# Function Junction, What�s your function? by w6WLescw

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```									Function Junction, What’s
• 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?
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
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).
Pledge Plans
Let’s use verbal            Jeff: y = 1.5x
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)
intercepts?
What do the points of
intersection mean?
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

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”
• 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 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
between the algebraic
equations and graphic
representations.
• What does
–4 < x < 4 and –3 < y < 3
• 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 Graphs to Stories
Let’s explain these graphs.
1. What happened when John and
his father raced?
profit graph.
3. Describe the raising of the flag
graph.
4. Describe the race between the
two swimmers.

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.

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