Lesson Plan: Angle Bisector as a Locus. The Incenter of a triangle.

Author:
Beatrice Tiron, Judith

Lesson Information

  • Subject: Mathematics
  • Grade Level: 6th
  • Duration: 50'
  • Technology setting: teacher computer with projector, student computers, tablets or smartphones

Topic

The interior bisector of an angle is the locus of the interior points of an angle, equidistant from its sides. The interior bisectors of a triangle meet at a point equidistant from the sides of the triangle, called incenter, which is the center of the inscribed circle in the triangle.

Learning Outcomes

  • Developing a positive attitude towards math classes, encouraging curiosity and creativity, communication and cooperation with colleagues;
  • Ability to methodically investigate mathematical objects and statements by going through the proposed experiments, formulating questions and hypotheses based on the observed results, then demonstrating hypotheses using mathematical language and formal logic;
  • Learning digital investigative techniques, especially dynamic geometry such as moving points and objects, changing positions and sizes, etc.
  • Familiarizing with the notion of the locus, the set of all points satisfying some certain condition;
  • Learning that the angle interior bisector is the locus of the angle interior points, equidistant from the sides of the angle;
  • Knowing that the all three interior triangle bisectors meet at a single point, called the incenter, equidistant from the sides of triangle;
  • Ability to construct the incenter of any triangle, using the geometry school kit.

Lesson Objectives and Assessment

Lesson Objectives By the end of the class, students will:
  • Individually investigate the bisector concept using Interactive Worksheets 1 and 2, formulate questions and answers, and issue assumptions about the Statements 1, 2, 3 and 4 (please, see below);
  • Participate in the front-to-class demonstration of Statements 1, 2, 3 and 4, and learn they are true, including by making drawings and demonstrations in their notebooks and by participating in discussions;
  • Be able to enunciate Statements 1, 2, 3 and 4, at least when a drawing can be used;
  • Be able to apply their knowledge to construct the interior bisectors and the incenter of any triangle.
Statements
  1. Any point on the bisector of an angle is equidistant from the sides of the angle.
  2. Any interior point of an angle, which is equidistant from the sides of the angle, belongs to its bisector.
  3. The (interior) bisector of an angle is the locus of all interior angle points equidistant from its sides.
  4. The all three interior bisectors of a triangle meet at a unique point, equidistant from the sides of the triangle, called the incenter (because it is the center of the circle inscribed in the triangle).
Lesson evaluation methods Observation, front-to-class dialogue, verification of the notes required in the interactive worksheets, checking the correctness of the drawings and observing the use of the geometry instruments, written verification - given any triangle, constructing the incircle (center of the circle inscribed in a triangle), with short explanations.

Teaching Strategies

Strategies and activities used in the lesson. Strategies
  • What kind of teaching-learning methods, techniques, and learning activities? The heuristic conversation, as a link between the activities described below, the investigation - including digital techniques, using Interactive Worksheets 1 and 2, modeling through drawing, structured dialogue, front-to-class demonstration, assessment and deepening the retention by completing some tasks at the end of the lesson.
  • What equipment, software, media, and materials will you need in order to carry out the instructional strategies? Teacher computer and projector, students computers, GeoGebra, Interactive Worksheets 1 and 2, blackboard/whiteboard, students' notebooks, writing instruments, and geometry kits.
  • How should the resources be arranged to support instruction and learning? On which phase of the lesson and for what are you going to use technology? The projector is used at the beginning of the activities that involve the interactive worksheets and the moments of the conclusions. Interactive worksheets will be used in 3 of the lesson moments to facilitate the investigation of concepts by students, the actual use of digital techniques will take about 15 minutes. In the rest of the time there will be talks, remarks, conclusions, demonstrations, work on notebooks.
Activities 1) Introduction of the theme and the working methods. The topic and goals are announced, and the interactive worksheets are described. The angle bisector definition and construction modalities are recapitulated;  2) Investigating statements (1), (2) and (3) using Interactive Worksheet 1, and then demonstrating them. Pupils open Interactive Worksheet 1, which shows an angle, its bisector, a variable point on the bisectors and the distances from that point to the sides of the angle. With the help of the projector, find out that they can change the size and position of the angle as well as the P-point on the bisectors, and are invited to write down all the comments and answers to the following questions on the notebooks:
  • If one changes the angle position (by moving the point O) or the angle measure (by moving point A), does [OP retain its bisector quality?
  • Given that PQ and PR are the distances from the point P to the sides of the angle, how they modify when point P moves along the bisector? What can be observed about these distances?
  • Is it enough to observe a fact, or it should be demonstrated (in order to be proved as true)?
  • What can be said about the triangles [POQ] and [POR]? Can you identify a case of congruence?
  • If we wouldn't know that the point P is on the bisector, still we would know that the distances PP 'and PP' are equals, what can be said about the triangles [POQ] and [POR] triangles]? Do they remain congruent?
Discussions are initiated, pupils write down their observations and advance hypotheses. Then, with the help and participation of students (one can go to the blackboard, for example), the statement (1), the statement (2) and, as a conclusion, the statement (3) will be frontally demonstrated and discussed. 3) Investigating the statement (4) and demonstrating it at the black/whiteboard. Students are given the task of constructing a triangle on the notebooks and the three bisectors of the triangle angles, then note what they notice (ideally, bisectors meet at a single point). Then they will open the Interactive Worksheet 2 and compare with their own drawings, write down what they notice on the notebooks, especially about the distances from the meeting point to the sides of the triangle, and discusses the following:
  • If there are differences between the construction showed in the interactive worksheet and their own constructions, how could it be explained?
  • What relationship can be observed between the distances from the meeting point of the bisectors at the sides of the triangle?
  • What results from the application of the statement (3) to this construction? Make a hypothesis statement and a strategy to demonstrate it.
Conclusions are drawn and a demonstration is made on the board with the participation of students. 4) Learning retention and transfer - making the final construction on the notebook, giving the homework. Students receive the final task of building a triangle and its incircle. It is required to build the bisectors with the help of the compass or protractor, and after obtaining the meeting point, the incircle, they will also draw the circle inscribed in the triangle. As a homework, students will be required to draw on their notebooks some different types of triangles (scalene, isosceles, equilateral, acute-angled, right-angled, obtuse-angled) and their inscribed circles. 5) Evaluating satisfaction about the lesson, discussing the lesson difficulty, suggestions for future, and end of the lesson.

Resources

Links to the interactive worksheets: Interactive Worksheet 1 - Angle Bisector: https://ggbm.at/xdVxJbAD Interactive Worksheet 2 - The Incenter: https://ggbm.at/fNcGfKZe Interactive Worksheet 3 - Angles measurement using the protractor: https://ggbm.at/YaZxUMFA

Technology Integration

The plan to minimize technology-related problems during the lesson.
  • Students do not need specific prior knowledge to be able to use technology adequately in the lesson, just basic computer skills;
  • A previous programming is required for computer lab use and 10 minutes of computer training so that students have access to previously prepared interactive plug-ins. If we do not have a good Internet connection, the files can also be used offline. Students could also use their smartphones;
  • If the computers would become malfunctioning or any other digital devices are not available, the lesson can continue using only blackboard and notebooks.

Interactive Worksheet 1 - Angle Bisector

Interactive Worksheet 2 - The Incenter

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