Overview | Content Knowledge | Essential Questions | Connection To Standards | Initiating Activity | Learning Experiences | Culminating Performance | Pre-Requisite Skills | Modifications | Schedule/Time Plan | Technology Use
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LU Title: Graphing Linear Equations |
Author(s): Franco A. DiPasqua |
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Grade Level: 9 |
School : Madison Oneida BOCES |
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Topic/Subject Area: Course 1 or Math A |
Address: 4937 Spring Road Verona, NY |
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Email: fadipas@aol.com |
Phone/Fax: 315-361-5715/ 315-361-5880 |
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This lesson clearly fits into the Course 1 /Math A requirement on teaching "Graphing Linear Equations". It consists of seven lessons that correlate with MST Standards 1(Analysis, Inquiry and Design), 3(Mathematics) and 5(Technology). Prior to this experience the student will need an understanding of equations and basic graphing techniques (drawing axes and plotting coordinates). It is important to note that the students for this lesson are in a vocational school setting and that interaction happens between their instructors and myself. So as I go through this unit's concepts, many of these vocational teachers are reinforcing the concepts through real-world applications. Also, as I teach my students this lesson, I am onsite looking at their class work. It is this constant checking and reinforcing that makes them proficient at graphing. |
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Declarative |
Procedural |
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Identify the "Standard Form" for graphing an equation.(i.e. y = mx + b) |
Given an equation, write it in "Standard Form" and find what the slope and y-intercept are. |
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Identify what the slope and y-intercept are from "Standard Form". |
Determine what a line should look like when given a specific slope and y-intercept. |
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Identify where to place the y-intercept and slope on the graph. |
Determine what happens to a line when a specific slope is changed to its negative reciprocal. |
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Identify what happens when the y-intercept and slope are changed. |
Determine what happens to a line when a specific y-intercept is changed and the slope remains the same. |
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CONNECTIONS TO NYS LEARNING STANDARDS
List Standard # and Key
Idea #: Write out related Performance Indicator(s) or Benchmark(s)
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MST Standard 1:Analysis, Inquiry and Design - the student will be given a linear equation and will have to manipulate its components to gather enough information to graph a line. MST Standard 3: Mathematics - the student will be exposed to identifying and working with linear equations, fractions, various mathematical operations, variables, x and y axes, slopes and y-intercepts. MST Standard 5: Technology - the student will be using a graphing calculator to study what happens when slopes and y-intercepts are manipulated. |
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The student will be given some data. That is, the teacher will role a toy car across the floor and state, "A car traveling at a constant rate of speed can travel 60 miles in 60 minutes, how far will it go after two hours?" This will be the opening discussion for showing the use of linear equations. |
LEARNING
EXPERIENCES
In
chronological order including acquisition experiences and extending/refining
experiences for all stated declarative and procedural knowledge.
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Title: Day 1- "What is Standard Form?" Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on teaching "Graphing Linear Equations". Prior to this experience the students will need an understanding of equations. Procedure: The students will have completed several lessons on basic equations with one unknown. Today, we will work on equations with two unknowns and writing them in the form y=mx+b. The primary purpose of this lesson is to show students how to get an equation into the form y = mx+b "Standard Form". For example: "Given 2x + 10 +y = 20, write the equation in Standard Form. So let's subtract two form both sides: 10 + y = 20 - 2x Now, subtract 10 from both sides: y = 10 - 2x. Now, make the "right" side have the "mx" come first: y = -2x +10. Here we have -2 is "m" and +10 is "b"." This is the type of dialogue I hope to open up with this lesson. It should help the students see how "moving" the equation around, places the equation into Standard Form - (why we need standard form will be explained to them later in the unit). There is an enclosed activity sheet the students will complete while working on this lesson, in order to grasp the procedure. (The sheet will be guided practice with the instructor.) Students must complete the homework sheet for tomorrow and will randomly display their work on the board at that time. Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time is already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Whiteboard, Markers Assessment Plan: Students will be assessed throughout the experience by their responses to what happens when the equations are manipulated and by their completion of the enclosed worksheet. Completion of the worksheet during class will show some competence on this topic. To assess the work of the students, I will be looking at their sheet and telling the students how they have met the defined standards. Also, they will have a handout of what I look for when grading these problems - there will be no formal grade for the worksheet done in class. Reflection: From doing this lesson, I know that this is a great way to teach students how to write an equation with two unknowns in Standard Form. This is the essential piece in being successful when graphing. Many students do not understand graphing because they never fully understood "Standard Form". _____________________________________________________________________ Title: Day 2- "What is Standard Form?"…continued Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on teaching "Graphing Linear Equations". Prior to this experience the students will have completed the lesson on writing an equation with two unknowns into the y= mx + b form. Procedure: The students will have completed a basic lesson on writing an equation with two unknowns into the y = mx + b form and they will have done a homework assignment. Today they we will go over the assignment in great detail. The students will be called upon randomly to put a problem on the board and explain it to the class. They can get their classmates to help out in the explanation. The teacher should facilitate this process. It is important that the teacher let the students develop discussion on these problems and to assist when difficult questions arise. If adequate time is spent on this development, then the rest of this unit should be "smooth" for the students. At the end of their presentation, they will be given a worksheet in which they will complete in-groups of two; this will be due tomorrow. The primary purpose of this lesson is to show students how to get an equation into the form y = mx+b "Standard Form" and to UNDERSTAND what they did. Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time is already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Whiteboard, Markers, Red Pens Assessment Plan: The teacher will go around in the beginning of class and make sure that the students have completed the homework assignment. At that point, some of the students will display their work on the board. The teacher will then have the students mark their own papers with "Red Pens", through the use of the point sheet given out at the previous lesson. This should help the students see "parts" they may be missing. Since some problems are longer than others, some problems will be worth more. The total assignment will be worth +19 points. When we are done going over the sheet the students will hand in their sheet with the total number of points they obtained on top. The teacher will then convert this to a grade and hand it back during the next lesson. Reflection: From doing this lesson, I know that this is a great way to teach students how to write an equation with two unknowns in Standard Form. By completing this lesson by empowering the students to grade their own work, it helps them "see" the point values for each problem. So now they will know what I will be looking for. _____________________________________________________________________ Title: Day 3- "What is y = mx + b?" Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on teaching "Graphing Linear Equations Using Slope and the Y-Intercept". Prior to this experience the students will have completed two lessons on writing an equation with two unknowns into the y= mx + b form. Procedure: Today they we will go over the group assignment from DAY 2 in great detail. The students will be called upon randomly to put a problem on the board and explain it to the class. They can get their classmates to help out in the explanation. The teacher should facilitate this process. It is important that the teacher let the students develop discussion on these problems and to assist when difficult questions arise. If adequate time is spent on this development, then the rest of this lesson should be "smooth" for the students. Once the sheet has been reviewed, begin to explain to the students why they have been placing equations into "Standard Form" - This process will help us graph these equations onto graph paper. Linear means "straight", so we will be drawing straight lines onto graph paper.
You find the Y-Intercept by looking at the equation after it has been put into Standard Form. The Y-Intercept is on the end of the equation. Ex. y = 2x +7, the Y-Intercept is +7 Ex. y = 2x - 7, the Y-Intercept is -7
The slope is the pitch of the line that will be graphed. Slope in mathematics is similar to a slope when you ski. The slope is found by using the number that is in front of the x, after the equation has been put into Standard Form. If the problem is y = 2x + 3, the slope is 2/1 If the problem is y = 2/3 x + 4, the slope is 2/3 ***The top number of the slope is how far "y" moves and the bottom number in the slope is how far "x" moves. If the "y" is negative, then we move down. If it is positive, we move up. If the "x" is negative, then we move left. If it is positive, we move up (this will more often be the case. Ex. y = -2/3 x +7, you start at +7 on the y-axis and move down two units and to the right three. You continue this process one more time, until you have three dots. Once the dots are down you connect them and draw a straight line through them. ****Once the line has been drawn, put arrows on each end, and write on the line, the equation you started with. It is important that you follow your previous graphing skills, i.e. Labeling the axes, Using a straight edge to draw your lines and Labeling 5 units on all sides of the graph. *After these 2 basic terms have been described, go back to yesterday's Groupwork and identify the Slope and Y-Intercept for each problem. *Then have the students graph the first problem and explain the point values to them for this part. The other 7 are for homework. Also, the students will have to fill out the question sheet. The primary purpose of this lesson is to show students how to use Standard Form to graph a line. Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time is already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Whiteboard, Markers, Graph Paper, Rulers Assessment Plan: The teacher will go around in the beginning of class and make sure that the students have completed the homework assignment. At that point, some of the students will display their work on the board. The teacher will then have the students mark their own papers with "Red Pens", through the use of the point sheet given out at the previous lesson. This should help the students see "parts" they may be missing. Since some problems are longer than others, some problems will be worth more. The total assignment will be worth +32 points. When we are done going over the sheet the students will hand in their score on a sheet of paper. They will keep their sheets for today's lesson. That is, they will identify what the slope and y-intercept is for each problem and they will graph these problems for homework. Reflection: From doing this lesson, I know that this is a great way to teach students how to graph linear equations using Standard Form. They will have ample opportunity in class to practice standard form, identifying slopes and y-intercepts and graphing the equations. _____________________________________________________________________ Title: Day 4- "What is y = mx + b?"…cont Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on teaching "Graphing Linear Equations Using Slope and the Y-Intercept". Prior to this experience the students will have completed seven graphs last nigh for homework and will have completed a writing assignment with any concerns and/or questions. Procedure: Today they we will go over the homework assignment from DAY 3 in great Detail (most likely this will take all period). The students will be called upon randomly (try to get different students involved than previous lessons) to put a problem on the board and explain it to the class. They can get their classmates to help out in the explanation. The teacher should facilitate this process. It is important that the teacher let the students develop discussion on these problems and to assist when difficult questions arise - the written responses of the students should be utilized at this time to "surface" any problems the students may be having. If adequate time is spent on this development, then the rest of graphing linear equations using "Standard Form" should be "smooth" for the students. Once the sheet has been reviewed, give out the homework assignment due for the next lesson. Also, introduce how the Graphing Calculator will be used in class at the next lesson. The primary purpose of this lesson is to reinforce the concept of how to use Standard Form to graph a line. Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time is already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Whiteboard, Markers, Graph Paper, Rulers Assessment Plan: The teacher will go around in the beginning of class and make sure that the students have completed the homework assignment. At that point, some of the students will display their work on the board. The teacher will then have the students mark their own papers with "Red Pens", through the use of the point sheet given out at the previous lesson. This should help the students see "parts" they may be missing. The total assignment will be worth +35 points. When we are done going over the sheet the students will hand in their work, indicating their total number of points for the assignment. Reflection: From doing this lesson, I know that this is a great way to reinforce how to graph linear equations using Standard Form. They will have ample opportunity in class to practice graphing in class, in front of their peers. This random display of their work will make them feel responsible to get their work done each day - no student wants to look unprepared in front of their classmates. _____________________________________________________________________ Title: Day 5- "Using Graphing Calculators to Enhance Your Graphing Skills." Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on teaching "Graphing Linear Equations". Prior to this experience the students will need an understanding of how to write linear equations into the standard y=mx+b form and graphing them on graph paper using the y-intercept and the slope. Procedure: The students will have completed several lessons on equations with two variables prior to this experience. They will also have been taught how to write an equation into the y=mx+b form and how to graph it on graph paper using the y-intercept and the slope. The primary purpose of this lesson is to show students what happens when the "m" and "b"is manipulated when using y=mx+b by completing the worksheet. For example: "Everybody take the equation y = 2x + 7 and type it into your calculator. (Note: I will have taught them how to use the various fields on the Graphing Calculator.) Now change the 7 to a 5, what happens? Now change the 5 to a -5, what happens? OK, now type y = -2x +7, what happens?" This is the type of dialogue I hope to open up with this lesson after going over the previous lesson's homework assignment. It should help the students see how changing "parts" of the equation effects the graph. This is clearly a faster, more resourceful way to teach this instead of hand graphing each equation separately. Afterall, the objective here is to teach the student what happens when the equation is manipulated and not on how to graph (it will be assumed that that has already been taught). Also note, the students need to be informed that we are at the end of the unit and that there will be an assigned review sheet tonight due at the next lesson and then a test the next day. Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time are already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Classroom Set of Graphing Calcultor (~15), Pencils, Notebooks, Board and Markers Assessment Plan: Students will be assessed throughout the experience by their responses to what happens when the equations are manipulated and by their completion of the enclosed worksheet. Completition of the worksheet during class will show competence on this topic. It is important to note, that the students will be working in groups of two when completing this lesson, but must hand in their own work with "original answers". While assessing the students work, I will be looking and telling the students how they have met the defined standards. Reflection: From doing this lesson, I know that this is a great way to utilize technology when teaching students how to graph equations. It gives them a different viewpoint on the mathematics in a more productive fashion. This is truly using technology to enhance student performance. Please note that these students first had to learn how to graph on paper before utilizing the technology. _____________________________________________________________________ Title: Day 6- "Review for the TEST." Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on assessing "Graphing Linear Equations". Procedure: The students will have completed several lessons on graphing linear equations Today, we will do formal review session. The primary purpose of this lesson is to review student performance with graphing linear equations. When the students enter the room, check to see they have their sheet completed. At that time have the students randomly put their work on the board. Then begin a classroom discussion on graphing. This is a great time to discuss:
Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time are already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Rulers, Graph Paper, Pencils, Review Answer Sheet Assessment Plan: Students will be assessed by the grade point sheet. Reflection: From giving this review session, I know that this is a great way to review the students' understanding of linear equations. It provides both the clear, precise mathematics and the written explanations that are needed on the State Assessment. It is vital that when the review is done that the teacher checks everyone's work carefully and questions everyone's under ____________________________________________________________________
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CULMINATING PERFORMANCE
Include rubric(s)
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Title: Day 7- "The TEST." Author: Franco DiPasqua Learning Context: The Commencement Level "Standards" that are going to be assessed are:
This lesson clearly fits into the Course 1 or Math A requirement on assessing "Graphing Linear Equations". Procedure: The students will have completed several lessons on graphing linear equations Today, we will assess them through a formal evaluation. The primary purpose of this lesson is to evaluate student's performance with graphing linear equations. The assessment should be handed out promptly when class starts. The students will be asked the day before to be sure to have all necessary supplies. They will have the entire class period to complete the assessment. This should give them adequate time to feel comfortable with doing the work. At the end of the assessment make sure that the student has an activity to work on quietly, while the others are still working. (i.e. a "launch to the next unit or a fun math puzzle) Instructional/Environmental Modifications: This lesson will be taught to students who already have IEP's (Individualized Education Plans) So things like calculator use and extended time are already built into the experience. In the event other modifications are needed, the student(s) resource teacher(s) will be notified, as needed. Time Required: This lesson will require about one 40 minute class period. Resources: Rulers, Graph Paper, Pencils, Tests Assessment Plan: Students will be assessed by the grade point sheet. Reflection: From giving this assessment, I know that this is a great way to assess students on their linear equation skills. It provides both the clear, precise mathematics and the written explanations that are need on the State Assessment. _____________________________________________________________________
Name: Date: Class: Mr.DiPasqua
Directions for TEST: Please complete the questions FULLY and COMPLETELY. Make sure you answer ALL parts. Draw each graph on separate side of the graph paper. 1. a. On the same set of axes, graph the following lines: ( 10 PTS) y = (1/2)x - 1 x + y = 8 (Put in "Standard Form" below)
b. At what coordinate do the two lines intersect? (1 PT) ______
c. What shape do the two lines form with the y-axis? (1 PT) _____________ d. What is the area of that shape? (2 PTS) _______________
2. a. On the same set of axes, graph the following lines: ( 12 PTS) y = - x + 2 3y - 2x = - 9 (Put in "Standard Form" below)
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SCORING RUBRIC: 1a. 4 pts for the standard form 2 pts for labeling the axes 2 pts for labeling the equation on the line 2 pts for correct y-intercepts 2 pts for correct slopes 1b. 1 pt for correct coordinate 1c. 1 pt for TRIANGLE as the answer 1d. 1 pt for correct formula 1 pt for correct answer (84 units squared) 2a. 4 pts for the standard form 2 pts for labeling the axes 2 pts for labeling the equation on the line 2 pts for correct y-intercepts 2 pts for correct slopes 2b. 2 pts for the correct answer of YES TOTAL POINTS IS 30. **** (ADD UP THE TOTAL EARNED POINTS, DIVIDE BY THE 30 AND MULTIPLY BY 100 TO ARRIVE AT A GRADE.) |
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The student has to be able to identify and manipulate basic equations. The student will also need the basic rules and techniques of drawing a set of coordinate axis and plotting coordinates. |
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*The students who have an IEP (Individualized Educational Plan) should be accommodated to the fullest. Calculator use and extended time will be built into his or her unit if needed. In the event other modifications are needed, the resource teacher will be contacted for additional assistance. |
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Day 1 & 2: "What is Standard From?" - 40 minutes Day 3 & 4: "What is y=mx+b?" - 40 minutes Day 5: "Using a Graphing Calculator to Enhance Your Graphing Skills." - 40 minutes Day 6: "Review of this Graphing Stuff." - 40 minutes Day 7: "The TEST." - 40 minutes |
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A graphing calculator will be used to show the student what happens when the slope and y-intercept are manipulated. This use of technology will be beneficial to show the student many more problems than hand graphing them on paper. (Be aware that this is to only be used after the student has demonstrated sufficient knowledge regarding graphing.) |