LU Title: Linear Equations

Author(s): Sherri B. Walczak

Grade Level: 9/10 Test A

School Address: Ilion Central School Weber Avenue Ilion, NY 13357

Subject Area: Mathematics/ELA

School Phone/Fax: (315) 895-7471

Declarative

Procedural

• Define terms involved in graphing linear equations

• Play the game "Battleship" to identify quadrants, and coordinate points

• Explain four types of slopes and be able to find the numerical value of a slope

• Transpose linear equations into slope-intercept form

• Explain the advantage of graphing a linear equation using slope-intercept form vs making a table

• Graph linear equations

• Understand a verbal sentence and transform it into an equation

• Graph a linear equation on a TI-82 calculator and traces for the y-intercept

• Identify lines that are parallel by use of their slopes

• Use a TI-82 calculator to construct models of parallel lines and to use inductive reasoning to make the conclusion that all slopes are equal

• Identify dependent and independent variables

• Use centimeter cubes to build several "buildings"

• Recognize patterns from the "buildings"
• Explain why the patterns are a direct variation
• Explain what a negative value means in a real-world problem
• Recognize that "real-data" is often not expressed in the second or third quadrant and can explain why
• Compare the graphs made manually to the graphs produced by a graphing calculator and explain the advantaged and disadvantages of each method

• Formulate expressions from patterns
• Formulate linear equations that are mathematical models of the physical "buildings"
• Explore a linear relationship using a real world problem. Students look for a pattern, write their own equation, and graph the data manually and using a TI-82 calculator
• Students use the "trace" mode on the graphing
• Write a brief report regarding the comparison of phone schemes

Benchmarks: Students use algebraic and geometric representations to describe and compare data.

Benchmarks: Students use deductive reasoning to construct and evaluate conjectures and arguments, recognizing that patterns and relationships in mathematics assist them in arriving at there conjectures and arguments.

Standard: Standard 1: Abstract and symbolic representation are used to communicate mathematically.

Standard: Standard 1: Deductive and inductive reasoning are used to reach mathematical conclusions.

What declarative knowledge should studentsbe in the process of acquiring & integrating? As a result of the unit, the student will know or understand…

What experiences or activities will be used to help students acquire & integrate this knowledge?

What strategies will be used to help students construct meaning, organize and/or store the knowledge?

Describe what will be done.

 Defines terms involved in graphing linear equations.

Recognize patterns from the "buildings"

Compare the graphs made by man to the graph produced by the graphing calculator to explain the advantages and the disadvantages of each method

 "Battleship" game

Build buildings using centimeter cubes

Students are given a graph for the USA today and asked to compare their models to that produces by the paper.

 Pairs Class discussion Graphing cartoon Worksheets for homework

Pairs discussion simulation

Pairs compare and contrast organizers guided questions

 Students sit in pairs. Each student prepares a coordinate plane. They put a barrier up between them and plot 4 treasures: a geegaw; 1x2 unit: a bauble; 1x3 unit, and a trinket; 1x4 unit. Students take turns naming coordinate points to try to find the other's treasures.

See initiating activity

Students each have a different problem to graph and upon completion they exchange with another group. After given a graph from USA Today, they use an organizer to compare and contrast the two. (Students usually do not have any written info on their graphs so the graphs give no valuable information at all.) The groups get together and discuss what would improve their work. They then make the changes to clarify their work. Of course, this can not be done on a graphing calculator.

What procedural knowledge will students be in the process of acquiring & integrating? As a result of this unit, students will be able to:

What will be done to help students construct models, shape & internalize the knowledge?

Describe what will be done.

 Transposes linear equations into slope-intercept form

Graphs linear equations

Graphs linear equations using a TI-82 calculator and traces to find the y-intercept

Uses graphing calculators to construct models of parallel lines to use inductive reasoning to make the conclusion that all slopes are equal

Uses centimeter cubes to construct "buildings"

Formulates expressions and linear equations from patterns.

Explores linear relationships using real-world problems.

 Worksheets guided practice individual practice

Guided practice individual practice

Demonstration with an overhead graphing calculator Individual students practice with a graphing calculator

Provide linear equations with the same slope to the students.

Provide a variety of buildings for the students to build

Group work Provide several examples from the "buildings"

Provide a variety of situations group work and discussion comparisons class discussions

 Lecture and model problems

Several examples will be demonstrated on the overhead

Students graph the equations and make conclusions by recognizing a pattern in the equations

Use problems from ALGEBRA THINKING: First Experiences

Demonstrate the first model and pattern and give several more for the students to practice

Students practice writing mathematical statements to express a real-world situation.

What knowledge will students be extending and refining? Specifically, they will be extending and refining their understanding of…

What reasoning process will they be using?

Describe what will be done.

 Graphing analysis of a real situation

Constructing models

Recognizing patterns

Representing a real-world situation as a mathematical model

Analysis of a systems of equations

Problem solving

Decision making

Investingating

 Comparing

• Classifying

 Inductive Reasoning

 Deductive Reasoning

 Error Analysis

 Analyzing Perspectives

• Constructing Support

 Abstracting

• Other:

 Each student is given a real-world problem situation where they are presented with three mobile phone network providers. They must determine which provider will be most effective for their individual needs.

A table will be filled in which will provide them with the information to find a pattern. The pattern will then be expressed in the algebraic form as a linear equation. Three equations will be written to describe the situation. Using a TI-82 calculator, students will plot the three equations, find the point of intersection, and explain, in writing, the significance of this point. On a separate sheet of paper, they will sketch the graph and write a brief report explaining which scheme will be the best for different customers. This information is based on the number of minutes customers spent on-line during a one month period.

Students work as individuals for this activity.

Planning Guide

Unit:

Step 1

Step 2

Step 3

What knowledge will students be using meaningfully? Specifically, they will be demonstrating their understanding of and ability to...........

What reasoning process will they be using?

Describe student's products and performances and the criteria for evaluation.

[ ] Decision Making
(selecting from seemingly equal alternatives or examining the decisions of others)
[ ] Problem Solving
(seeking to achieve a goal by overcomming constraints or lmiting conditions)
[ ] Invention
(creating something to meet a need or improve on a situation)
[ ] Experimental Inquiry
(generating an explanation for a phenomenon and testing the explanation)
[ ] Investigation
(resolving confusions or contradictions related to a historical event, a hypothetical past or future event, or to the defining characteristics of something)
[ ] Systems Analysis
(analyzing the parts of a system and how they interact)
[ ] Other:

Products/Performances

Criteria for evaluation

Score Point 4

Score Point 3

[ ]Algebraic concepts were applied to the problem and all values on the table are correct and the Most Expensive and Least Expensive were correctly identified. (All data was represented correctly on the graph).

[ ]All values were correct according to the students method (incorrect) and the Least Expensive and the Most Expensive were identified

[ ]A pattern was discovered and all three linear equations are written correctly

[ ]two equations are correct

[ ]A written explanation that compares the data and the student correctly identifies the best scheme for different times

[ ]Same explanation as 4 points but does not use any numerical values. Explains in words such as "talking the longest".

Score Point 2

Score Point 1

[ ]Major mistake such as finding the values by using an incorrect proportion and the values are correct according to their thinking; the Most Expensive and the Least Expensive are correctly identified according to their data

[ ]More than 3 mistakes on the table including the Most Expensive and the Least Expensive

[ ]one equation is correct

[ ]no equations are correct

[ ]Explained the scheme incorrectly with 1 mistake

[ ]Explanation is totally incorrect