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Title: What you know about the Bermuda Triangle?
Description:
The basis of this lesson is for students to form a connection between the importance of geometry as it relates to the Bermuda Triangle. Students should think about the geographical locations and how they form a particular type of triangle. Within the triangle exists a relationship between the angles and the length of the sides. As the students reflect prior concepts and skills should come to mind to help them solve the problem. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles.
This is a College and CareerReady Standards showcase lesson plan.
Standard(s): [MA2013] GEO (912) 33: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [GGPE7] [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]
Subject: Mathematics (9  12)
Title: What you know about the Bermuda Triangle?
Description: The basis of this lesson is for students to form a connection between the importance of geometry as it relates to the Bermuda Triangle. Students should think about the geographical locations and how they form a particular type of triangle. Within the triangle exists a relationship between the angles and the length of the sides. As the students reflect prior concepts and skills should come to mind to help them solve the problem. The ideas presented here can be used as the final lesson in a unit on triangles to summarize all that students have learned. Alternatively, the main problem from this lesson could be used as a final assessment activity for a unit on triangles.
This is a College and CareerReady Standards showcase lesson plan.
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Title: Is it a Triangle?
Description:
The purpose of this lesson is to help students investigate the relationships of the lengths of the sides of a triangle in order to discover the three triangle inequality theorems. Students will learn how to draw valid conclusions from the information obtained from the activity and apply those conclusions to real world geometry problems. This serves as an introduction into more advanced geometric theorems. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project, funded by the Malone Family Foundation.
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]
Subject: Mathematics (7  12)
Title: Is it a Triangle?
Description: The purpose of this lesson is to help students investigate the relationships of the lengths of the sides of a triangle in order to discover the three triangle inequality theorems. Students will learn how to draw valid conclusions from the information obtained from the activity and apply those conclusions to real world geometry problems. This serves as an introduction into more advanced geometric theorems. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project, funded by the Malone Family Foundation.
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Title: Investigation of Special Segments of Triangles
Description:
This lesson will enable students to investigate three special segments of triangles in a very concrete way. The students will fold paper triangles to create the segments. This lesson would be a great way for students to explore the properties of the segments and their intersections.
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) 26: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3]
Subject: Mathematics (9  12)
Title: Investigation of Special Segments of Triangles
Description: This lesson will enable students to investigate three special segments of triangles in a very concrete way. The students will fold paper triangles to create the segments. This lesson would be a great way for students to explore the properties of the segments and their intersections.
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Title: Beyond Triangula Island
Description:
Using this reproducible activity sheet, from an Illuminations lesson, students record their findings as they measure distances from interior points to sides of an equilateral triangle, a square, an octagon and a hexagon. Students answer questions based on their findings.
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]
Subject: Mathematics Title: Beyond Triangula Island
Description: Using this reproducible activity sheet, from an Illuminations lesson, students record their findings as they measure distances from interior points to sides of an equilateral triangle, a square, an octagon and a hexagon. Students answer questions based on their findings. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Inequalities in Triangles
Description:
In this Illuminations lesson, students use pasta to create models of triangles and nontriangles in order to investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle. There are also links to online activity sheets.
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) 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: Inequalities in Triangles
Description: In this Illuminations lesson, students use pasta to create models of triangles and nontriangles in order to investigate the relationship between the longest side of the triangle and the sum of the other two sides of the triangle. In addition, students measure the sides and angles of a scalene triangle and investigate the relationship between the location of the largest angle and largest side in a triangle. There are also links to online activity sheets. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Pieces of Proof
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This lesson offers an intermediate step of putting together the statements and reasons to build a formal geometry proof. There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. Students are given the parts of the proof and have to put them in a correct order, like a puzzle, rather than starting with a blank page to generate the statements and reasons themselves.
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) 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]
Subject: Mathematics Title: Pieces of Proof
Description: This lesson offers an intermediate step of putting together the statements and reasons to build a formal geometry proof. There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. Students are given the parts of the proof and have to put them in a correct order, like a puzzle, rather than starting with a blank page to generate the statements and reasons themselves. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Hospital Locator
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In this Illuminations lesson, students begin with a problem in a realworld context to motivate the need to construct circumcenters and then incenters of triangles. Students must make sense of these constructions in terms of bisecting sides and angles. There are links to student interactives and other resources.
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) 41: 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).* [GMG3]
Subject: Mathematics Title: Hospital Locator
Description: In this Illuminations lesson, students begin with a problem in a realworld context to motivate the need to construct circumcenters and then incenters of triangles. Students must make sense of these constructions in terms of bisecting sides and angles. There are links to student interactives and other resources. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Inequalities for Sides and Angles
Description:
This reproducible activity sheet, from an Illuminations lesson, guides students as they examine the relationship between the locations of sides and angles in a triangle.
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]
Subject: Mathematics Title: Inequalities for Sides and Angles
Description: This reproducible activity sheet, from an Illuminations lesson, guides students as they examine the relationship between the locations of sides and angles in a triangle. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: The Triangle Inequality
Description:
In this reproducible activity, from an Illuminations lesson, students investigate how three sides can be arranged to make both triangles and nontriangles.
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) 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: The Triangle Inequality
Description: In this reproducible activity, from an Illuminations lesson, students investigate how three sides can be arranged to make both triangles and nontriangles. 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.
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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: Area of a Triangle
Description:
This student interactive, from Illuminations, helps students understand and investigate how to find the area of a triangle by finding a way to make it look like a simpler shapone whose area formula is known. Students make a copy of the triangle, and then make a parallelogram using the triangle and its copy. Students then determine that the area of the triangle is onehalf that of the parallelogram.
Standard(s): [MA2013] (6) 21: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems. [6G1] [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) 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: IGD: Area of a Triangle
Description: This student interactive, from Illuminations, helps students understand and investigate how to find the area of a triangle by finding a way to make it look like a simpler shapone whose area formula is known. Students make a copy of the triangle, and then make a parallelogram using the triangle and its copy. Students then determine that the area of the triangle is onehalf that of the parallelogram. Thinkfinity Partner: Illuminations Grade Span: 3,4,5,6,7,8
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Title: Hospital Locator
Description:
In this student interactive, from an Illuminations lesson, students act as community planners, trying to place a new medical center equidistant from three cities.
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) 41: 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).* [GMG3] [MA2013] MI1 (912) 11: Describe the development of mathematical tools and their applications. (Alabama)
Subject: Mathematics Title: Hospital Locator
Description: In this student interactive, from an Illuminations lesson, students act as community planners, trying to place a new medical center equidistant from three cities. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Half Angle
Description:
This student interactive, from an Illuminations lesson, allows students to find the incenter of a triangle and see a visual explanation of why it occurs where it does.
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) 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: Half Angle
Description: This student interactive, from an Illuminations lesson, allows students to find the incenter of a triangle and see a visual explanation of why it occurs where it does. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Isosceles Triangle Investigation
Description:
This student interactive, from an Illuminations lesson, allows students to investigate the relationship between the area of the triangle and the length of its base.
Standard(s): [MA2013] GEO (912) 6: 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. [GCO6] [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) 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] AL2 (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] AL2 (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] AL2 (912) 30: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [FIF7] [MA2013] ALT (912) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] [MA2013] ALT (912) 22: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [ACED3] [MA2013] PRE (912) 16: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine odd, even, neither.)* [FIF4] (Alabama) [MA2013] ALT (912) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] [MA2013] ALT (912) 30: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [FIF7] [MA2013] ALT (912) 33: Write a function that describes a relationship between two quantities.* [FBF1]
Subject: Mathematics Title: Isosceles Triangle Investigation
Description: This student interactive, from an Illuminations lesson, allows students to investigate the relationship between the area of the triangle and the length of its base. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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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: IGD: Median
Description:
This student interactive, from Illuminations, helps students understand and investigate the median of a triangle. Students read a definition and then use the interactive tool to manipulate and explore the properties of triangle medians by changing the shape of a 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) 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]
Subject: Mathematics Title: IGD: Median
Description: This student interactive, from Illuminations, helps students understand and investigate the median of a triangle. Students read a definition and then use the interactive tool to manipulate and explore the properties of triangle medians by changing the shape of a triangle. 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
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Title: Triangle Island
Description:
This student interactive, from an Illuminations lesson, allows students to explore the distances from an interior point of a polygon to its sides and determine the sum of these distances.
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) 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 Island
Description: This student interactive, from an Illuminations lesson, allows students to explore the distances from an interior point of a polygon to its sides and determine the sum of these distances. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: IGD: Euler Line
Description:
This student interactive, from Illuminations, helps students understand and investigate Euler lines. Students read a definition and then construct the Euler line of a triangle using the interactive tool. They learn that the Euler line involves three points associated with triangles: the circumcenter, orthocenter, and centroid.
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) 18: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [GSRT5] [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: Euler Line
Description: This student interactive, from Illuminations, helps students understand and investigate Euler lines. Students read a definition and then construct the Euler line of a triangle using the interactive tool. They learn that the Euler line involves three points associated with triangles: the circumcenter, orthocenter, and centroid. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12

