Function Junction, What�s your function?

							Function Junction, What’s
your function?
• Function Junction, what's your function?
  Hooking up a numbers and making 'em run
  right.
  Algebraic equations, verbal descriptions,
  tables, graphs, and concrete, active
  representations.
• Milk and honey, bread and butter, peas
  and rice.
• y = 4x, y = 2x + 1, y = x2

       Hey that's nice!
             Function Junction:
             Beginning Ideas of Function

                               Lynn Stallings, Ph.D.
                                lstalling@kennesaw.edu
                              Kennesaw State University,
                                 Atlanta, Georgia, USA

National Council of Teachers of Mathematics 2004 Conference
Building with Toothpicks
• Look for patterns in the number of
  toothpicks in the perimeter of each
  shape.
• Shape 1, Shape 2, Shape 3, Shape n?




• Can you predict how many toothpicks would be
  in the perimeter of shape 7? Shape 10? Shape
  22?
• How did you do it?
Function Junction, What’s your
Function?
• Functions may be ―the most single important
  concept from kindergarten to graduate school‖
  (Harel & Dubinsky, 1992, The Concept of Function, MAA)
• ―Systematic experience with patterns can build
  up to an idea of function.‖ (PSSM, 2000)
• In the middle school, students ―can begin to use
  variables and algebraic expressions as they
  describe and extend patterns.‖ (PSSM, 2000)
• As students ―work with multiple representations
  of functions—including numeric, graphic, and
  symbolic—they will develop a more
  comprehensive understanding of functions.‖
  (PSSM, 2000)
Starring . . . PSSM Process Standard
              Representation
Also featuring Communications, Connections, Reasoning,
and Problem Solving
                        algebraic
                        equation

                                            table

      graph




        concrete or active            verbal
              context               description
        (real to students)

         Where do we typically start?
      What’s a good mnemonic for these?
Number Patterns from Cutting String
(or tearing paper)
Fold a strip of paper in half. Cut once.
How many pieces will you have?




Cut a second time. How many pieces?
Cut a third time. How many pieces?
   Number of cuts     1   2   3   Etc.

   Number of pieces
How’s that Function in Middle School?

Middle school students need
  developmentally appropriate instruction
  which includes
• Connections between function representations
  and what is real to them.
• Concrete and active experiences with patterns
  and actions that they can represent visually,
  orally, in diagrams, in tables, in graphs, and in
  symbols.
• Visual and active experiences building up to the
  abstraction of symbols.
• Experience interpreting various representations
  and the connections between them.
Number Patterns from Triangles




  Triangle 1 Triangle 2     Triangle 3
• Form these triangles.
• What relationship do you see between
  the triangle number and the area of the
  triangle formed?
from PSSM, 2000
ChitChat vs. Keep-in-Touch
• Two cellular phone plans:
• Keep-in-Touch - a basic fee of $20.00 a
  month, plus $0.10 per minute
• ChitChat - no monthly basic fee but
  charges $0.45 a minute
• Both companies use technology that
  allows them to charge for the exact
  amount of time used; they do not round
  up the time to the nearest minute as
  many competitors do.
• Compare their charges.
ChitChat vs. Keep-in-Touch
  L1 = number of minutes, L2 = Keep-in-Touch cellular
  phone rates, L3 = ChitChat rates
ChitChat vs. Keep-in-Touch
   L1 = number of minutes, L2 = Keep-in-Touch cellular
   phone rates, L3 = ChitChat rates


How to graph this data.
Press 2nd Y= (which is STAT PLOT).




Press ZOOM and select 9 ZoomStat.
Press Graph.
ChitChat vs. Keep-in-Touch
(cont.)

  What can we tell from a graph of the values in our
  table?
  Would it make sense to connect these points?

           Keep-in-Touch




                           ChitChat
ChitChat vs. Keep-in-Touch
(cont.)
What equations would connect these
dots?
                     Keep-in-
                     Touch
                                 (57.14, 25.71)




                                 ChitChat




  What does the point of intersection mean?
  What do the slopes and y-intercepts mean?
Guess the relationship . . .
If 8 is the input, what will the
output be?
What’s the relationship?
7 was the input more than once. What
was the output? Is that important?
               From Navigations
                                     Pledge Plans
               Pledge Plans for a 10K Walk-a-Thon
               • Jeff:               $1.50 per kilometer
               • Rachel:             $2.50 per kilometer
               • Annie:              $4.00 donation plus $0.75 per km

    Km          0     1       2       3      4       5       6        7      8       9       10


 Jeff's plan    $0   $1.50    $3     $4.50   $6    $7.50     $9     $10.50   $12   $13.50    $15


Rachel's plan   $0   $2.50    $5     $7.50   $10   $12.50   $15     $17.50   $20   $22.50    $25


Annie's plan    $4   $4.75   $5.50   $6.25   $7    $7.75    $8.50   $9.25    $10   $10.75   $11.50
From Navigations
                   Pledge Plans
What equation should                Jeff: y = 1.5x
 we use for each
 plan?                              Rachel: y = 2.5x
•   Jeff: $1.50 per kilometer       Annie: y = 4 + .75x
•   Rachel: $2.50 per kilometer
•   Annie: $4.00 donation plus      Press Y= and input the
    $0.75 per kilometer             three equations.
What should those
 equations represent?
•   The relationship between
    the number of kms (let’s call   Press WINDOW.
    it x) walked and the amount
    of money pledged (let’s call
    it y).
From Navigations
                Pledge Plans
Let’s use verbal            Jeff: y = 1.5x
  description to link the
                            Rachel: y = 2.5x
  graph, the algebraic
  equations, and the        Annie: y = 4 + .75x
  table                             Rachel    Jeff       Annie
What do the slopes
  mean?                                     (5 1/3, 8)
                        (2.3, 5.7)
What about y-
  intercepts?
What do the points of
  intersection mean?
From Navigations
           Walking Strides
• We need a designated walker and a
  timer.
• Our walker is going to walk 20 feet
  three different ways: with baby
  steps, regular steps, and
  exaggerated steps while our timer
  times it.
• We’ll record the data for 20 feet and
  then use it to figure out the rest.

                   Actions   graphs
From Navigations                       Actions         graphs

             Walking Strides
• How would we fill in
  the rest of the chart?
                                     Baby    Regular BIG
• Let’s let x = time and             Steps   Steps   Steps
  y = distance walked.
                           0 ft.
• Can we graph these?
                           20 ft.
  What equations
  could we use?            40 ft.
                           60 ft.
• What do the slopes
  mean?                    80 ft.
• What do y-intercepts     100 ft.
  mean? What would
  you do to change the
  y-intercept?
                                             Actions              graphs


  Match the Graph
Has anyone used the CBR before? How does it work? Would
you like to try some sample data first?
Given a graph, can we interpret it by moving in a way that will
replicate it. How would we replicate this one?
                        Actions   graphs

Match the Graph
• Press PRGM on your calculator. If
  you don’t see RANGER, follow the
  directions on the next two pages
  labeled “Getting Started with CBR”
  to load the program.
• The next page is “Notes for
  Teachers.”
• For right now, you can go on to the
  page titled “Activity 1: Match the
  Graph.” later you’ll want to go back
  to “Notes for Teachers.”
                                 Actions      graphs
Match the Graph
•   What happened?
•   What did slope mean here?
•   What about y-intercept?
•   What happened when
    – you were on a section of the graph with a
      positive slope?
    – you were on a section of the graph with a
      negative slope?
    – you were on a horizontal section of the
      graph?
    – you were on a vertical section of the graph?
                              Symbols   graphs

Transformation Creations
• ―Rain‖ is a way to
  practice your connections
  between the algebraic
  equations and graphic
  representations.
• What does
 –4 < x < 4 and –3 < y < 3
 tell you about the graph?
• How do we input that into
  the calculator?
                      Symbols   graphs

Transformation Creations
• How can you make ―Rain‖ on your
  calculators?




•How did you know?
From Navigations        Words    graphs

From Graphs to Stories
Let’s explain these graphs.
1. What happened when John and
    his father raced?
2. Describe the lemonade stand’s
    profit graph.
3. Describe the raising of the flag
    graph.
4. Describe the race between the
    two swimmers.
From Navigations                                                               Words     graphs

From Stories to Graphs
                    Sketch a graph for each of these
                    three stories. Note that number
                    two will vary the most.
                    Josephine’s walk                          Mowing the lawn

                    40                                                   100




                                                       Uncut Grass (%)
  Distance (feet)




                    20                                                   50




                         10   20   30   40   50   60                              Time
                         Time (seconds)
                          Words     graphs

More Graphs to Stories
The next pages include graphs to match to
   stories for more practice.

You can also make up your own. For
   example,
• the speed of students in the halls
   during class changes,
• the amount of studying done versus
   the number of days before the test,
• hyperactivity versus the number of
   days until school is out,
• What else?
PSSM standard on Algebra says . . .
In grades 6–8 all students should

Understand patterns, relations, and
 functions
• represent, analyze, and generalize a
  variety of patterns with tables, graphs,
  words, and, when possible, symbolic
  rules;
• relate and compare different forms of
  representation for a relationship;
• identify functions as linear or nonlinear
  and contrast their properties from tables,
  graphs, or equations.
PSSM standard on Algebra says . . .
In grades 6–8 all students should

Represent and analyze mathematical
  situations and structures using algebraic
  symbols
• explore relationships between symbolic
  expressions and graphs of lines, paying
  particular attention to the meaning of
  intercept and slope;
• use symbolic algebra to represent
  situations and to solve problems,
  especially those that involve linear
  relationships;
PSSM standard on Algebra says . . .
In grades 6–8 all students should

Use mathematical models to
 represent and understand
 quantitative relationships
• model and solve contextualized
  problems using various representations,
  such as graphs, tables, and equations.
Analyze change in various contexts
• use graphs to analyze the nature of
  changes in quantities in linear
  relationships.
Function Junction
• Function Junction, what's your function?
  Hooking up numbers and expressing patterns.
• Function Junction, how's that function?
  I got five great representations,
• That get most of my job done.
• Function Junction, what's your representation?
  I got graphs, symbols, tables, contexts, and verbal
  descriptions!
• They'll get you pretty far.
• Graphs:
  That's
• Symbols:
  That's where you say y=3x + 2
• Don’t forget to make a table,
  talk about it, and act it out!
• These representations will
  Get you pretty far into function understanding.