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Geometry Table of Contents

Chapter 1: An informal Introduction to Geometry

1A: Picturing and Drawing

Students examine techniques for drawing figures and specifying critical features of a drawing. Symmetry and connections to algebra are introduced.

1B: Constructing

Construction of a figure is contrasted to drawing, and some basic properties of triangles are investigated.

1C: Geometry Software

Geometric construction tasks invite students to investigate the tools and features of their geometry software. Students explore dynamic construction and find invariant measurements and relationships.

1D: Invariants - Properties and Values That Don't Change

Students examine algebraic invariants including difference and ratio, and geometric invariants including shape, concurrence, collinearity, measurement, and congruence.

1.0 Habits of Mind (Geom)

2A: The Congruence Relationship

Congruence is defined, and students figure out the minimum conditions they need to prove congruence for different kinds of figures including segments, angles, and triangles. They use basic postulates of triangle congruence to prove theorems.

2B: Proof and Parallel Lines

The deductive method of proof is explained and put into practice as students learn theorems about parallel and perpendicular lines and use them (along with previous results) to prove more theorems.

2C: Writing Proofs

Several techniques for finding a path from hypothesis to conclusion, presenting an argument, and writing formal proofs are examined and students evaluate and complete example proofs in addition to writing their own.

2D: Quadrilaterals and Their Properties

Quadrilaterals are defined and classified, and students examine their properties, develop conjectures, and prove theorems about them.

Chapter 3: Dissections and Area

3A: Cut and Rearrange

Algorithms for cutting a polygon and rearranging the pieces to form a specific type of polygon are developed, analyzed, and proven valid. Students use a dissection to prove the Midline Theorem.

3B: Area Formulas

Students use dissections to derive area formulas for various polygons including triangles, parallelograms, and trapezoids, given the formula for the area of a rectangle.

3C: Proof by Dissection

The Pythagorean Theorem is proven in several different ways using different dissections. Students apply the Pythagorean Theorem in different contexts and explore Pythagorean triples.

3D: Measuring Solids

Students use nets to find surface areas of solids and dissections to find volumes. Pyramids and prisms are defined and students make sense of formulas for their surface area and volume. Cones and cylinders are related to pyramids and prisms, as are the formulas for their surface area and volume.

Chapter 4: Similarity

4A: Scaled Copies

Similarity is introduced informally using the idea of scaled drawings. Students read scaled drawings and develop their own methods to make them. They also decide how to tell if one drawing is an accurate scaled copy of another.

4B: Curved or Straight? Just Dilate!

Students construct scaled drawings using both the Ratio and Parallel Methods. In the Ratio Method, they draw segments from a center of dilation to each critical point on the figure and then scale each segment from the center to a critical point along the rays. In the Parallel Method, students scale one critical point as in the Ratio Method, and locate other points in the scaled drawing by constructing parallels to corresponding segments in the original figure.

4C: The Side-Splitter Theorems

Students prove the Side-Splitter and Parallel Theorems and use them to show that the Ratio and Parallel Methods for scaling figures are equivalent.

4D: Defining Similarity

Similarity is formally defined, and students develop and prove tests for similarity in triangles.

Chapter 5: Circles

5A: Area and Circumference

Area and perimeter of curved figures are approximated using grids and lines, and students approximate the area and circumference of a circle by inscribing and circumscribing regular polygons.

5B: Circles and 3.141592653589793238462643383…

The formulas for area and circumference of a circle are presented and applied.

5C: Classical Results About Circles

Students prove results about arcs, chords, central angles, inscribed angles and polygons, secants, and tangents.

5D: Geometric Probability

Students investigate relationships between area and probability using Monte-Carlo techniques and approach a clearer understanding of measure.

Chapter 6: Using Similarity

6A: Similarity Proofs

Students use similarity in many ways including determining inaccessible distances through perspective and shadows. They prove some classical geometric results and develop and use the Arithmetic-Geometric Mean Inequality.

6B: Exploring Right Triangles

This investigation will introduce students to sine, cosine, and tangent, and they will use these ratios to determine missing sidelengths and angle measures in triangles, and to find areas. They will also extend the Pythagorean Theorem with the Law of Cosines.

6C: Volume Formulas

Chapter 3 introduced students to volume formulas for solids, and this investigation will go deeper by proving the volume formulas for prisms, cylinders, pyramids, cones, and sphere's using Cavalieri's Principle.

Chapter 7: Coordinates and Vectors

7A: Transformations

Reflections are performed with paperfolding and tracing on plain paper, with geometry software, and on the coordinate plane. Students compose reflections to create rotations and translations. They investigate and prove the properties of these transformations.

7B: Geometry in the Coordinate Plane

Students develop formulas for midpoint of a segment, distance between two points, and the equation of a circle on the coordinate plane using geometric methods. They also use slope relationships of parallel and perpendicular lines to prove geometric results. They extend their understanding to three-dimensional coordinates.

7C: Connections to Algebra

Vectors are introduced and students investigate their properties. Several geometric results are proven using vectors.

Chapter 8: Optimization

8A: Making the Least of a Situation

Students investigate ways to find the shortest path in various contexts. For example, they use reflection to turn piecewise linear paths into linear ones. They often use numerical examples to develop intuition about situations and then produce logical arguments to defend their solutions.

8C: Contour Lines

Students learn how to read, interpret, and create contour plots. Plots showing lines of constant elevation, distance, and angle measure are analyzed, and eventually abstracted to show lines of constant value for mathematical functions.

8B: Making the Most of a Situation

Students develop strategies to maximize various quantities, including length, area, and angle measures. They see the effect of constraints by working similar problems under different conditions.

8D: Advanced Optimization

Students examine what it means to reason by continuity – to find the limit of the value of one quantity while other quantities are increased or decreased as much as possible. They use this idea to find maxima and minima in geometric contexts. They also investigate the Isoperimetric Conjecture, finding the maximum area that can be enclosed by a closed curve of fixed perimeter.