Recent Changes

This study originated from the realization that many community college students have significant difficulties with the concepts related to linear functions and their graphs, even though these topics are included in the mathematics curriculum from about fourth grade onward. It is an attempt to identify possible gaps in the math standards in this area, as well as possible oversights in emphasis in classroom practice at various levels.
Several of these problems overlap with misconceptions about fractions. One of these is the unspoken assumption that the only numbers on the number lines are the integers. Therefore the act of connecting several points on the graph of a line is merely demonstrating that they all lie on the line, implying nothing about function values for non-integer inputs. Another misconception lies in seeing slope exclusively as a ratio of two numbers (rise over run) and not as a unit rate (a single number). These misconceptions can have consequences for future difficulties even with calculus.
with) including CaliforniaCalifornia State Standards(CSS)

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the function.
Minutes
1

36
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for the function..function. Assuming the the function.
Minutes
1
2. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image308.png}
3. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image311.png}
Conceptual Understanding Understanding//
1. Given the following matrices, determine if the corresponding systems are consistent or inconsistent. Identify consistent systems as dependent or independent.
Where appropriate, give the solution.
· “Understand that two lines with well-defined slopes are perpendicular if and only if the product of their slopes is –1.” (1)
· “Use the slope criteria for parallel and perpendicular lines to solve geometric problems.” (7)
the CCMS documendocument.

grade levels,
as
as described in California Standards
The
The following linear 3 – 77, the examples
For the Algebra 1, Algebra 2, Geometry, Mathematical Analysis and Linear Algebra standards, the examples are taken from the Cal-PASS deconstructed standards.
of California statesstate that “Emphasis
Notes refer to resources outside of the California State Standards, and comments refer to the opinions of the writers.
Grade 3:
Note: The CCM states that students “understand that the length of a number line (interval from 0 to 1) can be divided into parts of equal fractional length.”
Grade 5:
the results:
1.1 Use information taken from a graph or equation to answer questions about a problem situation.
1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.
4. Write the equation of the line passing through the points and
5. Write the equation of the line graphed below.
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image045.png}
Conceptual Understanding
Consider the table of values below.
f. When is it advantageous to use the substitution method? The matrix method? Give an example to illustrate your answers to both parts of this question.
g. Write the system of linear inequalities which describes the shaded region from the graph below (Inside the triangle is shaded):
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image156.png} {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image158.png}
Problem Solving/Application
1. Your family receives basic television and two movie channels for $32.30 a month. Your neighbor receives basic cable and four movie channels for $43.30 a month. What is the monthly charge for just the basic cable? (Assume that the movie channels have the same monthly cost.) What is the monthly charge for one movie channel? Why do parts a and b appear below? I suggest to cut them.

This study originated from the realization that many community college students have significant difficulties with the concepts related to linear functions and their graphs, even though these topics are included in the mathematics curriculum from about fourth grade onward. It is an attempt to identify possible gaps in the math standards in this area, as well as possible oversights in emphasis in classroom practice at various levels.
Several of these problems overlap with misconceptions about fractions. One of these is the unspoken assumption that the only numbers on the number lines are the integers. Therefore the act of connecting several points on the graph of a line is merely demonstrating that they all lie on the line, implying nothing about function values for non-integer inputs. Another misconception lies in seeing slope exclusively as a ratio of two numbers (rise over run) and not as a unit rate (a single number). These misconceptions can have consequences for future difficulties even with calculus.
several sources, beginning with the California(beginning with) including California State Standards. The sources are identified byStandards(CSS) the Common Core Standards for Mathematics (CCM), Statement on Competencies in Mathematics Expected of Entering College Students, and the National Common Core Standards for Mathematics. In addition, the Calpass deconstruction documents for Algebra I through Precalculus and the San Diego Department of Education's backward mapping project for algebra supplied the following abbreviations. (where areexample problems throughout the abbreviations?)document. We have

11/5/10
Concept Selection and Tracing of Development
these concepts."?) . .I agree this is confusing.
2. Identify the target concept.
3. Select sources for tracking the development of the concept.
7. Create a Conclusion which makes recommendation.
8. Create Table of Contents and Contributors page.
by "create inventions"?). inventions"?).I think we mean interventions.

Linear Equation Development through the grade levels,
as described in the California Standards
State Standards (years).(1997). For the deconstructed standards.
In the 2010 Statement on Competencies in Mathematics Expected of Entering College Students by the Intersegmental Committee of the Academic Senates of the California Community Colleges, the California State University and the University of California states that “Emphasis should be placed on various representations of functions – using graphs, tables, variables, and words – and on the interplay among the graphical and other representations…” Those standards with a * preceding them are referenced as key in the Statement on Competencies document.
Notes refer to resources outside of the California State Standards, and comments refer to the opinions of the writers.
Grade 3:
Algebra
Algebra and Functions functional relationships:
2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit).
2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4).
Note: The Common Core Standards for Mathematics (CCM) includes the ability to represent unit fractions on a number line. “Students … use fractions to represent parts of a whole or distances on a number line that begins with zero.”
Grade 4:
AF
AF 1.0 Students and sentences:
1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of the concept of a variable).
1.4 Use and interpret formulas (e.g., area = length × width or A = lw) to answer questions about quantities and their relationships.
3
4
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AF 2.0 Students know how to manipulate equations:
2.1 Know and understand that equals added to equals are equal.
2.2 Understand that the length of a horizontal line segment equals the difference of the x-coordinates.
2.3 Understand that the length of a vertical line segment equals the difference of the y-coordinates.
MG 3.03.0 Students demonstrate solve problems:
3.1 Identify lines that are parallel and perpendicular.
Number Sense (NS) 1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places.
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image004.png}
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Note: The CCMSCCM states that
Grade 5:
AF 1.0 Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results:
1.1 Use information taken from a graph or equation to answer questions about a problem situation.
Graph the following numbers on the number line: -5, 1, 5, {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image007.png} , 2.75, -3.5.
Note: The CCM includes the ability recognize that equivalent fractions correspond to the same point n a number line.”
Grade 6:
AF
AF 1.0 Students their results:
1.1 Write and solve one-step linear equations in one variable.
1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables.
The number of players on the football team increased by 3 equals 21. How many players were on the original team?
Note: The CCM states that students “understand that for a ratio a:b, the corresponding unit rate is a/b.” This concept is basic to the understanding of the slope of a line as a unit rate.
Grade 7:
*AF
*AF 1.0 Students and graphs:
1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
nonlinear functions:
3.1 Graph functions of the form y = nx2 and y = nx3 and use in solving problems.
3.2 Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths).
· “Understand that a straight line is a widely used model for exploring relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line…” (Statistics and Probability #3)
· Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.” (Statistics and Probability #4)
I standards:
4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
Computational and Procedural Skills
Add an application based on limiting cell phone costs for text messages or something similar.
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
Procedural Skills
Find the “x” and “y”-intercepts for the line defined by the following equation: 2x + 3y = 9
Use the “x” and “y”-intercepts to graph the line given by the equation above: 2x + 3y = 9
Graph the solution set for the following inequalities:
a) 2x – 3y < 6 b) {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image034.png} c) {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image036.png}
Conceptual Understanding
Sketch the graph of a line that has no x-intercept.
Identify the “x” and “y”- intercepts from the graph of the given line. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image038.png}
Y -3x + 5
Insert correct symbol in box.
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{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image040.png}
When is it advantageous to use the x and y-intercepts to graph the equation of a line? When would it perhaps be easier or better to use another graphing method? Give an example to illustrate your answers to both of these questions.
Problem Solving/Application
5. Write the equation of the line graphed below.
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image045.png}
Conceptual Understanding
Consider the table of values below.
Episode Number
b) Determine whether or not the data could be linear.
c) If the data is linear find and interpret the slope.
Problem Solving/Application
A taxi driver charges a $2.00 pick up fee plus $3 for each mile traveled.
If you are interested in calculating the cost of your trip, what quantities would be of interest?
Find an equation of a line, in slope-intercept form, through point {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image081.png} that is perpendicular to {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image084.png}
*9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
Procedural Skills
1. Solve systems of two equations in two variables using substitution:
Solve the system: {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image087.png} by the substitution method.
3. Identify the intersection of two shaded areas created by the graphs of linear inequalities in two variables as the solution to the system:
Solve the system of inequalities {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image093.png} graphically.
Conceptual Understanding
1. Interpret the solution as representing the intersection of two lines or the same line:
Suppose a system of two linear equations in two variables has a solution, (3,5). When graphing these two equations on the same Cartesian coordinate system, what would you expect the graph of the two lines to look like?
Solve the system: {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image099.png} by graphical method.
*16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
Conceptual Understanding
1. Given the following sets of ordered pairs, give the domain and range of the relation; determine whether the relation is or is not a function:
a. {(-4,3), (-2,1), (0,5), (-2,-8)}
d. (3, ?)
e. (0, ?)
Conceptual Understanding
1. State the domain and range of each relation:
a. { (2,4), (2,5), (4,6), (7,2), (5,10), (8,4), (3,6) }
4
3
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1
2
3
3
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image110.png} AppleMark
Problem
Problem Solving/Application
1. The table below shows the amount that a company charges for a bike rental. Identify the domain and range. Write a set of ordered pairs for the function. Assuming the cost of the bike rental is a linear function of the number of hours the bike was rented, find a formula for the function.
Time (hrs)
f. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image128.png}
g. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image131.png}
Conceptual Understanding
1. If you are looking at a graph, how do you determine whether or not it is the graph of a function?
2. Write a set consisting of three ordered pairs that is a relation, but not a function. Explain why the set of ordered pairs you wrote is not a function.
Algebra II:
2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
Procedural Skills
1. Solve the system by graphing.
a. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image134.png}
1. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image146.png} 2. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image149.png} 3. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image152.png} 4. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image155.png}
2.
Conceptual Understanding
a. If you graph two lines in the same coordinate plane, what are the possible outcomes?
b. A system of linear equations may have infinitely many solutions. Explain how this is possible.
7.0 Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.
(Here is another example of a possibly irrational slope. Students will need more than “rise over run’ in order to interpret this.
Mathematical Analysis:
5.0 Students are familiar with conic sections, both analytically and geometrically:
5.1 Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).
Computational and Procedural Skills
1. Graph each of the following conic sections and find the desired information.
A.
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image205.png}
Foci: _
Focus _
Directrix _
Opens
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image215.png}
1. Complete the square to determine whether the equation represents an ellipse, a parabola, or a hyperbola. If the graph is an ellipse, find the center, foci, vertices, and length of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation.
of conic:
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image220.png}
2. Determine the type of curve represented by the equation. Find the foci and vertices (if any), and sketch the graph.
C. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image284.png} , {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image286.png} , {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image288.png}
D. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image290.png} , {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image292.png} , {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image294.png}
Conceptual Understanding
1. Given the following parametrically defined equations, find the values of t that produce the graph in the given quadrant. What does this mean?
{file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image296.png}
C. Quadrant III
D. Quadrant IV
Linear Algebra
6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
Procedural Skills
Determine if the following systems are consistent or inconsistent. Identify consistent systems as dependent or independent.
1. {file://localhost/Users/calilinfor/Library/Caches/TemporaryItems/msoclip/0/clip_image305.png}

.
Color Codes
be blue.
1/31/11: Laura used red to edit the cover page and tracing development page and will use teal to edit the linear equation development page since red was already used there.

2. Identify the target concept.
3. Select sources for tracking the development of the concept.
(Pre-K-Statistics or 3rd-Calculus…)
5.3rd-Calculus…).
5. Collect standards/learning through the levels.
Isolatelevels:
Isolate CA Standards level and class class.
Collect standards and problems from backwards mapping documents such as the San Diego County Office of Education Algebra Backwards Mapping
Isolate Common Core Standards having to do with concepts in order of grade level and class.
Isolate CA and Core Standards of other disciplines for the concept and order in terms of grade level.
Review the NCTM documents to see if they have anything to contribute to the development of the concept.
and University Materials
ExamineMaterials.
Examine Cal-PASS Deconstructions for sample problems
Createproblems.
Create missing problem examples examples.
6. Make notations about what is missing in the development of the concept.
7. Make notations about common errors made by students due to the developmental gaps.
Document Creation, Editing, and Commentary
1. Remove from the documents any standards or problems that do not really trace the development of the concept.
of the concept concept.
3. Comment on gaps in development of the concept and issues with sample problems or standards
4. Reference ICAS Competencies in Mathematics http://icas-ca.org/competencies-in-mathematics and other ICAS Competencies that may apply in commentary
7. Create a Conclusion which makes recommendation.
8. Create Table of Contents and Contributors page.
reveal student errorerrors in understanding learning the concept.concept (Comment: What do we mean by "create inventions"?).