List of Unit Benchmarks
· MA.912.G.1.4 – Use coordinate geometry to find slopes, parallel lines, perpendicular lines and the equations of lines.
· MA.912.A.2.1 Create a graph to represent a real-world situation.
· MA.912.A.2.2 Interpret a graph representing a real-world situation.
· MA.912. A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
· MA.912.A.3.7 Rewrite equations of a line into standard form and slope-intercept form.
· MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x- and y- intercepts, two points, the slope and a point, the equation of the line in slope intercept form, standard form, or point-slope form.
· MA.912.A.3.9 Determine the slope, x-intercept, y-intercept of a line given its graph, its equation, or two points on the line.
· MA.912.A.3.10 Write an equation of a line given any of the following information: two points on a line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
· MA.912.A.3.11 Write an equation of a line that models a data set, and use the equation or graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Unpacked Benchmarks
Benchmark: MA.912.G.1.4 – Use coordinate geometry to find slopes, parallel lines, perpendicular lines and the equations of lines.
Rewritten: Use graphs of lines to find information about the line. We will find the slope, pairs of lines that never cross each other, pairs of lines that cross at a 90° angle. We will be able to represent the lines in an algebraic form.
T-chart:
Knowledge Skills
What a 90° angle is. How to add and divide.
What qualifies as a line. How to calculate slope
Slope is a constant ratio.
How to write an equation.
How to interpret a graph.
Meaning of task: Students will look at a given line on a graph and find a line that is perpendicular to it. Students will find slopes of lines and write equations of lines from looking at graphs. This will help students make the connection between a visual/pictorial representation and an algebraic form.
Essential Vocabulary:
Perpendicular, slope, intercepts, intersection, right angle, equation of a line
(We are not including parallel lines in our lesson.)
List of questions (of varying cognitive levels):
-What information can you list about the given graph?
-What is the slope of the given line?
-What information do you need to write the equation of a line?
-What is true about the equations of perpendicular lines?
-Given this line, how would you write the equation for a line perpendicular to it?
-Are these two lines perpendicular? Why or why not?
-What are the slopes of these two lines? Are they perpendicular to one another?
-Write equations for two lines that are perpendicular.
Essential Question:
Given this graphed line, draw a line that is perpendicular to it and write the equation for each line.
Benchmark: MA.912.A.2.1 Create a graph to represent a real-world situation.
Rewritten: Draw a picture of a line that represents a problem outside of the classroom.
T-chart:
Knowledge Skills
-Understanding that an equation of two -Plotting coordinate pairs on a graph.
variables produces coordinate pairs which
can be graphed. -Producing coordinate pairs from an equation.
-Making an equation from a real-world
situation.
Meaning of task: Students will be able to look at a data set of a real world situation and graph it on a coordinate plane. They will be able to graph a line given an equation of a real world situation and understand what the graph is telling them. This will help the students make meaning to what they are learning in math class to a real world situation.
Essential Vocabulary: graph, represent
List of questions (of varying cognitive levels):
- What information is needed to graph the line?
- Are you given enough information to graph the line?
- What does this graph mean?
Essential Question:
Given a data set of a real world situation, graph the line. What does the line represent?
Benchmark: MA.912.A.2.2 Interpret a graph representing a real-world situation.
Rewritten: Write down any information you can get from looking at the graph of a situation outside of the classroom.
T-chart:
Knowledge Skills
-Understanding what all of the components -Reading a graph.
of a graph represents of a real world
situation. (i.e. slope, intercepts, coordinates) -Writing an equation from a graph.
-Picking coordinate pairs from a graph.
Meaning of task: Students will be able to look at a graph of a real world situation and know what the graph represents. They will be able to know how the intercepts relate to the real world situation as well as the slope. They will also understand that some graphs of real world situations will only be in the first quadrant. This will help students see how the mathematics they are learning in the classroom is applied in the real world as well as the importance of the skills they are learning.
Essential Vocabulary: Intercept, Graph
List of questions (of varying cognitive levels):
- What information is needed to graph a line?
- What does the y-intercept represent on the graph?
- What does the x-intercept represent on the graph?
- What does the slope represent?
- Is the slope positive or negative? What does that mean?
- What is the equation of the line?
- What does the equation represent?
Essential Question:
Given a graph of a real world situation, describe what the intercepts represent and what the slope represents. Write the equation of the line.
Benchmark: MA.912. A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
Rewritten: Represent, using symbols and equations, real-world uses of linear equations.
T-chart:
Knowledge Skills
Manipulating equations. Representing words as symbols and equations.
Solving linear equations and inequalities. Knowledge of when a situation should be
represented by an equation or an inequality.
Meaning of task: To be able to analyze real-world problems, represent them mathematically and to find a solution. Many times students could find themselves doing things which math makes much easier to figure out, for example putting up a rectangular fence around a yard. With math the students will know if their perimeter has straight lines and if the opposite sides are of equal length, parallel, etc.
Essential Vocabulary: Inequality, equation, linear equation
List of questions (of varying cognitive levels):
- What kind of behavior do we notice here?
- Is there some kind of mathematical technique we can use to model this behavior?
- Why do we select this kind of equation or inequality to represent this behavior?
- What if a part of this problem changed?
Essential Question: How can we use the information fromthis situation to create an equation or inequality modeling the situation, and how does this model help us to solve this real-world problem?
Benchmark: MA.912.A.3.7 Rewrite equations of a line into standard form and slope-intercept form.
Rewritten: Change the equation of a line into various common forms, such as slope-intercept (y=mx+b) and standard form (Ax+By=C).
T-chart:
Knowledge Skills
Find slope. Knowledge of which for is the best to represent each line.
Manipulate equations.
Find intercepts. Realize that all forms of a line are
equal and represent the same line.
Meaning of task: Students will manipulate equations of lines into their various forms. This is important because many times information is given to use and we must alter it in some fashion before we can work with it. This requires making a choice about what needs to be changed and then how to change it.
Essential Vocabulary: Slope, intercept, standard form, slope-intercept form, equation of a line.
List of questions (of varying cognitive levels):
- Are these equations of the same line? Why or why not?
- What information can we gather from the equation of this line?
- What form is the equation of this line in now? Why do you say that?
Essential Question: Using the information given to us, which form can we put this line into? What information would we need to be able to put it into the other forms?
Benchmark: MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x- and y- intercepts, two points, the slope and a point, the equation of the line in slope intercept form, standard form, or point-slope form.
Rewritten: Graph a line given any possible information required to graph a line.
T-chart:
Knowledge Skills
What information is needed to graph a line. Reading an equation and plotting points on Knowledge of the below vocabulary. a graph.
Meaning of task: When students are able to translate an equation into a visual, graph, form it creates a connection between the two. The fact that students are able to graph lines when given a variety of information is useful since students will not always find their information in the same form.
Essential Vocabulary: Graph, table, x-intercept, y-intercept, slope, intercept, line, equation, points, slope-intercept form, standard form, point-slope form.
List of questions (of varying cognitive levels):
- Given a table of values, what does the line look like graphed?
- Graph the line given the two intercepts.
- Given two points,
- Given an equation of a line in slope-intercept form, what does the line look liked graphed?
- Given an equation of a line in standard form, what does the line look like graphed?
- Given an equation of a line in point-slope form, what does the line look like graphed?
- How do you know that? Why do you say that?
Essential Question: Graph the following lines when given the equations in different forms.
Benchmark: MA.912.A.3.9 Determine the slope, x-intercept, y-intercept of a line given its graph, its equation, or two points on the line.
Rewritten: Find the steepness of the line and where the line crosses the x and y axes when looking at a graphed line, a line represented in algebraic form, or given two points on the line.
T-chart:
Knowledge Skills
-Knowledge of what slope is. -How to calculate slope.
-Knowledge of what intercepts are. -How to perform basic arithmetic.
-How to read a graph.
Meaning of task: Students will find specific components of a line by looking at the graphed line/ the algebraic form of the line and from two given points. This will make connections for the students about how to find the same pieces of information in different ways.
Essential Vocabulary: Slope, x-intercept, y-intercept, intercept, line, equation, points.
List of questions (of varying cognitive levels):
-What is: slope, x-intercept, y-intercept?
-How do we find: slope, x-intercept, y-intercept?
-Is an intercept a point on a line?
-What are the intercepts of this line? (show graphed line)
-What is the slope of this line? (show graphed line)
-What types of slopes can we have? (pos., neg., zero, undef.)
-From this equation what is the slope and the intercepts? How did you find them?
-Using these two points, what will the x,y-intercepts be for this line?
-Using these two points, what is the slope of this line?
Essential Question: Write an equation of a line, find two points that lie on the line and plot them. With this information, what is the slope of the line and at which points with the line cross the x and y axes?
Benchmark: MA.912. A.3.10 Write an equation of a line given any of the following information: two points on a line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
Rewritten: If they give you either: two points on a line, the slope of a line and one point on it, or a graph of a line, you can write an equation to describe it. Also, be able to find a parallel or perpendicular line to the one they give you that goes through a certain point.
T-chart:
Knowledge Skills
Parallel lines have equal slopes. How to find the negative reciprocal of a number.
The slopes of perpendicular lines are How to find a slope by looking at a graph.
negative reciprocals.
How to find the slope by identifying the “m”
A number multiplied by its reciprocal is one. in an equation.
Meaning of task: Students will be given some information and then from there be able to write a parallel or perpendicular line through a given point to that given line. Students will find negative reciprocals and see that perpendicular lines will form 90 degree angles. This will help the students to make connections between reciprocals, how the function well together and how perpendiculars are useful.
Essential Vocabulary: Perpendicular, slope, parallel, reciprocal
List of questions (of varying cognitive levels):
- What is the slope of this line?
- What is the negative reciprocal of that number?
- What is the reciprocal of a number?
- What is true about the equations of perpendicular lines?
- Given this line, how would you write the equation for a line perpendicular to it?
- Are these two lines perpendicular? Why or why not?
- What are the slopes of these two lines? Are they perpendicular to one another?
- Write equations for two lines that are perpendicular.
Essential Question: Given this line and this point, find a line that is perpendicular to the first line that goes through this point.
Benchmark: MA.912.A.3.11 Write an equation of a line that models a data set, and use the equation or graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Rewritten: Write a line in algebraic form which represents a set of data. Use this algebraic form, or graph to form some conclusions. Explain what the data tells you about the slope. Notice that the slope is the change in y values over the change in x values.
T-chart:
Knowledge Skills
-What an equation is composed of. -Reading data and graphs.
-What slope is. -Drawing conclusions.
-How to find slope.
Meaning of task: Students will interpret data and transfer information in a table to a graph. This shows the same information in two different ways. Students will see that the difference in the data relates to the steepness of the graphed line; thus we can find slope with two given points or from looking at our graphed line.
Essential Vocabulary: Equation, data set, graph, predictions, slope, rate of change
List of questions (of varying cognitive levels):
-What is our date showing/telling us?
-What information do you need to write the equation of a line?
-Using your data, write the equation of the line.
-Graph the line.
-Looking at your equation and your graph, what do you notice? What do you think will happen?
-What is happening to the points on the line?
-If the points are changing by the same amount each time what does that tell us about our slope/line?
-How do we find what our slope is? Are there any other ways? What are they?
-What is our slope?
Essential Question: Looking at the given data and the graph of the data, what is happening? Why do you say that? How are the points changing?
· MA.912.A.2.1 Create a graph to represent a real-world situation.
· MA.912.A.2.2 Interpret a graph representing a real-world situation.
· MA.912. A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
· MA.912.A.3.7 Rewrite equations of a line into standard form and slope-intercept form.
· MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x- and y- intercepts, two points, the slope and a point, the equation of the line in slope intercept form, standard form, or point-slope form.
· MA.912.A.3.9 Determine the slope, x-intercept, y-intercept of a line given its graph, its equation, or two points on the line.
· MA.912.A.3.10 Write an equation of a line given any of the following information: two points on a line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
· MA.912.A.3.11 Write an equation of a line that models a data set, and use the equation or graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Unpacked Benchmarks
Benchmark: MA.912.G.1.4 – Use coordinate geometry to find slopes, parallel lines, perpendicular lines and the equations of lines.
Rewritten: Use graphs of lines to find information about the line. We will find the slope, pairs of lines that never cross each other, pairs of lines that cross at a 90° angle. We will be able to represent the lines in an algebraic form.
T-chart:
Knowledge Skills
What a 90° angle is. How to add and divide.
What qualifies as a line. How to calculate slope
Slope is a constant ratio.
How to write an equation.
How to interpret a graph.
Meaning of task: Students will look at a given line on a graph and find a line that is perpendicular to it. Students will find slopes of lines and write equations of lines from looking at graphs. This will help students make the connection between a visual/pictorial representation and an algebraic form.
Essential Vocabulary:
Perpendicular, slope, intercepts, intersection, right angle, equation of a line
(We are not including parallel lines in our lesson.)
List of questions (of varying cognitive levels):
-What information can you list about the given graph?
-What is the slope of the given line?
-What information do you need to write the equation of a line?
-What is true about the equations of perpendicular lines?
-Given this line, how would you write the equation for a line perpendicular to it?
-Are these two lines perpendicular? Why or why not?
-What are the slopes of these two lines? Are they perpendicular to one another?
-Write equations for two lines that are perpendicular.
Essential Question:
Given this graphed line, draw a line that is perpendicular to it and write the equation for each line.
Benchmark: MA.912.A.2.1 Create a graph to represent a real-world situation.
Rewritten: Draw a picture of a line that represents a problem outside of the classroom.
T-chart:
Knowledge Skills
-Understanding that an equation of two -Plotting coordinate pairs on a graph.
variables produces coordinate pairs which
can be graphed. -Producing coordinate pairs from an equation.
-Making an equation from a real-world
situation.
Meaning of task: Students will be able to look at a data set of a real world situation and graph it on a coordinate plane. They will be able to graph a line given an equation of a real world situation and understand what the graph is telling them. This will help the students make meaning to what they are learning in math class to a real world situation.
Essential Vocabulary: graph, represent
List of questions (of varying cognitive levels):
- What information is needed to graph the line?
- Are you given enough information to graph the line?
- What does this graph mean?
Essential Question:
Given a data set of a real world situation, graph the line. What does the line represent?
Benchmark: MA.912.A.2.2 Interpret a graph representing a real-world situation.
Rewritten: Write down any information you can get from looking at the graph of a situation outside of the classroom.
T-chart:
Knowledge Skills
-Understanding what all of the components -Reading a graph.
of a graph represents of a real world
situation. (i.e. slope, intercepts, coordinates) -Writing an equation from a graph.
-Picking coordinate pairs from a graph.
Meaning of task: Students will be able to look at a graph of a real world situation and know what the graph represents. They will be able to know how the intercepts relate to the real world situation as well as the slope. They will also understand that some graphs of real world situations will only be in the first quadrant. This will help students see how the mathematics they are learning in the classroom is applied in the real world as well as the importance of the skills they are learning.
Essential Vocabulary: Intercept, Graph
List of questions (of varying cognitive levels):
- What information is needed to graph a line?
- What does the y-intercept represent on the graph?
- What does the x-intercept represent on the graph?
- What does the slope represent?
- Is the slope positive or negative? What does that mean?
- What is the equation of the line?
- What does the equation represent?
Essential Question:
Given a graph of a real world situation, describe what the intercepts represent and what the slope represents. Write the equation of the line.
Benchmark: MA.912. A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear equations and inequalities.
Rewritten: Represent, using symbols and equations, real-world uses of linear equations.
T-chart:
Knowledge Skills
Manipulating equations. Representing words as symbols and equations.
Solving linear equations and inequalities. Knowledge of when a situation should be
represented by an equation or an inequality.
Meaning of task: To be able to analyze real-world problems, represent them mathematically and to find a solution. Many times students could find themselves doing things which math makes much easier to figure out, for example putting up a rectangular fence around a yard. With math the students will know if their perimeter has straight lines and if the opposite sides are of equal length, parallel, etc.
Essential Vocabulary: Inequality, equation, linear equation
List of questions (of varying cognitive levels):
- What kind of behavior do we notice here?
- Is there some kind of mathematical technique we can use to model this behavior?
- Why do we select this kind of equation or inequality to represent this behavior?
- What if a part of this problem changed?
Essential Question: How can we use the information fromthis situation to create an equation or inequality modeling the situation, and how does this model help us to solve this real-world problem?
Benchmark: MA.912.A.3.7 Rewrite equations of a line into standard form and slope-intercept form.
Rewritten: Change the equation of a line into various common forms, such as slope-intercept (y=mx+b) and standard form (Ax+By=C).
T-chart:
Knowledge Skills
Find slope. Knowledge of which for is the best to represent each line.
Manipulate equations.
Find intercepts. Realize that all forms of a line are
equal and represent the same line.
Meaning of task: Students will manipulate equations of lines into their various forms. This is important because many times information is given to use and we must alter it in some fashion before we can work with it. This requires making a choice about what needs to be changed and then how to change it.
Essential Vocabulary: Slope, intercept, standard form, slope-intercept form, equation of a line.
List of questions (of varying cognitive levels):
- Are these equations of the same line? Why or why not?
- What information can we gather from the equation of this line?
- What form is the equation of this line in now? Why do you say that?
Essential Question: Using the information given to us, which form can we put this line into? What information would we need to be able to put it into the other forms?
Benchmark: MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x- and y- intercepts, two points, the slope and a point, the equation of the line in slope intercept form, standard form, or point-slope form.
Rewritten: Graph a line given any possible information required to graph a line.
T-chart:
Knowledge Skills
What information is needed to graph a line. Reading an equation and plotting points on Knowledge of the below vocabulary. a graph.
Meaning of task: When students are able to translate an equation into a visual, graph, form it creates a connection between the two. The fact that students are able to graph lines when given a variety of information is useful since students will not always find their information in the same form.
Essential Vocabulary: Graph, table, x-intercept, y-intercept, slope, intercept, line, equation, points, slope-intercept form, standard form, point-slope form.
List of questions (of varying cognitive levels):
- Given a table of values, what does the line look like graphed?
- Graph the line given the two intercepts.
- Given two points,
- Given an equation of a line in slope-intercept form, what does the line look liked graphed?
- Given an equation of a line in standard form, what does the line look like graphed?
- Given an equation of a line in point-slope form, what does the line look like graphed?
- How do you know that? Why do you say that?
Essential Question: Graph the following lines when given the equations in different forms.
Benchmark: MA.912.A.3.9 Determine the slope, x-intercept, y-intercept of a line given its graph, its equation, or two points on the line.
Rewritten: Find the steepness of the line and where the line crosses the x and y axes when looking at a graphed line, a line represented in algebraic form, or given two points on the line.
T-chart:
Knowledge Skills
-Knowledge of what slope is. -How to calculate slope.
-Knowledge of what intercepts are. -How to perform basic arithmetic.
-How to read a graph.
Meaning of task: Students will find specific components of a line by looking at the graphed line/ the algebraic form of the line and from two given points. This will make connections for the students about how to find the same pieces of information in different ways.
Essential Vocabulary: Slope, x-intercept, y-intercept, intercept, line, equation, points.
List of questions (of varying cognitive levels):
-What is: slope, x-intercept, y-intercept?
-How do we find: slope, x-intercept, y-intercept?
-Is an intercept a point on a line?
-What are the intercepts of this line? (show graphed line)
-What is the slope of this line? (show graphed line)
-What types of slopes can we have? (pos., neg., zero, undef.)
-From this equation what is the slope and the intercepts? How did you find them?
-Using these two points, what will the x,y-intercepts be for this line?
-Using these two points, what is the slope of this line?
Essential Question: Write an equation of a line, find two points that lie on the line and plot them. With this information, what is the slope of the line and at which points with the line cross the x and y axes?
Benchmark: MA.912. A.3.10 Write an equation of a line given any of the following information: two points on a line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line.
Rewritten: If they give you either: two points on a line, the slope of a line and one point on it, or a graph of a line, you can write an equation to describe it. Also, be able to find a parallel or perpendicular line to the one they give you that goes through a certain point.
T-chart:
Knowledge Skills
Parallel lines have equal slopes. How to find the negative reciprocal of a number.
The slopes of perpendicular lines are How to find a slope by looking at a graph.
negative reciprocals.
How to find the slope by identifying the “m”
A number multiplied by its reciprocal is one. in an equation.
Meaning of task: Students will be given some information and then from there be able to write a parallel or perpendicular line through a given point to that given line. Students will find negative reciprocals and see that perpendicular lines will form 90 degree angles. This will help the students to make connections between reciprocals, how the function well together and how perpendiculars are useful.
Essential Vocabulary: Perpendicular, slope, parallel, reciprocal
List of questions (of varying cognitive levels):
- What is the slope of this line?
- What is the negative reciprocal of that number?
- What is the reciprocal of a number?
- What is true about the equations of perpendicular lines?
- Given this line, how would you write the equation for a line perpendicular to it?
- Are these two lines perpendicular? Why or why not?
- What are the slopes of these two lines? Are they perpendicular to one another?
- Write equations for two lines that are perpendicular.
Essential Question: Given this line and this point, find a line that is perpendicular to the first line that goes through this point.
Benchmark: MA.912.A.3.11 Write an equation of a line that models a data set, and use the equation or graph to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Rewritten: Write a line in algebraic form which represents a set of data. Use this algebraic form, or graph to form some conclusions. Explain what the data tells you about the slope. Notice that the slope is the change in y values over the change in x values.
T-chart:
Knowledge Skills
-What an equation is composed of. -Reading data and graphs.
-What slope is. -Drawing conclusions.
-How to find slope.
Meaning of task: Students will interpret data and transfer information in a table to a graph. This shows the same information in two different ways. Students will see that the difference in the data relates to the steepness of the graphed line; thus we can find slope with two given points or from looking at our graphed line.
Essential Vocabulary: Equation, data set, graph, predictions, slope, rate of change
List of questions (of varying cognitive levels):
-What is our date showing/telling us?
-What information do you need to write the equation of a line?
-Using your data, write the equation of the line.
-Graph the line.
-Looking at your equation and your graph, what do you notice? What do you think will happen?
-What is happening to the points on the line?
-If the points are changing by the same amount each time what does that tell us about our slope/line?
-How do we find what our slope is? Are there any other ways? What are they?
-What is our slope?
Essential Question: Looking at the given data and the graph of the data, what is happening? Why do you say that? How are the points changing?