[b]Branch of Mathematics: Geometry[/b]
[b]
Target Population:[/b]
• Age: 14-16 years of age
• Grade Level: 9th-10th grade
• Students could be in any level class: Math Tech, College Prep, or Honors for this activity. This activity could be modified to be successful at any aptitude for learning, but this activity is engaging for all students.
[b]Objectives:[/b]
Students will be able to:
• Understand the term angle sum and explain in writing why the angle sum of a triangle cannot exceed 180 degrees.
• Explain in writing how to classify an isosceles triangle.
• Describe why and show with GeoGebra what happens to the other angles when one angle is increased and decreased in an isosceles triangle.
• [b]Classify points as collinear.
• Explain in writing and show using GeoGebra why an angle in a triangle cannot be 0 degrees or 180 degrees.
• Make predictions about what would happen if we did this same task using an equilateral triangle and a scalene triangle.
• Create acute, right and obtuse triangles and explain in writing why they are classified as such.[/b]
*Objectives in bold are higher level, while objectives in regular text are lower level objectives.
[b]GeoGebra Applet Task Sheet
Investigation of the Triangle[/b]
[i]Guidelines[/i]:
• Engage/Explore: These two parts of the learning theory occur together in this activity. Students will be engaged while exploring the triangle using the sliders. Exploration occurs several times throughout the lesson, including at the beginning while students use the slider to observe what happens to the triangle, side lengths, angles, and angle sum when α changes in size. It occurs again when students use the check box to view the parallel line to investigate collinearity and why that cannot occur in a triangle. This exploration time is also engaging because students are working on their own, using technology, which captures their attention.
• Explain: This part of the lesson occurs when students answer the questions on the task sheet and write their explanations. If I were using this activity in a class I would also go over these questions as a class and have a whole-class discussion so that all students are engaged in the explanation and misconceptions can be corrected. I would assess how students are doing during this portion of the lesson as well.
• Extend: In this section of the lesson students are asked to predict what would happen if we used this activity with an equilateral or scalene triangle instead of an isosceles triangle. They are asked to extend what they now know is the case for an isosceles triangle to predict what would occur in these other two types of triangles. Students are also asked to create acute, obtuse, and right triangles using the slider tool. This extends their knowledge of these types of triangles to the actual creation of them.
[b]Task Sheet
[/b]
[i]Move the slider back and forth, and observe what happens to the triangle, side lengths, angles, and angle sum.[/i]
1. As you increase the size of α, what happens to the side lengths? Which sides increase, decrease, or stay the same? Why does this occur?
2. As you decrease the size of α, what happens to the side lengths? Which sides increase, decrease, or stay the same? Why does this occur?
3. What happens to the other two angles as α increases? As α decreases?
4. What is always true of the other two angles? What is always true of side lengths a and a’? How can we classify this type of triangle?
5. What happens to the angle sum as α increases? As α decreases? Why does this occur?
[i]Click the box that says show parallel line. Now we will again explore using the slider tool.[/i]
6. Move the slider all the way to the right. What happens to the size of the angles? What happens to the side lengths? Is this shape still a triangle? Explain why or why not.
7. Move the slider all the way to the left. What happens to the size of the angles? What happens to the side lengths? Is this shape still a triangle? Explain why or why not.
8. What happens to points A, A’, and B when α is 0 degrees or 180 degrees? Use the parallel line to help you answer this question.
[i]Definition: Two points are said to be collinear if they lie on a single straight line.[/i]
9. When are points A, A’, and B collinear?
10. What do you think would happen if we answered questions 1-9 using an equilateral triangle? With a scalene triangle?
11. Using the slider, create an acute triangle. Sketch it on your paper and label the angle sizes. Explain why it is acute. Repeat this process for both right and obtuse triangles.
