## How To Write Mathematics

Writing Mathematics is a key part of the introductory precalculus and calculus courses at the University of Michigan (Math 105/115/116). The objective of this website is to prepare students to organize and present mathematics in the rigorous manner expected of them in these courses.

## What Is "Restating the Problem"

Whenever it is possible, a solution should begin with a restatement of the problem.

You Restate The Problem by explaining what the problem is asking for as you understand it.

Often this includes introducing and assigning symbols to variables. Needless to say, copying the problem verbatim out of the text is not restating the problem. Two benefits of clearly restating the problem are:

• It helps you organize the way in which you present the solution.
• The grader can tell whether you understand the problem.

## Video Tutorial

Below is a video tutorial discussing how to restate a problem.

All problems are from:

Deborah Hughes-Hallett, Andrew Gleason, et al.: Calculus: Single Variable, Fourth Edition, Wiley, 2004

## Test Yourself

### Instructions

Read the following team homework problem. Below it are four possible restatements of the problem. Think about which one best fits the criteria described above and then check your answer.

### The Question

Each planet moves around the sun in an elliptical orbit. The orbital period, T, of a planet is the time it takes the planet to go once around the sun. The semimajor axis of each planet's orbit is the average of the largest and the smallest distances between the planet and the sun. Johannes Kepler (1571-1630) discovered that the period of a planet is proportional to the 3/2 power of its semimajor axis. What is the orbiting period (in days) of Mercury, the closest planet to the sun, with a semimajor axis of 58 million km? What is the period (in years) of Pluto, the farthest planet, with a semimajor axis of 6000 million km? The semimajor axis of the earth is 150 million km. [Hint: What is earth's period?]

### Possible Restatements Of The Problem

(a) Each planet moves around the sun in an elliptical orbit. The orbital period, T, of a planet is the time it takes the planet to go once around the sun. The semimajor axis of each planet's orbit is the average of the largest and the smallest distances between the planet and the sun. Johannes Kepler (1571-1630) discovered that the period of a planet is proportional to the 3/2 power of its semimajor axis. What is the orbiting period (in days) of Mercury, the closest planet to the sun, with a semimajor axis of 58 million km? What is the period (in years) of Pluto, the farthest planet, with a semimajor axis of 6000 million km? The semimajor axis of the earth is 150 million km. [Hint: What is earth's period?]

(b) The orbital period, T, of a planet is the time it takes the planet to go once around the sun. The semimajor axis, S, of each planet's orbit is the average of the largest and the smallest distances between the planet in the sun. In the figure, for Mercury, S=(A+B)/2. We are given that T=k*S3/2 where k is a constant of proportionality and that the semimajor axis of the earth is 150 million km. What is the orbiting period in days of Mercury with a semimajor axis of 58 million km? What is the orbiting period in years of Pluto, with a semimajor axis of 6000 million km?