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. Let the.

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.

Let the input number be listed at the bottom of the grid. The output numbers are listed along the left hand side.

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.

234-567-6543

0 1 2 3 4 5 6 7 8 99 8 7 6 5 4 3 2 1 0

765-432-3456

INPUTO

UTP

UT

Use the same coding.

Suppose a number has been coded as 876-545-6789

Use the Coding Chart un-code the coded cell phone number.

876-545-6789

123-454-3210

INPUTO

UTP

UT

0 1 2 3 4 5 6 7 8 99 8 7 6 5 4 3 2 1 0

Will every cell phone number receive a unique coded cell phone 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?

123-454-3210

876-545-6789

234-567-6543

765-432-3456

Output number = 9 - Input Number

Input Number = 9 – Output Number

0 1 2 3 4 5 6 7 8 99 8 7 6 5 4 3 2 1 0

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.

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 1 2 3 4 5 6 7 8 93 8 2 6 9 4 7 2 5 0

908-234-7531

035-269-2468

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 1 2 3 4 5 6 7 8 96 4 7 2 6 1 5 2 0

215-947-1836

Try coding a telephone number using this coding:

Can you describe the problems you would have with coding a telephone number using this coding system.

0 1 2 3 4 5 6 7 8 95 5 5 5 4 4 4 4 3 3

212-123-4567

555-555-4444

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

5

4

6

5

7

6

8

Predict what will happen when we put 10 in our Function Machine

3, 4, 5, 6, . . .

5, 6, 7, 8, . . .

10

12

Predict what number must be put in our Function Machine to get an output of 15

3, 4, 5, 6, . . .

5, 6, 7, 8, . . .

10

12 ,15

,13

What is the Function of this machine?

3, 4, 5, 6, . . .

5, 6, 7, 8, . . .

10

12 ,15

,13

Add 2

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:

Cut a number in half

20,12, 8, 4, 3,1, 0

Hold up your Communicators and let me check

your output values.Cut a number in half

20,12, 8, 4, 3,1, 0

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?

X 4

20,12, 8, 4,1, 0

Hold up your Communicators so

I can check your responses. X 4

20,12, 8, 4,1, 0

5,3, 2, 1,1/4, 0

How does a Function

Machine help you understand what a function

does in Mathematics?

Collecting data.

Find the area of a square given its side.

3

9

3

This time place your input and outputs in a table.

s

Find the area of a square given its side.

3

9

3 9

3

List the output values for the given input values.

s

Find the area of a square given its side.

3

16 25 36 4964 81

100121144

3 9

4 567 89101112

Can you tell me the area of a square whose side is 15?

s

Find the area of a square given its side.

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?

s

Find the area of a square given its side.

3 9

4 16

5 25

6 36

7 49

8 64

9 81

10 100

3

A function can be a descriptive relationship.

Place 4 additional number in your chart and in the function. Find the output values.

How far is a value from zero?

4 56 45 6

4 5 6-4 -5 -6

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?

Let’s dilate the shape by a factor of 2

What is the size of the new rectangle?What is its area?

:1”

2”

2”

4”

2 sq.in.

8 sq.in.

Place 2 input numbers in your chart and in the function. What are the output values?

s

What is the area of a 1 x 2 rectangle if it is dilated by a factor of …

28

1 2

:1”

2”

2 sq.in.

34

1832

Place 2 additional input numbers in your chart and in the function. What are the output values?

s

What is the area of a 1 x 2 rectangle if it is dilated by a factor of …

28

1 2

:1”

2”

2 sq.in.

34

1832

If a dilation causes a 1 x 2 rectangle to increase to an area of 200 what dilation took place?

s

What is the area of a 1 x 2 rectangle if it is dilated by a factor of …

2, 8, 18, 32

1 2

1, 2, 3, 4,

2 8

3 18

4 32

, 200

20010

1, 2, 3, 4, 5, 6

2, 2, 2, 2, 2, 2, 2

s2 3

2 1

2 2

2 4

2 5

2 6

What is the function of this machine?

s

1, 2, 3, 4, 5, 6

2, 2, 2, 2, 2, 2, 2

2 3

2 1

2 2

2 4

2 5

2 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

X 3

2, 3, 4

6, 9, 12

This time we will graph the input and output values.

Horizontal Axis: InputsVertical Axis: Outputs

X 3

2, 3, 4

6, 9, 12

What patterns do you see?Can you predict other points that will be on the graph?

X 3

2, 3, 4

6, 9, 12

Does the graph paint a picture?

2, 3, 4

4, 4, 4

Horizontal Axis: InputsVertical Axis: Outputs

Change to 4

2, 3, 4

4, 4, 4What picture is painted for this function?Write a sentence that describes the output values.

Change to 4

How does this template help you understand what a function is in mathematics?