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Linear Relations

Objectives: Student will be able to solve a system of linear equations by graphing

Download as Microsoft Word Document or Inspiration File:

Instructional Plan

Unit Plan – requires Inspiration

Instructional Plan Template

Mathematics Governor’s Institute 2005

Names of group members: Mary Cunnally, Robert Decker, Carol Wengerd

Topic/Theme: Linear Relations

Unit Themes - Relations and Functions

Solve Linear Equations

Graphing Linear Equations

Linear Inequalities

Solving a System of Equations

Selected Theme - Solving a System of Equations

Vocabulary List For Unit (Appendix A)

Level: Grade 8-9

Time Element: 90 minutes

NCTM Standards Addressed:

Algebra-

· Understand patterns, relations, and functions

· Represent and analyze mathematical situations and structures using algebraic symbols

· Analyze change in various contexts

Data Analysis and Probability-

· Select and use appropriate statistical methods to analyze data

Problem Solving

Reasoning and Proof

Communication

Connections

Representation

PA Math Standards Addressed:

2.2 Computation and Estimation

· C. Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities

· F. Demonstrate skills for using computer spreadsheets and scientific and graphing calculators

2.4 Mathematical Reasoning and Connections

· B. Construct valid arguments from stated facts

· E. Demonstrate mathematical solutions to problems

2.5 Mathematical Problem Solving and Communication

· B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas, and results

2.6 Statistics and Data Analysis

· B. Use appropriate technology to organize and analyze data taken from the local community

· D. Make predictions using interpolation, extrapolation, regression, and estimation using technology to verify them

2.8 Algebra and Functions

· D. Formulate expressions, equations, inequalities, systems of equations, systems of inequalities, and matrices to model routine and non-routine problem situations

· G. Analyze and explain systems of equations, systems of inequalities, and matrices

· H. Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets and other software

· J. Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane

· K. Select, justify, and apply an appropriate technique to graph a linear function in two variables, including slope-intercept, x- and y-intercepts, graphing by transformations, and the use of a graphing calculator

Math Assessment Anchors Addressed:

M11.C Geometry

· 3.1.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic expressions; slope formula provided on the reference sheet)

M11.D Algebraic Concepts

· 2.1.4 Solve systems of equations using graphing, substitution and/or elimination (use numbers that assess concept rather than computation or number sense).

· 3.2.1 Apply the formula for the slope of a line to solve problems (formula given on reference sheet).

· 3.2.3 Compute slope of a linear equation or graph

· 4.1.2 Graph linear equations in two variables

Reading Assessment Anchors Addressed:

R11.A.2 Demonstrate the ability to understand and interpret non-fiction including … textbooks …

Vocabulary:

System of linear equations Ordered pairs

Solution of a system Parallel lines

Intersecting Lines

Objectives:

Student will be able to solve a system of linear equations by

graphing

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

1. review graphing an equation of a line

2. review list of vocabulary words

3. introduce graphing of two lines on one coordinate plane

4. student worksheet

5. analyze results

6. generalize results

7. predict future solutions by looking at slope and y-intercept

Materials/Resources:

Graph paper Chalk board coordinate plane

Rulers Overhead projector

Graphing calculators Coordinate plane transparency

Calculator View Screen Handouts

Interdisciplinary Connections:

· Reading

Vocabulary Reading Strategy (List-Group-Label) (Appendix B)

· Technology

Graphing calculator, overhead projector, view screen

· Other

Assessment Strategies:

· Formative Evaluation (checking student understanding during the lesson):

At their seats, have students graph two lines and find the intersection. Have one student place the solution on the board/overhead transparency.

· Summative Evaluation (How will it be determined that the objectives were achieved?):

Written assessment containing examples of intersecting, parallel, and coincidental lines.

Correctives/Remediation:

Review: solving an equation for y

Slope

y-intercept

graph a line

Utilize one-on-one tutoring

Extensions/Enrichment:

Utilize an open-ended, free-response activity (Appendix C)

Mathematical Modeling problem (Appendix D)

Special Accommodations (special needs students)

· Description of the Special Needs student selected:

· Accommodations to use with this student:

Appendix A

Vocabulary List from State Glossary:

Coordinate Plane/graph

Line of Best Fit

Linear Function

Ordered Pairs

Origin

Rational Numbers

Real Number

Scatterplots

Appendix B

Vocabulary Reading Strategy (List-Group-Label)—used as a review after the topic has been taught.

Term: System of Linear Equations

Student Generated List:

Line	Intercept	Slope
Substitution	Parallel	Intersecting
Graphing	Perpendicular	Combinations
Ordered Pairs

Student Generated Groups:

Characteristics of Lines	Types of Lines	Methods of Solving
Slope	Parallel	Graphing
Intercept	Perpendicular	Substitution
Ordered Pairs	Intersecting	Combination

Appendix C

Open-Ended Question

Write a system of two linear equations with the given characteristics. Explain why you chose your system of linear equations.

one solution; perpendicular lines

no solutions; one equation is y = 2x + 5

one solution; (0, -4)

infinitely many solutions; one equation is y = 3x

Appendix C (page 2)

Solution Sheet for Open-Ended Question

Write a system of two linear equations with the given characteristics. Explain why you chose your system of linear equations.

one solution; perpendicular lines

any system of linear equations that would have negative reciprocal slopes

no solutions; one equation is y = 2x + 5

y = 2x + c where c is any real number. The system of linear equations must have the same slope to represent parallel lines which would not intersect.

one solution; (0, -4)

For example, y = -x - 4 and y = 2x - 4. The two equations would have to have the same y-intercept of (0, -4)

infinitely many solutions; one equation is y = 3x

One example would be 2y = 6x. The equations must be equivalent ones –that is, they must have the same equivalent slope and equivalent y-intercept.

Appendix C (page 3)

Score	In this item, the student –
4	Demonstrates a thorough understanding of the problem based on the equations written for the lines and provides an explanation.
3	Demonstrates a general understanding of the problem based on the equations written for the lines and provides an explanation with only minor errors or omissions.
2	Demonstrates a partial understanding of the problem by correctly performing a significant portion of the required task.
1	Demonstrates minimal understanding of the problem based on the equations written.
0	The response has no correct answer and insufficient evidence to demonstrate any understanding of the mathematical concepts and procedures as required by the task. Response may show only information copied from the question. 0 No response BLK – Blank, entirely erased or written refusal to respond OT – Off Task IL – Illegible LOE – Response in a language other than English

Appendix D

Car Rental Problem

Bernie’s “Good Egg” Car Rental Company charges $10 per day plus $0.15 per mile for a mid-size car rental. The same type of car can be rented at Beth’s Car Company for $12 per day plus $0.10 per mile.

A. Write linear equations representing the given information.

Use your graphing calculator to answer the following questions?

If you are going to travel 30 miles, which car company would be the least expensive? Why?

If you are traveling 95 miles, which car company is the least expensive? Why?

What mileage results in equal rental costs for the 2 firms? Explain your reasoning.

If you have $40 to spend, how many miles could you drive the car, in one day, if you rent from each company?

Appendix D (page 2)

Car Rental Problem Answer Key

Bernie’s “Good Egg” Car Rental Company charges $10 per day plus $0.15 per mile for a mid-size car rental. The same type of car can be rented at Beth’s Car Company for $12 per day and $0.10 per mile.

A. Write linear equations representing the given information.

Bernie’s Y = 10 + .15x

Beth’s Y = 12 + .10x

Use your graphing calculator to answer the following questions?

If you are going to travel 30 miles, which car company would be the least expensive? Why?

Bernie’s costs $14.50, Beth’s costs $15, so for 30 miles, Bernie’s is cheaper.

If you are traveling 95 miles, which car company is the least expensive? Why?

When traveling 95 miles, Bernie’s costs $24.25, while Beth costs $21.50, so

Beth’s would be cheaper.

What mileage results in equal rental costs for the 2 firms? Explain your reasoning.

The costs are equal when traveling 40 miles. Each firm would charge

$16. Looking at the graph, this is the point of intersection.

If you have $40 to spend, how many miles could you drive the car, in one day, if you rent from each company?

When spending $40, you could drive 200 miles if renting from Bernie,

and 280 miles when renting from Beth—this can be seen from looking

at a table of data from the two equations