Content Standard(s):
Mathematics 13 ) Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. [8-F3]
Example: The function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4), and (3,9), which are not on a straight line.
NAEP Framework
NAEP Statement::
NAEP Statement::
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 11. Find the position of pairs of integers and other rational numbers on the coordinate plane.
a. Identify quadrant locations of ordered pairs on the coordinate plane based on the signs of the x and y coordinates.
b. Identify (a,b ) and (a,-b ) as reflections across the x -axis.
c. Identify (a,b ) and (-a,b ) as reflections across the y -axis.
d. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane, including finding distances between points with the same first or second coordinate.
Unpacked Content
Evidence Of Student Attainment:
Students:
Create and interpret coordinate axes with positive and negative coordinates.
Given ordered pairs made up of rational numbers, locate and explain the placement of the ordered pair on a coordinate plane.
Given two ordered pairs that differ only by signs, locate the points on a coordinate plane and explain the relationship of the locations of the points as reflections across one or both axes.
Given real-world and mathematical problems where a coordinate graph will aid in the solution and given a graph of a real-world or mathematical situation, interpret the coordinate values of the points in the context of the situation including finding vertical and horizontal distances. Teacher Vocabulary:
Coordinate plane
Quadrants
Coordinate values
ordered pairs
x axis
y axis
Reflection Knowledge:
Students know:
Strategies for creating coordinate graphs.
Strategies for finding vertical and horizontal distance on coordinate graphs. Skills:
Students are able to:
Graph points corresponding to ordered pairs,
Represent real-world and mathematical problems on a coordinate plane.
Interpret coordinate values of points in the context of real-world/mathematical situations.
Determine lengths of line segments on a coordinate plane when the line segment joins points with the same first coordinate (vertical distance) or the same second coordinate (horizontal distance). Understanding:
Students understand that:
A graph can be used to illustrate mathematical situations and relationships. These representations help in conceptualizing ideas and in solving problems,
Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (number line). Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.6.11.1: Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
M.6.11.2: Demonstrate an understanding of an extended coordinate plane.
M.6.11.3: Draw a four-quadrant coordinate plane.
M.6.11.4: Draw and extend vertical and horizontal number lines.
M.6.11.5: Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
M.6.11.6: Recall how to graph points in all four quadrants of the coordinate plane.
M.6.11.7: Define ordered pairs.
M.6.11.8: Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
M.6.11.9: Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
M.6.11.10: Identify which signs indicate the location of a point in a coordinate plane.
M.6.11.11: Recall how to plot ordered pairs on a coordinate plane.
M.6.11.12: Define reflections.
M.6.11.13: Calculate the distances between points having the same first or second coordinate using absolute value.
Prior Knowledge Skills:
Model writing ordered pairs.
Identify the x- and y- values in ordered pairs.
Label the vertical axis (y).
Label the horizontal axis (x).
Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
Locate positive numbers on a vertical number line. Locate positive numbers on a horizontal number line.
Locate negative numbers on a horizontal number line.
Label x- and y-axis and zero on a coordinate.
Illustrate vertical and horizontal number lines.
Specify locations on the coordinate system.
Define x-axis, y-axis, and zero on a coordinate.
Define ordered pair of numbers.
Locate positive numbers on a horizontal number line.
Locate negative numbers on a horizontal number line.
Define symmetry.
Identify lines of symmetry on one-dimensional figures.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 6. Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y -intercept.
a. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.
b. Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.
c. Graph linear relationships, interpreting the slope as the rate of change of the graph and the y -intercept as the initial value.
d. Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts. [Grade 8, 9]
Unpacked Content
Evidence Of Student Attainment:
Students:
can analyze linear equations in the form
y=mx + b as representing a line
where m represents the rate of change, called the slope of the line when graphed. and b is the
initial value, called the y-intercept when graphed.
Can create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
Can explain why any two points on a given line will have the same slope.
Can graph linear relationships on a coordinate plane when given in multiple contexts. Teacher Vocabulary:
Slope
Rate of change
Initial Value
Y-intercept Knowledge:
Students know:
how to graph points on a coordinate plane.
Where to graph the initial value/y-intercept.
Understand how/why triangles are similar.
how to interpret y=mx equations. Skills:
Students are able to:
create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
point out similar triangles formed between pairs of points and know that they have same slope between any pairs of those points.
Show that lines may share the same slope but can have different y-intercepts.
Interpret a rate of change as the slope and the initial value as the y-intercept. Understanding:
Students understand that:
slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
When given an equation in the form y = mx + b it generally symbolizes that you will have lines with varying y-intercepts. even when the slope is the same.
you can use the visual of right triangles created between points on a line to explain why the slope is a constant rate of change. Diverse Learning Needs:
Mathematics 9. Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y -intercept.
a. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.
b. Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.
c. Graph linear relationships, interpreting the slope as the rate of change of the graph and the y -intercept as the initial value.
d. Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y -intercepts.
Unpacked Content
Evidence Of Student Attainment:
Students:
Analyze linear equations in the form y=mx + b as representing a line where m represents the rate of change, called the slope of the line when graphed. and b is the initial value, called the y-intercept when graphed.
Create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
Explain why any two points on a given line will have the same slope.
Graph linear relationships on a coordinate plane when given in multiple contexts. Teacher Vocabulary:
Slope
Rate of change
Initial Value
Y-intercept Knowledge:
Students know:
how to graph points on a coordinate plane.
Where to graph the initial value/y-intercept.
Understand how/why triangles are similar.
how to interpret y=mx equations. Skills:
Students are able to:
create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
point out similar triangles formed between pairs of points and know that they have the same slope between any pairs of those points.
Show that lines may share the same slope but can have different y-intercepts.
Interpret a rate of change as the slope and the initial value as the y-intercept. Understanding:
Students understand that:
Slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
When given an equation in the form y = mx + b it generally symbolizes that there will be lines with varying y-intercepts. even when the slope is the same.
Use of the visual of right triangles created between points on a line to explain why the slope is a constant rate of change. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.8.9.1: Define linear functions, nonlinear functions, slope, and y-intercept.
M.8.9.2: Recall how to solve problems using the distributive property.
M.8.9.3: Recognize linear equations.
M.8.9.4: Identify ordered pairs.
M.8.9.5: Recognize ordered pairs.
M.8.9.6: Define similar triangles, intercept, slope, vertical, horizontal, and origin.
M.8.9.7: Recognize similar triangles.
M.8.9.8: Generate the slope of a line using given ordered pairs.
M.8.9.9: Analyze the graph to determine the rate of change.
M.8.9.10: Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
M.8.9.11: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
M.8.9.12: Graph a function given the slope-intercept form of an equation.
M.8.9.13: Recognize that two sets of points with the same slope may have different y-intercepts.
M.8.9.14: Graph a linear equation given the slope-intercept form of an equation.
Prior Knowledge Skills:
Define ordered pairs.
Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Identify which signs indicate the location of a point in a coordinate plane. Recall how to plot ordered pairs on a coordinate plane.
Mathematics 12. Solve systems of two linear equations in two variables by graphing and substitution.
a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.
b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.
Unpacked Content
Evidence Of Student Attainment:
Students:
Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations. Teacher Vocabulary:
System of linear equations
Point of intersection
One solution
No solution
Infinitely many solutions
Parallel lines
Slope-intercept form of a linear equation
Standard form of a linear equation Knowledge:
Students know:
The properties of operations and equality and their appropriate application.
Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
Substitution techniques for algebraically finding the solution to a system of linear equations. Skills:
Students are able to:
generate a table from an equation.
Graph linear equations.
Identify the ordered pair for the point of intersection.
Explain the meaning of the point of intersection (or lack of intersection point) in context.
Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form. Understanding:
Students understand that:
any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.8.12.1: Define variables.
M.8.12.2: Recall how to estimate.
M.8.12.3: Recall how to solve linear equations.
M.8.12.4: Demonstrate how to graph solutions to linear equations.
M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
M.8.12.7: Define simultaneous.
M.8.12.8: Recall how to solve linear equations.
M.8.12.9: Recall properties of operations for addition and multiplication.
M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
M.8.12.11: Define point of intersection.
M.8.12.12: Recall how to solve linear equations.
M.8.12.13: Demonstrate how to graph on the Cartesian plane.
M.8.12.14: Identify ordered pairs.
M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
M.8.12.16: Create a word problem from given information.
M.8.12.17: Recall how to solve linear equations.
M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.
Prior Knowledge Skills:
Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
Demonstrate an understanding of an extended coordinate plane.
Draw a four-quadrant coordinate plane.
Draw and extend vertical and horizontal number lines.
Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
Recall how to graph points in all four quadrants of the coordinate plane.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 27. Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given real-world and mathematical problems that can be represented on a coordinate plane, apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths (distances between points) in right triangles within the problem contexts. Teacher Vocabulary:
Knowledge:
Students know:
The Pythagorean Theorem.
The operations and labeling within a coordinate system. Skills:
Students are able to:
Solve equations involving one variable and square root.
Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings on coordinate planes, equations).
Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse). Understanding:
Students understand that:
The properties of right triangles can be used to solve problems.
Theorems represent general relationships that are true for all shapes that fit certain criteria. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.8.27.1: Recall how to name points on a Cartesian plane using ordered pairs.
M.8.27.2: Recognize ordered pairs (x, y).
M.8.27.3: Solve problems using the Pythagorean Theorem, with or without a calculator.
M.8.27.4: Identify right triangles.
M.8.27.5: Demonstrate how to find square roots, with or without a calculator.
M.8.27.6: Solve problems with exponents, with or without a calculator.
Prior Knowledge Skills:
Define area, special quadrilaterals, right triangles, and polygons.
Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
Apply area formulas to solve real-world mathematical problems.
Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
Explain how to find the area for rectangles.
Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
Recognize and demonstrate that two right triangles make a rectangle.
Alabama Alternate Achievement Standards
AAS Standard:
