# Mathematics

## Course Summary

In Fishtank Math Geometry, students further their understanding of geometric relationships and learn to make formal mathematical arguments about geometric situations. This course, which follows the Common Core standards for Geometry and the Massachusetts Curriculum Frameworks, takes a somewhat different approach from more traditional Geometry classes in its heavy emphasis on transformation. Transformations are used to help students understand and prove congruence and other geometric relationships. There is also a strong emphasis on proofs: students learn to prove concepts and ideas they have been learning about for years. Class time focuses on six main topics 1) establishing criteria for congruence of triangles based on rigid motions; (2) establishing criteria for similarity of triangles based on dilations and proportional reasoning; (3) informally developing explanations of circumference, area, and volume formulas; (4) applying the Pythagorean Theorem to the coordinate plan; (5) proving basic geometric theorems; and (6) extending student work with probability. (See Massachusetts Curriculum Frameworks.) Because Fishtank Math seeks to offer students a pathway to study Calculus in their senior year, this Geometry course also covers advanced standards that are sometimes covered in advanced math and pre-calculus courses.

Foundations for Success:

High school geometry builds on geometry instruction that has occurred throughout elementary and middle school but with the key difference that students must prove and explain concepts they learned about in prior years. In elementary school, students learned about the attributes of shapes, compared and categorized these attributes, and learned to compose and decompose shapes. In middle school, students developed conceptual understanding of angle relationships in parallel line diagrams and angle relationships within and outside of triangles. They have also learned to describe geometric features, measure circumference and area of circles, and make observations and conjectures about geometric shapes using sound reasoning and evidence. Students have learned to “construct” a triangle using different side lengths and that the properties of a triangle are based on the relationship between the side lengths and the interior angle measures. These foundational understandings will be essential to students’ success in this course as they build chains of reasoning to explain, model and prove geometric relationships and situations.

## Units

Unit 1

19 Lessons

### Constructions, Proof, and Rigid Motion

Students use the properties of circles to construct and understand different geometric figures, and lay the groundwork for constructing mathematical arguments through proof.

Unit 2

18 Lessons

### Congruence in Two Dimensions

Students identify, perform, and algebraically describe rigid motions to establish congruence of two dimensional polygons, including triangles, and develop congruence criteria for triangles.

Unit 3

18 Lessons

### Dilations and Similarity

Students use constructions to explore dilations in order to define and establish similarity, and they prove and use similarity criterion and theorems in the solution of problems.

Unit 4

19 Lessons

### Right Triangles and Trigonometry

Students develop an understanding of right triangles through an introduction to trigonometry, building an appreciation for the similarity of triangles as the basis for developing the Pythagorean theorem.

Unit 5

15 Lessons

### Polygons and Algebraic Relationships

Students connect algebra to geometric concepts with polygons as they explore the distance formula, slope criteria for parallel and perpendicular lines, and learn to calculate and justify the area and perimeter of polygons.

Unit 6

18 Lessons

### Three-Dimensional Measurement and Application

Students extend their understanding of circles, volume, and surface area into modeling situations, formula analysis, and deeper conceptual understandings.

Unit 7

14 Lessons

### Circles

Students expand their knowledge of circles to establish relationships between angle measures in and around circles, line segments and lines in and around circles, and portions of circles as related to area and circumference.

Unit 8

10 Lessons

### Probability

Students formalize their understanding of compound probability, develop an understanding of conditional probability, and understand and calculate permutations and combinations.