We will explicitly define slope-intercept form. We have already examined slope, y- intercepts, and graphing from tables, now we are putting all of that together. This lesson focuses more upon the notation and traditional algebra problems, while future lessons will return to the problem-based curriculum. Algebra 1 Predicting Patterns & Examining Experiments Unit 3: Visualizing the Change Section 3: You ve Got to Start Somewhere
Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = 2x + 1 y = 1 2 x + 1 Feel free to make a table, if necessary. (Small group discussion) Tables are shown on the next slide.
Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = 2x + 1 y = 1 2 x + 1 0 1 1 2 2 3 3 4 0 1 1 3 2 5 3 7 0 1 1 1.5 2 2 3 2.5 Feel free to make a table, if necessary. (Small group discussion) For students to check their work.
(Large class discussion) Students should notice the same y-intercept for each graph and be able to deduce the +1 in the equation rendering this value. The differences exist in the slopes, stress not only steepness, but the constant change in the slope and the factor of x that gives that constant change. Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = 2x + 1 y = 1 2 x + 1 0 1 1 2 2 3 3 4 0 1 1 3 2 5 3 7 0 1 1 1.5 2 2 3 2.5 Feel free to make a table, if necessary.
Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = x + 4 y = x 2 Feel free to make a table, if necessary. (Small group discussion) Tables are shown on the next slide.
Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = x + 4 y = x 2 0 1 1 2 2 3 3 4 0 4 1 5 2 6 3 7 0-2 1-1 2 0 3 1 Feel free to make a table, if necessary. (Small group discussion) For students to check their work.
Compare and contrast the graphs of the three equations below, use the same grid for all three. y = x + 1 y = x + 4 y = x 2 0 1 1 2 2 3 3 4 0 4 1 5 2 6 3 7 0-2 1-1 2 0 3 1 Feel free to make a table, if necessary. (Large class discussion) Students should have observed that the lines are parallel (all have the same slope of one. The differences are the y-intercepts, which- once again- are the constant term in the equation.
What do the numbers in the equation represent... what effect do they have on the graph? y = # x + # slope y = mx + by-intercept transition slide
What do the numbers in the equation represent... what effect do they have on the graph? y = # x + # y = mx + b slope (the steepness y-intercept (the beginning) of the mountain) (Teacher Lecture) Slope-intercept form defined, next we will look at an analytic example.
Graph: y = 2x+8 (Individual work) Have students graph the equation by themselves. The next few slides show progressive hints for the graph.
Graph: y = 2x+8 slope y-intercept hint #1
Graph: y = 2x+8 slope y-intercept hint #2
Graph: y = 2x+8 slope y-intercept two steps down one step over hint #3
transition slide (next slide shows just the line) Graph: y = 2x+8
Solution. Graph: y = 2x+8
Susan s problem A problem in Susan s math book asks her to graph the line y=3x-5. However, there is a misprint in her book; it should read y=3x-15. Which statement describes how the graph of y=3x-5 compares to the graph of y=3x-15? a) the y-intercept of y=3x-5 is greater than the y-intercept of y=3x-15. b) the y-intercept of y=3x-5 is less than the y-intercept of y=3x-15. c) the slope of y=3x-5 is greater than the slope of y=3x-15. d) the slope of y=3x-5 is less than the slope of y=3x-15. (Think, pair, share) This is an SAT-type problem and is modeled after a problem from the Kansas high school mathematics standard M.10.2.3.K6. The answer is a (discuss why some students may incorrectly answer b ).
Which of the graphs below could fit the equation: y = -2x-1 (Think, pair, share) The answer is a.
(Small group discussion) This problem could be homework or an in-class mini-project. Students should work together and present solutions to the class. The next slide contains variations of the same table (with different numbers), if you would like to give each group different numbers Two trips A and a small started out from your town at 8:00 am. Their s, recorded at hourly intervals, are recorded in the tables at right. Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what s will each have driven 600 miles? 9:00 52 46 10:00 104 92 11:00 156 138 12:00 208 184
9:00 52 46 10:00 104 92 11:00 156 138 12:00 208 184 9:00 61 44 10:00 122 88 11:00 183 132 12:00 244 176 9:00 45 55 10:00 90 110 11:00 135 165 12:00 180 220 9:00 68 34 10:00 136 68 11:00 204 102 12:00 272 136 9:00 39 49 10:00 78 98 11:00 117 147 12:00 156 196 variations of the table in the previous problem
Two trips Solution Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what s will each have driven 600 miles? hours after 8:00 11:00 3 156 hours after 8:00 11:00 3 138 208 156 104 0 1 hr 52 miles 1 hr 46 miles 9:00 1 52 9:00 1 46 slope 10:00 2 104 = 52/1 mph 10:00 2 92 = 52 mph 52 slope = 46/1 mph = 46 mph 8 9 10 11 12 0 1 2 3 4 12:00 4 208 12:00 4 184 Solution - transition slide
Two trips Solution Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what s will each have driven 600 miles? hours after 8:00 11:00 3 156 12:00 4 208 hours after 8:00 11:00 3 138 12:00 4 184 208 156 104 0 1 hr 52 miles 1 hr 46 miles 9:00 1 52 9:00 1 46 slope 10:00 2 104 = 52/1 mph 10:00 2 92 = 52 mph m = 52 b = 0 y = 52x 52 slope = 46/1 mph = 46 mph m = 46 b = 0 y = 46x 8 9 10 11 12 0 1 2 3 4 600 miles 600 = 52x 600 = 52x 52 52 11.5 x 600 miles 600 = 46x 600 = 46x 46 46 13 x Solution - transition slide
Two trips Solution Plot this information on the same set of axes and draw two lines connecting the points in each set of data. What is the slope of each line? What is the meaning of these slopes in the context of this problem? What is the equation of each line? At what s will each have driven 600 miles? hours after 8:00 11:00 3 156 12:00 4 208 hours after 8:00 11:00 3 138 12:00 4 184 208 156 104 0 1 hr 52 miles 1 hr 46 miles 9:00 1 52 9:00 1 46 slope 10:00 2 104 = 52/1 mph 10:00 2 92 = 52 mph Solution - transition slide m = 52 b = 0 y = 52x 52 slope = 46/1 mph = 46 mph m = 46 b = 0 y = 46x 8 9 10 11 12 0 1 2 3 4 600 miles 600 = 52x 600 = 52x 52 52 11.5 x The will have driven 600 miles at 7:30 pm. The will have driven 600 miles at 9:00 pm. 600 miles 600 = 46x 600 = 46x 46 46 13 x
Homework Sid has a job at Milliken-Madison Motors. The salary is $1200 a month, plus 3% of the sales price of every or Sid sells (this is called a commission ). a) The total of the sales prices of all the vehicles Sid sold during the first month on the job was $72 000. What was Sid s income (salary plus commission)? b) In order to make $6000 in a single month, how much selling must Sid do? c) Write a linear equation that expresses Sid s monthly income y in terms of the value x of the vehicles Sid sold. d) Graph this equation. What is the meaning of its y-intercept, and the meaning of its slope? (Homework) The problem above is from the math curriculum at Exeter High School, which is an excellent source of a problem-based curriculum. More info can be found at: http:// www.exeter.edu/academics/84_9408.aspx