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Friday, September 16, 2011

Algebra 2: Graphing Absolute Value Functions

My goal with this was for students to understand "why the V shape" for absolute value functions. I think it will take three days. The TI-Nspire is used. This leads up to something very much like this, but with much more scaffolding ahead of time.

Phase 1: Learn lists and spreadsheets, data and stats skills on nspire:

1. Students watch and follow along on their handhelds "Data and Statistics: Adding and Rotating a Movable Line" tutorial.
2. Close file and do not save.
3. Send students document Squares.tns containing the case vs gray squares table from the beginning of the No Sleep Til Brookline problem set. Display these directions:
  • open file
  • add a Data & Statistics page
  • create a scatter plot of case vs gray squares
  • add a movable line, and try to get it through all the points
  • remove the movable line
  • menu, analyze, plot function
  • type in the equation you know fits the line
Phase 2: Understand why absolute value graph has a V shape and what it means
1. Write a number line on the whiteboard from -8 to +8 or even longer if possible.
2. Students line up against wall (however many will fit - maybe a subset of the class). Students note their position along the number line written on the whiteboard. This is your "x."
3. Choose whoever is at 2 to be the vertex. Let's call him Jake.
4. Give Jake something to hold up like a flyswatter.
5. When I say go, you are going to move away from the board. The rule is, however many floor tiles you are away from Jake on the number line, you are going to step that many floor tiles into the room away from the board. You'll move perpendicular to the wall. Take a moment to decide how many tiles you will move....Go.
6. Students move into a V shape.
7. Display in succession on projector:
jump up and down if you are a solution to |x - 2| = 5
jump up and down if you are a solution to |x - 2| > 5
jump up and down if you are a solution to |x - 2| < 5
jump up and down if you are a solution to |x - 2| = 3
jump up and down if you are a solution to |x - 2| = -3
8. Let's call the distance you stepped into the room y. What is the equation of x vs y?
9. Reset students back to line up against the board. (Or get a new group up there.)
10. Get the flyswatter away from Jake.
11. Our new function is |x + 3| = y.
12. Who gets the flyswatter? (Let's call her Jill.)
13. Your position is still your x. Decide what your y is and move there.
14. Display a few "jump up and down" questions. 
15. Note your position! Come to the smartboard and enter your name and position in the Lists & Spreadsheet.

Phase 3: Create Absolute Value Scatter Plot on TI-Nspire
16. Send everyone the lists&spreadsheet with name, position on NL
17. column 3 call it distJill
18. In formula cell, how can we calculate everyone's distance from Jill? Try difference...note problem with negatives.
19. Show how to enter absolute value function: template or abs(position - -3)
20. Show how to sort the whole spreadsheet from closest to farthest
21. Add a DataStats page
22. Can you get the dots to arrange themselves into the V-shaped graph?
Can you add the function that goes through all the dots?
Can you add a horizontal line function that represents "4 away from Jill"?
Can you shade the region that includes people that were within 4 spaces of Jill?
Can you add a vertical line (Plot Value) that represents the average distance from Jill?  

Phase 4: Apply skills to novel problem
For the past week, I left this sitting out on a desk in my room:

Something like 100 students entered their guess. I was able to copy the column containing their guesses from Google Docs into an Nspire lists and spreadsheets page. They'll get this Nspire file and these directions, and have the period to do what they can with it.

Phase 5: Transformations on the Absolute Value Function
Students will spend time playing with this Nspire file with this investigation, to understand how changes to the parent function transform the graph. To assess, they will try to match the pictures with a function.