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Unit 2 – Tools of Geometry

Last year we started with this unit. This year, we decided to start with our unit on Rigid Motions, since students have some familiarity with transformations from middle school. I noticed that students were much more willing to look for regularity in repeated reasoning and make sense of problems and persevere in solving them during the Tools of Geometry unit because they had some practice with those practices during the last unit.

CCSS-M Standards:

G-CO 1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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; the medians of a triangle meet at a point.

G-CO 12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 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-CO 13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-C 3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

I can statements:

Level 1: I can use basic terms in geometry: point, line, plane, polygon, inscribed, circumscribed.

Level 2: I can create basic geometric objects based on their definitions: median, angle bisector, altitude, perpendicular bisector.

Level 3: I can use tools and methods to perform formal constructions.

Level 4: I can solve problems using points of concurrency.

We learned about the tools of geometry through a few applications – Placing a Fire Hydrant & Locating a Warehouse, both of which come from Illustrative Mathematics. We used the Special Segments activity from Geometry Nspired, and we used The Shipwrecked Surfer problem published in the August 2006 Mathematics Teacher. I plan to write a separate post about each of the main lessons.

We spent the rest of our time on constructions – really paying attention to what is congruent as we did each construction. I wanted the constructions to come from a context, instead of just passing out a list of steps for a construction and asking students to do the construction because it was on our list of standards.

I have already looked at the student reflections for this unit. While the majority of students still believe that the practice they most used was make sense of problems and persevere in solving them, the second most used practice was use appropriate tools strategically (which, for a unit on constructions, means that students really thought about which practices they were using and didn’t just mark the first one on the list).

I asked students whether any of the content seemed like repeats of previously learned material.

  • Everything that I learned this unit was completely new to me.
  • I felt that I didn’t already know any of the material covered; it all seemed new to me.
  • I didn’t know any of this but I learned a lot!
  • Absolutely none of them were repeats for me. There were parts of each unit I’d already learned and heard about, but the overall idea of each lesson was new to me.
  • All of the lessons were excellent introductions into learning about and exploring circumcenters, centroids, orthocenters, and incenters. They required the usage of different techniques and critical thinking to achieve the answers. Finding these various properties of triangles required usage of terms and theorems from the past chapter about distance, perpendicular bisectors, justifications by measuring, etc.
  • Most of the information this unit was new. There were terms that I was already familiar with such as perpendicular and bisector, but I didn’t know all of the information I could gain just by saying that a line is a perpendicular bisector, an angle bisector, an altitude, or a median.

Which lesson helped you the most in this unit? Most students said that The Shipwrecked Surfer helped them the most.

  • The shipwrecked surfer really helped understand all of the different segments and points and centers. For us to have to bring together the median, altitude, angle bisector, and perpendicular bisector and their corresponding centers, so to say, showed me how they all work and when they can all be the same. It made more sense to see it like that.
  • I would say that 2A, 2B, and 2D helped the most because they were realistic situations.

What did you learn during this unit?

  • I learned how to use the incenter, circumcenter, and centroid to solve real-world problems.
  • I learned a lot that I didn’t know. I really enjoyed this unit because of this. I can easily find perpendicular bisectors, altitude, medians, and all the points of concurrency and other things we learned in this unit.
  • Other than the various terms and the application of said terms, I learned not to immediately give up when a problem doesn’t practically solve itself.
  • During this unit I learned many new mathematical terms. I also learned how to classify a triangle by using information other than the degree of it’s [sic] angles.
  • I have learned basically everything that the teacher has taught me. I didn’t know much of it, but I knew enough where I didn’t feel completely lost. I learned a lot of vocabulary words.

I think it is significant that the last student said he “learned a lot of vocabulary words”. When I talk about teaching geometry through exploration, I often have teachers ask about the vocabulary. For many, geometry is all about vocabulary – and assessing whether students know definitions and notation. The biggest difference in how I was taught geometry and how I teach geometry is that we don’t start with the vocabulary. We work through meaningful tasks (the students’ words – not just mine) in which the vocabulary will surface. When the vocabulary does surface, I make a big deal about it. I write the word down on the board. I suggest that they write the word down in their notes. I ask what the object will buy us mathematically, and I sometimes find myself saying how you would see the word defined or the object notated if you were to look in a textbook. I have a list of vocabulary in my lesson plans that needs to surface in the lesson, but that list shows up for the students one word at a time, as we get to it in our exploration, instead of as a list of words to define from the glossary before (or even worse, during) class.

And so the journey continues, right or wrong …

 

 
 

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Locating a Warehouse

Two more days of geometry have come and gone. On Monday, we proposed another task from Illustrative Mathematics, Locating Warehouse. In this task, students are asked to determine a location for a warehouse that is equidistant from three “roads”.

1. Hopefully, students are beginning to note that “solving a simpler problem” is a good problem-solving strategy. Where could you locate a warehouse that is equidistant from two roads? (These roads do intersect, but it also might be interesting to consider where the warehouse should be when they don’t intersect.)

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2. We are finding that some of our students have a weak geometry vocabulary. A few of our high school geometry students really didn’t know what we meant by the midpoint of a segment. We don’t know yet the results, but we are trying to present vocabulary in context. We are trying to anticipate the vocabulary words needed for a lesson so that as teachers, we can call out those words during the lesson. We are having students keep a list of terms as they happen in the lesson. And then we are trying to summarize important terms at the end of the lesson. We are trying to “attend to precision”, but we are doing it by example, instead of explicitly teaching vocabulary and notation from the traditional first chapter of a geometry textbook. So when a student said yesterday that any point on the perpendicular bisector of a line is equidistant to its endpoints, together we refined the statement to any point on the perpendicular bisector of a segment is equidistant to its endpoints. We all knew what he meant the first time, but the act of correcting the statement provided us an opportunity to enter into the mathematical practice of attending to precision.

After exploring two tasks that gave us a reason to know about the perpendicular bisector and the angle bisector, we got out the compasses and straightedges on Wednesday to construct them on paper. I have had students do the construction in the past, but this year, we paid more attention to the results. After the students constructed the perpendicular bisector, we asked the question “what things are equal”? Things might not have been the best word to use…but we wanted them to think past segments with equal measure and on to angles with equal measure, and possibly even congruent triangles. We then let them follow the directions themselves to construct the angle bisector (although some of them weren’t ready to follow the directions themselves) and again asked the question “what things are equal”. Performing the constructions in class took longer than expected, but we are trying to “ease the hurry syndrome” and let the needed exploration happen, even if it means we are already behind…

 
 

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Placing a Fire Hydrant

We are using both the PARCC Draft Model Content Framework for Mathematics and the high school sequence funded in part by the Bill and Melinda Gates Foundation and the Pearson Foundation to determine the order in which to present our geometry content.

One major difference for me is that these documents suggest starting geometry with constructions. I have had students do constructions at the end of the course the past few years, which I like, because they end up making a lot of connections about congruency and definitions. Starting the course by handing students a compass and a straightedge and directions to construct a perpendicular bisector seems like a more challenging way to get students engaged in learning geometry. So we are compromising. We gave students a problem Thursday that is from the Illustrative Mathematics site, about determining placement for a fire hydrant equidistant from 3 buildings (or “vertices”) that form an acute triangle. Actually, we didn’t use “equidistant” when we first presented the problem. We asked where they would place a fire hydrant, and let them come up with “equidistant” before making that part of the task.

When presenting tasks, we are trying to allow students to progress through the van Hiele model of geometry thought [1.]: visualization, analysis, abstraction (informal thought), deduction (formal thought), and rigor. So we started by giving students a paper copy of the location of the buildings and a ruler. Students played for a few minutes with their diagram. Many students used trial and error: they would place a point, measure the distant to each vertex, and erase and move as needed to make the point equidistant from the vertices. A few students began to think about the midpoint of each side of the triangle. In fact, they began to draw the medians of the triangle, found that they were concurrent, and decided that the fire hydrant should be located at the centroid. But when we asked them to verify that the centroid was equidistant to each vertex, they realized that the centroid wasn’t the point they needed.

In How to Solve It, George Polya suggests solving a simpler problem as a strategy for problem solving. In this case, that meant finding a point equidistant from two points instead of three. We had two students stand apart from each other and asked a third student to stand equidistant from those two students. In the first class, the third student stood at the midpoint of the “segment”. Then we asked a fourth student to stand equidistant from the two original “endpoints” but not on top of the midpoint. Ultimately, we had a line of students that formed the perpendicular bisector of the original segment.

At this point, we moved to TI-Nspire technology. Students used the perpendicular bisector tool on each side of the triangle and found the point of concurrency, the circumcenter. We used the circle tool to draw the circumscribed circle, and then we moved the vertices of the triangle around to determine whether the circumcenter would always be the best location for a fire hydrant (not necessarily, when the buildings are the vertices of an obtuse triangle).

Now we have a reason to construct a perpendicular bisector with a compass and straightedge. We will see how that goes today.

1. Crowley, M. “The van Hiele Model of the Development of Geometric Thought.” In M. Lindquist, ed., Learning and Teaching Geometry, K–12, 1987 Yearbook. Reston: National Council of Teachers of Mathematics, 1987.

 

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