Introduction to Geometry (Part 2)
Lesson 4 of 7
Objective: SWBAT identify different types of angles and angle pairs, and bisect line segments and angles.
More Work with Line Segments
In the opening activity, I focus on strengthening the students' understanding of line segments and their various representations.
I distribute activity 1 to all the students. These problems focus on several concepts:
- Students are asked to rewrite verbal statements using geometric symbols, and are asked to write statements verbally, given the geometric symbols. Symbols and vocabulary play such an important role in Geometry that I think it's really important to hit these concepts hard from the very beginning.
- Students are asked to measure line segments, to reinforce the segment addition postulate (I use "The whole is equal to the sum of its parts."), the definition of a midpoint, and the definition of congruent segments.
- I also introduce the students to their first construction, that of constructing the perpendicular bisector of a segment. They do not have the background yet to understand the theory behind the construction (we will do that a little later in the course), but I like to expose them to constructions early, to reinforce, in this case, the concept of a midpoint.
- The handout also allows the students to practice justifying statements with appropriate definitions and postulates.
The students work in their cooperative learning groups on the first question, consulting with each other when unsure of a concept or when looking for affirmation. When it looks like everyone is through, I call on two students; I ask one to read his or her statements aloud, while the other student comes to the board to write the statements using the geometric symbols. They then reverse their roles; the student at the board then reads the mathematical statement out loud, while the other writes the verbal statement on the board, along with its justification. (MP 2 and MP 3).
I purposely do not indicate units in this exercise. It is my hope that some will do their measurements in inches, some in centimeters. This will provide an opportunity to discuss the different units that are available for linear measurements, and will help to drive home the fact that the principles are the same, regardless of the units used. (This helps to set up our discussion later in the lesson on angle measures. I think it is important for students to realize that degrees are not the only unit available for angle measure.) This also provides an opportunity to discuss what an appropriate degree of precision might be and the limitations of the tools we are using to measure (MP 5 and MP 6)
We repeat this process with the second problem.
For the third problem, I distribute compasses and briefly explain the rules for constructions. I introduce the word bisect and then demonstrate the construction of a perpendicular bisector. (See this site for instructions on constructions.) I give the students time to try it on their own, and time to answer the questions beneath the construction. If I feel that the students are struggling with the symbols and vocabulary, we repeat the process of having students read aloud and write statements on the board; otherwise I call on individuals to read aloud each of the last three statements in the problem set.
[Sample answers to this activity can be found in sample answers to activity 1.]
In the opening month or two of Geometry, I expect students to learn certain definitions, postulates, and theorems, pretty much word for word, and provide lots of opportunities for students to practice them in problem sets and in class discussion. I have found that, by asking students to learn an initial body of justifications and familiarizing them with what precise and accurate statements sound like, the students soon become adept at developing their own justifications, without being made to memorize. An example of this is the definition of an angle bisector; after learning the definition for a midpoint most students are easily able to create for themselves the definition of an angle bisector.
An Introduction to Angles
Referring to the last problem in their previous activity, I ask the students, "Take a look again at that line segment that you just constructed. What did that line segment do? What did it create?" I'm hoping that, in addition to talking about the midpoint, someone will remark that the line segment formed right angles, which will be my segue into our discussion of angles.
I ask the students to tell me what they know about angles. This is a topic that I know they have studied in previous grades. If needed, I am prepared to ask questions such as, "How is an angle formed? How do we measure angles? What types of angles are there?" We discuss the units of measure for angles and I mention both degrees and radians, so that the students will realize that there are different units for angles, just as there are when measuring lengths of segments. As the students use the important geometric vocabulary, I write the words on the board. These words should include obtuse, acute, right, straight, degrees, radians, ray, vertex, and the interior and exterior of an angle. (More words for the word wall!)
Next I ask about how angles are named. In this discussion, I stress the role of the vertex of an angle when naming an angle, regardless of whether one letter or three letters are used in the name. I create diagrams on the board to illustrate situations in which one letter is sufficient when naming an angle, and situations in which three letters must be used. I ask several students, one at a time, to come to the board and act as the teacher. They draw a labeled diagram with several angles in it, and then call on their classmates to first name their angles in different ways and then to classify their angles.
When it is clear that everyone is comfortable naming angles, I ask a student to tell the class what it means to bisect a line segment, and pose the questions, "Can an angle be bisected? What is an angle bisector?" I ask a student to tell me the "official" definition of a midpoint, and then ask the class to tell me what, therefore, the "official" definition of an angle bisector might be.
The last topic we discuss are angle pairs. This discussion includes adjacent, complementary, supplementary, and vertical angles. These are all topics that are covered in the 7th grade curriculum, but I need to make certain that all the students are solid in their understanding of these topics. Some students confuse complementary and supplementary, and I point out that these terms go alphabetically; i.e. complementary comes first alphabetically and 90 comes first numerically, while supplementary and 180 appear later alphabetically and numerically. This usually seems to help.
I provide each student with a protractor and distribute the handouts entitled activity 2 and diagram for activity 2. I ask that the students work in their groups to complete numbers 1 through 10. As they work, I circulate through the room, observing and listening to their conversations. I explain that I will not take questions from a group unless the entire group has that question, because I think it is really important for them to talk to each other, to learn from each other, and to make the vocabulary their own by doing so. If a group does have a question, I will ask that they pose their question to the entire class, to see if any of their classmates might be able to answer their question. (MP 3)
When it appears everyone has satisfactorily completed 1 through 10, I hand out compasses to the students. The construction in number 11 is a review of the perpendicular bisector construction that the students did earlier in the class period. Number 12 is a new construction, that of an angle bisector. Before showing the students how to do this construction, I emphasize that constructions never (or hardly ever!) start a random point. I ask them where they started in the perpendicular bisector construction (at the two endpoints) and then ask them to use similar logic on the angle bisector construction. I pose these questions: Where is the only non-random point in your angle diagram? (the vertex) What could we draw from the vertex? (an arc) What did we just create by drawing an arc? (two new points) What can we draw from these two new points? (two intersecting arcs) I find that, instead of just demonstrating a construction, it really helps to have the students think and talk through these steps. Once they fully understand the concept of never drawing arcs from random points, the students are often able to figure out new constructions on their own.
I ask the students to work on the last problem in their groups. This problem can be done several different ways, using knowledge that the students acquired in previous grades. I walk around the room noting any different approaches, and, when everyone is finished, I ask students to explain/demonstrate their approach on the board or with a document camera. If everyone opts for the same technique, I will point out at least one other possibility. [I have included some of the possible approaches in Number 11 Solutions.]
I hand out the ticket out the door. On this, I have asked the students to look over the vocabulary terms on their anticipation guide (from the previous lesson), to choose 4 terms, and to explain to me either what they do know now that they didn't know before or what they don't understand about these terms. At this point in time, the students have been introduced to all of these terms.
I use the student responses to the ticket as a formative assessment. Their responses help me to know what I should and should not emphasize in upcoming lessons, and also communicates to students that they need to be responsible for thinking about and being aware of both their knowledge and their lack of knowledge throughout this course .