# Optimizing with Geometric Reasoning

There are a number of examples which illustrate how rate of change, and optimization using rate of change, can be made visible in geometric problems. We provide some examples of this general theme.

Overall, our claim is that:

(a) this can be done, over a wide range of grades;

(b) working these examples builds geometric reasoning which is helpful in further problem solving;

(c) these geometric connections 'make sense' (or is in make sensible?) the results which would also be found by calculation;

(d) In a number of problems, the dimension connections come alive: rate of change of volume appears as surface area(s), rate of change of area becomes perimeter, etc.

## The Popcorn Box Exploration

Here are some resources to support spatial reasoning explorations of a classical geometric optimization problem: the box of maximum volume formed from a square sheet of paper with squares cut from the corners and the sides folded up.

Here is a short summary of the key points

Here are some other supporting materials.

Class Handout used in a university capstone course for teachers (3 hrs of exploration). These were developed jointly with Ami Mamolo, in part to assist Data Collection.

2008 Draft article on this exploration. A revised version has been submitted for publication.

Companion pdf of photos of pairs to start an overview of the changes September 2016.

Companion GSP Sketch to support this exploration in April 2009.

Version of GSP pages from above - mounted on the web http://www.sfu.ca/~nathsinc/gsp/PopcornBoxBasic

A chart we used in a Grade 4 Classroom to move from individual pairs to an overview chart of the flow of changes in a set of models with cuts varying from 0 to 9 cm:

A powerpoint pdf which shows the pairwise comparisons needed to fill the chart.

Companion GSP Extended Sketch to explore extensions to other shapes.

A poster prepared by Robyn Ruttenberg on the connections which students encountered in the longer exploration:

## Maximizing a Triangle on a Circle

This is an exploration of the problem:

```  Given a three points on a circle, how should they be arranged to give the maximum area triangle?
```

This problem has a number of nice features:

(a) the analysis, with grinding calculation, is beyond most students - though they quickly guess the answer should be an equilateral triangle;

(b) when a key geometric strategy - variation of just one vertex to increase the area - students at many levels can get partial results based on simple geometric reasoning;

(c) moving from this partial information to the full answer can be done in several quite different ways.

(d) the 3-D extension is challenging: placing four points on a sphere so that the volume of the tetrahedron with these vertices is the maximum:

It is challenging and requires thinking carefully about how to use this first problem as a step in the larger problem.

GSP Sketch to support this exploration.

## Contact

For further information, contact Walter Whiteley: mailto://whiteley@mathstat.yorku.ca, 416-736-2100 ext 22598.