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Mathematics Governor’s Institute 2003

Problem- in-a-bag Template

 

(Download as Microsoft Word documents: Adopt a Highway, Question)

 

Title of Project:          Adopt – A – Highway

 

Team Members:        Gerald Burkepile, Gary Luft,  Pamela Mulhollem,  Lori Pawluck

 

Grade Level:              Algebra II

 

Concepts Used:          Distance, Rate, Time, Ratio, Problem Solving

 

PA Standards Addressed: 2.1A, 2.2A, 2.2F, 2.4E, 2.5A, 2.5B, 2.5C, 2.8D, 2.8G, 2.8H, 2.8I

 

NCTM Standards Addressed:

: Compute fluently and make reasonable estimates

: Represent and analyze mathematical situations and structures using algebraic symbols  

: Use mathematical models to represent and understand quantitative relationships

: Understand measurable attributes of objects and the units, systems, and processes of

measurement

: Apply appropriate techniques, tools, and formulas to determine measurements

 

Introduction / applications: This problem was developed to find the most efficient way to complete a local Adopt-A-Highway cleanup. Eight youth group volunteers are to clean a 2-˝ mile stretch of highway for the Adopt-A-Highway Program. They will work in pairs and can only pick-up one side of the road at a time. Two volunteers are dropped at the beginning, four volunteers are dropped at a point somewhere in the middle and the remaining volunteers start from the other end. This is accomplished by working from their drop off point toward the other groups until they meet and then turn and clean the opposite side of the road on their way back. (As illustrated in the diagram)   Pairs of volunteers walk at an average rate of 2 mph.  The van travels at an average rate of 15 mph. The van stops for 5 minutes at each drop for preparation after the first group begins.

 

Question: How far is the driver to go from the starting point before dropping off the group of four in order that all volunteers finish at the same time, and how long will it take? 

 

Model:

 

 

 

Resources and Materials (estimated cost): graphing calculator optional

 

Procedures & Activities: It may be helpful to build a model to aid students visualizing of the problem, however pencil and paper is sufficient.

 

Answer:

Step 1: A system of equations is needed to find the values of and  with  representing total time.

 

A.        Because these two groups cover the same distance in the same time.

 

B.         Since  using substitution

             

 

C.        Using the formula

time is divided by 2 because ˝  the time is spent on one side of the road.

 

D.       

           

 

E.

           

 

 

 

                       

Step 2:            Using the final equations from Step 1, B, D, and E, the following system is derived:    

                       

                       

 

            Using the matrix function on the graphing calculator to solve the system the following answers are obtained:

                                               

 

Step 3:  The drop off point for the group of 4 volunteers would be  miles from the starting point.

 

The time required to complete the adopt-a-highway program as shown in step 1C would be 0.79 hours.

 

 

 

 

 

 

 

Accommodations / Adaptations: This problem may be extended by changing the times, rates, or length of road.

 

 

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