Teaching Resources for Geometry
- Pythagorean Theorem - uses interactive activities to prove the Pythagorean Theorem.
- Computing Pi- The Greek mathematician Archimedes approximated pi by inscribing and circumscribing polygons about a circle and calculating their perimeters. Similarly, the value of pi can be approximated by calculating the areas of inscribed and circumscribed polygons. This applet allows for the investigation and comparison of both methods.
- Proportioner- This applet is useful in discovering ratios of various objects as well as comparing two objects, as well as finding perimeter and area of objects.
- Trigonometry- This web site has definitions, applets, and much more to help students learn about trigonometry and other connected concepts.
- The Math Page: Trigonometry- This resource has multiple concepts for geometry and trigonometry. The concepts are divided among chapters with links on common unknown concepts to help students understand the text. This resource also provides exercises that can be done by the students (answers are provided).
- Web Math - Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics. The site also offers specific sections on the conversion of units (applicable to the sciences).
- Discovery Math: Exploring Geometry - This lesson plan includes a project that allows students to deepen their understanding of simialar and congruent figures while also learning about surface area and geometric transformations.
Title Pythagorean Theorem URL http://mathematica.ludibunda.ch/pythagoras.html Materials needed Flash 5 or higher Learning Objectives Students will:
. prove the Pythagorean Theorem
Grade Levels Geometry CA 97 Standards Geometry: 14.0 Students prove the Pythagorean Theorem.
Geometry: 15.0 Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles CA Common Core State Standards Standards for Mathematical Practice:
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning8-Geometry: Understand and apply the Pythagorean Theorem. 8-Geometry-6:. Explain a proof of the Pythagorean Theorem and its converse. 8-Geometry-7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8-Geometry-8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Geometry-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. How could you use this resource? This site could be used either by a teacher in a whole-class demonstration, or by students in a lab setting to develop the meaning of the Pythagorean Theorem. Subsequent pages on the site could be used as a reference to a proof of the Pythagorean Theorem.
EL and Special Needs This site could be used either by a teacher in a whole-class demonstration, or by students in a lab setting to develop the meaning of the Pythagorean Theorem. Subsequent pages on the site could be used as a reference to a proof of the Pythagorean Theorem.
Lesson Plans Teacher Comments Cost Free Copyright unknown
Title Computing Pi URL http://illuminations.nctm.org/ActivityDetail.aspx?ID=161 Materials needed Computer
JavaLearning Objectives Students will:
- Investigate and compare two methods of approximating pi
Grade Levels Geometry CA 97 Standards Geometry: 19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
Geometry: 20.0 Students know and are able to use angle and side relationships in problems with special right triangles, such as 30-60-90 degree triangles and 45-45-90 triangles. Geometry: 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningGeometry-SRT.7; Explain and use the relationship between the sine and cosine of
complementary angles.Geometry-SRT.8; Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry-C.2; 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. Geometry-C.3; Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Geometry-C.4; Construct a tangent line from a point outside a given circle to the circle. How could you use this resource? Students cans use this resource as an exploratory tool to compare two methods of approximating pi. Teachers can use this resource as a method of demonstrating two methods used to approximate pi and discuss with the class the similarities and differences.
EL and Special Needs This is a visual tool that demonstrates the ideas of two methods used for approximating pi. The students can visually and instantly see multiple stages of approximating pi using the two methods. Students can also do the first few stages by and this tool used afterward to demonstrate the same activity to a different degree.
Lesson Plans Teacher Comments Cost Free Copyright (c) 2000-2010 NCTM
Title Proportioner URL http://seeingmath.concord.org/resources_files/Proportioner.html Materials needed Computer
JavaLearning Objectives Students will:
- Investigate and compare ratios, area, and perimeter
Grade Levels Grade 6, Grade 7, Geometry CA 97 Standards Grade 6: NS 1.2 Interpret and use ratios in different contexts to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
Grade 7: MG 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangle, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Geometry: 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6.RP.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 6.RP.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. 7. G.4: Know the formulas for the area and circumference of a circle and use them to solve problems: give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6: Solve real world and mathematical problems involving area, volume and surface area of two-and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. Geometry-GPE.7; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. How could you use this resource? Students can use this resource as a method of studying ratios and how ratios vary based on the two objects being compared. Students can also compare ratios, find the area, and find the perimeter of a given object. Teachers can use this resource as an introduction tool to ratios and geometric figures. Since the applet does the computations, it should be used to demonstrate the effects of ratios, perimeter, and area instead of how to find them.
EL and Special Needs This is a very visual applet that can help students link the concept to real life problems.
Lesson Plans Teacher Comments Cost Free Copyright (c) 2005 Concord Consortium
Title Trigonometry URL http://mathematica.ludibunda.ch/trigonometry.html Materials needed
- Java software
- Flash 5 or higher
Learning Objectives Students will be able to:
- discover how to use trigonometry applied to multiple concepts
Grade Levels Geometry, Trigonometry CA 97 Standards Geometry: 18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. Trigonometry: 2.0 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle ad are familiar with the graphs of the sine and cosine functions. Trigonometry 4.0 Students graph functions of the form f(t) = A sine(Bt + C) or f(t) = A cos(Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. Trigonometry 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningGeometry-SRT.6; 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. Geometry-SRT.7; Explain and use the relationship between the sine and cosine of complementary angles. Geometry-SRT.8; Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Functions-TF.2; Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Functions-TF.3; Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π+x, and 2π–x in terms of their values for x, where
x is any real number.Functions-TF.5; Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions-TF.8; Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. How could you use this resource? This resource could be used by a student as a research tool. The website has a lot of text with information that could be good if a student was doing a research-based assignment. The applets can be used to help gain further understanding of trigonometry concepts. A teacher can use this resource as a means of preparing a lesson or as a demonstration tool. EL and Special Needs This site doesn't address the needs of an EL student very well. However, it does provide definitions which are boldfaced, changed in color, or placed into boxes. The interactive activities on this site can be adopted into a teacher's lesson because they are visible and hands on. Lesson Plans Teacher Comments Cost Free Copyright Not available on web site
Title The Math Page: Trigonometry URL http://www.themathpage.com/aTrig/trigonometry.htm Materials needed Java software Learning Objectives Multiple learning objectives based on subject and subtopic.
Grade Levels Geometry, Trigonometry CA 97 Standards Multiple standards for Geometry from the Mathematics Framework for California Public Schools. Multiple standards for Trigonometry from the Mathematics Framework for California Public Schools. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningMultiple standards for Geometry and Trigonometry from the California Common Core State Standards. How could you use this resource? This resource can be used by students as a development tool to better comprehend trigonometric concepts. Teachers can use this resource as a way to find lesson information or to present trigonometric information to students. EL and Special Needs Many of the explanations for the trigonometric concepts have some color coding to help EL students and special needs students identify parts of the trigonometric concepts. Lesson Plans Teacher Comments Cost Free Copyright © 2001 - 2010 Lawrence Spector
Title Web Math URL Materials needed Computer
Learning Objectives Learning objectives vary by concept.
Grade Levels K-8, Algebra 1, Geometry, Calculus, Trigonometry CA 97 Standards Multiple standards from the Mathematics Framework for California Public School. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands. How could you use this resource? Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics.
The site also offers specific sections on the conversion of units (applicable to the sciences).
EL and Special Needs Visual graphics display concepts and are interactive. Lesson Plans Teacher Comments Cost Free Copyright (c) 2009 WebMath.com
Title Discovering Math: Exploring Geometry URL http://www.discoveryeducation.com/teachers/free-lesson-plans/discovering-math-exploring-geometry.cfm
Materials needed Computer
Tape, scissors, paper, cardboard or boxes to build city, compass, measuring tools
Learning Objectives Students will:
1. Create three-dimensional figures.
2. Construct a three-dimensional model of a city using similar and congruent figures and geometric transformations.
3.Create a two-dimensional representation of their city.
4. Find the surface area of their three-dimensional figures.
Grade Levels Grade 6, Grade 7, Geometry CA 97 Standards Grade 6 NS 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division.
Grade 6 MG 1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems: Grade 6 MG 1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base ⋅ height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid. Grade 7 AF 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs: Grade 7 MG 1.2 Construct and read drawings and models made to scale. Grade 7 MG 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale: Grade 7 MG 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Grade 7 MG 2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor. Grade 7 MG 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures: Grade 7 MG 3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Grade 7 MG 3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. Grade 7 MG 3.6 Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect). Geometry 9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders. Geometry 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Geometry 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6.G: (Cluster Statement) Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find the volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 7.NS. (Cluster Statement) Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. 7.G.1: Solve problems involving scale drawings of geometric figures, including actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6: Solve real world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. 7.EE. (Cluster Statement) Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.4a: Solve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution, identifying the sequence of the operations used in each approach. 8.G. Geometry Cluster Statement: Understand and apply the Pythagorean Theorem.
8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8-G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two- and three-dimensions.How could you use this resource? This resource could be used as a project to solidify the related concepts.
EL and Special Needs This resource uses manipulatives that allow students to better understand the concepts of surface area and similar and congruent figures. Lesson Plans Teacher Comments Cost Free Copyright (c) 2011 Discovery Education