High School Geometry

Before beginning this case study, try out this introduction to Geogebra. It is one of many free dynamic geometry tools available for the mathematics classroom that will allow you to explore characteristics of the centroid of a triangle (available at www.geogebra.org).

Go to Construct the Centroid and use the strategy of your choosing to construct the centroid of a triangle, that is, the point where the three medians of a triangle intersect.  Tutorial on how to use GeoGebra

Geogebra screenshot

  • Did you construct the centroid correctly before clicking on ‘Check Centroid’?
  • If not, how did you determine the error(s) in your sketch?
  • What happens to the centroid as you click on and drag point A to different locations?
  • Does the centroid ever lie outside the triangle?  Why or why not?

The primary technology tool used in the case that follows is Geometer’s Sketchpad, another dynamic geometry software tool that is sold commercially. As you explore each of these two programs, consider their different affordances and constraints.

Scenario

SurveyorStudents are commonly assigned problems in a high school geometry class that are unconnected to a real-world application and that do not require critical decision-making skills to solve.  However, knowledge of geometry is essential to many design and construction careers such as engineering, plumbing, architecture, and graphic design and is used to make vital decisions to ensure public safety and minimize costs.  Students need opportunities to experience how geometry can be used to make informed decisions about the world around them.

High school geometry students may explore various lines or segments in a triangle that are concurrent, such as angle bisectors or medians. The different points of concurrency studied in a high school geometry course are referred to as centers of a triangle. The properties of centers of a triangle make them useful locations in many design and construction problems, but they are not commonly explored in this way.

This rich media case features students in Nancee Garcia’s high school geometry course exploring centers of triangles using Geometer’s Sketchpad software to solve a real-world geometric design problem. Geometer’s Sketchpad allows students to model and investigate problems in a variety of ways. They can more efficiently and effectively engage in mathematical tasks, such as making conjectures, critiquing the reasoning of others, and testing mathematical ideas with the aid of Geometer’s Sketchpad.

Students sometimes incorrectly assume that all triangles have a single center point as seen in other shapes, such as regular polygons and circles. However, several different triangular centers can be located, including the incenter, centroid, orthocenter, and circumcenter.

Centers of triangles constructed using Geometer’s Sketchpad

Centers of triangles constructed using Geometer’s Sketchpad

When the various centers of triangles formed by the concurrent (intersecting) lines, such as angle bisectors, perpendicular bisectors, medians, and altitudes, are constructed using traditional construction tools like a straight edge and compass, many difficulties and misconceptions can arise for students.

Hand-drawn errors leading to incorrect conclusions.

Hand-drawn errors leading to incorrect conclusions.

For example, due to minor construction errors three medians may be constructed so that they appear not to be concurrent. Students may draw incorrect conclusions based on those errors.  If medians are not drawn to the center of each vertex and midpoint, the medians do not appear to be intersecting at the same point.  However, medians of a triangle are always concurrent.

Also, the multiple constructions and precise measurements required for this type of investigation can become tedious and lead to low levels of motivation when students must complete them by hand.  Although geometric construction and precise measuring skills by hand are certainly important to develop in students, these skills are neither the purpose nor learning objective of this particular investigation and may cause frustration or incorrect conclusions or may impede the intended goals of the lesson.

Solving a design problem such as this requires students to reason about the geometry and to make sense of the geometry in the context of the problem. Reasoning and sense making as described in the National Council of Teachers of Mathematics’ Focus in High School Mathematics: Reasoning and Sense Making in Geometry (King, Orihuela, & Robinson, 2010) requires developing reasoning habits by students. Reasoning and sense making are described as the purpose and means for mathematics learning, without which students view mathematics as an unrelated and complex set of rules to be memorized.

The Common Core State Standards include content objectives that emphasize the application of mathematical knowledge and also requires engagement in Mathematical Practices. In particular, students are challenged to “apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).” (Common Core:  Applying Geometric Knowledge)

Students cannot use mathematics effectively only by knowing mathematical facts and rules. They must also be able to apply the concepts.  However, high school geometry students often have few opportunities to apply their mathematical knowledge to meaningful and significant problems.   Problems that involve applying multiple mathematical concepts such as geometric design problems that minimize cost, time, and so forth, are even less frequently seen in the high school classroom. An advantage to using a dynamic geometry tool is that solving complex problems may be less time-consuming and more accessible to all students.

Common Core:  Standards for Mathematical Practice

The lesson featured in this case was designed to engage students in several of the mathematical practices.

  1. Make sense of problems and persevere in solving them.
  2. Construct viable arguments and critique the reasoning of others.
  3. Model with mathematics.
  4. Use appropriate tools strategically.
  5. Attend to precision.

What do you think?

How might you help students develop their skills to apply geometric concepts to the real world?
What difficulties might students face when engaging in activities that require geometric design skills?

 

Meet the Teacher

  • Nancee Garcia
  • Geometry, Discrete Mathematics, and IB Math Teacher
  • Auburn High School

I am a high school geometry and discrete mathematics teacher at Auburn High School in Auburn, Alabama.  I have taught geometry for 10 years, and I have also taught algebra, algebra II, IB math higher level and discrete mathematics.  I am currently also a doctoral student at Auburn University pursuing my Ph.D. in mathematics education, with a special interest in using technology in the mathematics classroom.

Auburn High School is a diverse school in the state of Alabama due to its proximity to Auburn University and large multinational corporations.  Of its 1,600 students, 27% are African- American, 8% are Asian, and 2% are Hispanic.  In addition, there are 42 different languages spoken among the student population. On average, 20% of my students are classified as special education and require special services.

My overarching philosophy of teaching is that all students must be given the opportunity to learn mathematics, and it is the responsibility of the teacher to adapt to the learning needs of students.  My goal for students when they leave my class is not to be able to recite a dozen formulas from memory but rather to have gained an appreciation for mathematics.  I want them to see the usefulness of mathematics, that it is used to explain the world around them and to help them make important decisions.  However, the bottom line is I must provide the opportunities for my students to see mathematics as useful in their life.

 

I believe that the most important skills students can gain in my class are critical thinking and problem solving skills.  My decision to use technology frequently in my classroom is in part based on my belief that students need to be comfortable using different types of technology for problem solving and investigating ideas.  Particularly in geometry, Geometer’s Sketchpad helps students to create examples and test their ideas such as exploring a shape to identify its properties or if the properties will change when the shape is manipulated.

 

The Activity

The lesson highlighted in this case, Building a Shopping Center, is the third in the lesson sequence.  The goal of the first lesson in the unit was for students to construct a centroid by finding the point of concurrency for the three medians of a triangle.

Students discovered that medians are always concurrent.  They also identified the center of gravity or balancing point of various triangles as the centroid and the properties related to measurements of segments formed by the construction.  The objective of the second lesson was to determine if the angle bisectors, perpendicular bisectors, and altitudes of a triangle are also concurrent.  While the first lesson was completed using traditional construction tools, the second was completed using Geometer’s Sketchpad.

In the third lesson, Building a Shopping Center, students worked in pairs and acted as members of a town planning committee to decide on the best place to build a shopping center that would serve three surround towns. Nancee explained to students that their arguments must be supported by mathematical findings and account for the various constraints of the problem.  Students were to use Geometer’s Sketchpad (GSP) to explore and test their ideas and also to create a model of the situation to support their argument.

Nancee informally assessed their understanding throughout by questioning their findings and listening to the dialog between pairs.  At the end of the lesson, students were formally assessed as they presented their final arguments.

Two class periods were required for this lesson. During the first day, Nancee introduced the activity to students and most of the investigation was completed.  The second day began with a summary of some findings from the first day, and then the pairs finished their investigations and prepared their arguments. The class ended with the presentations by the pairs.

To begin, Nancee projected the instructions on an electronic whiteboard and discussed the scope of the activity with the class. Students also had summary notes from the previous day’s lesson to serve as reminders of how each center was constructed.

Instructions for Building a Shopping Center

Students ended the activity by presenting to the class their decision about which was the best argument and why. Students displayed their work in GSP on the SMARTBoard as they presented. Nancee assessed students’ presentations and also assessed their understandings further by asking unexplored questions during their presentations.

TPACK Commentary: The objectives of the lesson provided a perfect opportunity for the use of technology, in particular the use of a dynamic software program, such as Geometer’s Sketchpad (GSP). The use of GSP enabled the students to make conjectures and test them out, and it allowed the students to have more precision than they would have had if they only used paper and pencil. The use of GSP and the multiple entry-level task enabled Nancee to create a lesson that was inquiry based and motivating for the students. She could utilize the software in a number of ways to help the students understand the problem and to help them make sense of their own responses. The task and GSP together enable Nancee to ask rich thoughtful questions that pushed students to think more deeply about the content of the problem and whether or not their solutions made sense.

What do you think?

What are the benefits of having students work in pairs on this project rather than individually or in small groups?

The Technology

Geometer’s Sketchpad is an effective tool for encouraging conjecturing, exploring, and reasoning by students. It includes tools to construct, draw, measure, and/or manipulate geometric objects such as points, lines, rays, and circles.  It also allows students to do the following:

  • Control and reason about the behavior of geometric objects
  • Utilize the constraints or properties of objects to make constructions,
  • Find patterns and properties and test them to determine their generalizability, and
  • Test and refine conjectures.

GSP has many advantages over more traditional methods of exploring geometric objects, such as performing manipulations not possible with ruler and compass (that is, objects retain properties related to the construction when manipulated) and making efficient and precise revisions and measures.

For this activity in particular, GSP allows students to construct each triangle center and find the measurement properties associated with it efficiently and accurately in order to investigate the best possible option for building the shopping center.  Without the technology, students would likely get frustrated with the large amount of constructions required to fully investigate each possibility and most likely would not explore different types of triangles.

Hand-drawn construction showing four centers of a triangle and the lines used to create the centers.

Hand-drawn construction showing four centers of a triangle and the lines used to create the centers.

Geometer’s Sketchpad construction showing four centers of a triangle with the lines creating the centers hidden.

Geometer’s Sketchpad construction showing four centers of a triangle with the lines creating the centers hidden.

 

Finally, students would find it difficult and confusing to construct all four centers on a triangle with pencil and paper in order to find the center that has the smallest sum when measuring each road from the center to each of the towns.  With GSP students can construct each of the centers and then hide the various lines used to make the construction.  They can then observe how the centers behave without the chaos of the 16 different lines needed to construct them.

This lesson addresses the following standards established by the International Society for Technology in Education:

Standards for Teachers

Facilitate and Inspire Student Learning and Creativity – Engage students in exploring real-world issues and solving authentic problems using digital tools and resources.

Standard 4 for Students

Critical Thinking, Problem Solving, and Decision Making:  Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources. Students:

  • a. Identify and define authentic problems and significant questions for investigation.
  • b. Plan and manage activities to develop a solution or complete a project
  • c. Collect and analyze data to identify solutions and/or make informed decisions
  • d. Use multiple processes and diverse perspectives to explore alternative solutions

What do you think?

  • What did you find insightful about Nancee’s decision to use Geometer’s Sketchpad for this lesson?
  • As you read through the lesson and the information about Nancee, what knowledge of students, pedagogy, technology, and content did she display in preparing, planning, and teaching this lesson

Classroom in Action

During the activity each pair of students had a sheet on which to record their ideas as they explored.  They were encouraged to construct each of the centers of the triangle they were familiar with and use measurements to determine if any of the centers could be used as the shopping center location described in the arguments from three committee members.

For example, students measured the distance from a center to each of the sides of the triangle in order to determine if the distances were equal as described by committee member B’s argument.  Several students had difficulty measuring the distance from a point to a line using a perpendicular segment.

A common misconception among geometry students

Vignette 1:  A window into student thinking

As you watch this next video, think about how Nancee recognizes the student’s mistake then helps the student to see her misconception and fix her errors.

What do you think?

  •  What might Nancee have observed that caused her to question the student’s answers?
  • What misconception did the student hold about the distance from a point to line?
  • How did Nancee’s questioning and the technology help the student to recognize her misconception?
  • In what other ways could Nancee have helped the student to understand and correct her misconception through the use of technology?
TPACK Commentary: In this vignette Nancee knew that the distances should be equal. The technology allowed her to see that the distances were measured incorrectly and to diagnose the misconception by examining the student’s sketch.  Nancee observed that the center appeared to be an incenter so she quickly scanned the picture to determine if the segments from the center to the sides of the triangle were perpendicular.  After realizing that they were not, she encouraged the student to consider how she might have measured incorrectly. One commonly noted advantage to using dynamic geometry software is that teachers are able to see evidence of student thinking and understanding.  For example, the choices of action that a student takes in a dynamic geometry environment often reflect the conceptions a student holds about the mathematics and the behavior of geometric objects in a constrained environment. Another important component of TPACK is that teachers have knowledge of which concepts are difficult to learn and ways that technology can help students to overcome the difficulties.  In this case, the concept of the distance from a point to a line caused many misconceptions.

 

Vignette 2:

One of the triangle centers, the orthocenter, can lie inside, outside, or on the triangle when the triangle is acute, obtuse, or right respectively.  Students had to consider two different constraints when deciding whether or not the orthocenter was a possible location for the shopping center.  The problem does not require a certain type of triangle but does require that the shopping center be located inside the triangle.

Watch the following video clip and pay attention to the ways that students communicate about and manage the various constraints in the problem that must be considered when finding a solution.

What do you think?

  • In what ways did the technology aide the students in making conclusions about the orthocenter?
  • In this interaction, what did you notice about the questions that Nancee asked the students? What did the questions require the students to do?
TPACK Commentary: Efficiency of computations and constructions is an important advantage when using technology to solve complex problems in mathematics. The ease with which the students were able to manipulate the triangle added complexity to the problem, since they were more likely to consider all types of triangles rather than just one acute, obtuse, or right triangle. Reasoning and making conclusions about “all possible” triangles (the general case) rather than one specific example (the particular case) is an important skill needed for constructing formal proofs. Students must have opportunities to move from reasoning about the particular case to the general case in order to develop the skills needed to engage in high-level proof and reasoning. This type of activity that acts as a bridge between informal and formal proof is all too often left out of geometry courses that include rigorous proof. 

 Vignette 3:  Thinking through results

When exploring the measures of the distances from the circumcenter to each town, the students in the next video questioned the reasonableness of their findings when the results did not match their predictions. Nancee encouraged students to think critically by encouraging them to reason about solutions given by the technology tool instead of simply accepting any result as true.

The students in the video are suppose to measure the distances from the circumcenter to each of the vertices of the triangle.  Instead, a student inadvertently measured the distance from a vertex to the midpoint of a side that was located extremely close to the circumcenter.

Watch the next video clip and pay attention to how Nancee guides the student, and listen to the student’s observations and reflections.

What do you think?

  • What specific pedagogical actions does Nancee take as a result of her technological, pedagogical, and mathematical knowledge?
  • How might you help students establish a routine of checking the reasonableness of technology-produced results?
TPACK Commentary: A common barrier to technology use in the mathematics classroom is the misconception that technology replaces student learning and understanding. In fact, the use of technology should enhance student learning and understanding. However, teachers must be aware of the ways that students can misuse technology and guide students to become more effective users. In this vignette, it is clear that Nancee has encouraged her students to verify and critique the results given by technology.

 

Vignette 4:  Exploring even more

The point can be found where the sum of the distances to the vertices of the triangle is the least, but it was not one of the four centers that students were exploring.  This figure shows that point, known as the Fermat point.

The fermat point

Some students, due to the relative ease of creating multiple sketches and testing conjectures, explored other points to find a better solution.  These students went above and beyond the requirements of the exploration, as their own curiosity challenged them to think about other possible solutions.

Watch the next video and notice how Nancee prompts the students to think about the feasibility of recommending a point that seems to be located in a random manner.

What do you think?

  • Why do you think Nancee encouraged the students to reconsider using their random point? How would you have responded to a student that seemed to have found a better solution?
  • How could Nancee have helped her students to discover the Fermat point?
TPACK Commentary: Teachers who use technology in the classroom to allow students to explore mathematical problems will inevitably be faced with questions and ideas from students that are unexpected, unfamiliar, or even not completely understood. Exploration of the random point did not fit in the time constraints of the problem. Instead of dismissing the findings of the students, Nancee acknowledged the point as a better mathematical solution but perhaps not a practical solution in the context of the problem.

Student Work

Students realized throughout day that one “best” solution was not well defined.  The committee members’ arguments focused on the roads that needed to be built, so the class decided through discussions that the best solution would mean that the cost of the roads that the towns had to build should be minimized.  Students summarized their work on their papers and on their Geometer’s Sketchpad sketch they prepared for their presentations.  The following is an example of each.

Student reference chart:

Student reference chart.

Exploration recording sheet:

Exploration recording sheet

Final presentation:

Final presentation

All of the groups came to the same conclusion, that the incenter was the best place to build the shopping center.  The incenter is the point out of the four centers examined that would result in the smallest sum of distances to the towns whether the roads are build directly to the towns (vertices of the triangle) or directly to the existing roads (sides of the triangle).

A Comparison of Incenter and Centroid:

A Comparison of Incenter and Centroid

The next video shows part of a group’s final presentation as they gave the rationale for the incenter.

In addition to assessing student work during the investigation, student understanding was assessed as they presented their findings to the class.  Students used the whiteboard to show their constructions and explain their findings.  Each group not only had to give the reasoning behind the point selected, but to provide a rationale for the points that they rejected.

Nancee challenged groups with alternate scenarios during their presentations in order to determine how well they understood the problem. For example, one group was not allowed to build the roads to the existing roads (the sides of the triangles).  In this case, they justified why the incenter was still the best solution.

What do you think?

  • What evidence do you see of students using their mathematical knowledge to solve meaningful problems?
  • How did Nancee use GSP as a tool to encourage her students to persevere when solving problems?

 

Teacher Reflection

It always feels like I run out of time. I wanted to explore some of the students’ questions further, but I did not have the time.  Most specifically, the random point that some pairs found that produced the smaller total sum of the lengths of the roads can be found systematically, and it is known as the Fermat point (when all angles are less than 120°). I was able to use Geometer’s Sketchpad to show the students this point.  I don’t know how I would use an exploration like Building a Shopping Center if I didn’t have the technology.

I also like how this activity allowed students to bring in their special knowledge.  For example, one student had experience with surveying tools and repeatedly mentioned using a surveying tool called butterflies that could be used to locate our seemingly random point.

Interestingly, characteristics of the location of the shopping center that would serve the towns equally after the construction process were not discussed by any of the students, but the problem could have been expanded to include these constraints.  For example, students could have considered that the shopping center needed to be built at a location equidistant from each town so that no town had a longer distance to travel.

What do you think?

  • How would you improve this activity or do it differently?
  • What other content could you use Geometer’s Sketchpad for in your classroom and how?
  • How could technology tools such as a dynamic geometry environment help all students to engage in complex problem solving?

 

 

Resources

King, J., Orihuela, Y., & Robinson, E. (2010). Focus in high school mathematics: Reasoning and sense making in geometry. Reston, VA: National Council of Teachers of Mathematics.