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Title: Arc Lengths and Sector Areas
Description:
Students will work in cooperative groups to discover the relationships between arc length, central angle measure, and circumference. They will also discover the relationship between circle area, central angle measures, and sector area. Students will share their discoveries and create formulas for calculating arc length and sector area. Then they will practice these skills in a journal entry and solving a reallife extension.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.
Standard(s): [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]
Subject: Mathematics (9  12)
Title: Arc Lengths and Sector Areas
Description: Students will work in cooperative groups to discover the relationships between arc length, central angle measure, and circumference. They will also discover the relationship between circle area, central angle measures, and sector area. Students will share their discoveries and create formulas for calculating arc length and sector area. Then they will practice these skills in a journal entry and solving a reallife extension.
This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.
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Title: Soda Cans
Description:
This reproducible activity sheet, from an Illuminations lesson, guides students through a simulation in which they try different arrangements to make the most efficient use of space and thus pack the most soda cans into a rectangular packing box.
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) 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) 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) 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) 36: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [GGMD3] [MA2013] GEO (912) 38: Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects. [GGMD4] [MA2013] GEO (912) 39: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [GMG1] [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] ALC (912) 7: Use analytical, numerical, and graphical methods to make financial and economic decisions, including those involving banking and investments, insurance, personal budgets, credit purchases, recreation, and deceptive and fraudulent pricing and advertising. (Alabama) [MA2013] ALC (912) 9: Analyze aesthetics of physical models for line symmetry, rotational symmetry, or the golden ratio. (Alabama) [MA2013] ALC (912) 10: Critique measurements in terms of precision, accuracy, and approximate error. (Alabama) [MA2013] ALC (912) 11: Use ratios of perimeters, areas, and volumes of similar figures to solve applied problems. (Alabama)
Subject: Mathematics Title: Soda Cans
Description: This reproducible activity sheet, from an Illuminations lesson, guides students through a simulation in which they try different arrangements to make the most efficient use of space and thus pack the most soda cans into a rectangular packing box. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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Title: Circle Packing and Curvature
Description:
In this lesson, one of a threepart unit from Illuminations, students investigate the curvature of circles. Students apply definitions and theorems regarding curvature to solve circle problems. In addition, there are links to an online activity sheet and other related resources.
Standard(s): [MA2013] (8) 24: Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve realworld and mathematical problems. [8G9] [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) 36: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [GGMD3] [MA2013] GEO (912) 39: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [GMG1] [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: Circle Packing and Curvature
Description: In this lesson, one of a threepart unit from Illuminations, students investigate the curvature of circles. Students apply definitions and theorems regarding curvature to solve circle problems. In addition, there are links to an online activity sheet 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.
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Title: Geometry Pad App
Description:
Create geometric shapes, explore/change their properties, and calculate metrics.
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) 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) 38: Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects. [GGMD4] [MA2013] GEO (912) 39: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [GMG1] [MA2013] ALT (912) 37: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. [FTF1]
Geometry Pad App
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Create geometric shapes, explore/change their properties, and calculate metrics.
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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.
Thinkfinity Learning Activities
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Title: Soccer Problem
Description:
This student interactive, from an Illuminations lesson, allows students to investigate a soccer problem by changing the location of a soccer player as well as the distance between the player and the goal posts. The angle changes as the player is moved, and students must therefore determine the player s position so that the angle is maximized.
Standard(s): [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) 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) 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) 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: Soccer Problem
Description: This student interactive, from an Illuminations lesson, allows students to investigate a soccer problem by changing the location of a soccer player as well as the distance between the player and the goal posts. The angle changes as the player is moved, and students must therefore determine the player s position so that the angle is maximized. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12

