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Course: Foundations and Pre-Calculus 10 Stage 1 ~ Desired Results Outcome: FM 10.9 Demonstrate understanding of the writing and application of equations of linear relations, given: ● a graph of a relation ● a point that satisfies a relation and the slope of the relation ● a point that satisfies the relation and the equation of a line parallel or perpendicular to the relation Key Indicators: . Develop, generalize, explain, and apply strategies for writing an equation for a linear relation using data obtained from a graph. 2. Develop, generalize, explain, and apply strategies for writing an equation for a linear relation when given: a point that satisfies the relation and the slope of the relation two points that satisfy the relation the coordinates of a point that satisfy the relation and the equation of a line parallel or perpendicular to the line 3. Compare and critique the structure and purposes of different forms of linear relations, including y = mx + b, Ax + By = C, and y – y 1 = m (x – x 1 ) (e.g., there is not way to write a vertical linear relation in the form y = mx + b 4. Graph and write equations for linear data generated within an experiment or collected from a situation. 5. Apply knowledge and skills of linear relations and their equations to solve situational questions. Enduring Understandings: Students will understand that… Essential Questions: What information do you need to write the equation of a line? The equation of a line can be determined from many different pieces of information. Students will know: ● Slope intercept form is y = mx + b ● Standard form is Ax + By = C ● When to use y = mx + b or y – y 1 = m(x – x 1 ) to write the equation of a line ● Equations of an oblique, vertical and horizontal line Students Will Do: Students will do: ● Analyze the graph of a linear function for slope and y-intercept ● Find the slope from a graph, from 2 given points, equations, from parallel and perpendicular equations ● Write the equation of a line, given the graph and slope ● Find y-intercept 2 Key Vocabulary: Common Misunderstandings: Prior Skills: Stage 2 ~ Assessment Evidence Performance Tasks: Matching Activity: Students are put into small groups and are given 3 different coloured decks of pre-made cards (one deck containing 10 equations in general form, one deck containing 10 in slope-intercept form and one containing 10 graphs). Their task is to match the two types of equations with the corresponding graph. HINT – they should have one of each colour per row. Evaluation - Rubric Summative: Unit Exam Performance Task Formative (Pre-Assessment, Self-Assessment, etc.) Textbook assignments Exit cards Introductory activities 3 Stage 3 ~ Learning Plan Learning Activities: Introduction: Do Math Lab 6.3 (p. 354) ~ discuss as a class Do Construct Your Understanding (p. 355) ~ with a partner Do Assess Your Understanding (p. 356) ~ individually Lesson 1: Writing an Equation Using Data Obtained from a Graph Give the students an equation (in slope-intercept form) and discuss the parts of the equation (y-intercept and slope). Then, as a class, graph the equation. ex) y = 2/3x - 2 Ask the students what they could do to come up with an equation, if they were only given the graph. Explain that by using the equation format y = mx + b, we can look at the graph and identify the slope and y-intercept and fill in those values to create the equation. Ask the students how we could check to see if it is correct. Explain that we choose a point on the line to use and substitute it into the equation to determine if it satisfies the equation. 4 Work through the following example: ex) Write an equation to describe the following function. Verify the equation. slope (m) = rise = 3 y-intercept = -2 run 2 *Since the line is falling from left to right, the slope is negative. **Now that we have acquired all of the necessary information, we can write the equation: y = mx + b y = -3 x - 2 2 Verify: Choose a point from the line (-2, 1) LHS RHS y_3x–2 2 1 _ 3 (-2) – 2 2 3–2 1 5 Discuss Example #3 on p. 360 Assign: p. 362 – 364 #12, 13, 14, 16, 17, 18 Lesson 2: Writing an Equation Given a Point that Satisfies and the Slope of a Relation Review the concepts from the previous day regarding equation formation and verifying. Ask the students how they could use any point on the line in the equation format y = mx +b ie) where would you insert those values? (x and y) Ask the students: “If you are only given a point that satisfies and the slope, what DON'T you know?” (y-intercept or b) Ask the students: “What variable in y = mx + b represents the y-intercept? (b) Work through an example together to illustrate a strategy to find the equation. ex) Find the equation of the line that passes through (-10, 4) and has a slope of -8. First, list what you do know: x = -10 y=4 m = -8 b = ??? Next, solve for what you don't know: y = mx + b 4 = (-8)(-10) + b 4 = 80 + b -76 = b Finally, insert values m and b into the slope-intercept format: y = mx + b y = -8x – 76 Discuss Example #2 on p. 368 Assign: p. 372 – 373 #5, 8, 9, 10, 18 6 Lesson 3: Writing an Equation Given Two Distinct Points that Satisfy a Relation Discuss the meaning of slope ~ the rise between two given points on a line run Introduce the equation format: y 1 –y 2 = m(x 1 –x 2 ) Discuss what we know vs. we don't know. ● We know 2 points that satisfy ● We know the slope (using the above formula) ● We DO NOT know the y-intercept (b) Ask the students: “How can we find b, once we know m and 2 points? (Encourage them to think back to when we were give only one point and the slope.) Explain that by knowing 2 points that satisfy, we can find slope. Then, once we know slope (and 2 points) we can find our y-intercept by isolating b. Illustrate steps through the following example: ex) Find the equation of the line that passes through the points (8, -4) and (-4, -13). Find slope: m=y 1 –y 2 = - 4 - (-13) = - 4 + 13 = 9 = 3 x 1 –x 2 8 - (- 4) 8 + 4 12 4 Using one of the given points and the slope, find y-intercept (b): m = ¾ y = mx + b x = 8 - 4 = ¾ (8) + b y=-4-4=6+b b = ?? -10 = b Now, insert values of slope and y-intercept into y = mx + b to create your equation: y = mx + b y = ¾ x – 10 Discuss Example #3 on p. 369 Assign p. 372 – 74 #11, 18, 19 7 Lesson 4: Writing an Equation Given a Point that Satisfies the Relation and the Equation of a Line Parallel or Perpendicular to the Relation Review the concept of slope of parallel lines and slope of parallel lines. ● If the lines are parallel, then their slopes are equal. ● If the lines are perpendicular, then their slopes are negative reciprocals of one another Explain that because we can identify (by sight) the slope of the parallel or perpendicular line, we can then determine the slope of the relation, given the above information. Ask the students: “How can we find the equation of a linear relation if we know the slope and a point on the line that satisfies it?” Illustrate with an example: ex) Find the equation of the line that passes through the point (2, -5) and is a) perpendicular to the line y = 2x – 12 slope = 2 ~ perpendicular slope = negative reciprocal So, slope is _ 1 2 List what we know and solve for what we don't know: m = _ 1 y = mx + b 2 -5 = (-1/2) (2) + b x = 2 -5 = -1 + b y = - 5 -4 = b b = ?? Now, use what we know to build the equation: y = mx + b y=_1x–4 2 b) parallel to the line y = 2x – 12 slope = 2 ~ parallel slope is equal So, slope is 2 List what we know and solve for what we don't know: m = 2 y = mx + b x = 2 -5 = (2)(2) + b y = -5 -5 = 4 + b b = ?? -9 = b 8 Now, use what we know to build the equation: y = mx + b y = 2x – 9 Pose the following question: “What would you do differently if you were given your equation in the following format, instead of the y = mx + b format? -2x + y – 9 = 0 **Discuss how you could change it into a format that meets the needs of the question. Discuss the commonalities between all of the equation formats ~ what are we always needing to find out before we can build an equation? (we are always finding m and b) Assign: p. 374 # 20 – 25 9 Lesson 5: Comparing and Critiquing the Structure and Purposes of Different Forms of Linear Relations Have the students recall the format we have been using and discuss why that format is useful for graphing and for creating an equation from a graph or from the other information we have been given throughout this unit. Introduce the other structures: 6x – 2y = 64 Standard 5x + 4y – 20 = 0 General y = 7x + 3 Slope-Intercept y – 3 = 5(x + 7) Slope-Point Discuss how one structure can be rearranged into another through the use of our equation solving methods. Explain how there is no way to write a vertical linear relation in the form y = mx + b. (Recall slope of a vertical line is infinity.) Discuss Examples #1 – 3 on p. 379 – 380 Assign: # 4 – 6, 9 – 13; 8, 24 Lesson 6: Graph and write equations for Linear Data Generated Within an Experiment or Collected from a Situation Discuss Example #4 on p. 382 Have the students complete “Check Your Understanding” and then discuss as a class. Assign: p. 384 # 10, 11, 16