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 setup 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 424=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
