation of Algebra, Analytic Geometry, and Trigonometry.

Essential Questions:

How can you identify the characteristics of equations and sketch their graphs?

How do you find and graph equations of lines, including parallel and perpendicular lines, using the concept of slope?

How can you evaluate and graph functions and their transformations?

Objectives:

Students will:

Sketch the graph of an equation.

Find the intercepts of a graph.

Test a graph for symmetry with respect to an axis and the origin.

Find the points of intersection of two graphs.

Interpret mathematical models for real-life data.

P.1: Graphs and Models

Additional Resource(s)

Larson Calculus Videos Section P.1

Visual Calculus Tutorials

Introducing Families of Functions

Algebra and Geometry Review Videos

Algebra II Activities Using the TI84

Glencoe Reading & Writing in the Mathematics Classroom

Literacy Skills and Strategies for Content Area Teachers

(Math, p. 22)

Graphic Organizers (dgelman)Graphic Organizers (9-12)

Preparation:

Algebra I, Algebra II & Geometry

A-CED.A.2

A-REI.A.3

F-IF.B.6

F-LE.A.2

G-GPE.B.5

A-CED.A.2




What is a linear function?

What are the different ways that linear functions may be represented?

What is the significance of a linear functions slope and y-intercept?

How may linear functions model real world situations?

How may linear functions help us analyze real world situations and solve practical problems?

Objectives:

Students will:

Find the slope of a line passing through two points.

Write the equation of a line with a given point and slope.

Interpret slope as a ratio or as a rate in a real-life application.

Sketch the graph of a linear equation in slope-intercept form.

Write equations of lines that are parallel or perpendicular to a given line.

P.2: Linear Models and Rates of Change



Khan Academy Videos: Linear Models

Khan Academy Videos: Slope & rate of Change


Functions and Graphs Review Videos


F-IF.A.1

F-IF.B.5

F-IF.C.7(b)

F-BF.A.1(b)

F-BF.B.3




How do functions model real world situations?

How do functions help us analyze real world situations and solve practical problems?

Objectives:

Students will:

Use function notation to represent and evaluate a function.

Find the domain and range of a function.

Sketch the graph of a function.

Identify different types of transformations of functions.

Classify functions and recognize combinations of functions.

P.3: Functions and Their Graphs








(Math, p. 22)


F-IF.B.5

F-BF.B..4




When and how is mathematical modeling used to solve real world problems?

When is it advantageous to represent relationships between quantities symbolically? Numerically? Graphically?

Objectives:

Students will

Fit a linear model to a real-life data set.

Fit a quadratic model to a real-life data set.

Fit a trigonometric model to a real-life data set.

P.4: Fitting Models to Data






Algebra Cheat Sheet

Chapter 1: Limits and Their Properties


Domain: Limits of Functions

Cluster: Understand the concept of the limit of a function.

F-LF.A.2 Estimate limits of functions (including one-sided limits) from graphs or tables of data. Apply the definition of limit to a variety of functions, including piece-wise functions.

Enduring Understandings:

The concept of a limit is one of the foundations of calculus.

The limit of a function is the value approached by f (x) as x approaches a given value or infinity.

The derivative is the instantaneous rate of change at a given point.

The integral is a function that can be used to determine the summation of an infinite set.

Differentiation and definite integration are inverse operations.


How does the derivative represent an instantaneous rate of change?

How does the integral represent the summation of an infinite set?

How do you determine that a function is continuous and/or differentiable?

Is there a way to visualize what a derivative is?

Objectives:

Students will

Understand what calculus is and how it compares with precalculus.

Understand that the tangent line problem is basic to calculus.

Understand that the area problem is also basic to calculus.

Estimate a limit using a numerical or graphical approach.

Learn different ways that a limit can fail to exist.

Study and use a formal definition of a limit.

1.1: A Preview of Calculus

1.2: Finding Limits Graphically and Numerically


Larson Calculus Videos Section 1.1

Calculus Tutorial Videos

Brightstorm: Finding Limits Graphically

Calculus Activities Using the TI-84


Chapter 1 Vocabulary:

Domain, range, independent, dependent variable, graph, function, absolute value, increasing, decreasing, linear , quadratic, polynomial, coefficients, degree, cubic , power, root,, reciprocal , rational, algebraic, trigonometric, exponential, logarithmic , translations, composite , limit, right-hand limit, left-hand limit, vertical asymptote, continuous at a point, discontinuity, removable discontinuity, jump discontinuity, horizontal asymptote, infinite limits, limits at infinity, intermediate value theorem.

Writing in Math

What is the definition of a limit?

What does it mean for a function to be continuous?

How do you find the slope of a line tangent to a curve?



F-LF.A.1 Calculate limits (including limits at infinity) using algebra.

Objectives:

Students will

Evaluate a limit using properties of limits

Develop and use a strategy for finding limits

Evaluate a limit using dividing out and rationalizing techniques

Evaluate a limit using the squeeze theorem

1.3: Evaluate Limits Analytically


Larson Calculus Videos Chapter 1.3


Brightstorm: Evaluating Limits Analytically



Domain: Continuity

Cluster: Develop an understanding of continuity as a property of functions.

F-C.A

1. Define continuity at a point using limits; define a continuous function.

2. Determine whether a given function is continuous at a specific point.

3. Determine and define different types of discontinuity (point, jump, infinite) in terms of limits.

4. Apply the Intermediate Value Theorem and Extreme Value Theorem to continuous functions.

Objectives:

Students will

Determine continuity at a point and continuity on a closed interval.

Determine one-sided limits and continuity on a closed interval.

Use properties of continuity.

Understand and use the Intermediate Value Theorem.

1.4: Continuity and One-Sided limits



Brightstorm: Continuity of a Function






(Math, p. 22)




F-LF.A.1 Calculate limits (including limits at infinity) using algebra.

Domain: Behavior of Functions

Cluster: Describe asymptotic and unbounded behavior of functions.

F-BF.A.1 Describe asymptotic behavior (analytically and graphically) in terms of infinite limits and limits at infinity.

Objectives:

Students will

Determine infinite limits from the left and from the right.

Find and sketch the vertical asymptotes of the graph of a function.

1.5: Infinite Limits






F-LF.A.1 (See above: Lesson 1.5)

F-BF.A.1 (See above: Lesson 1.5)

Objectives:

Students will

Determine (finite) limits at infinity.

Determine the horizontal asymptotes, if any, of the graph of a function.

3.5: Limits at Infinity


Larson Calculus Video Section 3.5




Chapter 2: Differentiation


Domain: Understand the Concept of the Derivative

Cluster: Demonstrate an understanding of the derivative.

D-CD.A

1. Represent and interpret the derivative of a function graphically, numerically, and analytically.

2. Interpret the derivative as an instantaneous rate of change.

3. Define the derivative as the limit of the difference quotient; illustrate with the sketch of a graph.

4. Demonstrate the relationship between differentiability and continuity.

Enduring Understandings:

The derivative is the instantaneous rate of change at a given point.

The integral is a function that can be used to determine the summation of an infinite set.

Differentiation and definite integration are inverse operations.


Why is the derivative important?

How is the average rate of change related to the instantaneous rate of change?

How is the derivative related to the tangent line to a curve?

What is the connection between differentiability and continuity?

Objectives:

Students will

Find the slope of the tangent line to a curve at a point.

Use the limit definition to find the derivative of a function.

Understand the relationship between differentiability and continuity.

2.1: The Derivative and the Tangent Line Problem





http://www.ima.umn.edu/~arnold/graphics.html


Chapter 2 Vocabulary:

Tangent line, position, velocity, acceleration, average rate of change, instantaneous rate of change, derivative, differentiable, constant rule, power rule, sum rule, constant multiple rule, logarithmic rule, exponential rule, product rule, quotient rule , chain rule, trigonometric rules, inverse trigonometric rule, implicit differentiation , chain rule, higher order derivatives, orthogonal, linear approximation, linearization, differentials

Writing in Math

What is the derivative of a function?

How do you find the derivative of a function?

What does it mean for a function to be differentiable?



(Math, p. 22)


Domain: Computing and Applying Derivatives

Cluster: Apply differentiation techniques.

D-AD.A

1. Describe in detail how the basic derivatives rules are used to differentiate a function; discuss the difference between using the limit definition of the derivative and using the derivative rules.

2. Calculate the derivative of basic functions (power, exponential, logarithmic, and trigonometric).

3. Calculate the derivatives of sums, products, and quotients of basic functions.

Objectives:

Students will

Find the derivative of a function using the Constant Rule.

Find the derivative of a function using the Power Rule.

Find the derivative of a function using the Constant Multiple Rule.

Find the derivative of a function using Sum and Difference Rules.

Find the derivative of the sine function and the cosine function.

Use derivatives to find Rates of Change.

2.2: Basic Differentiation Rules and Rates of Change






RESOURCE TOOLBOX

Textbook Resources

Larson/Edwards Calculus of a Single Variable 2010

Larson Calculus

Standards

Common Core Standards - Mathematics

Common Core Standards - Mathematics Appendix A

Edutoolbox.org (formerly TN Core)

The Mathematics Common Core Toolbox

Tennessees State Mathematics Standards

State Academic Standards (Calculus)

Videos

Larson Calculus Videos

KhanAcademy

Hippocampus

Brightstorm

Pre-Calculus Review

University of Houston Videos

Calculator


TICommonCore.com

Texas Instruments Education

Casio Education

TI Emulator

Interactive Manipulatives

http://www.ct4me.net/math_manipulatives_2.htm

Larson Interactive Examples

Additional Sites

http://www.freemathhelp.com/calculus-help.html

http://www.calculus.org/

http://www.calcchat.com/

http://functions.wolfram.com

http://www.opensourcemath.org

http://www.analyzemath.com/Graphing/piecewise_functions.html

http://www.onlinemathlearning.com/math-word-problems.html


Lamar University Tutorial

Algebra Cheat SheetTrigonometry Cheat SheetOnline Algebra and Trigonometry TutorialStudy Tips for Math Courses

Literacy



(Math, p. 22)


Mathematical Practices

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quatitatively

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

3. Construct viable arguments and crituqe the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

Focus

Coherence

Rigor

The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.

Thinking across grades:

Conceptual understanding:

The Standards call for conceptual understanding of key concepts such as limits and differentiation. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures.

Procedural skill and fluency:

The Standards call for speed and accuracy in calculation. While high school standards for math do not list high school fluencies, there are fluency standards for mathematical subjects outside of calculus

Application:

The Standards call for students to use math flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and access content.

The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding on to foundations built in previous years. Each standard is not a new event, but an extension of previous learning. .

Shelby County Schools 2016/2017Revised 6/29/165 of 13

Documents

Grade 8 - Shelby County Schools Q1 16-17... · Web viewThe Standards for Mathematical Practice describe varieties of expertise, ... arnold/graphics.html. Visual Calculus Tutorials