Reservoir Problems

You work for the World Food Organization (WFO) and have been asked to locate a new reservoir that two villages will use. Optimally you want to locate the reservoir so that it is equidistant from the villages. Where should the reservoir be placed?
Since you were so successful with your first task, you have now been asked to locate a new reservoir that three villages will use. Describe how you will locate the best spot for the reservoir.
You are helping the WFO to locate another reservoir, but this time four villages will use the reservoir. Describe how you will locate the best spot for the reservoir.

David Henderson's Proof as a Convincing Communication that Answers -- Why?

I believe that much of the problem of teaching proofs is that we do not give a useful definition of what we mean by "proof". There is a formal definition of proof but, in my experience, this is not what most mathematicians use; and I find that the formal notion of proof deadens the class and the students learning. I propose the following definition as being closer to the way that mathematicians actually work and closer to how we want our students to work.

A proof is a convincing communication that answers -- Why? It is not a formal proof -- computers can now find and check these. What we need are alive human proofs that are:

communications -- When we prove something we are not done until we can communicate it to others and the nature of this communication, of course, depends on the community to which one is communicating and is thus in part a social phenomenon.

convincing -- A proof "works" when it convinces others. Of course some persons become convinced too easily so we are more confident in the proof if it convinces some one who was originally a skeptic. Also, a proof that convinces me may not convince you or my students.

answer -- Why? -- The proof should explain, especially it should explain something that the hearer of the proof wants to have explained. I think most people in mathematics have had the experience of logically following a proof step by step but are still dissatisfied because it did not answer questions were of the sort: "Why is it true?" "Where did it come from?" "How did you see it?" "What does it mean?"

Standards for School Mathematics

Instructional programs from prekindergarten through grade 12 should enable all students to...

Related Content Standards

Geometry Standard

analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;

specify locations and describe spatial relationships using coordinate geometry and other representational systems;

apply transformations and use symmetry to analyze mathematical situations;

use visualization, spatial reasoning, and geometric modeling to solve problems.

Measurement Standard

understand measurable attributes of objects and the units, systems, and processes of measurement;

apply appropriate techniques, tools, and formulas to determine measurements.

Related Process Standards

Problem Solving Standard

build new mathematical knowledge through problem solving;

solve problems that arise in mathematics and in other contexts;

apply and adapt a variety of appropriate strategies to solve problems;

monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof Standard

recognize reasoning and proof as fundamental aspects of mathematics;

make and investigate mathematical conjectures;

develop and evaluate mathematical arguments and proofs;

select and use various types of reasoning and methods of proof.

Communication Standard

organize and consolidate their mathematical thinking through communication;

communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

analyze and evaluate the mathematical thinking and strategies of others;

use the language of mathematics to express mathematical ideas precisely.