Linear Equations as Models

How do rates determine when you'll get there?
How does your starting amount determine when you'll get there?
How steep is too steep?

Enduring Understandings: Many situations in life can be modeled by linear equations. Linear equations can be used to solve problems and predict results.

 

GLE’s: 1.3.3, 1.4.4, 1.4.5, 1.5.1, 1.5.2, 1.5.4, 1.5.6, 2.2.1, 2.2.2, 3.2.2, 5.1.2, 5.2.1

In this unit we will carefully study linear equations as models for situations.
We will focus on

• slope as a rate of change
• direct variations
• slope-intercept form of an equation for a line
• standard form of an equation of a line
• solving systems of linear equations with algebra
• solving systems of linear equations by graphing

Along the way we will practice skills including

• finding slope between two points
• converting between different rates
• writing the equation of a line given two points
• rewriting equations of lines into different forms

Here is a quick overview of the five sections of this unit.

Section 1: Linear Modeling Overview.

Ideas	Given a linear situation, we will review how to find graphical, numerical, and algebraic models while focusing on rates and starting amounts. In this section we will introduce and study the form y = mx + b. We will introduce the terms “slope” and “vertical intercept”.
Objectives	When finished with this section, the student will be able to write the equation for a linear model given a situation.
Investigations & Practice	Linear Equation Overview.doc

 

Section 2: Slope as a rate of change.

Ideas	In this section we will focus on the meaning of slope as a rate of change, how to calculate slope, and how to convert rates of change into different units. Horizontal and vertical lines have contextual meanings for their slope.
Objectives	When finished with this section, the student will be able to describe the meaning of slope as a rate of change with appropriate units of measure for a given context or graph. When finished with this section, the student will be able to find the rate of change (slope) of a situation from its context, from a graph model, and from a table model. When finished with this section, the student will be able to convert a rate of change into different units.
Investigations & Practice	Slope as a Rate of Change1.doc Slope as a Rate of Change2.doc Slope as a Rate of Change3.doc practice 7.3.doc

 

Section 3: Writing Linear Equation Models

Ideas	Focus on Vertical Intercept Find the vertical intercept given two points in a linear context. Find the equation of a line given two points.
Objectives	When finished with this section, the student will be able to write the equation for a linear model given two data points of a situation.
Investigations & Practice	Writing Linear Equation Models #1 Writing Linear Equation Models #2 Writing Linear Equation Models #3 Tables to Eq1.doc Tables to Eq2.doc practice 8.4.doc (skills practice writing equations of lines)

 

Section 4: Linear models for “codependent” variables

Ideas	“Standard form” Ax + By = C Converting between forms (standard and slope-intercept)
Objectives	When finished with this section, the student will be able to graph a linear equation that is in standard form. When finished with this section, the student will be able to translate between standard and slope-intercept forms of equations.
Investigations & Practice	Codependent variables 1 Codependent variables 2 practice 8.2, skill practice 1 (graphing from standard form) practice 8.2.2, skill practice 2 (rewriting) practice 8.2.3, skill practice 3 (graphing and rewriting)

 

Section 5: Linear Systems

Ideas	Graphing linear systems Solving linear systems graphically and algebraically Graphing systems of linear inequalities
Objectives	When finished with this section, the student will be able to graph a system of linear equations. When finished with this section, the student will be able to solve a system of linear equations graphically. When finished with this section, the student will be able to solve a system of linear equations algebraically by substitution.
Investigations & Practice