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Discovering the Properties of a Function

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.

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Discovering the Properties of a Function

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  1. Discovering the Properties of a Function

  2. 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 OUTPUT INPUT 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 765-432-3456

  3. 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 OUTPUT INPUT 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 123-454-3210

  4. 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? 234-567-6543 123-454-3210 Output number = 9 - Input Number Input Number = 9 – Output Number 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 765-432-3456 876-545-6789

  5. 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? 908-234-7531 0 1 2 3 4 5 6 7 8 9 3 8 2 6 9 4 7 2 5 0 035-269-2468

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

  7. 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. 212-123-4567 0 1 2 3 4 5 6 7 8 9 5 5 5 5 4 4 4 4 3 3 555-555-4444

  8. 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.

  9. For a coding chart to be useful there must be one unique output for each input and one unique input for each output.

  10. What is the function of a coding chart? How does this activity help you understand what a function does?

  11. What is a function? It could be thought of as a Function Machine

  12. 4 3 5 6 Suppose we put some numbers in our Function Machine 5 6 7 8

  13. Predict what will happen when we put 10 in our Function Machine 3, 4, 5, 6, . . . 10 5, 6, 7, 8, . . . 12

  14. Predict what number must be put in our Function Machine to get an output of 15 3, 4, 5, 6, . . . 10 ,13 5, 6, 7, 8, . . . 12 ,15

  15. 3, 4, 5, 6, . . . 10 ,13 What is the Function of this machine? Add 2 5, 6, 7, 8, . . . 12 ,15

  16. You just discovered the Function of one specific Function Machine

  17. 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

  18. 20,12, 8, 4, 3,1, 0 Hold up your Communicators and let me check your output values. Cut a number in half 10,6, 4, 2, 1.5,0.5, 0

  19. You can find the input values given the description of the Function Machine and the output values.

  20. What input values must be put in this Function Machine to obtain the following output values? X 4 20,12, 8, 4,1, 0

  21. 5,3, 2, 1,1/4, 0 Hold up your Communicators so I can check your responses. X 4 20,12, 8, 4,1, 0

  22. How does a Function Machine help you understand what a function does in Mathematics?

  23. Find the area of a square given its side. 3 3 9 Collecting data.

  24. Find the area of a square given its side. 3 3 9 3 9 This time place your input and outputs in a table.

  25. Find the area of a square given its side. 3 9 16 25 36 49 64 81 100 121 144 4 5 6 7 8 9 10 11 12 3 List the output values for the given input values.

  26. Find the area of a square given its side. 3 9 • 16 5 25 6 36 7 49 3 8 64 9 81 10 100 Can you tell me the area of a square whose side is 15?

  27. Find the area of a square given its side. 3 9 • 16 5 25 6 36 7 49 3 8 64 9 81 10 100 Can you tell me what square will have an area of 400?

  28. A function can be a descriptive relationship.

  29. 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.

  30. A function can be a description about a geometric relationship.

  31. Suppose we start with a rectangle that measures 1” by 2”. What is its area? 8 sq.in. 2” 4” Let’s dilate the shape by a factor of 2 : 1” 2 sq.in. What is the size of the new rectangle? What is its area? 2”

  32. What is the area of a 1 x 2 rectangle if it is dilated by a factor of … 1 2 2 8 3 4 18 32 2 sq.in. : 1” 2” Place 2 input numbers in your chart and in the function. What are the output values?

  33. What is the area of a 1 x 2 rectangle if it is dilated by a factor of … 1 2 2 8 3 4 18 32 2 sq.in. : 1” 2” Place 2 additional input numbers in your chart and in the function. What are the output values?

  34. What is the area of a 1 x 2 rectangle if it is dilated by a factor of … 1, 2, 3, 4, 1 2 • 2 8 3 18 4 32 200 10 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?

  35. 2, 2, 2, 2, 2, 2, 2 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?

  36. 2, 2, 2, 2, 2, 2, 2 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.

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

  38. Functions can also paint a picture

  39. This time we will graph the input and output values. 2, 3, 4 X 3 6, 9, 12 Horizontal Axis: Inputs Vertical Axis: Outputs

  40. 2, 3, 4 X 3 6, 9, 12 What patterns do you see? Can you predict other points that will be on the graph?

  41. 2, 3, 4 X 3 6, 9, 12 Does the graph paint a picture?

  42. 2, 3, 4 Change to 4 4, 4, 4 Horizontal Axis: Inputs Vertical Axis: Outputs

  43. 2, 3, 4 Change to 4 4, 4, 4 What picture is painted for this function? Write a sentence that describes the output values.

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

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