Discovering the Properties of a
Function
Suppose you would like to use
a secret code to hide your cell
phone telephone list from
your friends.
234-567-6543
Shade in the red squares to
show a matching of the input
and output values.
Fill in the chart at the bottom
of the page to show the
connection between the input
and output values.
Code your telephone number
or the cell phone number on
the screen.
OUTPUT
Let the input number be listed
at the bottom of the grid. The
output numbers are listed
along the left hand side.
INPUT
0 1
9 8
2
7
3
6
4
5
765-432-3456
5
4
6
3
7
2
8 9
1 0
Use the same coding.
876-545-6789
Suppose a number has been
coded as 876-545-6789
OUTPUT
Use the Coding Chart un-code
the coded cell phone number.
INPUT
0 1
9 8
2
7
3
6
4
5
5
4
123-454-3210
6
3
7
2
8 9
1 0
Will every cell phone number
receive a unique coded cell
phone number?
234-567-6543
123-454-3210
Output number = 9 - Input Number
Will every coded cell phone
number match with exactly
one original cell phone
number?
Can you find a rule that will
convert every original cell
phone number to a coded cell
phone number using the
coding chart at the right?
Can you find a rule that will
convert every coded cell
phone number back to its
original cell phone number?
Input Number = 9 – Output Number
0 1
9 8
2
7
765-432-3456
3
6
4
5
5
4
6
3
7
2
8 9
1 0
876-545-6789
Create a new coding system that
can be used to code a cell phone
number that is not as systematic as
the last coding chart.
908-234-7531
Step 1: Be sure that the coding
system you create will produce a
unique coded number.
Show your coding chart.
Try coding a telephone number by
using the one at the right.
Step 2: Can you undo the coded
telephone number using your
chart?
What is unique about your coding
system that makes it possible to
complete step 1 and 2?
0
3
1
8
2
2
3
6
4 5
9 4
035-269-2468
6
7
7 8 9
2 5 0
215-947-1836
Use the chart at the
right to decode a
telephone number
that has been coded
215-947-1836.
What’s the problem
with this coding
chart?
0
6
1
2
4
3 4
7 2
5 6
6 1
7
5
8
2
9
0
Try coding a telephone
number using this
coding:
212-123-4567
Can you describe the
problems you would
have with coding a
telephone number
using this coding
system.
0
5
1
5
2
5
3 4
5 4
5
4
6
4
555-555-4444
7 8
4 3
9
3
Can you create a
different coding that
could be used to create
a way to hide your cell
phone numbers?
What will have to be
true with your coding
to make it possible to
code your telephone
numbers and decode
your telephone
numbers.
For a coding chart to be
useful there must be
one unique output for
each input and one
unique input for each
output.
What is the function of a
coding chart?
How does this activity help
you understand what a
function does?
What is a
function?
It could be
thought of
as a
Function
Machine
Suppose
we put
some
numbers
in our
Function
Machine
3 456
5 6 78
Predict
what will
happen
when we
put 10 in
our
Function
Machine
3, 4, 5, 6, . . . 10
5, 6, 7, 8, . . . 12
Predict what
3,
4,
5,
6,
.
.
.
10
,13
number
must be put
in our
Function
Machine to
get an
output of 15
5, 6, 7, 8, . . . 12 ,15
3, 4, 5, 6, . . . 10 ,13
What is the
Function of
this
machine?
Add 2
5, 6, 7, 8, . . . 12 ,15
You just discovered the
Function of one specific
Function Machine
The function
of the
machine could
be described
in words and
you have to
show various
inputs and
outputs:
20,12, 8, 4, 3,1, 0
Cut a number in
half
Hold up your
Communicators
and let me check
your output values.
20,12, 8, 4, 3,1, 0
Cut a number in
half
10,6, 4, 2, 1.5,0.5, 0
You can find the input values
given the description of the
Function Machine and the
output values.
What input
values must be
put in this
Function
Machine to
obtain the
following
output values?
X4
20,12, 8, 4,1, 0
5,3, 2, 1,1/4, 0
Hold up your
Communicators so
I can check your
responses.
X4
20,12, 8, 4,1, 0
How does a
Function
Machine help
you understand
what a function
does in
Mathematics?
Find
the
area of
a
square
given
its side.
3
3
9
Collecting data.
Find
the
area of
a
square
given
its side.
3
3
9
s
3
9
This time place your input and outputs in a
table.
Find
the
area of
a
square
given
its side.
s
3
9
4
5
6
7
8
9
10
11
12
16
25
36
49
64
81
100
121
144
List the output values for the given input
values.
3
Find
the
area of
a
square
given
its side.
s
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
3
Can you tell me the area of a square whose
side is 15?
Find
the
area of
a
square
given
its side.
s
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
3
Can you tell me what square will have an
area of 400?
A function can be a
descriptive relationship.
How
far is a
value
from
zero?
4
5
6
-4
-5
-6
4
5
6
4
5
6
Place 4 additional number in your chart and in the
function. Find the output values.
A function can be a
description about a
geometric relationship.
Suppose we start with a
rectangle that measures 1”
by 2”. What is its area?
2”
8 sq.in.
4”
Let’s dilate the shape by a factor of 2
1”
2 sq.in.
:
2”
What is the size of the new
rectangle?
What is its area?
What is
the area
of a 1 x 2
rectangle
if it is
dilated by
a factor of
…
s
1”
1
2
3
4
2
8
18
32
2 sq.in.
:
2”
Place 2 input numbers in your chart and in the
function. What are the output values?
What is
the area
of a 1 x 2
rectangle
if it is
dilated by
a factor of
…
s
1”
1
2
3
4
2
8
18
32
2 sq.in.
:
2”
Place 2 additional input numbers in your chart and in
the function. What are the output values?
What is
the area
of a 1 x 2
rectangle
if it is
dilated by
a factor of
…
1, 2, 3, 4,
s
1
2
2
8
3
18
4
32
10
200
2, 8, 18, 32 , 200
If a dilation causes a 1 x 2 rectangle to increase to an
area of 200 what dilation took place?
2, 2, 2, 2, 2, 2, 2
s
2
1
2
2
2
3
2
4
2
5
2
6
1, 2, 3, 4, 5, 6
What is the function of this machine?
2, 2, 2, 2, 2, 2, 2
s
2
1
2
2
2
3
2
4
2
5
2
6
1, 2, 3, 4, 5, 6
We would say this is not a function because the one input
value has many output values.
How does this template help you understand what a
function is in mathematics?
Functions can also paint a
picture
This time we
will graph
the input
and output
values.
2, 3, 4
X3
6, 9, 12
Horizontal Axis: Inputs
Vertical Axis: Outputs
2, 3, 4
X3
6, 9, 12
What patterns do you see?
Can you predict other points that will be on
the graph?
2, 3, 4
X3
6, 9, 12
Does the graph paint a picture?
2, 3, 4
Change
to 4
4, 4, 4
Horizontal Axis: Inputs
Vertical Axis: Outputs
2, 3, 4
Change
to 4
4, 4, 4
What picture is painted for this function?
Write a sentence that describes the output
values.
How does this template help you understand what a
function is in mathematics?