# Review of Dr. Math® Introduces GEOMETRY and

Dr. Math® Presents More GEOMETRY by The Math Forum, Drexel University

Enjoyable reading. Excellent down-to-earth explanations. Right price.

I don't know how else to put this book in words. Just like the algebra books, Dr. Math's geometry books are **very easy reading, with light tone, clear layout, and humorous cartoons** by Jessica Wolk-Stanley. But the best part is of course that the **explanations to math questions are superb, accurate, and clear**.

Dr. Math Introduces GEOMETRY is meant for middle school, and Dr. Math Presents More GEOMETRY is for high school. These two books are meant for study companions or supplements for a geometry curriculum. They contain questions/answers that have been collected from Dr. Math's archives - asked by real students over several years, and answered by experts at the Math Forum.

The questions are chosen and organized so that together with the answers, the material covers most of the geometry curriculum in a coherent way. Please note though that the books do not contain practice exercises.

Dr. Math books don't really read like a textbook but something much better; the reader feels much more involved in the text. I feel the authors and the compilers have done an excellent job in readibility of all of the Dr. Math books.

Description of Dr. Math^{®} Introduces Geometry (middle school)

Description of Dr. Math^{®} Presents More Geometry (high school)

## Dr. Math® Introduces Geometry

**Introduction to Two-Dimensional Geometric Figures** not only shows the figures but concentrates on helping the student learn and remember all those difficult terms and words in geometry (obtuse, reflex, scalene, supplemental angles, alternate exterior angles etc.). The terms become so much easier when you know where they come from. For example, "isosceles" comes from "iso-" and "skelos" and means same-legged. Dr. Math also presents mnemonic tricks to remember supplemental/complementary angles, or the 45-45-90 and 30-60-90 triangles.

The chapter (as do all of them) ends with web links where you can learn more or explore and investigate with interactive models.

**Areas and Perimeters of Two-Dimensional Geometric Figures** explains problems that involve both area and perimeter. I was delighted at the answer of why increasing the perimeter doesn't necessarily increase the area, and at the visual proof of the area formula for a trapezoid. (I always enjoy learning new proofs.) Also explained are some confusing points, such as the difference between length times width and base times height, or "meters squared" versus "square meters."

The third chapter, **Circles and Pi**, first goes to the nitty-gritty of radius and diameter – you may not have thought of these two this profoundly. (We actually use the word 'radius' to mean two different things.) Dr. Math is not afraid to use common language such as "pie are square" alongside A = πr^{2}. There is a proof for the area formula, and an excellent question on philosophical lines: how could one length of something be measured as an exact number and another not (relating to the fact that either the radius or the circumference must be irrational)? School mathematics does not always explain such things.

In the next chapter, **Introduction to Three-Dimensional (3-D) Geometric Figures**, we first learn the basics of polyhedra and platonic solids. The section on surface area emphasizes how to "figure it out" without memorizing formulas. It also includes a very important question about 2 cups (of rice) and cubic inches both being measures of volume. Also touched on are the nets of solids.

**Symmetry** deals with geometrical transformations, different types of symmetry, and briefly on tessellations. This section includes lots of pictures; yet it felt more like being on an introductory level.

I was happy to see that Dr. Math explained the proof for the existence of only three regular polygon tessellations. We need students to read proofs, too. (This wasn't the only proof in the book, by the way.)

## Dr. Math® Presents More Geometry

The first part, **Points, Lines, Planes, Angles, and Their Relationships**, talks about parallel lines and angles, gives a short explanation of non-Euclidean geometries, and touches on the distance formula and coordinate system.

The part **Logic and Proof** gives numerous examples of how to build two-column proofs, and tackles many common questions, misunderstandings, and difficulties students typically have about proving.

Dr. Math explains how to think of the proof as a 'bridge', how to get started, how mathematicians often build the proof 'backwards' and only later write it neatly, etc. I want to emphasize that the examples here aren't just the proofs themselves but include the thought processes of how a math doctor finds the arguments that make up the proof, and how it is later written down into the rigid two-column format. Excellent chapter.

There is a lot to learn about triangles, and the part **Triangles: Properties, Congruence and Similarity** again presents an excellent collection of questions-answers to cover these things. It is like a teacher talking personally to you, sometimes not giving you quite all the answers, while still explaining the basics of how the answer is found.

The fourth part has to do with **Quadrilaterals and other Polygons**. You learn about interior and outerior angles of polygons, diagonals and heights, area and perimeter.

The problems here are very varied: what happens to the area and perimeter if a rectangle is "pressed on its corners," if a parallelogram is tilted, proving diagonals perpendicular, or proving diagonals of a trapezoid congruent, and so on. And there is help for learning the areas of different shapes – how to derive the formulas instead of memorizing them.

Lastly, **Circles and Their Parts.** Tangent problems seem so easy with the diagrams and explanations of Dr. Math. The connection between the perpendicular bisector of a chord and the center of circle is used and made clear in several problems. A few proofs here are just a bit more challenging, for example the construction of a line tangent to two circles, or the problem involving "power of a point" concept.

All in all, these geometry companion books are packed full of excellent easy-reading explanations and are WELL worth their low price (around $8-$10). Highly recommended.

Dr. Math Introduces GEOMETRY and Dr. Math Presents More GEOMETRY by The Math Forum. Prices around $12 per book, or $4-7 used. Kindle versions available.

*Review by Maria Miller*