The Discovering Geometry textbook is currently published by Kendall Hunt Publishing. In August 2011, Key Curriculum Press sold their high school mathematics textbooks to Kendall Hunt Publishing. One year later, Key Curriculum was purchased by McGraw Hill.

In August 2012, all rights for Michael Serra's books reverted back to him except Discovering Geometry. They are published by Playing It Smart, which is owned by Angie and Michael Serra and available on this website.

Kendall Hunt offers a free online trial of Discovering Geometry on their website. 

Discovering Geometry is an inquiry based learning program that was developed before the CCSS were mandated, yet it is relevant today at every level. DG helps to develop the following skills:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

More information can be found here.

In Discovering Geometry we begin with students discovering their vocabulary. Students perform investigations that lead to an in-depth understanding of the definitions of the basic terms. For example, in one investigation students observe samples of trapezoidal and non-trapezoidal figures. Student groups then discuss the characteristics of trapezoids, and create their own definition. Finally, each group tests and fine-tunes that definition against the examples, presents their definitions to the rest of the class, and a consensus is reached. Each definition, along with its illustration, is added to the students’ vocabulary lists in their notebooks.

Discovering Geometry offers a number of features including:

• Hands-on activities

• Constant review

• Careful slow development of proof

• Algebra review lessons

In addition, supplementary materials are available:

1. Condensed lessons for make-up work

2. Practice Your Skills with Answers

3. DG Investigation videos

4. More Projects!

Almost every lesson involves students working in groups and performing geometric investigations with manipulatives. The investigations use a wide range of readily available manipulatives including compasses, straightedges, rulers, protractors, scissors, heavy stock paper, graph paper, patty paper, square dot paper, isometric dot paper, glue sticks, tape, string, wooden or plastic cubes, mirrors, and pattern blocks.

We recommend commercially available geometric solids for discovering volume and surface area formulas for prisms, pyramids, cylinders, cones, and spheres. Students typically use graphing calculators and geometry software, such as The Geometer’s Sketchpad, for investigations and extra explorations. Projects and Explorations in the text and supplementary materials also encourage the use of hands-on materials. Students build arches out of Chinese take-out cartons, kaleidoscopes out of paper towel tubes, and clinometers and hypsometers out of cardboard and straws.

Moving students from the concrete to the abstract is at the heart of Discovering Geometry. We begin with students discovering the basic vocabulary and properties of geometry through hands-on investigations. Students work with the tools of geometry, build models, perform geometric investigations, and then make and test conjectures about their experiences. As they gain confidence with the terms and concepts of geometry, we gradually begin to ask them to explain what they did and why they think their discovery is true. See comments on van Hiele levels and development of proof here.

Right from the start (Chapter 0: Geometric Art) students use a ruler and compass to create geometric art. In Chapters One and Two students use a ruler, protractor, and compass to perform investigations and discover geometric properties (vertical angles and the properties of parallel lines). In Chapter Three we introduce the classic Euclidean compass and straightedge constructions. This is done inductively with students discovering the construction methods rather than the constructions being a set of rules or steps to memorize. The Euclidean constructions are blended with paper folding constructions (Patty Paper® Geometry) to reinforce the properties being discovered. We also recommend that teachers use The Geometer’s Sketchpad at this time, as a third way to reinforce the concepts students are learning. The use of the compass and straightedge as tools for construction exercises and discovery continue throughout the text.

Discovering Geometry's primary focus is getting students to work together to discover and discuss geometry concepts in a student-centered environment. In Discovering Geometry, students are guided to discover both the basic vocabulary of geometry, and the construction techniques they will need to understand geometric properties.

At the same time students are inductively discovering geometric properties, we are carefully developing a full breadth of inductive and deductive reasoning skills. There is a heavy emphasis on using higher order thinking as opposed to memorization and drill. In this investigative approach to geometry, reasoning, concepts, and skills are developed with two basic principles in the mind: the research of the van Hieles (van Hiele Model), and the work of Michael de Villiers and others on purposes of proof.

At the beginning of the course we assume a low level of geometric reasoning in students (van Hiele research suggests that over 70% of all high school students enter geometry at level 0 or 1 on the van Hiele scale). Through this concrete, hands-on discovery approach we attempt to move the students up the stages to higher levels of reasoning over the course of the term.

We also recognize that one of the traditional purposes of a proof in geometry, to convince someone that a particular property is true, is not the best use of proofing at the early stages. With most beginning geometry students, conviction comes upon seeing and doing, not through proof. The proof does not convince, in fact, it often creates more doubt. (See Rethinking Proof by Michael de Villiers.) In the early stages, a better purpose for proof is as a means of explaining why something they discovered through hands-on investigations is indeed true. Therefore, proof as a means of explanation is our initial and primary focus. In addition to the more typical exercises, where students apply the concepts discovered, we also utilize three unique types of exercises that ask students to explain their reasoning. 

1. In Angle Chase, students work in pairs to find the measure of an angle and then explain why it is correct. (In DG3 see page 188 exercise #14, page 224 exercise #20, and page 292 Exercise #25. In DG4 see page 190 exercise #14, page 226 exercise #20, and page 297 Exercise #25). 

2. In “What’s Wrong With This Picture?”, students develop proof and reasoning skills by explaining why a figure is incorrect. (In DG3 see page 251 exercise #26, page 323 exercise #16, and page 436 exercise #21. In DG4 see page 253 exercise #26, page 328 exercise #16, and page 452 exercise #21).

3. In the third type of exercise we simply ask students to explain why something is true (In DG3 see exercises page 436 exercise #20, page 466 exercise #18, page 472 exercise #22. In DG4 see exercises page 452 exercise #20, page 482 exercise #18, page 488 exercise #22). In the early instances of these exercises, we ask students to explain why something is true for a particular case (e.g. numbers are provided as in exercise #20 page 436 in DG3). Later, we ask students to explain their reasoning in more general cases (e.g. letters rather than numbers are provided as in DG3 exercise #22 page 472, DG4 exercise #22 page 488).

Proof is developed as a means of communication and discovery as students progress through Discovering Geometry. Eventually, as students move to more sophisticated reasoning abilities, we introduce geometry as a mathematical system in the final chapter. Systematization then becomes the purpose of proof in Chapter 13. 

In Discovering Geometry we develop reasoning skills in four general ways:

First, when students perform investigations and make conjectures in the form of formulas (e.g., a formula for the number of diagonals of an n-gon, or a formula for the sum of the measures of the interior angles of an n-gon), students must use inductive reasoning while doing geometry. This occurs throughout Discovering Geometry.  Chapter Two: Reasoning In Geometry, introduces this type of reasoning skill which is used throughout the text. This is the type of reasoning mostly commonly used by mathematicians and scientists, and it is essential to an investigative discovery approach to learning.

The second general way is in the development of logical arguments and proof, or deductive reasoning. This development is based on classroom experiences and research, workshops on reasoning and proof, and in particular, the research of the van Hieles, and de Villiers.

• Deductive reasoning is introduced in Chapter Two: Reasoning In Geometry, and it begins informally with exercises that ask students to “explain why.” This occurs in numerous examples, exercises, and special activities mentioned above and in sections called “Take Another Look.”

• Examples of algebraic, paragraph, and flow chart proofs are modeled early. The concepts of conditional statements, converses, and counterexamples are emphasized and reinforced. (It is recommended, however, that students in a regular geometry course not be graded on their proofing abilities at this early stage of geometric reasoning.)

• By Chapter Four: Discovering and Proving Triangle Properties, students are asked to follow the reasoning in paragraph proofs and flow chart proofs. They are eventually asked to create their own flow chart proofs, testing the level of their reasoning.

• As students move through the traditional synthetic geometry they move to higher levels of geometric reasoning and critical thinking. Their “explaining why” becomes more fully developed, and they are expected to present more detailed paragraph and flow chart proofs. At the same time, students are developing a firmer grasp of the content of geometry.

• As a transition from inductive reasoning and informal explanation, more formal deductive reasoning is introduced through a series of three Logic Explorations. These explorations begin with the logical reasoning found in Sherlock Holmes stories. This is followed with four properties of symbolic logic: Modus Ponens, Modus Tollens, Law of Syllogism, and Law of the Contrapositive. Students then create direct, conditional, and indirect proofs using these logic properties.

• Finally, students develop geometry as a mathematical system (de Villiers Systematization). After establishing definitions, properties of algebra and congruence, and the postulates of geometry, students begin to use these premises to establish theorems and to see the connections between groups of theorems.

The third general way in which higher order thinking is developed in Discovering Geometry is in solving word problems. After students have completed their investigations and made conjectures, they are asked to apply the new properties to real applications and novel situations. Many problems in the exercise sets come from recreational mathematics or are modeled after SAT-type questions and mathematics competitions.

Finally, a fourth way in which reasoning is fine-tuned in Discovering Geometry is in the puzzles called Improving Algebra Skills, Improving Reasoning Skills, and Improving Visual Thinking Skills. There are 14 Improving Algebra Skills, 22 Improving Reasoning Skills, and 44 Improving Visual Thinking Skills puzzles spread throughout the text. They occur after each lesson in every chapter. These puzzles employ a wide variety of higher order thinking skills and problem solving skills. Students are often found working these puzzles without even being assigned!