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Title: Construct This!
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Students will construct equilateral triangles, squares, and regular hexagons inscribed in circles using the free GeoGebra computer program.
Standard(s): [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13]
Subject: Mathematics (9  12)
Title: Construct This!
Description: Students will construct equilateral triangles, squares, and regular hexagons inscribed in circles using the free GeoGebra computer program.
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Title: All in the Family
Description:
In this Illuminations lesson, students use a dynamic geometry applet to conjecture about the relationships between characteristics of a square: side length, diagonal length, perimeter, and area. Students use graphs to represent functional relationships between two characteristics, such as diagonal length as a function of perimeter. In addition, there are links to online activities, reproducibles, and other related resources.
Standard(s): [MA2013] GEO (912) 11: 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. [GCO11] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13]
Subject: Mathematics Title: All in the Family
Description: In this Illuminations lesson, students use a dynamic geometry applet to conjecture about the relationships between characteristics of a square: side length, diagonal length, perimeter, and area. Students use graphs to represent functional relationships between two characteristics, such as diagonal length as a function of perimeter. In addition, there are links to online activities, reproducibles, and other related resources. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Geometry StepbyStep
Description:
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
Standard(s): [MA2013] GEO (912) 1: 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. [GCO1] [MA2013] GEO (912) 5: 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. [GCO5] [MA2013] GEO (912) 7: 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. [GCO7] [MA2013] GEO (912) 8: Explain how the criteria for triangle congruence, anglesideangle (ASA), sideangleside (SAS), and sidesideside (SSS), follow from the definition of congruence in terms of rigid motions. [GCO8] [MA2013] GEO (912) 9: 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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [GCO9] [MA2013] GEO (912) 10: Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, 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, and the medians of a triangle meet at a point. [GCO10] [MA2013] GEO (912) 11: 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. [GCO11] [MA2013] GEO (912) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [GCO12] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13] [MA2013] GEO (912) 16: Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. [GSRT3] [MA2013] GEO (912) 17: Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4] [MA2013] GEO (912) 18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [GSRT5] [MA2013] GEO (912) 19: 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. [GSRT6] [MA2013] GEO (912) 20: Explain and use the relationship between the sine and cosine of complementary angles. [GSRT7] [MA2013] GEO (912) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [GSRT8] [MA2013] GEO (912) 22: (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [GSRT10] [MA2013] GEO (912) 23: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).
[GSRT11] [MA2013] GEO (912) 24: Prove that all circles are similar. [GC1] [MA2013] GEO (912) 25: 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. [GC2] [MA2013] GEO (912) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3] [MA2013] GEO (912) 27: (+) Construct a tangent line from a point outside a given circle to the circle. [GC4] [MA2013] GEO (912) 28: 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. [GC5] [MA2013] GEO (912) 29: 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. [GGPE1] [MA2013] GEO (912) 35: Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [GGMD1] [MA2013] GEO (912) 37: Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama)
Geometry StepbyStep
http://agutie.homest...
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
Learning Activities
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Title: Geometry StepbyStep
Description:
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
Standard(s): [MA2013] GEO (912) 1: 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. [GCO1] [MA2013] GEO (912) 5: 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. [GCO5] [MA2013] GEO (912) 7: 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. [GCO7] [MA2013] GEO (912) 8: Explain how the criteria for triangle congruence, anglesideangle (ASA), sideangleside (SAS), and sidesideside (SSS), follow from the definition of congruence in terms of rigid motions. [GCO8] [MA2013] GEO (912) 9: 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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [GCO9] [MA2013] GEO (912) 10: Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, 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, and the medians of a triangle meet at a point. [GCO10] [MA2013] GEO (912) 11: 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. [GCO11] [MA2013] GEO (912) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [GCO12] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13] [MA2013] GEO (912) 16: Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. [GSRT3] [MA2013] GEO (912) 17: Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4] [MA2013] GEO (912) 18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [GSRT5] [MA2013] GEO (912) 19: 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. [GSRT6] [MA2013] GEO (912) 20: Explain and use the relationship between the sine and cosine of complementary angles. [GSRT7] [MA2013] GEO (912) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [GSRT8] [MA2013] GEO (912) 22: (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [GSRT10] [MA2013] GEO (912) 23: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).
[GSRT11] [MA2013] GEO (912) 24: Prove that all circles are similar. [GC1] [MA2013] GEO (912) 25: 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. [GC2] [MA2013] GEO (912) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3] [MA2013] GEO (912) 27: (+) Construct a tangent line from a point outside a given circle to the circle. [GC4] [MA2013] GEO (912) 28: 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. [GC5] [MA2013] GEO (912) 29: 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. [GGPE1] [MA2013] GEO (912) 35: Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [GGMD1] [MA2013] GEO (912) 37: Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama)
Geometry StepbyStep
http://agutie.homest...
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
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Title: Geometry StepbyStep
Description:
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
Standard(s): [MA2013] GEO (912) 1: 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. [GCO1] [MA2013] GEO (912) 5: 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. [GCO5] [MA2013] GEO (912) 7: 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. [GCO7] [MA2013] GEO (912) 8: Explain how the criteria for triangle congruence, anglesideangle (ASA), sideangleside (SAS), and sidesideside (SSS), follow from the definition of congruence in terms of rigid motions. [GCO8] [MA2013] GEO (912) 9: 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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [GCO9] [MA2013] GEO (912) 10: Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, 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, and the medians of a triangle meet at a point. [GCO10] [MA2013] GEO (912) 11: 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. [GCO11] [MA2013] GEO (912) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [GCO12] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13] [MA2013] GEO (912) 16: Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. [GSRT3] [MA2013] GEO (912) 17: Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4] [MA2013] GEO (912) 18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [GSRT5] [MA2013] GEO (912) 19: 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. [GSRT6] [MA2013] GEO (912) 20: Explain and use the relationship between the sine and cosine of complementary angles. [GSRT7] [MA2013] GEO (912) 21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [GSRT8] [MA2013] GEO (912) 22: (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [GSRT10] [MA2013] GEO (912) 23: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).
[GSRT11] [MA2013] GEO (912) 24: Prove that all circles are similar. [GC1] [MA2013] GEO (912) 25: 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. [GC2] [MA2013] GEO (912) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3] [MA2013] GEO (912) 27: (+) Construct a tangent line from a point outside a given circle to the circle. [GC4] [MA2013] GEO (912) 28: 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. [GC5] [MA2013] GEO (912) 29: 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. [GGPE1] [MA2013] GEO (912) 35: Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [GGMD1] [MA2013] GEO (912) 37: Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama)
Geometry StepbyStep
http://agutie.homest...
This site provides a mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", and more.
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Title: IGD: Simson Line
Description:
This student interactive, from Illuminations, helps students understand and investigate a triangle s Simson line, which is the line passing through the feet of the perpendiculars dropped from P to the three sides of a triangle. Students construct the Simson line for a given triangle using the interactive applet.
Standard(s): [MA2013] GEO (912) 10: Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, 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, and the medians of a triangle meet at a point. [GCO10] [MA2013] GEO (912) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [GCO12] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13] [MA2013] GEO (912) 17: Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4] [MA2013] GEO (912) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3] [MA2013] MI1 (912) 11: Describe the development of mathematical tools and their applications. (Alabama)
Subject: Mathematics Title: IGD: Simson Line
Description: This student interactive, from Illuminations, helps students understand and investigate a triangle s Simson line, which is the line passing through the feet of the perpendiculars dropped from P to the three sides of a triangle. Students construct the Simson line for a given triangle using the interactive applet. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Triangle Classification
Description:
This student interactive, from an Illuminations lesson, allows students to turn a segment into a right triangle, an isosceles triangle, and an obtuse triangle.
Standard(s): [MA2013] (7) 12: Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. [7G2] [MA2013] GEO (912) 10: Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, 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, and the medians of a triangle meet at a point. [GCO10] [MA2013] GEO (912) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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. [GCO12] [MA2013] GEO (912) 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13] [MA2013] GEO (912) 17: Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4]
Subject: Mathematics Title: Triangle Classification
Description: This student interactive, from an Illuminations lesson, allows students to turn a segment into a right triangle, an isosceles triangle, and an obtuse triangle. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12

