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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.


  1. Nice post. Perhaps you could consider getting your students to probe the concept of absolute value functions further by attempting to graph the sum of two separate mod functions, for example: |x-2|+|x+3|, and subsequently extending this to three or more such functions. Peace.

  2. supposedly i get a sabbatical next year through Math For America. Instead of taking math classes at Cal, I want to come be a student in your classes!

  3. I love this Kate! Keep up the good work! I'm sure the students are finally going to get why it's V shaped!

  4. Cool stuff. The visual equations are a nice touch, and hopefully you got a chance to refer back to the equations and inequalities solved in one variable.

    Not a fan of Phase 5, since none of the things pictured would actually be modeled by absolute value functions -- especially the pictures that are in 3D, since any 2D function graph would be of the projection of the actual 3D object. (For example, the 60-degree angle formed at the pool triangle's corner looks more like a 90-degree angle in the picture.)

    For later work you might talk about how the absolute value concept in |x-5| = 2 is similar to the center-and-radius concept in (x-5)^2 + (y-3)^2 = 2^2. If you do complex numbers, you can use that to explain why |z| is the notation for the magnitude of a complex number: it's the distance from zero.

  5. I am not a fan of phase 5 either! I struggle with attaching meaning to more complicated absolute value functions.

  6. Don't attach meaning to more complicated absolute value functions, because there isn't any. Any modeling that would be done with a complicated absolute value function is, in reality, done with a piecewise linear function instead.

    The closest thing there is was suggested by Mr Koh: the sum of two or more absolute value functions models total distance along a number line to several points, and you get to answer interesting questions like "What point on the number line has the shortest total distance to 1, 3, and 11?"

    Otherwise it's useless, throw it out in favor of general graph-movin' skillz and you get the same and more mileage :)

  7. Can I add functions on a graphing funtions calculator?
    I have the ti-84 plus. I have f(x)=-1/3x + 2. And g(x)=-1/4(x-2)^2 + 3. I need to find (f + g) Once I graph y1 and y2 can I enter something on y3 to find y1+y2?

    If not can you solve this? I keep getting an answer with a vertex of (-4/3,40/9) but it seems wrong. Thanks.

  8. You can find Y1 and Y2 in VARS, Functions, Y-vars, I believe. So you can put Y1+Y2 right into Y3.

  9. How to print Olympic Logo using C without Graphics Functions & header files included?
    Our Practical exam consisted of a question which wanted us to print Olympic logo without any graphical header files included. It means we have to print logo using basic concepts like loop and if statements.

  10. Hi Nick - Comp Sci homework help is kind of beyond the scope of this blog. There are lots of places online with people better equipped to answer your question.


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