Activities for building geometric connections (handout) MCTM Conference 2013 Cheryl Tucker Minneapolis Public Schools Tucker.cherylj@gmail.com (Many materials are from Geometry Connections, CPM, used with permission)
Kaleidoscope: Each team will need hinged mirrors and a piece of colored paper. Your Task: Place a hinged mirror on a piece of paper so that its sides intersect the edge of the paper. Explore what shapes you see when you look directly at the mirror, and how those shapes change when you change the angle of the mirror. Discuss the following questions with your team. Be ready to share your responses with the class. Team Questions: 1. What happens to the shape you see as the angle formed by the mirror gets bigger (wider)? What happens as the angle gets smaller? 2. Does it matter if the sides of the mirror intersect the edge of the paper the same distance from the point where the mirrors are hinged? 3. What happens to the shapes if they are not equal? What is the smallest number of sides the shape you see in the mirror can have? What is the largest? 4. Find a way to form a regular hexagon (a shape with 6 equal sides and equal angles). 5. Describe how you set the mirrors to form a hexagon.
Next, try this with the class: Use a protractor to measure the angle at the hinge. Put the protractor on the top of the hinge so it does not interfere with what you see. Specific shapes may be formed with specific angle measures. Some students may note that the product of the angle measure and the number of the sides is always 360 degrees!
Height Lab Materials: small weight or washer, string - one piece 3 ft., another 1 ft., tape - 2 pieces per team, rulers - 1 per team Objective: I will investigate heights of triangles. 1. Tape a 15 cm section of the long string along the edge of a desk or table. Be sure to leave long ends of string hanging off each side. 2. Tie one end of the short string to the weight and the other to the end of a pencil. Mark a distance 14 cm from the pencil toward the weight by placing a piece of tape on the string. 3. Bring the loose ends of string up from the table and cross them. Put the pencil with the weight over the crossing of the string. Cross the strings again on top of the pencil. Now, with your team, build and sketch triangles that meet these 3 conditions: Student jobs are: *Hold the pen with the weight. *Make sure that the height stays the same and all strings stay taught. *draw accurate sketches. *hold the strings that make the sides of the triangles. Find and draw a picture of: 1. The height of the triangle is inside the triangle. 2. The height of the triangle is a side of the triangle. 3. The height of the triangle is outside of the triangle. Questions: Do all three triangles have the same base and height? Do they look the same? Do they have the same area? What follows? Use index cards to find height of triangles, finding areas of triangles.
Heights For figures (a) through (d), draw a height to the side labeled base. One possible way to do this (there are many more!) is shown below. 2-89, part (b)
Pantograph Lesson Intro: Before computer and copy machines existed it sometimes took hours o enlarge documents or to shrink text on items such as jewelry. A pantograph device was once used to duplicate written documents and rtistic drawings. You will now employ the same geometric principles by using rubber bands to draw enlarged copies of a design. 1. Draw a simple shape. 2. Take either the 2 or 3 rubber band chain, putting a pencil in 1 end and placing it on the stretch point.. 3. Stretch until the first knot is on the design. Keep knot on the design while using marker to draw the new shape. Marker is in the other end of the rubber band chain. Questions to ask: 1. What do the shapes have in common? 2. What is different about the shapes? The teacher is leading to: Figures will have the same shape if their angles are equal in measure and their sides grow in a common pattern.
You are getting sleepy Prior learning consists of triangle similarity. This can be used as an assessment to determine understanding of sides being proportional and similarity relationships. Legend has it that if you stare into a person s eyes in a special way, you can hypnotize them into squawking like a chicken! Here is how it works. Place a mirror on the floor. Your victim has to stand exactly 200 cm away from the mirror and stare into it. The only tricky part is that you need to figure out where you have to stand so that when you stare into the mirror, you are also staring into your victim s eyes. If your calculations are correct and you stand at the EXACT distance, your victim will squawk like a chicken! Directions: 1. Choose a member of your team to hypnotize. 2. Measure heights of both yourself and your victim (eye height) 3. Draw a diagram to represent this situation. 4. Label all lengths you can on the diagram. (Remember - victim stands 200 cm from mirror!) 5. Find any equal angle pairs(remember what you know about how images reflect off mirrors.) 6. What is the relationship between the 2 triangles? How do you know? 7. Moment of truth check it out! Squawk!!
Making Predictions and Investigation Results Today you will investigate what happens when you change the attributes of a Möbius strip. As you investigate, you will record data in a table. You will then analyze this data and use your results to brainstorm further experiments. As you look back at your data, you may start to consider other related questions that can help you understand a pattern and learn more about what is happening. This Way of Thinking, called investigating, includes not only generating new questions, but also rethinking when the results are not what you expected. 1-9. On a piece of paper provided by your teacher, make a "bracelet" by taping the two ends securely together. Putting tape on both sides of the bracelet will help to make sure the bracelet is secure. In the diagram of the rectangular strip shown at right, you would tape the ends together so that point A would attach to point C, and point B would attach to point D. Now predict what you think would happen if you were to cut the bracelet down the middle, as shown in the diagram at right. Record your prediction in a table like the one shown below. Experiment Prediction 1-9 Cut bracelet in half as shown in the diagram. 1-10 1-11 1-12a 1-12b 1-12c 1-12d Now cut your strip as described above and record your result in the first row of your table. Make sure to include a short description of your result.
1-10. On a second strip of paper, label a point X in the center of the strip at least one inch away from one end. Now turn this strip into a Möbius strip by attaching the ends together securely after making one twist. For the strip shown in the diagram above, the paper would be twisted once so that point A would attach to point D. The result should look like the diagram at right. A Möbius Strip Predict what would happen if you were to draw a line down the center of the strip from point X until you ran out of paper. Record your prediction, conduct the experiment, and record your result. 1-11. What do you think would happen if you were to cut your Möbius strip along the central line you drew in problem 1-10? Record your prediction in your table.cut just one of your team's Möbius strips. Record your result in your table. Consider the original strip of paper drawn in problem 1-9 to help you explain why cutting the Möbius strip had this result. 1-12. What else can you learn about Möbius strips? For each experiment below, first record your expectation. Then record your result in your table after conducting the experiment. Use a new Möbius strip for each experiment. What if the result from problem 1-11 is cut in half down the middle again? What would happen if the Möbius strip is cut one-third of the way from one of the sides of the strip? Be sure to cut a constant distance from the side of the strip. What if a strip is formed by 2 twists instead of one? What would happen if it were cut down the middle? If time allows, make up your own experiment. You might
change how many twists you make, where you make your cuts, etc. Try to generalize your findings as you conduct your experiment. Be prepared to share your results with the class. 1-13. LEARNING REFLECTION Think over how you and your study team worked today, and what you learned about Möbius strips. What questions did you or your teammates ask that helped move the team forward? What questions do you still have about Möbius strips? What would you like to know more about?
Final Exam Circle Fold 1. Make a circle with a 3- inch radius. Cut it out 2. How can you find the center if you don t know it? a. 2 quarter folds b. If you can t fold it? Fold on any chord. Bisect it and crease. Repeat and the point of intersection is the center O. Why: diameter is perpendicular to a chord at its midpoint 3. Fold any point A on the circle to the center of the circle. Crease. Label the endpoints of the crease (chord) AB. 4. Fold chord AC where C is determined by folding this section down so another point on the chord intersects (touches) the center O. You now have 2 chords equal in length with a common endpoint and 2 points on the circle intersecting at the center 5. Do this same procedure a third time, creating an equilateral triangle. a. Questions should be asked at this time what are the angle measures? Is it regular? Etc 6. Bisect 1 side by folding it in half. Make a small crease at the midpoint. a. If folded completely, what kind of a triangle do you have? Special 30-60- 90. Leg lengths, area 7. Fold the vertex of the side opposite down to the midpoint. Crease. a. What shape do you have? Area? Base angles? Characteristics? Etc 8. Fold the 2 single triangles on top of the others to form 1 single smaller triangle. Open it up. a. Questions regarding similar shapes, areas, ratio of sides, perimeters, area 9. Next, with 1 triangle folded down and 1 folded over, what shape do you have? a. Trapezoid questions are next! 10. Make a 3 dimensional shape. What is it a triangular pyramid? Or a tetrahedron. a. Ask a few volume questions. 11. Open up so you have the original triangle. Fold 1 vertex to the center of the circle. Crease. Repeat with the other 2 vertices. What shape? Hexagon a. Questions regarding hexagon, symmetry, central angles, area, area of regular polygons. 12. Gently squeeze the shape so that the 3 small triangles of each side overlap (on top of each other). a. What shape is this? A truncated tetrahedron! Why is it not a pyramid? Similarities? Differences? 13. Lastly, I have my students autograph the large triangle (the base). I use a small amount of tape and then collect them. I have a couple of volunteers put 20 of them together to make an icosahedron. Then I hang it from my ceiling, recalling their class forever!!