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Write-up


Title
Number Machine


Problem Statement
I have a "number machine" that always affects in the same way whatever number I put in it. For example, when I put in 1, the machine gives me 6; when I put in 3, it gives me10; when I put in 6, it gives me 16; and when I put in 9, it gives me 22. What will the machine give me if I put in 100?

 

Plans to Solve/Investigate the Problem

The problem can be set-up initially in a table to identify a pattern. Then an equation can be determined.

 

Investigation/Exploration of the Problem

 

The following is a table reflecting number input and output and the rule/linear equation.

 

Input

Output

Rule

1

6

2(input#) + 4, or y = 2x +4

3

10

 

6

16

 

9

22

 

 

 

 Extensions of the Problem

My "number machine" gave me 42. What number did I put in my machine?   Based on the above equation, the problem could be solved by subtracting 4 from 42, then dividing 38 by 2 or 42-4=38/2=19

I purchased a new "number machine." This one gives me 1 when I put in 1;3 when I put in 2; 6 when I put in 3; 10 when I put in 4; and 15 when I put in 5. What number will I get when I put in 10? 20? 1000?  A table will help identify the pattern.  See the following:

Input

Output

Observations/Assumptions/Rules

1

1

There is a unique relationship between input and output.

2

3

The pattern between input and output when both are known appears to be new input plus old output.

3

6

Rule:  n(n+1)  or n2+n

               2            2

4

10

 

5

15

 

6

21

 

7

28

 

8

36

 

9

45

 

10

55

 

11

66

 

12

78

 

13

91

 

14

105

 

15

120

 

16

136

 

17

153

 

18

171

 

19

190

 

20

210

 

The equation should be used to calculate the answer to “What number will I get when I put in 1000?”

 

Author & Contact
Susan Kelly
mailto:skelly@rockdale.k12.ga.us