### 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:

### 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.