2016 Alabama Course of Study: Mathematics Standards
The grade that I would associate the concepts discussed in the project would be 10th grade geometry. Looking through grades K-8 in the course of study, there were no standards that jumped out as being directly related to our topic. Additionally, content in geometry is usually segregated into high school in the form of a 10th grade geometry class. As a result, the most likely grade for this material to appear is in the 10th grade. Below is a list of standards and concepts that I identified as being relevant to how I solved the problem detailed in my project.
- Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]
- Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. [G-CO12]
- Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13]
This problem is a great example of a high demand task that uses a Dynamic Geometry Environment (DGE) to draw out deep connections in geometric concepts. It’s a problem that requires a lot of cognitive effort on the part of the students in order to answer the questions appropriately (National Council of Teachers of Mathematics, 2014, 18). In addition, the problem asked students to create conjectures about their observations of the Geogebra constructions, an application of SMP 3. Students make these conjectures and then very informally prove them by utilizing their construction. Another factor that made it a high level task was in being able to combine the previous constructions for circumcenter, incenter, orthocenter, and centroid and draw more conclusions based on their relationships with different triangles and angles (National Council of Teachers of Mathematics, 2014, 18).
I also like this problem because it was a great example of how to use a mathematical action technology in the classroom in order to promote student learning in geometry. A mathematical action technology is a described as a technological tool that can perform mathematical tasks and respond to the user’s actions in mathematically defined ways (Dick & Hollebrands, 2011, xii). I think a typical, instrumental lesson about the different centers of a triangle would have students just look at the definitions for those centers. In this problem however, we were given what made each center, but then had to see how it behaved in different environments and what that meant for it’s significance. Finding this significance and behavior became much easier and more efficient as a result of the Geogebra constructions. Rather than having to draw 30 different triangles with each center, we were able to create one construction and freely manipulate it for different types of triangles and angles. Having this technology available that allows for a greater exploration of geometric concepts makes it easier for teachers to ask questions that promote a relational understanding of the material rather than being forced to ask more instrumental ones (Dick & Hollebrands, 2011, xvi). The problem that we analyzed in this project is a great example of that in that all of the questions posed by the problem are ones that ask for students to have a deeper understanding of the mathematics behind the problem in order to answer. Every part of this problem has students make their own conjectures, and then ask why these conjectures are important to the significance of each center.
Below is my outline for a similar exploration that might be used in a geometry class. The method for my exploration would be a Geogebra activity sheet.
- Have students construct a generic triangle in Geogebra.
- Students will add angles and side measurements.
- Have students add the perpendicular bisectors of each side to their triangle construction
- What do you notice about the point where the perpendicular bisectors meet?
- What happens if we turn our general triangle into an isosceles triangle, an equilateral triangle, etc.
- What can you say about the behavior of this intersecting point at different angle measures.
- This point is called the circumcenter. It is the center of the circumcircle, a circle that goes through all three vertices. Have students rename this point on their construction to “circumcenter”.
- Have students delete/hide the perpendicular bisectors of their construction.
- Have students add angle bisector’s to their triangle construction.
- What do you notice about the point where the angle bisectors meet?
- What happens if we turn our general triangle into an isosceles triangle, an equilateral triangle, etc.
- What can you say about the behavior of this intersecting point at different angle measures.
- This point is called the incenter. It is the center of the incircle, the largest circle that will fit inside of the triangle. Have students rename this new intersecting point on their construction to “incenter”.
- Have students delete/hide the angle bisectors of their construction.
- Have students add the medians to their triangle construction.
- What do you notice about the point where the medians meet?
- What happens if we turn our general triangle into an isosceles triangle, an equilateral triangle, etc.
- What can you say about the behavior of this intersecting point at different angle measures.
- This point is called the centroid. It is the point of the triangle where all mass seems to act. If you were wanting to balance this triangle on the tip of your pencil, the point which would achieve equilibrium would be the centroid. Have students rename this new intersecting point of their construction to “centroid”.
- Have students delete/hide the medians of their construction.
- Have students add the altitudes to their triangle construction.
- What do you notice about the point where the altitudes meet?
- What happens if we turn our general triangle into an isosceles triangle, an equilateral triangle, etc.
- What can you say about the behavior of this intersecting point at different angle measures.
- This point is called the orthocenter. Depending on the angle, it is not always within the triangle. Have students rename this new intersecting point of their construction to “orthocenter”. Have students remove the altitudes on their constructions so that just the four labeled centers are left.
- Now that students have all four centers on their construction, have students examine their collective behaviors relative to each other by changing the general triangle into an isosceles triangle, an equilateral triangle, etc.
- What centers always stay within the triangle, which do not?
- How does changing the angles of your Geogebra construction show this?
- What connections do you see between certain angles and the position of the different centers in your construction. Explain your reasoning.
References
- Alabama Course of Study: Mathematics. Alabama State Dept. of Education, 2016.
- Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to Support Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics.
- National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
Morrell Math Corner 