Triangles at Work
Triangles do their share of work in construction as well as in measurement of distances, areas of irregular figures, and even the circumference of the Earth. These exploratory problems will give your students an insight into how triangles work in the everyday world.
Triangles are used in finding the area of irregular figures, which can be broken into rectangles and triangles and then handled in a piecemeal fashion. Also, the formulas for the areas of regular hexagons, octagons, and so forth are based on their triangular component. The formula for the area of a triangle is memorized by students in elementary school. What this applet brings to their learning is a way to consider the area of a triangle visually. Students can actually derive the formula themselves when the exercise is limited to the "easy" level, which shows only right-angled triangles. Higher levels of difficulty challenge students to find the area of scalene triangles, given helpful hints. MSP
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Is it true that a triangular structure is stronger than other types? It is, after all, used in bridges, tunnels, and other loadbearing structures. In this activity, students build four beams (triangle, rectangle, corrugated, and cylinder) out of paper and test their strength. Patterns for the paper beams are included. MSP
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In a letter to Dr. Math, a student asks, “How can I measure a tree using its and my shadows?” What follows is an explanation of how to draw the similar triangles and how to calculate the ratio of the shadow lengths. This is no better than the explanation you have given in class, but sometimes one more voice can make the difference for the struggling student. MSP
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An out-of-the-ordinary question on the same subject: “What time of day is best to use a shadow to measure the height of a building by using triangles?” Dr. Math answers this one again, bringing in the critical consideration of error estimate. The question is offered here as a real-world problem for your class. MSP
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Looking for a challenging problem? "If you're flying a kite 75 yards away from 70-foot tall trees, how much line would it take to get to the top of the trees?" Your students can apply the Pythagorean theorem here, but the next level of the problem requires basic ideas of trigonometry: "What if the kite moved up so that it was 65 yards away? What would the angle be?" There are more levels to the problem, plus solutions given by students.
This is an awesome Internet project! The site presents all the information you need to re-create the measurement of the circumference of the Earth as done by the Greek librarian Eratosthenes more than 2,000 years ago. Every step in Eratosthenes’ method is explained, illustrated, and diagrammed; applets and handouts reinforce the learning. You can sign up your class to participate in the annual Noon Day Project. Students from around the country take shadow measurements at high noon on a designated day in March; these measurements are posted online and used to calculate the circumference. A teacher’s guide outlines the procedure, from posting measurements to finding a class estimate of the Earth’s circumference. MSP
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Copyright
January 2007 — The Ohio State University. This material is based upon work
supported by the National Science Foundation under Grant No. 0424671. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views of
the National Science Foundation.
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This work is licensed under a
Creative Commons License.
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