ExploreLearning Gizmos: Math & Science Virtual Labs and Simulations

G.LP: Logic and Proofs

G.LP.1: Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates, methods of reasoning, and theorems). Understand the differences among supporting evidence, counterexamples, and actual proofs.

Biconditional Statements
Conditional Statements
Investigating Angle Theorems
Isosceles and Equilateral Triangles
Parallel, Intersecting, and Skew Lines
Proving Triangles Congruent

G.LP.2: Use precise definitions for angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, and plane. Use standard geometric notation.

Chords and Arcs
Circumference and Area of Circles
Classifying Quadrilaterals
Classifying Triangles
Investigating Angle Theorems
Parallel, Intersecting, and Skew Lines
Parallelogram Conditions

G.LP.3: State, use, and examine the validity of the converse, inverse, and contrapositive of conditional (“if – then”) and bi-conditional (“if and only if”) statements.

Biconditional Statements
Conditional Statements

G.LP.4: Understand that proof is the means used to demonstrate whether a statement is true or false mathematically. Develop geometric proofs, including those involving coordinate geometry, using two-column, paragraph, and flow chart formats.

Congruence in Right Triangles
Investigating Angle Theorems
Isosceles and Equilateral Triangles
Proving Triangles Congruent

G.PL: Points, Lines, and Angles

G.PL.1: Prove and apply theorems about lines and angles, including the following:

G.PL.1.a: Vertical angles are congruent.

Investigating Angle Theorems

G.PL.1.b: When a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent.

Constructing Parallel and Perpendicular Lines
Triangle Angle Sum

G.PL.1.d: Points on a perpendicular bisector of a line segment are exactly those equidistant from the endpoints of the segment.

Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

G.PL.2: Explore the relationships of the slopes of parallel and perpendicular lines. Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and equations.

Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

G.PL.3: Use tools to explain and justify the process to construct congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, and parallel and perpendicular lines.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines

G.PL.4: Develop the distance formula using the Pythagorean Theorem. Find the lengths and midpoints of line segments in the two-dimensional coordinate system.

Distance Formula

G.T: Triangles

G.T.1: Prove and apply theorems about triangles, including the following:

G.T.1.a: Measures of interior angles of a triangle sum to 180°.

Polygon Angle Sum
Triangle Angle Sum

G.T.1.b: The Isosceles Triangle Theorem and its converse.

Isosceles and Equilateral Triangles

G.T.1.c: The Pythagorean Theorem.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard

G.T.1.f: The Angle Bisector Theorem.

Segment and Angle Bisectors

G.T.2: Explore and explain how the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) follow from the definition of congruence in terms of rigid motions.

Congruence in Right Triangles
Proving Triangles Congruent

G.T.3: Use tools to explain and justify the process to construct congruent triangles.

Constructing Congruent Segments and Angles

G.T.4: Use the definition of similarity in terms of similarity transformations, to determine if two given triangles are similar. Explore and develop the meaning of similarity for triangles.

G.T.5: Use congruent and similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles.

Congruence in Right Triangles
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

G.T.6: Prove and apply the inequality theorems, including the following:

G.T.6.a: Triangle inequality.

Triangle Inequalities

G.T.6.b: Inequality in one triangle.

Triangle Inequalities

G.T.7: Explore the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Understand and use the geometric mean to solve for missing parts of triangles.

Similarity in Right Triangles

G.T.8: 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.

Sine, Cosine, and Tangent Ratios

G.T.9: Use trigonometric ratios (sine, cosine, tangent and their inverses) and the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles.

Cosine Function
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

G.T.10: Explore the relationship between the sides of special right triangles (30° - 60° and 45° - 45°) and use them to solve real-world and other mathematical problems.

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios

G.QP: Quadrilaterals and Other Polygons

G.QP.1: Prove and apply theorems about parallelograms, including those involving angles, diagonals, and sides.

Parallelogram Conditions
Polygon Angle Sum
Special Parallelograms

G.QP.2: Prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids. Include coordinate proofs of quadrilaterals in the coordinate plane.

Classifying Quadrilaterals

G.QP.3: Develop and use formulas to find measures of interior and exterior angles of polygons.

Polygon Angle Sum

G.QP.4: Identify types of symmetry of polygons, including line, point, rotational, and self-congruences.

Holiday Snowflake Designer
Quilting Bee (Symmetry)
Rotations, Reflections, and Translations

G.QP.5: Compute perimeters and areas of polygons in the coordinate plane to solve real-world and other mathematical problems.

Distance Formula

G.QP.6: Develop and use formulas for areas of regular polygons.

Area of Parallelograms
Area of Triangles
Perimeter and Area of Rectangles

G.CI: Circles

G.CI.1: Define, identify and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, congruent circles, and concentric circles.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

G.CI.2: Derive the fact that the length of the arc intercepted by an angle is proportional to the radius; derive the formula for the area of a sector.

Radians

G.CI.3: Explore and use relationships among inscribed angles, radii, and chords, including the following:

G.CI.3.a: The relationship that exists between central, inscribed, and circumscribed angles.

Chords and Arcs
Inscribed Angles

G.CI.3.b: Inscribed angles on a diameter are right angles.

Inscribed Angles

G.CI.4: Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents).