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are the contributions of the Summer 2000 participants.|
Geometry Activities |
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are the contributions of the Summer 2000 participants.
Pythagoras' Theorem states that given a right triangle with legs a and b and hypotenuse
c,  Numbers that fit this pattern are called Pythagorean triples. The chart below shows several of these Pythagorean triples. Study the pattern in the chart to find the next two sets of triples. | ||||||||||||||||||||||||||||||||||||||||||||
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Will any of the triples satisfy Fermat's Last Theorem - a3 + b3 = c3?
What about a4 + b4 = c4? The following formulas can be used to generate numbers that will satisfy Pythagoras' Theorem. Be sure that x > y and that one of them is odd and the other is even.
Here is an example: Let x=7 and y=6.
Tip: To get numbers a, b, and c which have no factors in common, make sure your numbers x & y have no common factors, for example 5 & 6 or 5 & 8, but not 5 & 10. | ||||||||||||||||||||||||||||||||||||||||||||
| Contributed by Charlene Evans | ||||||||||||||||||||||||||||||||||||||||||||
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Reference: Collins, William, et al. Mathematics: Applications and Connections, Course 3. Glencoe/McGraw Hill. Ohio. 1998 | ||||||||||||||||||||||||||||||||||||||||||||
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Materials
Have students solve three problems that have only the value for diameter given.
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| Contributed by David Leib | ||||||||||||||||||||||||||||||||||||||||||||
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Purpose: Students will be able to use algebraic concepts while investigating nature. Objective: Students will use geometry and the concept of ratios and apply them towards measuring items found in nature. Materials: Five petaled flowers, Apples, Sand dollars, Starfish. Focus: Show the film: Donald Duck in Mathmagic Land (this can be obtained through Library Media Services, USD 259) Procedure:
Assessment: Have the students report what they found from the variety of materials used. Did they find the Golden Triangles? What helped? What needs corrected?
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| Contributed by Jennifer Garretson | ||||||||||||||||||||||||||||||||||||||||||||
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For this activity, you will need five strips of paper per student. Each strip should be approximately 3 x 14 inches. Each student will also need about two feet of tape either transparent tape or making tape will work), a pair of scissors and a pen or pencil.
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| Contributed by Steve Bixler | ||||||||||||||||||||||||||||||||||||||||||||
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References: This project designed by: pkelley@informns.k12.mn.us | ||||||||||||||||||||||||||||||||||||||||||||
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Purpose: Using tangrams to introduce the Pythagorean Theorem. Objective: Students will use tangrams as an introduction to the Pythagorean Theorem. Materials: Tangram set (4 in. x 4 in. square), paper, pencil To ensure each student has enough pieces for some of these activities you may allow students to work in pairs or provide each student with more than one set of tangrams. Introducing Pythagoras
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6 (Behold! Your students have just discovered the Pythagorean Theorem!!!) This is the perfect opportunity to share with your students the history of Pythagoras and how he developed this theorem.
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| Contributed by Angela Ceradsky | ||||||||||||||||||||||||||||||||||||||||||||
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Title of Lesson: Euler's formula, toothpicks and gumdrops. Subject Area: Geometry, Pre-Algebra Grade Level: Eighth Description or Outcome Statement: With information discovered about planar networks and three-dimensional objects, students will be able to demonstrate how to satisfy Eulers's Formula. Objectives:
Verbal questions and answers and a ten-question pencil and paper quiz will be given to assess the student's comprehension of the material presented.
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| Contributed by Jan Swanson | ||||||||||||||||||||||||||||||||||||||||||||
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This activity is to teach the student how to calculate volume and surface area of a cylinder. This activity can easily be linked to history by discussing how the ancients may have measured volume or even length before a standardized measurement system existed. Procedure:
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| Contributed by Lindsay Eastridge | ||||||||||||||||||||||||||||||||||||||||||||
Creating a Truncated Tetrahedron.
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I acquired this activity during a workshop for USD 259 in Wichita, KS called "Monday Night Math". I have taught this to classes from third grade to the college level. This activity helps build, reinforce and can even be used to introduce vocabulary in math. This activity can be done with various grade levels by simply altering the vocabulary to make it appropriate to the skill level you are working with. This is a great listening / fine motor skill activity. (The key vocabulary words are in parenthesis.) Materials: one 7 inch paper circle, pencil, & ruler Look at the shape you are holding. Describe it. It is a (circle). Look at the outer edge of you circle. What is the distance around the outside of the circle called? (Circumference) Fold your circle directly in half and crease it well. Open the circle, the crease you made is the (diameter) of the circle. Hold the circle at the ends of the crease. Fold your circle in half again, but this time match up the end points of the crease. Open your circle, is this also a diameter? How do you know? Do the lines (intersect)? Yes. Is there something special about the way these lines intersect? They create four 90o (or right) angles. This special type of intersection is called (perpendicular). Place a dot, no bigger than the width of a pencil, at the point where the creases connect. This is called the (center) of the circle. Using your pencil, trace one of the lines from the center to the edge of the circle. This line from the center is called a (radius). Fold in one of the outer, curved edges of the circle until it just touches the dot in the middle. Crease it well. Open the fold and look at the crease you just made. Is it a diameter? Is it a radius? Why or why not? This line is called a (chord). Look at the curved part of the circle between the points where this line touches the outside of the circle. This is called an (arc). Can you find other arcs on your circle? Take the opposite side of your circle and fold it so that the curved part just touches the center and the bottom forms a perfect point. Your circle will look like an ice cream (cone). Crease it well. Fold the top of your ice cream cone down until the curved part just touches the center of the circle. The top corners should make perfect points, crease well. Now describe the shape you have. (Triangle) Do you notice anything special about this triangle? Look at all of the (angles), they are the same as well as all of the sides are the same. This triangle is called an (equilateral and/or acute triangle). You could also use (equiangular). Fold the new triangle in half by matching up two of the points. Crease well. The new cease splits the triangle in half, this line is called the (height or altitude). Can you figure out anything else about this triangle? It is a (right triangle). Open the right triangle up to the equilateral triangle. Take the top corner of the big triangle and fold it. By folding along the crease of the height you can match the top point up to the bottom crease line. On the inside you will now see three smaller triangles. Turn the paper over so that you do not see the creases. What is this shape called? Since it has four sides it can be classified as a (quadrilateral). Since this quadrilateral has two sides that are (parallel) and two that are not it is also called a (trapezoid). Turn it back over so that you now see all of the creases. Fold one of the outer triangles in so that it lies directly on top of the center triangle. Turn it back over and describe the shape you now see. It is not a kite, kites fly in the sky. It is not a diamond, I wear diamonds on my fingers. In mathematics this shape is called a (rhombus). Turn your shape back over and fold the last outer triangle over onto the center one again. You should now have a smaller equilateral triangle. Open up all three of the small triangles. Bring the three loose points together so that you now have a (pyramid). At this point you can discuss (faces) (edges) (points) (vertices) (base) and the fact that this is a (triangular pyramid) and not a squared pyramid like those built in Egypt. Open your pyramid back up to the large equilateral triangle. Fold over one of the points so that it just touches the dot in the middle. What shape have you re-created? The trapezoid though not the traditional shape it can still be identified as a trapezoid. Fold one more of the points in so that it just touches the dot in the middle. Now what shape do you have? (Pentagon) Even though it is not the traditional shape you are accustomed to, it still has five sides, therefor it is still classified as a pentagon. Now fold in the last point. What shape is it now? (Hexagon) Discuss (plane) figures. Turn to the other side and fit one of the corners into a flap on the opposite side of the triangle. You may have to try more than one. Choose the one that makes the best fit. Slide the last corner under/inside the others. You have now created a (truncated tetrahedron)! For any of the shapes during this activity you can have students calculate the surface area, volume, perimeter and/or area.
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| Contributed by Angela Ceradsky | ||||||||||||||||||||||||||||||||||||||||||||
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This activity can be completed in one class period. It may take another class period to answer questions and draw conclusions depending on how long your class period is. It is great for spring when students and teachers have spring fever. Its purpose is to find the optimum angle to achieve the greatest distance. Depending on what grade level you teach, this activity could be modified. History: The simple protractor is an ancient device. The first complex protractor was created for plotting the position of a boat on navigational charts. Called a three-arm protractor or station pointer, it was invented in 1801, by Joseph Huddart, a U.S. naval captain. The center arm is fixed, while the outer two are rotatable, capable of being set at any angle relative to the center one. Materials:
Attach the garden hose to a tap and adjust the flow of water so that it is at constant pressure. Begin with an angle of 0 degrees to the ground, measure and record the distance the stream travels in the horizontal direction along the ground. Repeat for angles of 20o, 30o, 45o, 70o, and 75o. Questions and Conclusions:
URL: http://inventors.about.com/science/inventors/libr.../blmeasurement.htm?terms=protractor+histor 6/14/00 8:30 pm
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| Contributed by Amy Troutman | ||||||||||||||||||||||||||||||||||||||||||||
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Approximately 4000 years ago people were first beginning to understand pi. It was believed that it’s value was about three. Today we use a better approximation of pi. It is 3.14. Pi a irrational number. That means that it does not repeat in a pattern. Pi is defined as being the ratio between the circumference of a circle and the diameter of the same circle. Now, here is a little activity to help you determine pi. Materials:
First on a sheet of paper make 4 columns labeled: object, diameter, circumference, and ratio. (See example below.) Wrap the string around an object. This measures the circumference of the object. Measure the string using the meter stick. Write the value in the column labeled circumference. Next, measure the distance straight across the object. Put this value in the column labeled diameter. Now take the circumference divided by the diameter. Put this in the column labeled ratio. Is this close to pi? Make sure you take the ratio out to three or four decimal places.
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| Contributed by Jeremy Troutman | ||||||||||||||||||||||||||||||||||||||||||||
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Grade: 5 or 6 Time: 20-30 minutes lessons Materials Needed: A mixture of polygon shapes such as geoblocks, paper, pencil and colored pencils. Lesson 1 Focus: Put various shapes on the overhead. Have a class discussion about the similarities and differences of the shapes. Inform students that the shapes are polygons (a close plane figure). Introduce the concept of tiling and how there can be no gaps or overlaps. Demonstrate the concept on the overhead using the square. Activity: Have students predict which shapes will tile and which will not. Make a chart showing the results of the students predictions. Give each table (or student if possible) a set of shapes. Have students individually discover whether or not a shape will tile. Conclusion: Discuss as a class which shapes actually tile and which do not. Compare discoveries to the charted predictions. Closure: Have students define polygon and what they have learned about them in a short journal entry. Lesson 2 Focus: Review the concepts the polygon and tiling. Introduce the term tessellation and how it relates to lesson 1. Discuss the concepts of slide, rotation (turn), and reflection (flip) and demonstrate on the overhead. Activity: Have students choose two or more shapes. Have students color each individual shape the same color. Have students outline the shapes with a black marker if it is a slide. Outline the shape with a blue marker if it is a rotation. A reflection should be outlined in red marker. Conclusion: Have students raise their hands as they are finished. Instructor will check results. Closure: Discuss what various shapes tessellate together charting results. Allow students to journal their findings. Lesson 3 Focus: Review previous lessons. Discuss how students will be applying their knowledge of the tessellation technique to finding the concept in art. Activity: Provide a number of pieces by various artists, include a few pieces by M. C. Escher. Have students study the different artworks to find whether or not the artist uses the tessellation technique. Conclusion: Have students discuss what they found in the pieces. Hypothesize why the artists used or did not use the tessellation technique. Closure: Have students record their findings in their journals.
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| Contributed by CiCi Naifeh | ||||||||||||||||||||||||||||||||||||||||||||
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Area of Geometric Shapes Purpose: To give the student the opportunity to practice calculating the area of different geometric shapes Materials Needed: Used magazines, scissors, rulers, glue, construction paper, and calculators
Procedure:
To emphasize the process of problem solving (solve the problem by using a method you already know about), give students a simplistic picture taken from coloring books and ask them to estimate the area of the picture. By using the above method, the students should take the picture apart by shapes and calculate the area of each shape and then sum the area shapes to estimate the area of the picture. Practical Application: If one is going to paint a room, the total area of all the walls needs to be determined in order to purchase the right amount os paint. Volume of Geometric Shapes Purpose: To give students the opportunity to visual understand three dimensional geometric shapes and practice calculating volume Materials needed: Homemade playdoh ingredients, hot plate, cooking pot, spoon, rulers, and calculators
Procedure:
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| Contributed by Judy Lasater | ||||||||||||||||||||||||||||||||||||||||||||
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Objectives:
Materials: For each student or group: one piece of graph paper, taped or glued onto corrugated cardboard or a stiff, foam board; a closed loop of string, length 20 centimeters; approximately six straight pins or thumbtacks per students or group; pencil and paper for recording. Historical note: According to Roman mythology, Dido was the daughter of a king of Tyre. Her brother Pygmalion murdered her husband and she fled, fearing for her own safety. Dido crossed the Mediterranean with all her wealth and some companions, and landed in the North African kingdom of Libya. Here she sought some land as a place of refuge for her group. The king agreed to give her as much land she could encompass with the hide of a bull. Now, Dido knew something about geometry, and she wanted as much land as possible. She had the hide of a bull cut into thin strips, and then joined to form one, long strand. What shape do you think she enclosed? Set-up: Students may do this activity individually or in groups. Possible group roles for four members are two students to make shapes, one to calculate areas, and one to record shapes, dimensions and areas. Each group will need one set of materials as described above: graph paper on a board, thumbtacks, and string. Activity: Students are to use the string and thumbtacks to lay out plane figures on the grid. Students will need to calculate the area of each figure they make, so the shapes chosen will depend on students’ background knowledge. Suggested figures are a square, rectangles, triangles, and a circle. Remind students that they may form more than one rectangle and more than one triangle. Students should calculate areas for each figure formed, and decide what shape they think Dido should choose. Note: if this is used as an activity to introduce areas of plane figures, students may find areas by counting squares. As an enrichment activity, students should use area formulas. The following table, or similar, may be provided for record-keeping:
Discussion questions:
In the end Dido measured off a half-circle shape, joining one point on the coast to another. This way water formed one edge of her claim, and she had the added advantage of access to the sea. She only needed to use the hide to enclose land, so the area she marked off was as large, and desirable, as possible. The King of Libya was true to his word, and gave her the land. The reputation of the clever Dido, future Queen of Carthage, was established. Have students untie their string loops and use their 20-cm strands to form semicircles at the edges of their grids. Calculate the area of the semicircle formed in this way to find out just how clever Dido was. Related Activity: The following trick may be a fun introduction to the activity. Bring a piece of paper, perhaps the size of a piece of printer paper, to class. Ask your students, “Do you think I can cut a hole in this sheet of paper and push (fill in the name of some student in the class) through it?” When the students express their skepticism, cut a large hole in the paper, put your arm through the hole, and give the chosen student a gentle shove. Next bring out a fresh sheet of paper, in which there is a very small hole. Ask “Do you think I can get (same student) through a sheet of paper with this hole?” The general reaction will be that you probably can’t, though some students will be on the lookout for another trick. To show them you can do what you said, bring out a paper with the same, small hole, and slits cut as shown in the diagram below. You may use the sheet you already have and cut slits while they wait, but it is time-consuming. Open the slits to form one, huge paper loop, and put it right over the student’s head. You have done it!
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| Contributed by Laurie Kiss | ||||||||||||||||||||||||||||||||||||||||||||
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Sometimes students have trouble relating mathematics to the “real world”; therefore, this activity in geometric textile design will emphasis the purpose of geometric shapes and patterns in conjunction with artistic design, computer applications, and career opportunities. Purpose: For each student to design their own geometric textile square that will be made into a quilt. Materials Needed: Geometrically designed fabric squares (5 inches x 5 inches), template paper, colored pencils, white fabric squares (5 inches x 5 inches), scraps of cardboard, masking tape and fabric paint Procedure:
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| Contributed by Judy Lasater | ||||||||||||||||||||||||||||||||||||||||||||
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A polyhedron is a geometrical figure which is the three-dimensional version of the plane polygon (two-dimensional). There are five regular polyhedra. They are: the tetrahedron (4 faces), the hexahedron or cube (6 faces), the octahedron (8 faces), the dodecahedron (12 faces), and the icosahedron (20 faces). Below are patterns for two polyhedra that can be reproduced and transformed into three-dimensional figures. Click on the figures to open a window with the figure in it for ease of printing.
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| Contributed by Susan Eastman | ||||||||||||||||||||||||||||||||||||||||||||