- Course type: Self-paced
- Available Lessons: 25
- Average Lesson Length: 8 min
Eligible for Certificate: Yes
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Watch a preview:chapter 1 / lesson 1Transformations: How to Shift Graphs on a Plane
Course SummaryFun lessons in this course are designed to strengthen your students' high school geometry knowledge while ensuring your instruction aligns with Common Core State Standards. Use mini quizzes and practice exams to assess their comprehension of the geometry concepts you cover.
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- Comprehensive test covering all topics
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About the Course
Developed by educators, parents and other community members, the Common Core State Standards for math define the knowledge and understanding students should have of specific mathematical concepts based on grade level. In addition to knowing how to calculate and write different types of equations, students who meet these standards should also be able to grasp the tenets and formulas that govern these mathematical elements. This course brings students the information that conforms to the standards through engaging and brief video lessons, making the concepts easy to understand and retain. Each lesson illustrates a basic component with examples, clear explanations and simple language. Lessons in the collection cover the following concepts:
- Geometric congruence, transformations and rigid motions
- Geometric proofs, postulates and theorems of shapes
- Tools used in constructing geometric shapes
- Identification of similarities and congruence
- Trigonometric components: sine, cosine, tangent and Pythagorean Theorem
- Equations for properties of circles, angles and squares
- Conic Sections: parabolas, hyperbolas and ellipses
- Units of measurement: circumference, volume, area and density
Math can be difficult for some students to understand, and our instructors work to make lessons understandable yet challenging at every level. To make learning more interactive, each chapter in the collection includes helpful tips that you can implement within your lesson plan to keep students interested and engaged. Each lesson also incorporates a short quiz to gauge students' comprehension of the material.
Geometry Definitions (CCSS.Math.Content.HSG-CO.A.1)
Standard: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
The lesson for this standard introduces students to basic geometry vocabulary and definitions.
Standard: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
With this lesson, students can learn about transformations and how to shift graphs on a plane.
Rotations & Reflections (CCSS.Math.Content.HSG-CO.A.3)
Standard: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
This lesson explains rotations and reflections of quadrilaterals and regular polygons.
Definitions of Rotations, Reflections & Translation (CCSS.Math.Content.HSG-CO.A.4)
Standard: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Use these lessons to define rotation, reflection and translation in geometry.
Transformed Figures (CCSS.Math.Content.HSG-CO.A.5)
Standard: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
These lessons show students how to draw transformed figures and identify sequences of transformations that make one figure into another.
Rigid Motions (CCSS.Math.Content.HSG-CO.B.6)
Standard: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
These lessons cover the use of rigid motions to transform figures and explain rigid motions in the definition of congruence.
Corresponding Pairs (CCSS.Math.Content.HSG-CO.B.7)
Standard: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
This lesson discusses the congruence and corresponding pairs of sides or angles.
ASA, SAS, SSS and Congruence (CCSS.Math.Content.HSG-CO.B.8)
Standard: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Use the lesson on ASA, SAS, SSS and congruence to explain how to prove triangles are congruent.
Theorems of Lines and Angles (CCSS.Math.Content.HSG-CO.C.9)
Standard: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Students learn what postulates and theorems are, and the important postulates and theorems about lines and angles. They also explore how to create a proof in geometry and how to prove theorems about lines and angles.
Theorems of Triangles (CCSS.Math.Content.HSG-CO.C.10)
Standard: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
This lesson shows how to prove theorems about triangles.
Theorems of Parallelograms (CCSS.Math.Content.HSG-CO.C.11)
Standard: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
After students learn how to prove theorems about parallelograms, they can also engage in a practice lesson.
Geometric Constructions (CCSS.Math.Content.HSG-CO.D.12)
Standard: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
The lessons for this standard explain how to make geometric constructions using various tools and allow students to practice constructing geometric shapes.
Constructing Shapes (CCSS.Math.Content.HSG-CO.D.13)
Standard: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
This lesson shows how to construct various shapes, such as equilateral triangles, squares, and regular hexagons inscribed in a circle.
Properties of Dilations (CCSS.Math.Content.HSG-SRT.A.1a and CCSS.Math.Content.HSG-SRT.A.1b)
Standards: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
The lesson on the properties of dilations shows how larger and smaller object retain the same shape.
Similar and Congruent Shapes (CCSS.Math.Content.HSG-SRT.A.2)
Standard: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Use these lessons to explain how to identify similar triangles and determine the properties of congruent and similar shapes. Students can also learn about similarity transformations.
Similar Transformations & the AA Criterion (CCSS.Math.Content.HSG-SRT.A.3)
Standard: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
These lessons discuss similar triangles and the AA criterion, as well as show applications of similar triangles.
Similarity Theorems (CCSS.Math.Content.HSG-SRT.B.4)
Standard: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
This lesson covers the process for proving similarity in triangles through theorems.
Proving Relationships (CCSS.Math.Content.HSG-SRT.B.5)
Standard: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
These lessons prove relationships in figures using congruence and similarity and provide a practice lesson for students.
Trigonometric Ratios & Similarity (CCSS.Math.Content.HSG-SRT.C.6)
Standard: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
With these lessons, students learn about sine and cosine, ratios and similarity, and can practice finding trigonometric ratios.
Complementary Angles (CCSS.Math.Content.HSG-SRT.C.7)
Standard: Explain and use the relationship between the sine and cosine of complementary angles.
This lesson explains the sine and cosine of complementary angles.
Pythagorean Theorem (CCSS.Math.Content.HSG-SRT.C.8)
Standard: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
These lessons introduce students to the Pythagorean Theorem and show how it's used in trigonometry. They illustrate how to solve right triangles and provide an in-depth look at problem solving.
Using Sine to Find the Area (CCSS.Math.Content.HSG-SRT.D.9)
Standard: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
This lesson explains how to use A = ½ ab sinC to find the area of a triangle.
Laws of Sine and Cosine (CCSS.Math.Content.HSG-SRT.D.10)
Standard: Prove the Laws of Sines and Cosines and use them to solve problems.
These lessons introduce the Law of Sines and the Law of Cosines.
Applying the Laws of Sine and Cosine (CCSS.Math.Content.HSG-SRT.D.11)
Standard: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Students can learn the application of the Law of Sines and the Law of Cosines in finding measurements for triangles.
Similarity of Circles (CCSS.Math.Content.HSG-C.A.1)
Standard: Prove that all circles are similar.
This lesson explores the similarity of circles and shows students how to prove all circles are similar.
Inscribed Angles, Radii & Chords (CCSS.Math.Content.HSG-C.A.2)
Standard: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
These lessons cover the inscribed angles, radii and chords of circles and provide practice problems.
Inscribed and Circumscribed Triangles & Quadrilaterals (CCSS.Math.Content.HSG-C.A.3)
Standard: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
This lesson explains how to construct the inscribed and circumscribed circles of triangles. Students also learn about the properties of angles in quadrilaterals.
Tangents to Circles (CCSS.Math.Content.HSG-C.A.4)
Standard: Construct a tangent line from a point outside a given circle to the circle.
This lesson shows students how to form a tangent line to a circle.
Arc Length & Radian / Area of a Sector (CCSS.Math.Content.HSG-C.B.5)
Standard: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
These lessons cover arc length and radians, and explain how to find the area of a sector.
Equations of Circles & Squares (CCSS.Math.Content.HSG-GPE.A.1)
Standard: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Use these lessons to show students how to derive the equation of a circle, complete a square, and find the center and radius of a circle by completing a square.
Focus & Directix of a Parabola (CCSS.Math.Content.HSG-GPE.A.2)
Standard: Derive the equation of a parabola given a focus and directrix.
These lessons identify the focus and directix of a parabola and show how to find the equation of a parabola from the focus and directix.
Foci of Ellipses & Hyperbolas (CCSS.Math.Content.HSG-GPE.A.3)
Standard: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
This collection of lessons discusses foci of ellipses and hyperbolas, and explains how to derive the equation of an ellipse or hyperbola from the foci. Students can also practice with conic sections.
Standard: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, the square root of 3) lies on the circle centered at the origin and containing the point (0, 2).
This lesson teaches students how to use coordinates to prove geometric theorems.
Slope Criteria (CCSS.Math.Content.HSG-GPE.B.5)
Standard: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
These lessons prove the slope criteria and explain how to solve geometric problems using the slope criteria.
Partitioning Segments (CCSS.Math.Content.HSG-GPE.B.6)
Standard: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Use this lesson to show students how to find a point that partitions a segment.
Distance Formula & Coordinates (CCSS.Math.Content.HSG-GPE.B.7)
Standard: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Lessons cover how to use the distance formula. Students can also learn how to use coordinates to find perimeters of polygons and areas of triangles and rectangles.
Circumference, Area & Volume Formulas (CCSS.Math.Content.HSG-GMD.A.1)
Standard: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Lessons for this standard explain how to calculate for circumference, area and volume using formulas. Students learn how to find the perimeter of triangles, rectangles, quadrilaterals and irregular shapes, as well as determining the area of circles, triangles, quadrilaterals and complex shapes.
Calculating Volumes of Shapes (CCSS.Math.Content.HSG-GMD.A.2)
Standard: Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
These lessons cover Cavalieri's Principle and how to calculate the volumes of basic shapes, such as cylinders, cones, spheres, prisms and pyramids.
Applied Problems Involving Volume (CCSS.Math.Content.HSG-GMD.A.3)
Standard: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
This lesson teaches how to apply volume formulas for various geometric shapes in order to solve problems.
3-Dimensional & 2-Dimensional Objects (CCSS.Math.Content.HSG-GMD.B.4)
Standard: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
These lessons discuss cross sections of three-dimensional objects and rotations of two-dimensional objects.
Real-World Objects (CCSS.Math.Content.HSG-MG.A.1)
Standard: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)
Students can learn how to define real-world objects with geometric shapes.
Standard: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
This lesson defines density and explains how it can be used on modeling.
Solving Design Problems (CCSS.Math.Content.HSG-MG.A.3)
Standard: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
This lesson explains how to use geometric formulas to determine sizing constraints and resolve problems with design issues.
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