Algebra for All 2010 - 2011

Mile Markers

The mile markers you have seen along the expressway date back to the Roman Empire. Mile markers provide reference points along the road. They are used to by maintenance engineers and emergency services to direct them to specific points where they are needed.

In the US, mile markers generally tell how far it is to the southern or western border of that state. For example there is a Saginaw exit from I-75 at mile marker 150. That exit is 150 miles from the Michigan-Ohio border.

Mile markers can also be used to find the distance you have traveled from a starting point. For the table below, use mile marker 150 as your starting point.

Expand the table below.

Distance Traveled Find some additional values for your table. from Mile Marker 150 What patterns do you see in the table? Mile Distance Marker Traveled 200 175 150 0 140 100

Use the information in the table to create a graph.

Write a rule that you can use to find the distance traveled (from marker 150) using the mile marker at your current location.

Can you write your rule in another way?

Kristin Chatas [email protected] Leo Paveglio [email protected] Algebra for All 2010 - 2011

Teacher Notes

Students are likely to express their rule in two forms: d = |150 – m| or d = | m – 150| Quantitatively they are equivalent, however when your students are in their physical science classrooms they will learn displacement is defined as d = end point – starting point, and distance is the absolute value of displacement. The discussion is worth having to help students clarify their understanding of absolute value.

A discussion centered around a third column for the table that shows the calculations that students actually used to complete their table can be a foundation for defining the function as a piecewise function.

Distance Traveled from Mile Marker 150 Mile Distance Marker Traveled 200 200 – 150 175 175 – 150 150 0 150 – 150 140 150 – 140 100 150 - 100 d = m – 150, if m > 150 d = 150 – m, if m < 150

This foreshadows the definition of absolute value.

Kristin Chatas [email protected] Leo Paveglio [email protected]