Module 4: In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. The module opens with a modeling challenge, one that reoccurs throughout the lessons, to use coordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the plane—the room in which it sits. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the coordinates of points that divide given segments in specific length ratios, and so on), students will describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region

Unit 1: Segments in the Coordinate System                                           (G.GPE.7)

Students impose a coordinate system and describe the movement of the robot in terms of line segments and points.  This leads to graphing inequalities and discovering regions in the plane can be defined by a system of algebraic inequalities.   Students then program the robot to move on lines cutting through these regionsrotating and rotate 90°clockwise or counterclockwise about an endpoint.

M4.U1 Lessons:

M4.U1.L1
M4.U1.L2
M4.U1.L3
M4.U1.L4

Unit 2: Parallel and Perpendicular Segments                                     (G.GPE.4.5)

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines.  Students write equations for parallel, perpendicular, and normal lines.  Additionally, students will  and study the proportionality of segments formed by diagonals of polygons.

M4.U2 Lessons:

M4.U2.L5
M4.U2.L6
M4.U2.L7
M4.U2.L8

Unit 3: Calculating Regions             (G.GPE.7)

Students sketch the regions, determine points of intersection (vertices), and use the distance formula to calculate perimeter and the “shoelace” formula to determine area of these regions.  Students return to the real-world application of programming a robot and extend this work to robots not just confined to straight line motion, but motion bound by regions described by inequalities and defined areas.

M4.U3 Lessons:

M4.U3.L9
M4.U3.L10
M4.U3.L11

Unit 4: Proportional Segments
                                                      (G.GPE.4, 6)

Proving Properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties—those that have been accepted as true and those that are new—of parallelograms and triangles 

​M4.U4 Lessons:

M4.U4.L12
M4.U4.L13
M4.U4.L14
M4.U4.L15
Module 5: This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year.  It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story.  This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.

Unit 1: Central and Inscribed Angles                
​                                                             (G.C.2, 3)

​M5.U1 Lessons:
This unit leads students first to Thales' theorem (an angle drawn from a diameter of a circle to a point on the circle is sure to be a right angle), then to possible converses of Thales' theorem, and finally to the general inscribed-central angle theorem.  Students use this result to solve unknown angle problems.  Through this work, students construct triangles and rectangles inscribed in circles and study their properties.
M5.U1.L1
M5.U1.L2
M5.U1.L3
M5.U1.L4
M5.U1.L5
M5.U1.L6

Unit 2: Arcs and Sectors                (G.C.1, 2, 5)

This unit defines the measure of an arc and establishes results relating chord lengths and the measures of the arcs they subtend.  Students build on their knowledge of circles from Module 2 and prove that all circles are similar.  Students develop a formula for arc length in addition to a formula for the area of a sector and practice their skills solving unknown area problems.

​M5.U2 Lessons:

M5.U2.L7
M5.U2.L8
M5.U2.L9
M5.U2.L10

Unit 3: Secants and Tangents         (G.C. 2, 3)

​M5.U3 Lessons:
In this unit students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even two tangent lines), meeting a point inside or outside of a circle.  They establish the secant angle theorems and tangent-secant angle theorems.  By drawing auxiliary lines, students also notice similar triangles and thereby discovere relationships between lengths of line segments appearing in these diagrams.
M5.U3.L11
M5.U3.L12
M5.U3.L14
M5.U3.L15
M5.U3.L16

Unit 4: Equations for Circles and Their Tangents                              (G.GPE1, 4)

This unit brings in coordinate geometry to establish the equation of a circle.  Students solve problems to find the equations of specific tangent lines or the coordinates of specific points of contact.  They also express circles via analytic equations.

M5.U4 Lessons:

M5.U4.L17
M5.U4.L18
M5.U4.L19

Unit 5: Cyclic Quadrilaterals and Ptolemy's Theorem                        (G.C.3)

The module concludes with Unit 5 focusing on the properties of quadrilaterals inscribed in circles and establishing Ptolemy's theorem.  This result codifies the Pythagorean theorem, curious facts about triangles, properties of the regular pentagon, and trigonometric relationships.  It serves as a final unifying flourish for students' year-long study of geometry.

M5.U5 Lessons:

M5.U5.L20
M5.U5.L21