Unit 1: Foundations of Geometry

This unit will review solving linear equations and introduce students to the foundational concepts of geometry and to different types of angle relationships, specifically vertical angles, adjacent angles, complementary angles, supplementary angles, and linear pairs of angles. Students will learn to identify these angles, understand their properties, and apply these concepts to solve geometric problems. Students will learn about basic geometric terms and properties and develop skills in using formulas in Geometry including distance and midpoint. Students will be introduced to geometric transformations and their properties. Students will explore translations, reflections, and rotations, and learn how these transformations affect geometric figures in the coordinate plane.

Algebra I Review: Solving linear equations

  • Solve one step, two step and multi-step equations including equations with variables on both sides
  • Check solutions of linear equations using substitution

Understanding Points, Lines and Planes

  • Identify, name, and draw points, lines, segments, rays, and planes
  • Apply basic facts about points, lines, and plane

Measuring Segments

  • Understand the Segment Addition Postulate and use it to determine the length of a segment by adding the lengths of its smaller segments
  • Define a segment bisector and use it to determine the midpoint and lengths of segments

Angles and Angle Measurement

  • Name and classify angles
  • Find the measure angles using a protractor and using the Angle Addition Postulate
  • Apply angle bisectors to find angle measures

Pairs of Angles

  • Identify and find the measure of complementary angles, supplementary angles, linear pair angles, adjacent angles, and vertical angles

Midpoints and the Distance Formula

  • Find the midpoint of a line segment
  • Use the Distance Formula and the Pythagorean Theorem to find the distance between two point

Transformations in the Coordinate Plane

  • Identify reflections, rotations, and translations
  • Apply translation, reflection, and rotation transformations to geometric figures on the coordinate plane, accurately determining the new coordinates of points and describing how the figures are affected by each transformation

Unit 2: Parallel and Perpendicular Lines

In this unit on parallel and perpendicular lines, students explore the definitions, properties, and geometric implications of these fundamental concepts. Parallel lines, which never intersect and have equal slopes, and perpendicular lines, which intersect at right angles with slopes that are negative reciprocals, form the basis of various geometric relationships and theorems. Key angle relationships, such as corresponding, alternate interior, and consecutive interior angles, emerge when parallel lines are intersected by a transversal, aiding in the solving of complex geometric problems. Understanding these principles is essential for real-world applications, including architectural design and urban planning, and serves as a foundation for more advanced geometric studies.

Lines and Angles

  • Identify parallel, perpendicular, and skew lines
  • Identify the angles formed by two lines and a transversal

Measures of Angles Formed by Parallel Lines and Transversals

  • Find the measure of the alternate interior, alternate exterior, corresponding and same side interior angles

Converse Postulates for Parallel Lines and Transversals

  • Use the Converse Postulates to prove that two lines are parallel

Perpendicular Lines

  • Prove and apply theorems about perpendicular lines

Slope

  • Find the slope of a line from a two points using the slope formula
  • Find the slope of a line in the coordinate plane using rise/run

Unit 3: Triangle Congruence

In the unit on triangle congruence, students explore the criteria and methods for determining when two triangles are congruent. Triangle congruence is established when two triangles have the same size and shape, which can be proven using specific congruence postulates: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles. These criteria allow students to identify and prove the congruence of triangles based on the corresponding parts being equal, which in turn helps solve geometric problems and understand the properties of geometric figures. Understanding triangle congruence is fundamental for further studies in geometry, including similarity, parallel and perpendicular lines, and polygon properties.

Congruence and Transformations

  • Draw, identify, and describe transformations in the coordinate plane
  • Use properties of rigid motions to determine whether figures are congruent and to prove figures congruent

Classifying Triangles

  • Classify triangles by their angle measures and side lengths
  • Use triangle classification to find angle measures and side lengths

Angle Relationships in Triangles

  • Find the measures of interior and exterior angles of triangles
  • Apply theorems about the interior and exterior angles of triangles

Congruent Triangles

  • Use properties of congruent triangles to determine corresponding parts, side lengths and angle measures
  • Prove triangles congruent by using the definition of congruence

SSS and SAS

  • Apply SSS and SAS to construct triangles and solve problems
  • Prove triangles congruent by using SSS and SAS

AAS, ASA, HL

  • Apply ASA, AAS, and HL to construct triangles and to solve problems
  • Prove triangles congruent by using ASA, AAS, and HL

Isosceles and Equilateral Triangles

  • Prove theorems about isosceles and equilateral triangle
  • Apply properties of isosceles and equilateral triangles

Unit 4: Polygons and Quadrilaterals

In the unit on polygons and quadrilaterals, students delve into the properties, classification, and characteristics of these geometric figures. A polygon is a closed, two-dimensional shape with straight sides, and its classification depends on the number of sides and angles. The focus on quadrilaterals, a specific type of polygon with four sides, includes exploring various types such as parallelograms, rectangles, squares, rhombuses, and trapezoids. Key concepts include understanding the properties of sides, angles, and diagonals. The unit also covers theorems related to angle sums, congruence, and similarity within polygons and quadrilaterals. This foundational knowledge is essential for solving geometric problems and understanding more complex shapes and structures in higher-level mathematics.

Properties and Attributes of Polygons

  • Classify polygons by their sides and angles
  • Find and use measures of interior and exterior angles of polygons

Properties of Parallelograms

  • Apply properties of parallelograms
  • Use properties of parallelograms to solve problems

Conditions for Parallelograms

  • Prove that a given quadrilateral is a parallelogram

Properties of Special Parallelograms

  • Apply properties of parallelograms to find missing values in special parallelograms (rhombus, rectangles, and squares)

Conditions for Special Parallelograms

  • Apply properties of parallelograms to prove special parallelograms are rhombus, rectangles, or squares

Properties of Kites and Trapezoids

  • Use properties of kites and trapezoids to solve problems

Unit 5: Similarity

In the geometry unit on similarity, students explore the concept of similar figures, focusing on their properties and the criteria for determining similarity. Two figures are considered similar if they have the same shape, but not necessarily the same size, which means their corresponding angles are equal and their corresponding sides are proportional. Key topics include the properties of similar triangles, the AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) similarity criteria, and the concept of scale factor. Students learn to apply these concepts to solve problems involving indirect measurement, proportional reasoning, and real-world applications such as map scaling and model building. Understanding similarity is crucial for advancing to topics like trigonometry and further studies in geometric transformations.

Dilations

  • Dilate figures on and off the coordinate plane
  • Understand how distances and lengths in a dilation are related to the scale factor and center of dilation

Similarity Transformations

  • Understand that two figures are similar if there is a similarity transformation that maps one figure to the other
  • Identify a combination of rigid motions and dilation that maps one figure to a similar figure
  • Identify the coordinates of an image under a similarity transformation

Using Proportional Relationships

  • Use ratios to make indirect measurements
  • Use scale drawings to solve problems

Unit 6: Right Triangles and Trigonometry

In the unit on right triangles and trigonometry, students investigate the relationships between the angles and sides of right triangles and learn to apply trigonometric ratios. Key concepts include understanding the Pythagorean Theorem, which relates the lengths of the sides of a right triangle, and using it to solve for unknown side lengths. Students also explore the primary trigonometric ratios—sine, cosine, and tangent—and how these ratios are used to find missing angles and sides in right triangles. The unit covers solving real-world problems involving right triangles, such as calculating heights and distances. Additionally, students learn about special right triangles (30-60-90 and 45-45-90) and their unique properties. This foundational knowledge prepares students for more advanced studies in trigonometry and its applications in various fields, including physics and engineering.

Trigonometric Ratios

  • Define and calculate sine, cosine, and tangent ratios
  • Use trigonometric ratios to solve problems

Law of Sines

  • Understand why the Law of Sines applies to any triangle
  • Use the Law of Sines to solve problems

Law of Cosines

  • Develop and understand the Law of Cosines
  • Use the Law of Cosines to solve problems

Angles of Elevation and Angles of Depression

  • Distinguish between angles of elevation and depression
  • Use trigonometric ratios and the Laws of Sines and Cosines to solve problems

Unit 7: Perimeter, Circumference and Area

In the unit on area, perimeter, and circumference, students learn to measure and calculate the dimensions of various geometric figures. Perimeter is the total distance around the edge of a polygon, while area measures the surface enclosed within a shape. Students explore formulas for finding the perimeter and area of common shapes such as rectangles, squares, triangles, and circles. For circles, students focus on understanding the concepts of circumference, which is the distance around the circle, and the area, which covers the space within. This unit emphasizes applying these formulas to solve practical problems and real-world scenarios, enhancing spatial reasoning and measurement skills essential for more advanced geometry and other mathematical applications.

Developing Area Formulas for Triangles and Special Quadrilaterals

  • Develop and apply the formulas for the areas of triangles and special quadrilaterals.
  • Solve problems involving perimeters and areas of triangles and special quadrilaterals.

Area of Circles and Regular Polygons

  • Develop and apply the formulas for the area and circumference of a circle.
  • Develop and apply the formula for the area of a regular polygon

Area of Compound Figures

  • Use the Area Addition Postulate to find the areas of composite figures.
  • Use composite figures to estimate the areas of irregular shapes

Unit 8: Volume

In the unit on three-dimensional figures, students explore the properties, classification, and measurement of solid shapes such as prisms, cylinders, pyramids, cones, and spheres. The unit covers key concepts including volume, which measures the space a figure occupies, and surface area, which is the total area of all the faces or surfaces of a figure. Students learn to apply formulas to calculate the volume and surface area of various three-dimensional figures and use these concepts to solve real-world problems, such as determining the amount of material needed to construct or cover an object. This unit builds spatial reasoning skills and provides a foundation for more advanced studies in geometry and related fields.

Three Dimensional Figures and Cross Sections

  • Use Euler’s Formula to calculate the number of vertices, faces, and edges in polyhedrons
  • Describe cross sections of polyhedrons
  • Describe rotations of polygons about an axis

Volume of Prisms and Cylinders

  • Understand how the volume formulas for prisms and cylinders apply to oblique prisms and cylinders
  • Model three-dimensional figures as cylinders and prisms to solve problems

Pyramids and Cones

  • Understand how the volume formulas for pyramids and cones apply to oblique pyramids and cones
  • Model three-dimensional figures as pyramids and cones.

Spheres

  • Use Cavalieri’s principle to show how the volume of a hemisphere is related to the volume of a cone and a cylinder
  • Calculate volumes and surface areas of spheres and composite figures

Original document