Linear Equation Development through the grade levels,

Linear Equation Development through the grade levels,
as described in the California Standards
The following linear development is based on the California State Standards (1997). For the grades 3 – 7, the examples are taken from the Backwards Mapping project from the San Diego County of Education Department. Those items marked AF are from the Algebra and Functions component. Those marked MG are from the Measurement and Geometry component.
For the Algebra 1, Algebra 2, Geometry, Mathematical Analysis and Linear Algebra standards, the examples are taken from the Cal-PASS 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 state 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 and Functions (AF) 2.0 Students represent simple 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 1.0 Students use and interpret variables, mathematical symbols, and properties to write and simplify expressions 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.
1.5 Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given.

AF 1.5 Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given. Note: this is the first time students see a number and a variable next to each other with no operation symbol. They need to understand that the implied operation is multiplication. CST (2)	AF 1.5 In an equation in two variables, find the value of the first variable when given the value of the second variable.	AF 1.5 Complete the table using the Rule: y = 2x + 4	Input (x)	Output (y)
0
1
2
3
4

AF 2.0 Students know how to manipulate equations:

2.1 Know and understand that equals added to equals are equal.

2.2 Know and understand that equals multiplied by equals are equal.


Measurement and Geometry (MG) 2.0 Students use two-dimensional coordinate grids to represent points and graph lines and simple figures:

2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line).



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.


3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and 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.

NS 1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places.	NS 1.9 Plot decimals and mixed numbers on a number line.	NS 1.9 What value is represented by the point P on this number line?

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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:
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.
1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.
1.3 Know and use the distributive property in equations and expressions with variables.
1.4 Identify and graph ordered integer pairs in the four quadrants of the coordinate plane.
1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

NS 1.5 Identify and represent on a number line decimals fractions, mixed numbers and positive and negative integers. (Note does this mean no negative decimals or fractions?)

NS 1.5 Graph decimals fractions, mixed numbers and positive and negative integers.

NS 1.9 Graph the following numbers on the number line: -5, 1, 5, , 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 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret 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.
1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator.
Comment: Solving an equation is not directly related to the concept of a line, however for the student to understand what they are doing, a linear model is helpful.

AF 1.1 Write and solve one-step linear equations in one variable.	AF 1.1a Write one-step linear equations in one variable.	AF. 1.1 a Which algebraic equation best describes the total growth (T) in height of pine trees over a three-year period, if g equals the rate of growth in centimeters per year? A) T = 3g B) T = 3 + g C) T = D) T =
	AF 1.1b Solve one-step linear equations in one variable	AF 1.b Write an equation for the following problem and solve. 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 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, 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.

AF 3.0 Students graph and interpret linear and some 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).
3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph.
3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of quantities. (Comment: The ratio of circumference to diameter is an irrational number.)
Note: According to the CCM, by grade 7 students should understand that “fractions and their opposites form a system of numbers called the rational numbers, represented by points on a number line.” In addition they “prove that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.” It is in grade 7 that students “plot proportional relationships on a coordinate plane where each axis represents one of the two quantities involved, observe that the graph is a straight line through the origin, and find unit rates from a graph. Explain what a point (x, y) means in terms of the situation, with special attention to the points (0,0) and (1, r) where r is the unit rate.”

The State of California has deemed that all students will take algebra in Grade 8 and hence the Algebra I standards are reflective of what is taught. The CCM separates Grade 8 from the High School curriculum. For grade 8 the following concepts are included as core:
· “Students use linear equations … to represent, analyze and solve a variety of problems. … they understand that the slope (m) of a line is a constant rate of change.”
· “Understand that the sloe of a non-vertical line in the coordinate plane has the same value for any two district points used to compute it.” (Linear equations in two variables #3)
· “Construct a function to model a linear relation ship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship; from two (x, y) values, including reading these from a table; or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.” (Functional relation ships between quantities #6)
· “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)


Algebra 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
1. Simplify the following expressions:



a.

b.


c.

d.



*5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
1. Solve the following equations and inequalities:

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. 
   

1. Solve and justify each step.

1. The length of a rectangle is six less than twice the width. Its perimeter is 36 inches. Find the dimensions of the rectangle.

1. The sum of 32 and twice a number is at most 118. Determine all numbers with this property?

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).
Computational and Procedural Skills

1. Find the “x” and “y”-intercepts for the line defined by the following equation: 2x + 3y = 9
2. Use the “x” and “y”-intercepts to graph the line given by the equation above: 2x + 3y = 9
3. Graph the following lines using the method of your choice. Identify and label the “x” and “y” intercepts for each graph if they exist:

a) 3x – 5y = 10
b)
c) y = 2
d) x = 3.5
e) 2x + 4y = 3
f)

1. Graph the solution set for the following inequalities:

a) 2x – 3y < 6 b) c)

Conceptual Understanding

1. Sketch the graph of a line that has no x-intercept.
2. Identify the “x” and “y”- intercepts from the graph of the given line.
3. Can a line have more than one x-intercept? Explain your answer using a diagram.
4. The solution to an inequality has been graphed correctly below. Insert the correct inequality symbol in the inequality below to match the graph of the solution. (Everything else about the inequality is correct – it just needs the correct symbol).

Y -3x + 5 Insert correct symbol in box.

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1. 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

1. The graph displayed below is the graph of the following equation: , where “x” represents the amount of time that has passed since a 5 gallon fish tank sprung a leak, and “y” represents the number of gallons of water in the tank after the leak.

a) What is the significance of the “x-intercept” in this situation? What information is given to us by this point?

b) What is the significance of the “y-intercept” in this situation? What information is given to us by this point?

c) What is the rate of leaking in gallons per minutes. (rate = slope)

1. The cost of a trash pickup service is given by the following formula: y = 1.50x + 11, where “x” represents the number of bags of trash the company picks up, and “y” represents the total cost to the customer for picking up the trash.

a) What is the “y-intercept” for this equation?

b) What is the significance of the “y-intercept” in this situation? What does it tell us about this trash pickup service?

c) Draw a sketch of the graph which represents this trash pickup service.


7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Computational and Procedural Skills
1. Given the equation of a line, determine whether or not the following points lie on the line: , , and .

2. Write the equation of the line with slope and y-intercept of –2

3. Write the equation of the line with a slope of 4 and passing through the point

4. Write the equation of the line passing through the points and

5. Write the equation of the line graphed below.




Conceptual Understanding


Consider the table of values below.

Episode Number	Number of Survivors
1	25
2	23
3	21

a) Identify the input and output variables
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?
d) Identify the input and output quantities.
e) Make a table for inputs 0, 1, and 2.
f) Identify and interpret the meaning of the slope.
g) Identify and interpret the meaning of the vertical axis intercept.
h) Write an equation describing the output as a function of the input.
i) Use your equation to determine the cost of a 12-mile taxi ride.


*8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
a) Recognize parallel lines as having the same slope and different y-intercepts graphically.

Graph: and on the same rectangular coordinate system. Does it appear the two lines are parallel? Explain!
(Do students have a geometric understanding of the meaning of parallel lines? If not, is this topic revisited in a geometry class, perhaps in the context of similar triangles?)
b) Recognize parallel lines as having the same slope and different y-intercepts algebraically.

Algebraically, demonstrate the two lines whose equations are given by: and are parallel.

c) Recognize perpendicular lines algebraically by intersecting at angles.

Graph: and on the same rectangular coordinate system. Does it appear the two lines are perpendicular? Explain!

d) Recognize perpendicular lines algebraically as having slopes whose product is –1.
(Is there any way to demonstrate this fact geometrically at this time? Will it be done at some time? Should we add an exercise that looks at it heuristically?
Algebraically, demonstrate the two lines whose equations are given by: and are perpendicular.

e) Understand the relationship between the slopes of parallel lines.

See a above

f) Determine the equation of a line parallel to a given line and passing through a point not on the line.

Find an equation of a line, in slope-intercept form, through point that is parallel to

g) Determine the equation of a line perpendicular to a given line and passing through a point not on the line.

Find an equation of a line, in slope-intercept form, through point that is perpendicular to

*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.

Computational and Procedural Skills

1. Solve systems of two equations in two variables using substitution:
Solve the system: by the substitution method.


2. Solve systems of two equations in two variables using elimination:
Solve the system: by the elimination 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 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?

Suppose a system of two linear equations in two variables has infinite solutions in the form (x,y). 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: by graphical method.


Solve the system: 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)}

b. {(3,7), (1,4), (0,-2), (-1,-1), (-2,5)}

2. Given the following figures, give the domain and range of the relation; determine whether the relation is or is not a function:
a. b.

3. Define relation and function. Compare the two definitions. How are they alike? How are they different?

*17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
Computational and Procedural Skills
1. Complete each ordered pair so that it is a solution to 3x + y = 10. Then identify the domain and range of the resulting set of ordered pairs.



a. (1, ?)

b. (2, ?)

c. (?, 4)

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) }


2. Express the relation in each mapping, table, or graph as a set of ordered pairs and then state the domain and range of each:




a.

							7 6 5 4 3

		1 2 3 4 5

X	y
3	4
5	6
7	8
7	6
5	4
3	3

1. 
   

AppleMark

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)	1	2	3	4	5	6
Cost ($)	20	24	28	32	36	40

2. The table below shows the per-minute rate for a cell phone. Identify the domain and range. Write a set of ordered pairs for the function. Assuming the cost is a linear function of the number of minutes, find a formula for the function.

Minutes	1	2	3	4	5	6
Cost ($)	2.00	2..25	2.50	2.75	3.00	3.25

3. The table below shows the distance that a car travels over time. Identify the domain and range. Write a set of ordered pairs for the function. Use the table to write an equation for this function (assume the function is linear).

Time (hrs)	1	2	3	4	5	6
Distance (miles)	50	100	150	200	250	300

*18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

Computational and Procedural Skills
1. Determine whether or not the below relations are functions. Justify your answer.

a.

b.



c.

d.

e.

f.

g.



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.

3. Draw a graph that is both a relation and a function. Write a sentence that states why the graph you drew represents both a relation and a function.

Geometry:
*7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

Algebra II:
2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
Computational and Procedural Skills
1. Solve the system by graphing.



a.

b.



2. Solve the system by substitution:


a.

b.





i.Solve the system by matrices (Elimination):


1. 2. 3. 4.
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.

c. Does every system of linear equations have a solution? Explain.

d. After a solution of a system of linear equations is found, why should the solution be checked algebraically?

e. If the solution exists, what is the solution of a system of linear inequalities?

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):

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.
a. Write a system of two equations using two variables.
b. Solve and put answer in a complete sentence.

2. The senior class has a carnival to raise money for a senior trip. Student tickets are $6 and adult tickets are $11. Since 324 people were in attendance, the senior class raised $2,359. How many of the people in attendance were adults?


24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
Computational and Procedural Skills
1. Let and ; find :

2. What is the inverse of ?

Conceptual Understanding

1. Show that and are inverse functions. Graphically what is the relationship? What does the original function have to be ( 1 : 1 ) in order to have the inverse also be a function?
Problem Solving/Application
1. A department store is having a 20%-off-everything sale. You also have a $10 coupon for any purchase.
a. Write the function M that represents the sale price of an item after the 20% discount, and a function K that represents the price of an item after the $10 coupon:

Determine which is the best deal for you, discount then coupon, or coupon then discount, when buying an item costing $25.

25.0 Students use properties from number systems to justify steps in combining and simplifying functions.

Computational and Procedural Skills
1. What property of real numbers enables you to simplify: (Note that the suggested identity does not hold if x= -1)
to ?
(This function has an error in the numerator. The “=” should be a “+”.
It would be wise to explain what the simplification provides for the original f(x). The two functions do not have the same domain. Would it be wise to use g(x) for the linear equation or R(x) for the original f(x)? (Brenda Kracht)
2. What properties of real numbers enable you to simplify: (Note that the suggested identity does not hold if x=3)
to ?

Conceptual Understanding
1. Give examples of how the properties of equality are used to solve equations.
Ex: ;
by the multiplicative property of equality This is the distributive property
and by the subtraction property of equality Change to addition prop.
and by the division property of equality. Change to multiplication prop.
2. Explain when does not exist is not defined and why.
Problem Solving/Application
Due to the nature of this standard, there are no application problems. Justifying something is an end in itself, allowing use of the justified item in the standard repertoire.

Trigonometry
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).
5.2 Students can take a geometric description of a conic section—for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6—and derive a quadratic equation representing it.

Computational and Procedural Skills
1. Graph each of the following conic sections and find the desired information.

A.
Foci: _	Asymptotes: _
B.
Center:_	Eccentricity: _

2. Find the focus and directrix of the parabola . Graph the parabola. Does it open up, down, left or right?

Focus _ Directrix _ Opens

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.

Type of conic:

2. Determine the type of curve represented by the equation. Find the foci and vertices (if any), and sketch the graph.


A.

B.
C.

D.
E.



3. Find an equation for the conic section with the given properties.
A. The parabola with focus and directrix .
B. The ellipse with center , foci and , and major axis of length 10.
C. The hyperbola with vertices and asymptotes .
D. The hyperbola with center , foci and , and vertices and .
4. Find an equation of a conic section with the given properties.
A. The ellipse with foci and with one vertex on the x-axis.

B. The ellipse with vertices and , and passing through the point .

5. Find an equation of a conic section with the given properties.
A. The ellipse with foci and with one vertex on the x-axis.

B. The ellipse with vertices and , and passing through the point .

7.0 Students demonstrate an understanding of functions and equations defined parametrically and can graph them.
Computational and Procedural Skills
Graph the following parametrically defined equations:


A. ,
A better choice would be using squares! ....result would be linear.
B. ,

C. , ,

D. , ,



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?

[image]

A. Quadrant I

B. Quadrant II

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.
Computational and Procedural Skills

Determine if the following systems are consistent or inconsistent. Identify consistent systems as dependent or independent.



1.

2.

3.



Conceptual 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.
a) b) c)
2. If you solve a system of linear equations without using matrices, how do you know if the system is consistent or inconsistent? If consistent, how do you know if the system is dependent or independent.
8.0 Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.


Calculus: __

4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:
4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.

4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
From the National Common Core Mathematics Standards for high school, the following standards reflect the linear understanding expected beginning with the understanding that “linear functions with a constant term of zero, describe proportional relationships.
In addition the following are from the Linear, quadratic and exponential models standards:
· “Understand that a linear function defined by f(x) = mx + b for some constants m and b, models a situation in which a quantity changes at a constant rate, m relative to another.” (1)1
· “Understand that linear functions grow by equal differences over equal intervals…” (4)
· “Arithmetic sequences can be seen as linear functions.” (5)
· “Construct a function to describe a linear relationship between two quantities. Determine the rate of change and constant term f a linear function from a graph, a description of a relationship, or from two (x, y) values (including reading from a table).” (10)
· “Interpret the rate of change and constant term of a linear function or sequence in terms off the situation it models, and in terms of its graph or table of values.” (13)
The following are from the Statistics and Probability standards:
· “Functional models may be used to approximate data; if the data approximately linear, the relationship may be modeled with a regression line.” (Introduction)
The following are from Geometric Properties with Equations standards:
· “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 number in () identifies the corresponding standard in the CCMS document.