Solving Linear Equations and Their Graphs
Chapter 6, Algebra 1 Solving Linear Equations and their Graphs
I. Rate of Change and Slope
Focus on the TEKS you must put in the actual objects from the released tests.
In algebra 1, students develop the concept of slope as rate of change and determine slopes from graphs, tables and algebraic representations A.6.A. Analysis of TAKS Results shows that students struggle with this expectation.
In their first lessons, students solve for the rate of change using numbers and graphs. In Lessons 6-2, and 605, students find slopes using various slope forms and explore the parallel and perpendicular lines in Lesson 6. Place special emphasis on these lessons because they all focus on A.6.A.
Also, emphasize interpretation of the meaning of slope and intercepts. A.6.B. Students determine and analyze slope in different situations in Lessons 6-1, 6-3 and 6-5 (a.6.B).
Overview of Big Ideas Affecting Slope and Linear Equations
In the previous chapter, students learned the connection between a function and its graph. In this chapter, students focus on the connection between a linear equation and its graph. This topic is introduced first by a discussion of rate of change that leads to the definition of the slope of a line. The idea of slope is then expanded to include the SIF of a linear equation, and the use of this form to graph an equation. Using these skills, students write linear functions and interpret graphs to solve real world problems. Students then learn to write a linear equation in standard form, followed by graphing and writing linear equations when given various kinds of information about a line.
The characteristics of the slopes and graphs of both parallel and perpendicular lines are discussed. Students apply the preceding skills as they write the equation of a trend line and a line of best fit. Graphing a linear equation is extended to graphing and translating absolute value equations.
I. Rate of Change and Slope A.6.A A.6.B Obj. 3
The steepness of a road going up a hill and the pitch of a roof can both be expressed as the ratio of the rise over the run, or the slope. Students may at first confuse the order of the ration, since they are accustomed to putting x first when writing the coordinates of an ordered pair. This may lead them to place the difference in x values, or run, incorrectly in the numerator. Emphasize that the rise is the vertical change and the run is the horizontal change.
II. Slope Intercept Form A.2.A, Obj. 2, a.6.a. a.6.D, a.6.E, A.6.G Obj 3 a.4 C.
The lesson begins by showing students that all linear functions are related. That is linear functions are a family of functions where f(x) = x is the linear parent function.
Remind students that in Chapter 5, they learned that a function in the form of y = kx is a direct variation. Have student compare y = kx to y = mx + b . Lead students to understand that the slop of a linear function through the origin is a constant of variation.
Students may find it helpful to express slope as change in y/change in x, where x delta means change in as the change in y divided by the change in x.
III. Applying Linear Functions
A.1.E Obj. 1, a.2.C Obj. 2; a.6.B a.6F, Obj 3. a.5.B
Building on the concepts and skills discussed in the previous two lessons, in this section students use linear functions to model real world situations. They analyze application problems, write linear functions and create graphs to represent these functions. Point out to students that importance of choosing a reasonable domain to represent the original situation.
IV. Standard Form
Tell students that a linear equation in standard form is useful in making quick graphs. Suggest that an efficient way of graphing an equation in this form is to create a table of values for the equation by substituting 0 for x and finding the corresponding y value. Then substituting 0 for y and finding the corresponding x value. If more points are needed, it may be convenient to evaluate the equation using 1 for x and then for y.
Explain to students why finding a third point is helpful.
V. Point Slope From and Writing Linear Equations
A.5.A, A.6.A. a.6.B a.6.D Obj 3
Point out to students that using the point slope form of a linear equation is a generalization of the more specific slope-intercept form, in which the point is specifically designated to be the y-intercept. Students can use the point slope form to write equations that model sets of data.
VI. Parallel and Perpendicular Lines
A.6.A, A.6.D Obj 3.
Two lines in the same plane that have the same slope must either be parallel or coincide. In the latter case, the two equations are equivalent, as for example, x + 1 = y and 3 x = 3y – 3. (Student may not realize that this is just another case of equivalent equations having exactly the same solution set.) The family of lines with a given slope, m = 5, for example, consists of infinitely many lines, all which are parallel.
Discuss with the students the slopes of a family of lines with one common point. Graph examples of such lines. By finding the slope of each line, show students that for two perpendicular lines that the slope of one line is the negative reciprocal of the other.
VII. Scatter Plots and Equations of Lines TEKs a.1.B Obj 1 A.2.C a.2.D obj 2.
Real-world data, when graphed, rarely falls exactly along a line. Data are classified as linear if the points approximate a line on a graph. When this is the case, there are techniques to approximate an equation for a line of best fit. The graphing calculator makes this tasks easy (input TI-84 activities), by correlating equations of “best fit” to actual entered data.
Because real-world data many not be linear, the range (the difference between the value of the greatest point and the least) is called a measure of scatter. Measures of scatter indicate whether the measurements in a distribution are bunched together or scattered apart. In statistics, other measures of scatter include deviation, variance, and standard deviation.
VIII. Graphing Absolute Value Equations
TEKS Prepares for TEKS 2.A.4.A, 2.A.4.B
Translation is one of three commonly used transformations: translation or slide, rotation or turn, and reflection or flip. These three transformations are called isometries or rigid motions, because the size and the shape of the figure remain unchanged.
Dilation, such as enlarging a figure using copying machines, is another transformation but it is not rigid motion because distances within the figure are changed. Students learn that graphs of various absolute value equations can be thought of as translations of the equation y = |x|. Remind students that they studied absolute value equations in Chapter 4.
I. Rate of Change and Slope
Focus on the TEKS you must put in the actual objects from the released tests.
In algebra 1, students develop the concept of slope as rate of change and determine slopes from graphs, tables and algebraic representations A.6.A. Analysis of TAKS Results shows that students struggle with this expectation.
In their first lessons, students solve for the rate of change using numbers and graphs. In Lessons 6-2, and 605, students find slopes using various slope forms and explore the parallel and perpendicular lines in Lesson 6. Place special emphasis on these lessons because they all focus on A.6.A.
Also, emphasize interpretation of the meaning of slope and intercepts. A.6.B. Students determine and analyze slope in different situations in Lessons 6-1, 6-3 and 6-5 (a.6.B).
Overview of Big Ideas Affecting Slope and Linear Equations
In the previous chapter, students learned the connection between a function and its graph. In this chapter, students focus on the connection between a linear equation and its graph. This topic is introduced first by a discussion of rate of change that leads to the definition of the slope of a line. The idea of slope is then expanded to include the SIF of a linear equation, and the use of this form to graph an equation. Using these skills, students write linear functions and interpret graphs to solve real world problems. Students then learn to write a linear equation in standard form, followed by graphing and writing linear equations when given various kinds of information about a line.
The characteristics of the slopes and graphs of both parallel and perpendicular lines are discussed. Students apply the preceding skills as they write the equation of a trend line and a line of best fit. Graphing a linear equation is extended to graphing and translating absolute value equations.
I. Rate of Change and Slope A.6.A A.6.B Obj. 3
The steepness of a road going up a hill and the pitch of a roof can both be expressed as the ratio of the rise over the run, or the slope. Students may at first confuse the order of the ration, since they are accustomed to putting x first when writing the coordinates of an ordered pair. This may lead them to place the difference in x values, or run, incorrectly in the numerator. Emphasize that the rise is the vertical change and the run is the horizontal change.
II. Slope Intercept Form A.2.A, Obj. 2, a.6.a. a.6.D, a.6.E, A.6.G Obj 3 a.4 C.
The lesson begins by showing students that all linear functions are related. That is linear functions are a family of functions where f(x) = x is the linear parent function.
Remind students that in Chapter 5, they learned that a function in the form of y = kx is a direct variation. Have student compare y = kx to y = mx + b . Lead students to understand that the slop of a linear function through the origin is a constant of variation.
Students may find it helpful to express slope as change in y/change in x, where x delta means change in as the change in y divided by the change in x.
III. Applying Linear Functions
A.1.E Obj. 1, a.2.C Obj. 2; a.6.B a.6F, Obj 3. a.5.B
Building on the concepts and skills discussed in the previous two lessons, in this section students use linear functions to model real world situations. They analyze application problems, write linear functions and create graphs to represent these functions. Point out to students that importance of choosing a reasonable domain to represent the original situation.
IV. Standard Form
Tell students that a linear equation in standard form is useful in making quick graphs. Suggest that an efficient way of graphing an equation in this form is to create a table of values for the equation by substituting 0 for x and finding the corresponding y value. Then substituting 0 for y and finding the corresponding x value. If more points are needed, it may be convenient to evaluate the equation using 1 for x and then for y.
Explain to students why finding a third point is helpful.
V. Point Slope From and Writing Linear Equations
A.5.A, A.6.A. a.6.B a.6.D Obj 3
Point out to students that using the point slope form of a linear equation is a generalization of the more specific slope-intercept form, in which the point is specifically designated to be the y-intercept. Students can use the point slope form to write equations that model sets of data.
VI. Parallel and Perpendicular Lines
A.6.A, A.6.D Obj 3.
Two lines in the same plane that have the same slope must either be parallel or coincide. In the latter case, the two equations are equivalent, as for example, x + 1 = y and 3 x = 3y – 3. (Student may not realize that this is just another case of equivalent equations having exactly the same solution set.) The family of lines with a given slope, m = 5, for example, consists of infinitely many lines, all which are parallel.
Discuss with the students the slopes of a family of lines with one common point. Graph examples of such lines. By finding the slope of each line, show students that for two perpendicular lines that the slope of one line is the negative reciprocal of the other.
VII. Scatter Plots and Equations of Lines TEKs a.1.B Obj 1 A.2.C a.2.D obj 2.
Real-world data, when graphed, rarely falls exactly along a line. Data are classified as linear if the points approximate a line on a graph. When this is the case, there are techniques to approximate an equation for a line of best fit. The graphing calculator makes this tasks easy (input TI-84 activities), by correlating equations of “best fit” to actual entered data.
Because real-world data many not be linear, the range (the difference between the value of the greatest point and the least) is called a measure of scatter. Measures of scatter indicate whether the measurements in a distribution are bunched together or scattered apart. In statistics, other measures of scatter include deviation, variance, and standard deviation.
VIII. Graphing Absolute Value Equations
TEKS Prepares for TEKS 2.A.4.A, 2.A.4.B
Translation is one of three commonly used transformations: translation or slide, rotation or turn, and reflection or flip. These three transformations are called isometries or rigid motions, because the size and the shape of the figure remain unchanged.
Dilation, such as enlarging a figure using copying machines, is another transformation but it is not rigid motion because distances within the figure are changed. Students learn that graphs of various absolute value equations can be thought of as translations of the equation y = |x|. Remind students that they studied absolute value equations in Chapter 4.
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