Bringing Algebra Together with Geometry

1. Bringing Algebra Together with Geometry Jim Rahn www.jamesrahn.com James.rahn@verizon.net

2. Reasoning and sense making not only underlie students’ effective use of mathematics, they are also an important means through which students learn mathematics. • Reasoning and sense making should be a part of the high school mathematics classroom every day.

3. We need to develop a reasoning habit or a productive way of thinking that becomes common in the processes of mathematical inquiry and sense making.

4. These reasoning habits are divided into four broad categories: • analyzing a problem, • implementing a strategy, • seeking and using connections, • and reflecting on a solution to a problem.

5. Although many of the more-specific reasoning habits, such as “seeking patterns and relationships” or “considering the reasonableness of a solution,” may seem familiar, they often receive limited attention in the classroom.

6. In reflecting on a solution to a problem, students might “reconcile different approaches,” including those proposed by their classmates. • This reasoning habit may help students recognize which approaches are most useful in various contexts and may also build a deeper appreciation for “elegant” proofs that provide insight into why a particular result may be true.

7. Reasoning with algebraic symbols suggests that the curriculum be focused on the following: • Meaningful use of symbols • Mindful manipulation • Reasoned solving • Connections between algebra and geometry • Connections between expressions and functions

8. Formulas used throughout Geometry and Algebra

9. Conceptual Understand leads toProcedural Understanding

10. Slope • Draw the line that passes through • (0,3) and (2,7) • (0,3) and (5,8) • (0,3) and (8,7)

11. Slope • As you move to the right from (0,3) on each of these lines, what happens to both the x and y values of the coordinates on all three lines? • Would you describe these lines as having positive or negative slope?

12. Slope • What can you tell me about the steepness of each line? • Explain how you determined the steepness of the line.

13. Slope • Draw the line that passes through • (0,6) and (2,5) • (0,6) and (5,1) • (0,6) and (6,-6)

14. Slope • As you move to the right from (0,6) on each of these lines, what happens to both the x and y values of the coordinates on all three lines? • Would you describe these lines as having positive or negative slope?

15. Slope • What can you tell me about the steepness of each line? • Explain how you determined the steepness of the line.

16. The time and position of two people is described in the chart. Create a graph of their position.

17. Draw a line that represents the position of person 1.

18. What is happening to the position of Person 1? How is this reflected in the graph of these points? What is the steepness of this set of points? What does it mean when a slope is positive?

19. Draw a line that represents the position of person 2.

20. What is happening to the position of Person 2? How is this reflected in the graph of these points? What is the steepness of this set of points? What does it mean when a slope is negative?

21. Plot the points and draw a line through the points. (-8,-2) (-4,0) (0,2) (2,3) (6,5) • How steep is this line? We call this steepness the slope of the line. • Draw a line parallel to this line that passes through (0,5). Explain how you were able to complete this task. • What is the slope of the second line?

22. Plot the points and draw a line through the points. (-8,-2) (-4,0) (0,2) (2,3) (6,5) • How steep is this line? We call this steepness the slope of the line. • Draw a line parallel to this line that passes through (0,5). Explain how you were able to complete this task. • What is the slope of the second line?

23. Study the line at the right. • Create a second line that is parallel to this line that passes through (-8, -4). Name at least 3 other points that are on this new line. Explain how your know these points are on the new line.

24. Create a line through (0,2) and (4,10). • Use the edge of a sheet of paper to create a line perpendicular to the given line at the point (2,6). • Draw the line and find its slope or steepness. Explain what you found out about the slope of the perpendicular line. • Describe how the two slopes are related.

25. Create a line through (4,4) and (0,5). Then draw a perpendicular line to this line that passes through (4,4). • What did you notice about the slope of this perpendicular line?

26. Understanding the Distance Formula

27. (-1,-4) and (1, -4) • (-2,1) and (2, 1) • (-4, 6) and (8, 6) • Graph these pairs of points to create three line segments

28. Find the length of the three line segments. • Explain how your found their lengths.

29. (1,-4) and (1, 6) • (-2,1) and (-2, -3) • (-4, 6) and (-4, -2) • Graph these pairs of points to create three line segments.

30. Find the length of the three line segments. • Explain how your found their lengths.

31. (6,3) and (2,6) • Graph this pair of points and create the line segment between them.

32. Draw a line that shows the vertical change and a line that shows the horizontal change. • Explain how you found its lengths.

33. (3,3) and (-7,-2) • Graph this pair of points and create the line segment between them. • .

34. Draw a line that shows the vertical change and a line that shows the horizontal change. • Explain how your found their lengths.

35. Finding the midpoint of a segment

36. Graph each pair of points and draw the line segment between them: (-1,-4) and (1, -4) (-2,1) and (2, 1) (-4, 6) and (4, 6)

37. Find the coordinates of each midpoint. • Explain how your found their lengths.

38. Graph each pair of points and draw the line segment between them: • (1,-4) and (1, 6) • (-2,1) and (-2, -3) • (-4, 6) and (-4, -6)

39. Find the coordinates of the midpoint for each segment. • Explain how your found their lengths.

40. Graph the pairs of points and draw the line segment between them: • (-6,8) and (6,2) • (-4,1) and (-2, -3) • (-6,-8) and (4, -6)

41. Draw each line segment and locate the midpoint for each segment. • Explain how you found the coordinates of the midpoint.

42. If one endpoint of a segment is at (2,5) and the midpoint is at (-3,2) find the coordinates of the other endpoint.

43. How will the graph help you solve this problem?

44. Using slope within geometric problems

45. Graph the quadrilateral whose vertices are (2,-3), (-2,1), (1,5) and (5,1).

46. Use slope and the distance formula to decide what type of quadrilateral you have graphed.

47. Graph the points (1,0), (7,-2), (4,-5), and (-6, -7).

48. Use slope and the distance formula to find the type of figure graphed by connecting the points. • Find the midpoints of two sides that are not parallel. • Connect the midpoints. Show and defend two properties of this line segment.

49. Graph the points (1,0), (6,0), (6, -5), and (-5, -11). • Use slope and the distance formula to find the type of figure graphed by connecting the points. • Write the equation of the line that represents the each diagonal. • Show that the midpoint of the shorter diagonal is a solution of the equation that represents the longer diagonal.

50. Graph the points (1,0), (6,0), (6, -5), and (-5, -11). • Use slope and the distance formula to find the type of figure graphed by connecting the points. • Write the equation of the line that represents the each diagonal. • Show that the midpoint of the shorter diagonal is a solution of the equation that represents the longer diagonal.