Function Junction, Whats your function

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Function Junction, Whats 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!

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Function Junction Beginning Ideas of Function

• Lynn Stallings, Ph.D.
• lstalling_at_kennesaw.edu
• Kennesaw State University,
• Atlanta, Georgia, USA

National Council of Teachers of Mathematics 2004
Conference
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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?

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Function Junction, Whats 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 functionsincluding numeric, graphic, and
  symbolicthey will develop a more comprehensive
  understanding of functions. (PSSM, 2000)

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Starring . . . PSSM Process Standard Representat
ionAlso featuring Communications, Connections,
Reasoning, and Problem Solving
Where do we typically start?
Whats a good mnemonic for these?
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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?

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Hows 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.

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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?

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from PSSM, 2000ChitChat 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.

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ChitChat vs. Keep-in-Touch
L1 number of minutes, L2 Keep-in-Touch
cellular phone rates, L3 ChitChat rates
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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.
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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
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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?
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Guess the relationship . . .
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If 8 is the input, what will the output be?
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Whats the relationship?
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7 was the input more than once. What was the
output? Is that important?
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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

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From Navigations Pledge Plans

• What equation should we use for each plan?
• Jeff 1.50 per kilometer
• Rachel 2.50 per kilometer
• Annie 4.00 donation plus 0.75 per kilometer
• What should those equations represent?
• The relationship between the number of kms (lets
  call it x) walked and the amount of money pledged
  (lets call it y).

Jeff y 1.5x Rachel y 2.5x Annie y 4
.75x Press Y and input the three
equations. Press WINDOW.
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From Navigations Pledge Plans

• Lets use verbal description to link the graph,
  the algebraic equations, and the table
• What do the slopes mean?
• What about y-intercepts?
• What do the points of intersection mean?

Lets use verbal description to link the graph,
the algebraic equations, and the table What do
the slopes mean? What about y-intercepts? What do
the points of intersection mean?
Jeff y 1.5x Rachel y 2.5x Annie y 4
.75x
Rachel
Jeff
Annie
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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.
• Well record the data for 20 feet and then use it
  to figure out the rest.

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From Navigations Walking Strides

• How would we fill in the rest of the chart?
• Lets let x time and y distance walked.
• Can we graph these? What equations could we use?
• What do the slopes mean?
• What do y-intercepts mean? What would you do to
  change the y-intercept?

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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?
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Match the Graph

• Press PRGM on your calculator. If you dont 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 youll want
  to go back to Notes for Teachers.

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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?

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Transformation Creations

• Rain is a way to practice your connections
  between the algebraic equations and graphic
  representations.
• What does
• 4 lt x lt 4 and 3 lt y lt 3
• tell you about the graph?
• How do we input that into the calculator?

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Transformation Creations

• How can you make Rain on your calculators?

• How did you know?

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From NavigationsFrom Graphs to Stories

• Lets explain these graphs.
• What happened when John and his father raced?
• Describe the lemonade stands profit graph.
• Describe the raising of the flag graph.
• Describe the race between the two swimmers.

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From NavigationsFrom Stories to Graphs

• Sketch a graph for each of these three stories.
  Note that number two will vary the most.
• Josephines walk Mowing the lawn

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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?

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PSSM standard on Algebra says . . . In grades
68 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.

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PSSM standard on Algebra says . . . In grades
68 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

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PSSM standard on Algebra says . . . In grades
68 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.

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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.
• GraphsThat's
• SymbolsThat's where you say y3x 2
• Dont forget to make a table,
• talk about it, and act it out!
• These representations will Get you pretty far
  into function understanding.