## Problem Statement

The “One Mile at a Time” problem asked us to solve for the speed of an individual under certain circumstances. First, it asked us for the speed when, after an hour of their trip, the travelers saw a milepost with a two digit number. One hour later, they saw another milepost with the same two digits but backwards. After another hour, they saw another milepost with a three digit number using the same milepost as the first milepost except with a zero between the two digits.

## Process

We started by trying random numbers like 28 but then realized we should go in an order to see if we could find a pattern. we started with 01 and went in increasing order. We would write out what the numbers would be for each milepost and then figure out the miles traveled between each milepost. We knew the mileage had to be the same between the first and second milepost as the mileage between the second and third milepost. We started realizing that the mileage between each milepost was always a multiple of 9 but didn’t understand how that tied in. When finding the algebraic expression, we remembered that each distance traveled was a multiple of 9 and tried to include the number 9 in my algebraic terms. We saw that 16 was pretty close to 18, a multiple of nine so we set up 16 as (9*2)-2 and tried something similar for the next to mileposts. We realized they were both 2 miles short of a multiple of nine and went on to expand 61 to (9*5)-2 and 106 to (9*12)-2. Then we saw the pattern that each time you are multiplying 9 by 5 more than the last. |

## Solution

As we were listing the numbers, we reached 16 making the second milepost 61 and the third milepost 106. They each had a distance of 45 between each milepost. Our answer for the algebraic terms is

9(2+(5(P-1)))-2=M. “P” being the post number, and “M” being the miles on the post.

9(2+(5(P-1)))-2=M. “P” being the post number, and “M” being the miles on the post.

## Self Assessment

For this problem we mainly used hypothesizing and experimenting, because we hypothesized that if we kept with my pattern, eventually some numbers would match up and then experimented with the numbers. We also did a lot of experimenting when we were trying to find an algebraic explanation for the solution.

## Problem Assessment

I thought this was a really hard problem to start because when I tried the first number, I thought that it wasn’t possible. But once I started doing it in order, I realized that it was just a matter of time until one of the numbers matched up. I was really confused with the second part of explaining it in algebraic terms because there are dozens of ways to represent any number using different equations and didn’t think I could find one that could help me see something in common between the numbers on each milepost.