Understand the Slope-Intercept Equation for a Line
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- Ariel Laura Atkins
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1 Lesson Part : Introduction Understand the Slope-Intercept Equation for a Line Focus on Math Concepts CCLS 8.EE..6 How can ou show that an equation in the form 5 mx b defines a line? You have discovered in previous lessons that linear equations follow the format 5 mx b. These equations are written in slope-intercept form, because ou can identif the slope (m) and the -intercept (b) from the equation. If the -intercept (b) is 0, then the equation simplifies to 5 mx. Now ou will see wh linear equations can be written this wa b examining slopes and similar triangles. Think How does slope relate to triangles? To understand slope, it helps to understand similar triangles. Similar triangles are scale drawings of one another the have the same shape but can have a different size. The corresponding sides of similar triangles are proportional, and the corresponding angles have equal measures. Triangles AC and DEF are similar triangles. E A C D 6 F m/a 5 m/d m/ 5 m/e m/c 5 m/f A 5, or DE 4 8 C EF 5 5, or 0 AC 5, or DF 6 Each side of triangle AC is half the length of the corresponding side of triangle DEF. Each side of triangle DEF is twice the length of the corresponding side of triangle AC. 0
2 Part : Introduction Lesson Think A coordinate grid can be used to compare similar triangles. You can examine the corresponding sides and angles of similar triangles drawn along a non-vertical line on a coordinate grid. oth triangles AC and ADE contain /DAE, and m/dae 5 m/dae. ecause /CE and /DEA are both right angles, their measures are also equal. The sum of the angle measures in an triangle is 808. So the 5 measures of /ADE and /AC must also be equal. 4 D Since all three corresponding angles have the same measure, the triangles are similar. That means that the A E C sides are proportional: DE AE 5 C AC. proportional: DE C 5 AE AC. The rise is also run You could draw other triangles like these along the line. The rise ratio will alwas be equal run and the triangles will alwas be similar. O x Think You can calculate the slope (m) of a line using an two points on the line. Use the formula m 5 with different pairs of points. x x A(0, 0) and D(, 4): 4 0 5, or 0 4 D(, 4) and (4, 8): 8 4 5, or 4 4 A(0, 0) and (4, 8): 8 0 5, or Reflect What do the similar triangles tell ou about the slope of a line segment between an two points on a non-vertical line?
3 Part : Guided Instruction Lesson Explore It Follow the directions to write the equation of a line through the origin. Fill in the blanks as ou go. You learned on the previous page that the slope between an two points on a non-vertical line is the same. Let m 5 slope and (x, ) 5 an point on the line. Fill in the blanks to show how to find the slope of the line using (0, 0) and (x, ).. m 5 0 Simplif the equation.. m 5 4 What is the next step in rewriting the equation if ou want to isolate on one side? 4. m 5 5 Simplif. 6 Rewrite the equation with on the left side How do ou know that the graph on the previous page represents a proportional relationship? 8 Explain the reasoning used in problems 6 to find a general equation for all proportional relationships.
4 Part : Guided Instruction Lesson Talk About It Solve the problems below as a group. 9 What is the slope of the line in this diagram? What is the -intercept? 0 Compare the slope and -intercept of this diagram with the one in the introduction. How are the similar? How are the different? D A E C O x Write the coordinates for each labeled point in the diagram. Compare these coordinates to the ones in the diagram in the introduction. What do ou notice? How does this affect the position of the line and triangles on the grid? How do ou know that the graph on this page represents a linear function that is not a proportional relationship? Tr It Another Wa 4 Use the -intercept (0, b) and an other point on the line (x, ) to derive the general form of a linear equation 5 mx b. Look at the steps in Explore It to guide ou. 5 How is our equation in problem 4 different from 5 mx? What does this mean?
5 Part : Guided Practice Lesson Connect It Talk through these problems as a class, then write our answers below. 6 Compare: Look at these equations. Do ou think the are all linear equations? Can the all be written in the form 5 mx b? If so, show how. 5 x 5 x x Analze: Alana used the table of values to find the slope of the graph for this function. Analze her work and explain wh ou do or don t agree with her. m 5 6 5, or x Verif: Explain how to find the slope and -intercept b just looking at the equation 5 x. Then graph the equation and verif our answers x 4
6 Part 4: Common Core Performance Task Lesson Put It Together 9 Use what ou have learned to complete this task. A Show how to find the slope of a line that passes through the points in the table. x C Graph the data in the table. Using the graph, show how to find the slope in a different wa than ou did in part A. Show how to write an equation for the table and graph. Verif that the equation works with the table and the graph x 5
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