1. Quadratics: How a Function-Based Approach Can Help http://generative.edb.utexas.edu/materials/aisd09/aisd10.htm

2. Today FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves. Go over FBA activities that can be used “tomorrow” in class to introduce quadratics Make connection from solving with quadratics back to solving with linear functions.

3. Big Picture (FBA)

4. Aug Feb Today

5. But first some data about the kinds of algebra-related challenges the function-based approach can address The survey was given to 80 Algebra 1 students at a local high school on November 24, 2009. All students were taught by the same teacher. 63 of the 80 students completed the survey. Research from Prudie Cain

10. Good news Sizable Fraction (~1/3) Select “Solving” Strategies Most seem unaware of how calculator (FBA) might help More likely to view paper version as useful See these strategies as meaningful

11. Good news Still sizable fraction (~1/3) Select “Solving” Strategies And FBA is seen as even less useful Although seemingly more useful or acceptable if done with paper and pencil

12. Good news And FBA is EVEN LESS USEFUL for solving than for simplifying Still sizable fraction (~1/3) seems to select an inappropriate or irrelevant strategy Yet don’t seem to be as able to connect to potentially relevant sub- strategies Although seemingly more appropriate if done with paper and pencil

13. Good news Still sizable fraction (~1/3) seems to select an inappropriate or irrelevant strategy But somehow it is more likely to be useful if you graph or create tables using pencil and paper And FBA not seen as useful Again, seemingly not connected to potentially relevant sub- strategies

14. Simplifying and Solving are Confused Function-Based Approach (1)Can Help (see images) (2)Can Help in Ways that Will Help with Future Challenges

15. Today FBA can help with distinguishing when/how to use rules for simplifying from rules for solving in a way that students can check themselves. Go over FBA activities that can be used “tomorrow” in class to introduce quadratics Make connection from solving with quadratics back to solving with linear functions.

16. Big Picture (FBA)

17. Aug Feb Today

18. For lines, solving for x means getting to a machine that gives out what you put in, knowing what you get out and thereby knowing what you put in. x 79.8568 x ? 3 ? IN OUT (Must be 3) 79.8568 e.g., X = 3

19. unsolving_lines_00.ggb [1] unsolving_lines_01.ggb [2] [3] 2ndTRACE CALC GRAPH unsolving_lines_02.ggb ENTER 2ndMODE QUIT [1] [2] [3] 2ndPRGM DRAW ENTER GeoGebra Type 3 or x Calculator

20. Double-click on object to change color or thickness (style) Click to turn on Enter each pair (side) in solving Example of “unsolving”; Can do (almost) anything to both sides [e.g., multiply both sides by x] Repeat [1] - [3] or (because after using 5:intersect the x is set to the x-value of the intersection) Can turn off functions by driving over equal sign and pressing ENTER If intersection happens off the screen use WINDOW unsolving_lines_03.ggb Each transformation changes lines but preserves solution

21. Unsolving … Do whatever you want to both sides. unsolving_lines_03.ggb Check intersection (same x-value?) Draw vertical line at this value Put new pair of functions on the board

22. For quadratics, solving for x means getting to a place where what we know about roots helps us know the original intersection(s). Thankfully, doing the same thing to both sides does the same sort of thing to the original functions (replaces the original pair with functions that are NOT equivalent but do have the same solution set). (Use other activities to help understand the importance of roots)

23. [1] [2] [3] solving_quads_00.ggb solving_quads_01.ggb Calculator: Follow earlier steps and use 5:intersect and draw 4:Vertical

24. solving_quads_02.ggb Draw vertical lines (& adjust appearances)

25. What Do We Notice? Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens? solving_quads_03.ggb Why is this good?: We know how to find roots. solving_quads_03plus.ggb Just like for lines, enter each pair (side) in solving Use 5:Intersect to find x-values for drawing vertical lines

26. Where could we go from here? Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values. solving_quads_04.ggb Challenge

28. handouts

29. unsolving_lines_00.ggb [1] unsolving_lines_01.ggb [2] [3] 2ndTRACE CALC GRAPH unsolving_lines_02.ggb ENTER 2ndMODE QUIT [1] [2] [3] 2ndPRGM DRAW ENTER GeoGebra Type 3 or x Calculator

30. Double-click on object to change color or thickness (style) Click to turn on Enter each pair (side) in solving Example of “unsolving”; Can do (almost) anything to both sides [e.g., multiply both sides by x] Repeat [1] - [3] or (because after using 5:intersect the x is set to the x-value of the intersection) Can turn off functions by driving over equal sign and pressing ENTER If intersection happens off the screen use WINDOW unsolving_lines_03.ggb Each transformation changes lines but preserves solution

31. [1] [2] [3] solving_quads_00.ggb solving_quads_01.ggb Calculator: Follow earlier steps and use 5:intersect and draw 4:Vertical

32. What Do We Notice? Like for lines, can we do (almost) ANYTHING to both sides? [Unsolving] Try it. What happens? solving_quads_03.ggb Why is this good?: We know how to find roots. solving_quads_03plus.ggb Just like for lines, enter each pair (side) in solving Use 5:Intersect to find x-values for drawing vertical lines

33. Where could we go from here? Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values. solving_quads_04.ggb Challenge

34. Where could we go from here? Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values. solving_quads_04.ggb Challenge

35. Where could we go from here? Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values. solving_quads_04.ggb Challenge

36. Where could we go from here? Assignment: Get parabola and line to intersect in second quadrant. [1] plot each pair of functions in solving [2] confirm that intersections with original functions have the same x values as each step [3] confirm that what comes out of the quadratic formula are these x values. solving_quads_04.ggb Challenge