Share My Lesson: The Slope of a Line
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The document outlines a detailed lesson plan on the concept of slope for students, including objectives, materials, and teaching strategies. It emphasizes the importance of understanding slope in relation to linear functions and provides various examples and practice problems. Additionally, it highlights resources for teacher preparation and student engagement, aligning with national educational standards.
Share My Lesson: The Slope of a Line
- 1. 21st Century Lessons The Slope of a Line Primary Lesson Designer(s): Corey Cheever 1
- 2. Access 300,000+ free lesson plans like this one on Share My Lesson, developed by the American Federation of Teachers and TES Global: sharemylesson.com. 2
- 3. 3 This project is funded by the American Federation of Teachers.
- 4. 4 *1st Time Users of 21st Century Lesson: Click HERE for a detailed description of our project. 21st Century Lessons – Teacher Preparation • Spend AT LEAST 30 minutes studying the Lesson Overview, Teacher Notes on each slide, and accompanying worksheets. • Set up your projector and test this PowerPoint file to make sure all animations, media, etc. work properly. Please do the following as you prepare to deliver this lesson: • Feel free to customize this file to match the language and routines in your classroom.
- 5. 5 Lesson Objective Students will be able to identify the slope of a line, and graph a line with a given slope. Lesson Description The Do Now will remind students about the order of operations when dealing with negative numbers and fraction bars. Then, the students will see a demonstration of positive, negative, zero, and undefined slope. During the exploration, students will find slope by definition (rise/run), and the practice will turn towards the slope formula. Finally, the homework assignment investigates slope with regards to geometry. Lesson Overview (1 of 4)
- 6. 6 Lesson Vocabulary Slope Lattice Point Vertical Horizontal Materials Graph Paper Pencil Ruler Common Core State Standard CCSS.MATH.CONTENT.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis. Lesson Overview (2 of 4)
- 7. 7 Scaffolding During the practice, all questions are covered on the power point slide. The teacher may choose to have the students work first, then go over the answers, alternate between the students seeing the answers and trying on their own, or working with the students side by side. Enrichment The last question of the practice asks students to give a conjecture about slope and collinear points. This is a great opportunity for students to experience a low threshold, high ceiling question. This concept is also further explored in the homework. Online Resources for Absent Students https://www.engageny.org/resource/grade-8-mathematics- module-5 Lesson Overview (3 of 4)
- 8. 8 Lesson Overview (4 of 4) Before and After Before: Students need to have a strong understanding of adding and subtracting integers, the order of operations, and plotting points on a coordinate grid. After: Students will be able to find the slope of any line, and graph a line with a given slope. This prepares them to deal with functions in the form: y = mx + b. Topic Background Slope is an overlapping concept throughout each grade in the Common Core standards. With regards to functions, it is very important to have a deep understanding of slope before starting to discuss linear functions. Slope can be found everywhere in the real world, and this lessons includes multiple examples of slope in everyday life.
- 9. Warm Up OBJECTIVE: Students will be able to define slope and find the slope of a line given two points on the line or given the graph of the line. LANGUAGE OBJECTIVE: SWBAT define and describe the following words - Slope, Lattice Point, Horizontal, Vertical. Agenda 9 Evaluate each of the following expressions. 1. 𝟕−𝟑 𝟓−𝟑 = 2. 𝟒−(−𝟐) 𝟒−𝟑 = 3. 𝟒−𝟏𝟎 𝟐−(−𝟏) = 4. −𝟐−𝟑 −𝟏−(−𝟒) = 6 -2 2 2 4 1 6 3 6 3 5
- 10. Agenda: 1) Warm Up – Arithmetic Review (Individual) 2) Launch – What is slope? (Class) 3) Explore – Which points to pick? (Partner) 4) Summary – The Slope Formula (Class) 5) Practice – Applying our new knowledge (Small Group) 6) Assessment – Exit Slip (Individual) 10 OBJECTIVE: Students will be able to define slope and find the slope of a line given two points on the line or given the graph of the line. LANGUAGE OBJECTIVE: SWBAT define and describe the following words - Slope, Lattice Point, Horizontal, Vertical.
- 11. Launch – What is slope? Agenda 11 Our friend Luis is riding his bike. He goes up two different hills. Which hill will be harder for Luis to pedal up? HILL #1 HILL #2
- 12. Launch – What is slope? Agenda 12 Later on, Luis is going down two hills. Which hill will he gain more speed on? HILL #3 HILL #4
- 13. Launch – What is slope? Agenda 13 The measure of how hard it is for Luis to pedal up the hill, or how much speed Luis gains down hill is known as slope. Here is the formal definition for slope. Slope of a line – The ratio of vertical distance between two points and horizontal distance between the same two points. Here is an easy way to remember slope: Vertical (rise) Horizontal (run)
- 14. Launch – What is slope? Agenda 14 The harder it is to pedal up hill for Luis, the larger the slope. SLOPE = POSITIVE
- 15. Launch – What is slope? Agenda 15 The more speed Luis gains going downhill, the more negative the slope is SLOPE = Negative
- 16. Launch – What is slope? Agenda 16 When Luis is riding on flat ground, there is no slope, or a slope of zero. SLOPE = 0
- 17. Launch – What is slope? Agenda 17 Luis can’t ride his bike straight up or down, so the slope is undefined. SLOPE is undefined
- 18. Launch – What is slope? Agenda 18 Here are some real world examples of slope. A wheelchair ramp can have a slope of 1/12 at the most. This means for every 1 foot a person goes up, they travel 12 feet across. This is a small positive slope. 12 feet 1 foot
- 19. Launch – What is slope? Agenda 19 The steepest stairs in the world can be found in Huashan mountain, in China. The slope here would be a large, positive number.
- 20. Launch – What is slope? Agenda 20 The “steeps” of San Fransico include some of the biggest hills in a U.S. city. The slope shown in the picture to the left would be negative.
- 21. Launch – What is slope? Agenda 21 Let’s find the exact slope of a line on a coordinate axis. Step 1 – Pick two points on the line that that have integer coordinates. These are called lattice points. Step 2 – Draw a vertical line from the LEFT dot to the same height as the RIGHT dot. Count the distance. This is the RISE. 2 1 Step 3 – Draw a straight across to the right dot. Count the distance. This is the RUN. Step 4 – Divide the RISE by the RUN to get the SLOPE. SLOPE = RISE/RUN = 2/1 = 2
- 22. Launch – What is slope? Agenda 22 Let’s find the exact slope of a line on a coordinate axis. VERY IMPORTANT: Remember, always go from left to right. If your vertical line goes down, your RISE is negative. If you can’t remember, think of Luis Riding his bike! -2 1 SLOPE = RISE/RUN = -2/1 = -2
- 23. Explore – Which Two Points to Pick? Agenda 23 Here is the main question: Does it matter which two points we pick to find the slope of a line?
- 24. Explore – Which Two Points to Pick? Agenda 24 Remember, similar triangles are triangles whose corresponding angles are congruent. Also, the ratios of corresponding sides of similar triangles are equal! 3 5 6 10 We will need to remember what the term similar triangles means to answer our question. Here is a definition that you will need to use during the exploration.
- 25. Explore – Which Two Points to Pick? Agenda 25 ? 5 16 10Here are two more similar triangles. Can you find the missing side on this triangle?! 10 16 5 ? 10 16 5 8
- 26. Explore – Calculating Slope Agenda 26 Work with your partner. We will investigate the slope of a line a little more closely. You will get a worksheet and a ruler. You should: -Remember rise/run goes left to right -Plot the points carefully -Be careful with positives and negatives! 1-Partners 2-Share Out 3-Discussion In 10 minutes you will be asked to stop to discuss! Click on the timer!
- 27. Explore – Which points to pick? Agenda 27
- 28. Explore – Student Share Out Agenda 28 Discussion - (5 Min) Why were we able to find the length of the missing side? Click here to see an interactive display! Did anyone find the missing side?
- 29. Summary 29 Agenda No matter what two points we pick, we will create a triangle will its two sides in the same ratio. Therefore, we can pick any two points we want! (𝑥1, 𝑦1) (𝑥2, 𝑦2) The vertical distance, or the RISE is the difference between the two y values: 𝑦2 − 𝑦1. The horizontal distance, or the RUN is the difference between the two x values: 𝑥2 − 𝑥1. So, the slope is: 𝑆𝐿𝑂𝑃𝐸 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑥2 − 𝑥1
- 30. Summary 30 Agenda This is known as the slope formula. (𝑥1, 𝑦1) (𝑥2, 𝑦2) Given two points with coordinates 𝑥1, 𝑦1 and 𝑥2, 𝑦2 the slope of the line that goes through the two points is given by: 𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1 Note that an “m” is usually used to represent slope.
- 31. Practice 31 Agenda 1. Find the slope between the two lines using the slope formula. a) (3, 5) and (-2, 3) The slope is 2/5. Every time the line rises two units, it goes to the right five units!
- 32. Practice 32 Agenda 1. Find the slope between the two lines using the slope formula. b) (4, 24) and (6, 36) The slope is 6. Every time the line rises 6 units, it goes to the right one unit!
- 33. Practice 33 Agenda 1. Find the slope between the two lines using the slope formula. c) (2, 1) and (2, -4) The slope is undefined. The line goes straight up and down.
- 34. Practice 34 Agenda 1. Find the slope between the two lines using the slope formula. d) (3.2, -2) and (3, -1) The slope is -5. Every time the line goes DOWN 5 units, it goes to the right one unit.
- 35. Practice 35 Agenda 2a. Find the slope of the line. 1. Pick two lattice points: (0, 2) and (4, 4). 2. Use the slope formula: The slope of this line is ½. Every time the line goes up one unit, it goes to the right by two.
- 36. Practice 36 Agenda 2b. Find the slope of the line. 1. Pick two lattice points: (0, 2) and (1, -1). The slope of this line is -3. Every time the line goes DOWN three units, it goes to the right by one. 2. Let’s do this one by counting: 1 -3 Rise = -3 Run = 1 Slope = Rise/Run = -3/1 = -3
- 37. Practice 37 Agenda 3a. Find missing side of the triangle. The slope of the line is 2/3. So, this triangle must have sides in the ratio of 2/3. The length of the missing side is 4! 4
- 38. Practice 38 Agenda 3b. Find the missing side of the triangle. The slope of the line is -5/2. So, this triangle must have sides in the ratio of -5/2. The length of a triangle cannot be negative, so the length of the missing side must be 8! 8
- 39. Practice – Challenge Problem 39 Agenda The red triangle is 8 𝑏𝑦 3. The blue triangle is 5 𝑏𝑦 2. 8 3 ≠ 5 2 Therefore, the triangles are not similar. Since the triangles are not similar, the larger shape is actually a quadrilateral, not a triangle!
- 40. Assessment – Exit Ticket 40 Agenda
- 42. The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in urban and turnaround schools, by bringing together teams of exemplary educators to develop units of high-quality, model lessons. These lessons are intended to: •Support an increase in student achievement; •Engage teachers and students; •Align to the National Common Core Standards and the Massachusetts curriculum frameworks; •Embed best teaching practices, such as differentiated instruction; •Incorporate high-quality multi-media and design (e.g., PowerPoint); •Be delivered by exemplary teachers for videotaping to be used for professional development and other teacher training activities; •Be available, along with videos and supporting materials, to teachers free of charge via the Internet. •Serve as the basis of high-quality, teacher-led professional development, including mentoring between experienced and novice teachers. 21st Century Lessons The goal… 49
- 43. Directors: Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee Ted Chambers - Co-director of 21st Century Lessons Tracy Young - Staffing Director of 21st Century Lessons Leslie Ryan Miller - Director of the Boston Public Schools Office of Teacher Development and Advancement Wendy Welch - Curriculum Director (Social Studies and English) Carla Zils – Curriculum Director (Math) Shane Ulrich– Technology Director Marcy Ostberg – Technology Evaluator 21st Century Lessons The people… 50
- 44. Visit sharemylesson.com to download this and other high- quality PreK-12 lesson plans shared by education professionals and content partners. 51
Editor's Notes
- #10: (Time on this slide – 3 min) 0-3 In-Class Notes This do now is here to make sure students are prepared to complete the arithmetic required for the slope formula. Preparation Notes Make sure students remember how to handle a “double negative.” –(-4).
- #11: (Time on this slide – 1 min) 3-4
- #12: (Time on this slide – 1 min) 4-5 In-Class Notes This slide is meant to be a simple answer, and calling on one student should suffice.
- #13: (Time on this slide – 1 min) 5-6 In-Class Notes This slide is meant to be a simple answer, and calling on one student should suffice. Preparation Notes
- #14: (Time on this slide – 2 min) 6-8 In-Class Notes The previous two slides were meant to ease the students into understanding slope. We will return to Luis as an example in a second, but this slide is meant as a more formal definition for slope, in mathematical terms. Preparation Notes Make sure every student understands this definition before moving on. It may be a good idea to have the students draw down the picture here, as well as the definition.
- #15: (Time on these 4 slides – 2 min) 8-10 In-Class Notes In order to use this method to think about slope, make sure the students are measuring only from left to right! You may want to have them think about what happens if you measure from right to left, but at this point in their understanding of slope it may just confuse them. Preparation Notes Make sure to state that slope is measured from left to right!
- #16: (Time on these 4 slides – 2 min) 8-10 In-Class Notes In order to use this method to think about slope, make sure the students are measuring only from left to right! You may want to have them think about what happens if you measure from right to left, but at this point in their understanding of slope it may just confuse them. Preparation Notes Make sure to state that slope is measured from left to right!
- #17: (Time on these 4 slides – 2 min) 8-10 In-Class Notes In order to use this method to think about slope, make sure the students are measuring only from left to right! You may want to have them think about what happens if you measure from right to left, but at this point in their understanding of slope it may just confuse them. Preparation Notes Make sure to state that slope is measured from left to right!
- #18: (Time on these 4 slides – 2 min) 8-10 In-Class Notes In order to use this method to think about slope, make sure the students are measuring only from left to right! You may want to have them think about what happens if you measure from right to left, but at this point in their understanding of slope it may just confuse them. Preparation Notes Make sure to state that slope is measured from left to right!
- #19: (Time on Launch– 2 min) 10-12 In-Class Notes You may want to mention the use of the 1:12 as a ratio instead of a fraction in the diagram. This helps remind students that slope is a ratio. Preparation Notes These slides just serve to give the students a concrete idea of slope in the real world.
- #20: (Time on Launch– 2 min) 10-12 In-Class Notes Preparation Notes These slides just serve to give the students a concrete idea of slope in the real world.
- #21: (Time on Launch– 2 min) 10-12 In-Class Notes A good question to ask would be: “if we looked at the hill from the other side of the street, would it still be positive, or become negative?” Preparation Notes These slides just serve to give the students a concrete idea of slope in the real world.
- #22: (Time on Launch– 2 min) 10-12 In-Class Notes It may be a good idea to have the students think about the slope quickly on their own before going through the animations. Preparation Notes This is the first actual example of calculating slope. Make sure the students understand before moving on, in order to complete the exploration.
- #23: (Time on Launch– 2 min) 10-12 In-Class Notes Again, it may be useful to have the students think of the answer on their own before going through the animations. It also may be worthwhile to recall the definition of the word integer with the students. (The counting numbers, zero, and the negative of the counting numbers.) Preparation Notes This slide is here to help students avoid making the mistake of confusing positive and negative slope.
- #24: (Time on this slide – 2 min) 12-14 In-Class Notes If there is confusion of similar triangles after just one slide, try to do a couple other examples on the board. The students must know what similar triangles are before moving on to the exploration. Preparation Notes The standard that addresses slope in the eighth grade requires students to show the slope is the same at every point of a line by using similar triangles. This slide will remind them what a similar triangle is, and the exploration will give them the tools they need to master the standard.
- #25: (Time on this slide – 2 min) 12-14 In-Class Notes If there is confusion of similar triangles after just one slide, try to do a couple other examples on the board. The students must know what similar triangles are before moving on to the exploration. Preparation Notes The standard that addresses slope in the eighth grade requires students to show the slope is the same at every point of a line by using similar triangles. This slide will remind them what a similar triangle is, and the exploration will give them the tools they need to master the standard. The students will not see the connection between slope and similar triangles until they begin the exploration.
- #26: (Time on this slide – 2 min) 12-14 In-Class Notes If there is confusion of similar triangles after just one slide, try to do a couple other examples on the board. The students must know what similar triangles are before moving on to the exploration. Preparation Notes The standard that addresses slope in the eighth grade requires students to show the slope is the same at every point of a line by using similar triangles. This slide will remind them what a similar triangle is, and the exploration will give them the tools they need to master the standard. The students will not see the connection between slope and similar triangles until they begin the exploration.
- #27: Online timer link on slide – (10 min) (14-24) Preparation Notes The standard that addresses slope in the eighth grade requires students to show the slope is the same at every point of a line by using similar triangles. This slide will remind them what a similar triangle is, and the exploration will give them the tools they need to master the standard.
- #28: (Time on this slide – 10 min) 14-24 Preparation Notes The standard that addresses slope in the eighth grade requires students to show the slope is the same at every point of a line by using similar triangles. This slide will remind them what a similar triangle is, and the exploration will give them the tools they need to master the standard.
- #29: (Time on this slide – 5 min) 24-29 In-Class Notes The answer that you will be looking for is since every triangle drawn between two points is similar, the RATIO of the triangle sides is the same, therefore the RATIO of rise and run is the same. Try to extract the words ratio, sides, rise and run. The link brings you to a web page with an applet that allows you to move around the line and points on the line to see different slopes. This truly brings the concept of similar triangles and slope alive for the students. Preparation Notes
- #30: (2- min) 29-31 In-Class Notes It may be helpful to do a quick example right after this explanation with the class before moving on to the independent practice. Preparation Notes Now since the students understand WHY the two points picked do not matter, we can show them a slope formula that will always work for every linear function.
- #31: (1- min) 31-32 In-Class Notes It may be helpful to do a quick example right after this explanation with the class before moving on to the independent practice. Preparation Notes Now since the students understand WHY the two points picked do not matter, we can show them a slope formula that will always work for every linear function.
- #32: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #33: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #34: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #35: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #36: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #37: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #38: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #39: (20- min) 32-52 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #40: (3- min) 52-55 In-Class Notes Have the students work in groups, or with partners. Answers are provided in the slides for each problem. Teachers may choose to go over each answer, or only some depending on time and the students levels of progress. Preparation Notes The emphasis of these problems are to utilize the slope formula. The last problem is a challenge, but it is recommended that each student tries. This last problem is much more conceptual, and focused on similar triangles than the practice problems proceeding it.
- #41: (5 - min) 55-60 In-Class Notes There are multiple answers for the last problem, but make sure the students are drawing a line, and not just two points or a triangle. Preparation Notes At this point of practicing slope, allow the students to find the slope any way that they feel most comfortable.