Lesson Plan by Mark A. Hewitt Lesson: Point­Slope Form Length: 52 Minutes Age or Grade Intended: 9 th grade (Algebra I) Standard: Standard 4 — Graphing Linear Equations and Inequalities Students graph linear equations and inequalities in two variables. They write equations of lines and find and use the slope and y­intercept of lines. They use linear equations to model real data. A1.4.1 A1.4.2 A1.4.3 Graph a linear equation. Example: Graph the equation 3x – y = 2. Find the slope, x­intercept, and y­intercept of a line given its graph, its equation, or two points on the line. Example: Find the slope and y­intercept of the line 4x + 6y = 12. Write the equation of a line in slope­intercept form. Understand how the slope and y­intercept of the graph are related to the equation. Example: Write the equation of the line 4x + 6y = 12 in slope­intercept form. What is the slope of this line? Explain your answer. We have now worked with linear graphs, slopes, y­intercepts, x­intercepts and the slope intercept and standard forms of linear equations. In this lesson, we will look at the point­slope form of linear equations and graphing lines in the point­slope form. In addition, the lesson will review and incorporate the concepts from the first 4 lessons. NOTE: Since these lessons follow the progression of the text rather than the sequence of the standards, the information contained in the standards, while all addressed, are presented in a different order from the standards. Performance Objectives: 1. All students in the class, given a linear equation in point­slope form will graph the equation with 75% accuracy, by correctly graphing 3 out of 4 assigned problems. 2. All students in the class, given the coordinates of a point and the slope of a line through that point, will write an equation in point­slope form for the line through the point, with the given slope, with 75% accuracy, by correctly providing the answer for 3 out of 4 assigned problems. 3. All students in the class, given the coordinates for two points will write an equation for the line through the point in standard form with 75%
Lesson Plan 5 accuracy, by correctly providing the answer to 3 out of 4 assigned problems. (This is a review of the standard form.) 4. All students in the class, given the coordinates for two points will write an equation for the line through the point in point­slope form with 75% accuracy, by correctly providing the answer to 3 out of 4 assigned problems. (These are the same points and the same problems assigned in number 3, above, which asks the students to write each answer in both standard form and point­slope form.) Advanced Preparation by Teacher: The teacher should have the following: 1. An overhead graph with several equations written below the graph in point­slope form (from the text). Below the equations in point­slope form, the overhead will have the equations in slope­intercept and standard forms; these should be covered until needed. 2. Extra copies of the handout passed out in the prior lesson, in case students do not have them available; and, 3. A handout for any gifted and talented students in the classroom that asks the student to convert an equation from slope­intercept form to point­slope and standard form, using variables only. In addition, the students have been assigned to read pages 304 through 307 of their textbook. (ALGEBRA I (2004) Pearson Prentice Hall, Bellman, Bragg et. al.) Procedure: Introduction/Motivation: To begin the class, all students will be asked to close their eyes, and imagine that they are standing on a hill that travels up and below them for as far as they can see, with its slope never changing. When they have all been silent for a moment, ask them now to imagine that they are a “point” on the slope. With their eyes still closed, ask them if anyone can state how this demonstration relates to a line. (Gardner’s spatial and logical­mathematical intelligences; Bloom’s evaluation – this is not a specific accommodation for the gifted and talented students, however, it is expected that this will only be answered without further prompting by a gifted and talented student, or one with exceptional spatial abilities who has a strong understanding of linear equations.) Next, ask the class if they are the point, and the hill represents the slope, how many lines go up and down the hill through them. (Gardner’s spatial and logical­mathematical intelligences; Bloom’s application) Have the students open their eyes, and explain for those who may not have understood, that if you know a point, which they represented in the demonstration, and you know the slope that passes through that point, which the hill stood for, there is exactly one line that is defined by that point and slope.
Lesson Plan 5 Today we will learn how to write the equation for a line, and graph it, when we know one point on the line and the slope of the line. Step­by­Step Plan: The students will first participate in the discussion set forth above. Next have the students ask any questions that they have regarding the previous days homework. At the conclusion of answering the questions, collect the homework. Reference the graph of the equations in point­slope form. Explain to the students how an equation appears in point­slope form, and which elements of the equations represent the x­coordinate, the y­coordinate and the slope. Move to the board and write several points with slopes on the board and ask the students to assist in writing the equations in point­slope form. (Bloom’s comprehension) Ask the students to copy down the equations that have been written on the board. Ask several students to come forward and convert the equations on the board to slope­intercept form, and have the remainder of the class do so in their notebooks. (Bloom’s Synthesis). Have several other students come forward to the board and convert the equations to standard form. (Bloom’s Synthesis). Return to the overhead, and reveal the equations in point­slope and standard form, and work through with the students who the conversions were made. The next step is again a directed discussion – a portion of this is a review from prior lessons. Lead the students through the following questions. This will call for a variety of Bloom’s taxonomical levels, so the questions may vary as the students provide their responses.*
1. 2. 3. What is the y­intercept? Knowledge What is the x­intercept? Knowledge How is a slope determined (in terms of shift up and to the right)? Comprehension 4. What information do we need to have to write the equation of a line in point­slope form? Knowledge 5. How would we write the equation? Comprehension 6. So what general statement can we now make when we know the a point on a line and its slope? Synthesis Answer: with this information, we can graph any linear equation. 7. How do we convert an equation in point­slope form to slope­intercept form? Application 8. How do we convert an equation in point­slope form to standard form? Application
*
The questions should be in the form “Who can tell me . . .”, “Jill, can you explain . . .”, “Does anyone see the relationship between . . .” or other appropriate forms of questioning that involves as many of the students as possible in the discussion.
Lesson Plan 5 Ask the entire class to respond to the next questions: 9. How many different ways have we learned to write an equation for a line? Knowledge 10. What are they? Knowledge (agree as a class on the 1 st , 2 nd and 3 rd methods – do not just have all students shout out three answers without relating them and confirming them.) Closure:
At the close of the presentation, give the students an opportunity to ask any additional questions. Remind the students that there will be a quiz on this material the next class day. Ask them to think about the various representations of the concepts that we have used, such as the interception for the intercept and the hill for a slope. Tell them that if they encounter any problems with the quiz, that they should just think about the questions in the ways that we have all used to envision the concepts. Assign problems 2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, and 28 on page 307 of the text as class work/homework. Assist the students as needed in solving these problems during the remainder of the class period. Adaptation/Enrichment (For Gifted and Talented student(s)): Those students who have been determined to be gifted and talented will be given the opportunity to complete an additional homework assignment. The incentive for completing the assignment will be the opportunity to disregard one quiz score (which may only be used once per semester), waiving a homework assignment, or whatever school wide incentives may be available, such as school dollars for rewards, extra computer time, etc. The assignment will be to create conversions from slope­intercept form (y = mx + b) into point­slope form and standard form. Give the hint that slope may be written in the form m = r1/r2 for the rise over the run. (Bloom’s synthesis) Although other students in the class may be capable of completing this task, the assignment will only be given (discretely as the remainder of the class is working on the homework/class work assignment) to those students specifically designated as gifted and talented. This will reinforce to the gifted and talented students that it is important for them to have a complete understanding of a topic, and that there are additional incentives involved in using their talents in expanding upon what the lessons have presented. Self­Reflection: N/A
Lesson Plan 5