Graphs & Functions (Algebra 1)
TEKS Math Background: Algebra 1 Chapter 5. Graphs and Functions
In Algebra 1, students describe functional relationships for given problem situations and write equations to answer questions arising form the situations (A.1. (C). Analysis of TAKS results shows that students struggle with this new student expectation.
[Camille, on blog put images of each of the problems the state isolated as representative of the TEKS.]
Students write function rules using tables and apply this skill to write function rules for given situations. We should place special emphasis on mastery of this skill, because it focuses on (A.1) (c).
• Also new for Algebra 1 is analyzing data and representing situations involving inverse variation A.11.B. Students use data given in tables and graphs to determine inverse variation… A.11.B.
Overview of Big Ideas behind Graphs * Functions in Algebra 1:
In this chapter, students further their knowledge of functions by building on the exploration of patterns in Chapter 1. Students move from the specific case of equations in one variable, found in Chapter 3, to the study of functions in two variables. Functional relationship relates the value of one variable, such as y, to the value of another variable, such as x. Functional relationships can be represented visually by graphs.
Students begin the chapter by graphing situations and interpreting the situations represented by a graph. They then use the vocabulary and mathematical notation for functions, and model functions with rules, tables and graphs. Students examine two particular functions called a direct variation and an inverse variation. Students will use what they learn about inverse variation in this lesson when they study rational functions in Chapter 12. Finally, they explore the special patterns in arithmetic sequence.
Relating Graph to Events
TEKS A.1.A A.!D. Obj. 1 A.2.C Obj. 2
The lesson will help students see how they can use mathematics to make a general representation of a specific relationship that occurs in their daily lives. Through graphing, students explore the idea of change, such as changes in speed, altitude, etc. (Fast and Furious). In addition to change, student investigates relationships between quantities by using graphs. Students will continue their work on graphs in their lessons on ….
Relations and Functions
TEKS A.1.D Obj. 1 a.4.A Obj. 2 A.4.C.
Building on their understanding of functions discussed in Chapter 1, students
Further their knowledge of functions by determining whether a relation is a function. Students should learn to correctly read f(x), as “F of x.” This will help them distinguish function notation from the more familiar notation used to show the multiplication of two variables, such as ab or a(b). Explain that the context usually makes the difference clear, and certain letters, such as f, g, h and k, are most commonly used for function notation.
Function Rules, Tables and Graphs: TEKS. A.1.A Obj 1. A.2. B Obj 2. A.5 c a.6 E
Help students see that the rule f(x) = x + 3 represents an infinite number of points, each of which can be written as an ordered pair in the form (x, x + 3). Similarly, the straight line that is the graph of y = x + 3 includes all these points, since the line extends forever. However, a table represents only a sample of points from this solution set—enough points to determine and check the straight line that contains all the point sin the solution set. Using these three views of a function, students can show whether a given function models discrete or continuous data.
Make sure students understand that graphs represent the relationship between two quantities. Although in this lesson the quantities are abstract variables in equations. Remind students (initially) that they learned that graphs can present the relationship between two real quantities, such as speed and time. Help students think of real world situations that could be represented by the graph of a quadratic equation, such as the flight of a projectile.
Writing a function Rule (TEKS A.1C Obj. 1 A.2B A.3B Obj 2
When students write a rule for a table of values, they must make sure that the apparent pattern does indeed hold true for each pair of points listed in the table. While two points are enough to determine a line, at least a third point should be used as a check to find and correct errors in calculation.
The fact that two points determine a line may not at first be obvious to students until they try to find a second and different line that includes the same two points.
Direct Variation
TEKs a.1.e Obj. 1
Variation is a statement of how one quantity changes relative to the way another quantity changes. A variation may be expressed as an equation that states a constant relationship between the values of two variables.
In direct variation, as the absolute vale of one variable increases, the abs. value of the second variable also increases proportionally. Thus for example in y = - 2x, as x increases from 10 to 20, the absolute value of y increases from 20 to 40. This means that ratio of y to x remains equal to the constant k, or y/x = k. Another way to express this direct relationship is with the proportion y1/x1 = y2/x2. So, in this example, -20/10 = -40/20.
Tell students that there are other kinds of variation, such as inverse variation, which will be discussed in the next lesson.
Inverse Variation TEKS a.1D Obj 2. A.11 B
Students extend their knowledge of variation studied in prior lessons to the study of inverse variation. In an inverse variation, as the value of one variable increases, the value of the other variable decreases so that the product of the two variables remains constant. Another way to express this inverse relationship is with an equation in the form of xy = k.
Make sure that students understand the difference between a direct variation and an inverse variation. They should recognize how the graphs of each reflect their respective definitions.
Describing Number Patterns
TEKS A.1.D Obj 1. a.3B, A.4.A Obj 2
An inductive argument reasons from observations of particular instances, (e.g. this crow is black, that crow is black) to a generality (all crows are black). Not all conclusions reached in this way are correct, but this kind of reasoning is very often used in daily life, and is often effective.
Students can use inductive reasoning to identify patterns, or sequences in sets of numbers. An arithmetic sequence is formed by adding a specific number of to each term after the first. Another type of sequence, the geometric sequence (see changed TEKS), will be introduced later. Deductive reasoning, as a next stage, using formal logic, will confirm or discredit their conclusions.
In Algebra 1, students describe functional relationships for given problem situations and write equations to answer questions arising form the situations (A.1. (C). Analysis of TAKS results shows that students struggle with this new student expectation.
[Camille, on blog put images of each of the problems the state isolated as representative of the TEKS.]
Students write function rules using tables and apply this skill to write function rules for given situations. We should place special emphasis on mastery of this skill, because it focuses on (A.1) (c).
• Also new for Algebra 1 is analyzing data and representing situations involving inverse variation A.11.B. Students use data given in tables and graphs to determine inverse variation… A.11.B.
Overview of Big Ideas behind Graphs * Functions in Algebra 1:
In this chapter, students further their knowledge of functions by building on the exploration of patterns in Chapter 1. Students move from the specific case of equations in one variable, found in Chapter 3, to the study of functions in two variables. Functional relationship relates the value of one variable, such as y, to the value of another variable, such as x. Functional relationships can be represented visually by graphs.
Students begin the chapter by graphing situations and interpreting the situations represented by a graph. They then use the vocabulary and mathematical notation for functions, and model functions with rules, tables and graphs. Students examine two particular functions called a direct variation and an inverse variation. Students will use what they learn about inverse variation in this lesson when they study rational functions in Chapter 12. Finally, they explore the special patterns in arithmetic sequence.
Relating Graph to Events
TEKS A.1.A A.!D. Obj. 1 A.2.C Obj. 2
The lesson will help students see how they can use mathematics to make a general representation of a specific relationship that occurs in their daily lives. Through graphing, students explore the idea of change, such as changes in speed, altitude, etc. (Fast and Furious). In addition to change, student investigates relationships between quantities by using graphs. Students will continue their work on graphs in their lessons on ….
Relations and Functions
TEKS A.1.D Obj. 1 a.4.A Obj. 2 A.4.C.
Building on their understanding of functions discussed in Chapter 1, students
Further their knowledge of functions by determining whether a relation is a function. Students should learn to correctly read f(x), as “F of x.” This will help them distinguish function notation from the more familiar notation used to show the multiplication of two variables, such as ab or a(b). Explain that the context usually makes the difference clear, and certain letters, such as f, g, h and k, are most commonly used for function notation.
Function Rules, Tables and Graphs: TEKS. A.1.A Obj 1. A.2. B Obj 2. A.5 c a.6 E
Help students see that the rule f(x) = x + 3 represents an infinite number of points, each of which can be written as an ordered pair in the form (x, x + 3). Similarly, the straight line that is the graph of y = x + 3 includes all these points, since the line extends forever. However, a table represents only a sample of points from this solution set—enough points to determine and check the straight line that contains all the point sin the solution set. Using these three views of a function, students can show whether a given function models discrete or continuous data.
Make sure students understand that graphs represent the relationship between two quantities. Although in this lesson the quantities are abstract variables in equations. Remind students (initially) that they learned that graphs can present the relationship between two real quantities, such as speed and time. Help students think of real world situations that could be represented by the graph of a quadratic equation, such as the flight of a projectile.
Writing a function Rule (TEKS A.1C Obj. 1 A.2B A.3B Obj 2
When students write a rule for a table of values, they must make sure that the apparent pattern does indeed hold true for each pair of points listed in the table. While two points are enough to determine a line, at least a third point should be used as a check to find and correct errors in calculation.
The fact that two points determine a line may not at first be obvious to students until they try to find a second and different line that includes the same two points.
Direct Variation
TEKs a.1.e Obj. 1
Variation is a statement of how one quantity changes relative to the way another quantity changes. A variation may be expressed as an equation that states a constant relationship between the values of two variables.
In direct variation, as the absolute vale of one variable increases, the abs. value of the second variable also increases proportionally. Thus for example in y = - 2x, as x increases from 10 to 20, the absolute value of y increases from 20 to 40. This means that ratio of y to x remains equal to the constant k, or y/x = k. Another way to express this direct relationship is with the proportion y1/x1 = y2/x2. So, in this example, -20/10 = -40/20.
Tell students that there are other kinds of variation, such as inverse variation, which will be discussed in the next lesson.
Inverse Variation TEKS a.1D Obj 2. A.11 B
Students extend their knowledge of variation studied in prior lessons to the study of inverse variation. In an inverse variation, as the value of one variable increases, the value of the other variable decreases so that the product of the two variables remains constant. Another way to express this inverse relationship is with an equation in the form of xy = k.
Make sure that students understand the difference between a direct variation and an inverse variation. They should recognize how the graphs of each reflect their respective definitions.
Describing Number Patterns
TEKS A.1.D Obj 1. a.3B, A.4.A Obj 2
An inductive argument reasons from observations of particular instances, (e.g. this crow is black, that crow is black) to a generality (all crows are black). Not all conclusions reached in this way are correct, but this kind of reasoning is very often used in daily life, and is often effective.
Students can use inductive reasoning to identify patterns, or sequences in sets of numbers. An arithmetic sequence is formed by adding a specific number of to each term after the first. Another type of sequence, the geometric sequence (see changed TEKS), will be introduced later. Deductive reasoning, as a next stage, using formal logic, will confirm or discredit their conclusions.
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