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Hitting the Slopes

Hitting the Slopes. An adventure on the bunny hill of linear equations!. Start. What would you like to learn about?. 1 Calculating the slope of the line, given two points. 2 Solving for y (slope-intercept form) and graphing the line. 3 Determining the equation of the line

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Hitting the Slopes

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  1. Hitting the Slopes An adventure on the bunny hill of linear equations! Start

  2. What would you like to learn about? 1 Calculating the slope of the line, given two points 2 Solving for y (slope-intercept form) and graphing the line 3 Determining the equation of the line from the graph

  3. The definition of slope Given two points, (x1,y1) and (x2,y2), the slope of a line is determined by this equation: where m = slope. We think of slope as the change in y divided by the change in x. Look at an example

  4. Example of finding slope. Find the slope of the line between the points (0,6) and (5,10). • Remember the formula. • Identify which numbers represent the given variables. • Substitute into the formula and simplify. The slope of the line extending through the points (0,6) and (5,10) is . Take a Quiz

  5. Definition of Slope Quiz Click on the correct answer below: Slope is a number that represents the sum of y divided by the sum of x Slope is a number that represents the change in x divided by the change in y Slope is a number that represents the change in y divided by the change in x

  6. Hooray! You are correct, congratulations! Slope is a number that represents the change in y divided by the change in x. We can also think of it as: Since the y-axis is the vertical axis on the coordinate grid, we think of the change in y as “rising” and since the x-axis is the horizontal axis, we think of the change in x as “running”. Try calculating slope

  7. Oops! Remember, slope is defined as the change in y divided by the change in x. It is important to understand that change means difference. In math, the term difference tells us to subtract. The answer you chose contained the word “sum”, which means to add. Still a bit confused, take me back to the definition. I got it! Take the quiz Again.

  8. Oops! You are close, but let’s recall the definition: Slope is the change in y divided by the change in x. We think of slope as . Since the y-axis is the vertical axis on the coordinate grid, we think of the change in y as “rising”. Since the x-axis is the horizontal axis on the coordinate grid, we think of the change in x as “running”. So, it’s important to rise first and then run! Still a bit confused, take me back to the definition. I got it! Take the quiz Again.

  9. Calculating Slope What is the slope of the line between (5,2) and (10,1)? *Be sure to click on the blue part of the button, not in the white box*

  10. Congratulations! Yes, the correct slope is . In this case, the slope was in fraction form, but sometimes this won’t always be the case. If, for example, the slope of a line was 4. We should think of it as a fraction:

  11. Uh Oh! Let’s recall the steps to finding slope: • Remember the formula. • Identify which numbers represent the given variables. • Finally, substitute the numbers into the formula and simplify! Still confused, take me back to the beginning. I got it! Return to the quiz.

  12. The Graphs of Slopes Let’s graph the previous examples. Example 1: The slope between (0,6) and (5,10) is . Example 2: The slope between (5,2) and (10,1) is . Example 3: The slope between (-1,2) and (-2,-2) is 4.

  13. Plotting Points & Graphing Lines • In order to graph the lines, we must first plot each of the points. • Then draw a line through the points, adding arrows at each end to represent a line (rather than a line segment). • Now, start at the bottom of the two points. If you use the slope to “rise and run”, you should end at the second point. run 5 rise 4 run 1 run -5 rise 1 rise 4 Take a quiz

  14. Graph Quiz Given the slope, click on the correct graph: *Be sure to click on the blue button, not the graph when choosing an answer*

  15. You’re Right! Yes, if you start at the bottom point, then rise 3 and run 7 you will end at the other point! Think about this… You could also start at the top point, run -3 and rise -7 (which means go down three and left 7). This works because . Both fractions simplify to .

  16. Not Quite! Since all of the points were already plotted and the lines were graphed, we just need to focus on how to get from one point to another. Remember to rise from the bottom point and then run to the other point. The number of places you “rise” is the numerator of the slope. The number of places you “run” is the denominator. In the quiz, remember to run 3 and rise 7. I’m still a bit confused, return to the example. I got it! Take the quiz again.

  17. run 12’’ rise 8’’ Where you see Slope outside of the Classroom Engineers & carpenters consider slope when determining the pitch of a roof (or how steep it is). A pitch of 8/12 means that the roof rises 8” for every 12” (or 1’) it runs. See more examples

  18. Where you see Slope outside of the Classroom Ski hills have a variety of slopes. The incline of slopes vary from gentle, like a bunny hill, to steep, like a black diamond.

  19. Congratulations, you have successfully completed Part 1! What is your next step? Stop (I will complete the next section(s) another time) Continue to next section (Part 2-solving for y and graphing)

  20. Equations in Slope-Intercept Form y-intercept slope y-intercept y-intercept slope slope The three equations above are all in slope-intercept form. The slope is the number in front of the x-variable, and the y-intercept is the number after the x-variable.

  21. Slope-Intercept Form An equation is written in slope-intecept form if it is of the form: y=mx+b • The equation must be solved for y. • m is the slope of the line, we think of slope as a fraction, . • The y-intercept is represented by the variable, b. • b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or zero (y=mx) • The y-intercept is the point (0,b), where the graph of the line crosses the y-axis.

  22. Solving for y. If an equation is not in y-intercept form, follow these steps to solve for y: • Is y positive? • Is y by itself? • Is y on the left side of the equal sign? • Is the equation exactly in y=mx+b form? If you answered no to any of the questions, manipulate the equation so the answer becomes yes, and then move on to the next question. Take a Quiz

  23. Slope-Intercept Form Quiz Which of the following equations are in slope-intercept form? a) b) c) d) a a and b a and c a and d All of the above

  24. Absolutely!! Correct, in both equations y is: • positive • by itself • on the left of the equal sign • and, in the form of y=mx+b and

  25. You’re So Close! You are correct that is in slope-intercept form. However, this is not the only equation that is. Ask yourself the 4 key questions again: What other equation(s) can you answer yes to all four questions? • Is y positive? • Is y by itself? • Is y on the left side of the equal sign? • Is the equation exactly in y=mx+b form? Still confused, return to examples & explanation. I got it! Take the quiz again.

  26. Not Quite Let’s examine the four key questions again: • Is y positive? Yes, y is positive in each equation. • Is y by itself? Take a look at each equation again and . Is y all alone on one side of the equal sign? NO! In which equation is this false? • Is y on the left side of the equal sign? Yes, in both equations y is on the left side of the equal sign. • Is the equation exactly in y=mx+b form? No, both equations are not in slope-intercept form. Which one isn’t? Still confused, return to examples & explanation. I got it! Take the quiz again.

  27. Not Quite Let’s examine the four key questions again: • Is y positive? Yes, y is positive in each equation. • Is y by itself? Take a look at each equation again and . Is y all alone on one side of the equal sign? NO! In which equation is this false? • Is y on the left side of the equal sign? Yes, in both equations y is on the left side of the equal sign. • Is the equation exactly in y=mx+b form? No, both equations are not in slope-intercept form. Which one isn’t? Still confused, return to examples & explanation. I got it! Take the quiz again.

  28. Uh Oh! Although more than one equation is in slope-intercept form, not all of the equations are. Ask yourself the four key questions of determining if an equation is in slope-intercept form: • Is y positive? • Is y by itself? • Is y on the left side of the equal sign? • Is the equation exactly in y=mx+b form? Still confused, return to examples & explanation. I got it! Take the quiz again.

  29. Slope-Intercept Form Remember, an equation is written in slope-intecept form if it is of the form: y=mx+b • The equation must be solved for y. • m is the slope of the line, we think of slope as a fraction, . • The y-intercept is represented by the variable, b. • b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or zero (y=mx) • The y-intercept is the point (0,b), where the graph of the line crosses the y-axis.

  30. Graphing Lines from equations in Slope-Intercept Form Once an equation is in slope-intercept form, graphing the line is a breeze! y=mx+b First, use the y-intercept to plot the point (0,b). This shows where the line crosses the y-axis (where the x-value is zero). Next, use the slope to “rise and run” to another point on the graph. See an example

  31. Example of Graphing • Identify the y-intercept. • The number after the x is +3, so the y-intercept is a positive 3. The graph intersects the y-axis at (0,3). • Plot the y-intercept, then use the slope to find another point. • Plot (0,3) on the graph. Then, from (0,3) rise -3 (go down 3) and run 1 (go right 1). Next, connect the points to draw a line. • Identify the slope and think of it as a fraction. • The number before the x is -3, so the slope of the line is -3. We think of it as: Take a Quiz

  32. Graph Quiz Which of the following is the correct process for graphing the equation ?

  33. Oh So Smart! Absolutely! You chose the correct sequence of steps to plot the line . • First, plot the y-intercept (0,-2) • Second, use the slope to “rise 3 and run 4” • Finally, connect the points with a line. (Remember a line extends forever in both directions, so arrow heads are required!)

  34. Not Quite! It is very tempting to start at the origin (0,0) and plot the slope from there. But remember, the origin isn’t always a point on a graph. In this case, the line does not pass through the origin so you cannot use it as your starting point! The y-intercept (0,-2) should be your starting point. Still confused, return to examples & explanation. I got it! Take the quiz again.

  35. Congratulations, you have successfully completed Part 2! What is your next step? Stop (I will complete the next section(s) another time) Continue to next section (Part 3- finding the equation from the graph) Review the previous section (Part 1- calculating slope between two points)

  36. Review So far you have learned… • how to calculate the slope between two points using . • that the y-intercept is the point, (0,b) where a graph of a line crosses the y-axis. • using the slope to “rise & run” from the y-intercept will give another point on the graph, and by connecting the two points the line can be drawn.

  37. Determining the Equation from the Graph of the Line Now, using what you have learned previously, you will learn how to determine an equation from a graph.

  38. Finding the Equation in Slope-Intercept Form • Identify the two points plotted on the graph. • (-4,-1) and (2,8) • Using the formula from Part 1-Calculating Slope from Two Points, find the slope of the line. • Look at the graph to determine where the line intersects the y-axis. • The line & y-axis intersect at (0,5) • Substitute into the slope-intercept formula Take a Quiz

  39. Equation Quiz Which equation matches the graph of the line:

  40. You’re Great! Yes, the two points plotted on the graph are (-2,2) and (6, -2). By calculating the slope, you find: and the y-intercept is (0,1). So, by substituting the slope and y-intercept, you find the equation:

  41. Uh Oh! Review the steps necessary to find the equation of the line: • Identify the two points plotted on the graph. • Using the formula from Part 1-Calculating Slope from Two Points, find the slope of the line. • Look at the graph to determine where the line intersects the y-axis. • Substitute into the slope-intercept formula Still confused, return to examples & explanation. I got it! Take the quiz again.

  42. Congratulations! You have successfully completed this section. Think back to the barn example. Let’s determine the equation for the slope of the left pitch of the barn roof, assuming the peak of the barn roof is the point (0,0). Since the roof peaks at (0,0) we can substitute the y-intercept is y=mx+0. From the peak, for every 8” down (rise of -8), there is a run of 12” back (run -12). Let’s substitute the slope: Now simplify the equation: x-axis y-axis

  43. Congratulations, you have successfully completed Part 3! What is your next step? Stop-I’m the Slope Master! (I have completed all three sections) Review the previous section (Part 2- solving for y and graphing the line) Review the first section (Part 1- calculating slope between two points)

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