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The items marked with are the contributions of the Summer 2000 participants.

Pythagoras' Theorem states that given a right triangle with legs a and b and hypotenuse
c, ^{2} + b^{2} = c^{2}.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 - a^{3} + b^{3} = c^{3}?
What about a 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.
^{2} - y^{2} b = 2xy c = x^{2} + y^{2}
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 | ||||||||||||||||||||||||||||||||||||||||||||

Reference: Collins, William, et al. Mathematics: Applications and Connections, Course 3. Glencoe/McGraw Hill. Ohio. 1998 | ||||||||||||||||||||||||||||||||||||||||||||

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- A selection of different sized coffee can lids, jar lids
- string, pencil, paper, calculators (optional), rulers
Objectives- Calculate the value for (Pi)
- Determine the universality of (Pi)
- Determine the irrationality of (Pi)
Procedure- Give students 1 lid each.
- Measure the radius of the lid.
- Measure the circumference of the lid(using the string).
- Divide the circumference by its radius squared.
- Have students contruct a table comparing the radii, circumfrences and values for (Pi).
Assessment
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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- Tell your students that now they have watched the film, you have some of the natural items shown for them to investigate.
- Pass out Apples, that have been cut in half to show a cross section of the seedbed (this should appear to be a star), Sand dollars, Starfish, and any five petaled flowers that are available.
- Using the five tips the students need to draw a pentagon.
- Have the students then draw two diagonals from each tip. This star shape that appears is a pentagram.
- Students then need to choose a triangle, measure the angles, and they should find that they have produced a Golden Triangle. Continue measuring triangles that are formed by the diagonal lines. Students should be able to find 20 distinct Golden Triangles.
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. - First, make a Mobius Strip. Use one strip of paper. Put the two ends together, and give one end a half—twist. Tape the paper together this way, being sure to tape across the entire strip.
- Draw a line down the middle of the strip, continue drawing until you meet up with your starting point. Could you have drawn this line without lifting your pencil from the paper? Without turning the paper over? Could an ant walking along your line walk until he met his starting point, without walking over an edge of the paper? How many sides does a Mobius strip have? What do you think will happen if you cut along that line?
- Cut along the line. What happens? What would happen if you cut in half again? Try it and see.
- Make another Mobius Strip. If you cut 1/3 of the way, what will happen? Do it. Note: By this time, many students will give up on the “guessing” part of the activity, or else give absurd opinions.
- Make another strip, but use a full twist on the end this time, instead of a half-twist. Cut in half. Note what happens. Then, cut again.
- Put two of your strips together——one should be a Mobius strip, the other a regular loop. Tape them together at right angles, bisecting each other. What happens if you cut along the middle of both? Do it.
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Contributed by Steve Bixler | ||||||||||||||||||||||||||||||||||||||||||||

References: This project designed by: pkelley@informns.k12.mn.us | ||||||||||||||||||||||||||||||||||||||||||||

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(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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- TLW be able to determine the number of vertices, edges and regions on planar networks.
- TLW be able to determine the number of vertices, edges and regions on a three-dimensional object.
- TLW be able to demonstrate how to satisfy Euler's Formula.
Materials and Resources:
- Blackboard
- Chalk
- Toothpicks
- Gumdrops
Procedure:
- Motivation-Who can tell me who might use Geometry, besides math teachers? Some responses might include architects, carpenters, tile layers, scientists working on molecular models, moms and dads. Architects and their works we probably think of more than others. What structures can you think of that almost shout Geometry at you? Baseball Diamonds? Basketball court? Golden Gate Bridge? How about EPCOT Center in Florida?
- Today we are going to introduce you to vertices (or points), edges (or lines), and areas (or faces) of three-dimensional objects by letting you construct your own. What do you suppose would be a good way to construct these? At this point we will take suggestions and explore them, such as attempting to draw a three-dimensional object on the chalkboard. After exploring a few options, we will introduce the toothpicks and gumdrops.
- Teacher modeling or demonstration. We will build a three-dimensional object of our own and point out the vertices, edges and areas that comprise our object.
- Check for understanding. After the students have a chance to build their own three-dimensional objects, we will questions students about what they have learned. Such as what are the vertices, edges and areas and how many are included in your object.
- Guided practice or activity. At this point we will introduce Euler's Formula (
*V - E + F = 2*) and demonstrate how to satisfy it using our model. - We will then instruct the students to satisfy Euler's Formula using their model.
Assessment Plan: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.
- Show students a piece of 8 ½" x 11" piece of paper. Roll the piece of paper into a tall cylinder, then tape it. Roll a different piece of paper, the same size, into a shorter, wider cylinder. Ask the students which they think would hold more (has the larger volume).
- After breaking students into groups, give each group a piece of paper. Have students measure the length and width of the paper to figure the surface area of the cylinder (excluding tops and bottoms).
- Have each group calculate the volumes of their cylinders (V = Bh; with B = Area of Base) as tall, narrow ones and shorter, wider ones.
- Now, to show them how to figure which 8 ½ x 11 cylinder holds more. Use some sort of dry good (such as beans or marshmallows) to measure how many "cups" fit into each cylinder. This is where the history of unstandardized measurement and how it was used could be discussed.
- Next, have the students figure the actual volume of the cylinders.
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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.)
Look at the shape you are holding. Describe it. It is a (
Look at the outer edge of you circle. What is the distance around the outside of the circle called? ( Fold your circle directly in half and crease it well.
Open the circle, the crease you made is the ( 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 ( ^{o} (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 (
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 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 (
Look at the curved part of the circle between the points where this line touches the outside of the circle. This is called an (
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 (
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. ), they are the same as well as all of the sides are the same. This triangle is called an (angles). You could also use (equilateral and/or acute triangle).
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 ( ). Can you figure out anything else about this triangle? It is a (altitude).
right triangleOpen 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 ( ) and two that are not it is also called a (parallel).
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 ( 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 ( ) (faces) (edges) (points) (vertices) and the fact that this is a (base) and not a squared pyramid like those built in Egypt.
triangular pyramidOpen 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? (
Now fold in the last point. What shape is it now? ( ) figures.
plane
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 ( 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.
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.
- Protractor
- Pencil
- Paper
- Meter Stick or tape measure
- Garden hose with nosel attached
Procedures:
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 20
- Which angle allowed you to achieve the maximum distance?
- Sketch the path the water traveled in and describe the shape of the path.
- Can you think of a method to determine the maximum height the water achieved at the optimum angle? Briefly describe your method.
- If you were to increase the pressure on the water hose, what effect if any would it have on the angle you would use to achieve the maximum distance at the new pressure?
- What connections can you make to the real world from this experiment?
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.
- Tape measure
- String
- Meter stick
Objects to be measured:
- Fruit can
- Soup can
- Coffee can
- Juice can
- Oatmeal Box
- V-8 can
- Paint can
Directions:
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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Contributed by CiCi Naifeh | ||||||||||||||||||||||||||||||||||||||||||||

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- Have students cut out different geometric shapes (triangle, rectangle, parallelogram, trapezoid, circle, circular ring) and glue on a separate piece of paper.
- Calculate the area of each of the shapes using the correct area formula.
Part 2-
- Have students design their own picture using the geometric shapes on a sheet of paper. (May take some time, so could assign as a homework project.)
- Using construction paper, have the students make their pictures and glue on a separate sheet of paper.
- After completing the picture, have students calculate the area of their design by calculating the area of each individual piece and summing to find total area.
Part 3-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.
- Students will enjoy making their own playdoh with the following recipe:
- ½ cup salt
- 1 cup flour
- 2 teaspoon cream of tartar
- 1 cup water
- 1 tablespoon cooking oil
- Food coloring
[You will need to make some ahead of time to determine the amount needed for your classroom.] - After students have made their playdoh, let them play with making geometric shapes (cone, cylinder, sphere, box) and draw the shape they make on a piece of paper with the measurements of each shape.
- Calculate the volume of each geometric shape.
Part 2-
- Have students design their own three dimensional object using their knowledge of geometric shapes. (May take some time, so could assign as a homework project.)
- On a separate sheet of paper, have students draw their object with dimensions.
- Estimate the volume of their object by calculating individual volumes for the shapes and summing the volumes.
Practical Application: A manufacturer needs to determine the volume of a container a product is to be shipped in.
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Contributed by Judy Lasater | ||||||||||||||||||||||||||||||||||||||||||||

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- - areas of plane figures
- - areas of plane figures with equal perimeters
Level: General math and pre-algebra
The following table, or similar, may be provided for record-keeping:
- What areas did you find for:
- squares?
- rectangles?
- triangles?
- circles?
- Which shape gave you the largest area?
- Which shape do you think Dido should lay out?
Closing: Discuss the fact that, with constant perimeter, the greatest area is obtained from a circular shape. Ask the students for other places they think this knowledge might be useful. One idea that might come up is fencing in a pasture.
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.
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.
- Give each student a square of a geometrically designed fabric square and a piece of template paper (a piece of paper with two squares drawn on it-5 inches x 5 inches).
- Have students transfer the geometric design from the fabric on to their paper and then color the design using the fabric as an example.
- The second square on the template paper is to draw their own geometric textile design and color their design (use only colored pencils in which you have the same color of fabric paints).
- Give each student a square of white fabric and have them transfer their design onto the fabric (I suggest you tape the fabric down to a piece of cardboard and use a pencil to transfer the design).
- Once the design is complete in pencil, the student uses the fabric paints to complete the design.
- Collect all the design squares and sew together to make a geometric textile quilt to display in your room.
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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 | ||||||||||||||||||||||||||||||||||||||||||||