Introduction to Geometry
Lesson 3 of 7
Objective: SWBAT work with the undefined terms and the various segment relationships.
Going Over the Homework
The Do Now for today is to get out the Homework assignment. This assignment is intended to review the area and perimeter concepts that were covered in the Dig In! lesson, and also provides practice with the rule for the lengths of the sides of an isosceles right triangle.
The students and I discuss the answers to the homework problems and the various approaches that they used to find the perimeter and area of the figures.
In the previous lesson, my students developed their own set of class norms, rules for their interactions with each other. I will take this time to discuss my rules. I am not a big fan of listing and elaborating all sorts of rules and regulations - it's just not me, I guess. However, I do stress a few items that are of particular importance to me:
- "If you can't say anything nice, don't say anything at all."
I post this rule in a prominent place in my classroom, and it is a mantra that gets repeated often.
- "When Mrs. Menzie speaks, people listen."
This is another rule that I post on the wall. I generally don't lecture much, and try hard to have my students communicate with one another throughout much of the class period. However, when I have something to say to the class, I'd like my students to stop talking to one another and listen.
- On the other hand, I explain that I like to hear students.
When I ask a question, unless directed at a specific student, I love to hear 20 voices responding! I love it when students ask questions about the mathematics and the learning that is happening (rather than asking something like, "Can I go to the bathroom?"). Therefore I explain my expectations for a class environment of collaborative discussion and conversation and emphasize that in my classroom there will not be the typical "raise your hand before speaking" rule.
- I explain that when my students enter the room, I'd like them to immediately sit in their seats (for Cooperative Learning), take out their math folders, and put everything else away.
In Geometry we seem to accumulate so much stuff on the desktops - calculators, colored pencils, compasses, rulers, etc. - that it drives me nuts when there are also backpacks, cell phones, handbags, and other clutter sitting on top of the desks, too! I'm thinking about experimenting this year with having students deposit all their non-geometry belongings in a specific place in the room, away from where they sit and work.
- The only supplies I require are a math folder (one with pockets on both sides) and a writing implement, preferably a pencil.
Over the million or so years I've been teaching, I think I've tried just about every approach to keeping student work organized - 3-ring binders with dividers, spiral notebooks, folders that are kept in the classroom, etc. These days, in a high-needs district with a lot of low-income families, I have settled on just asking for folders with 2 pockets. On one side, I have the students keep their work from the unit that we are working on; in the other pocket they store the important, colored sheets of paper that I hand out. These usually help to summarize concepts in a unit, or list important information that we will use throughout the year. The fact that these sheets are colored (usually fluorescent!) make them easy to find and harder to lose, and, at the end of the year, they make good study guides for the Regents test (our summative State test).
I hand out an Anticipation Guide that encompasses a lot of the vocabulary and concepts in this unit. Many of the concepts are not new to the students; however some are new and unfamiliar.
I ask the students to do the true/false questions on the first page by themselves. Then I ask them to discuss their responses to these questions within their groups. I stress that they don't need to agree with each other, but that they do need to hear and respect each other's point of view. (MP3)
We then repeat the process with the vocabulary on the second page. In this latter section, the students indicate their familiarity with each term.
As a class, we talk only briefly about the material on the anticipation guide. Instead, I explain that this exercise helps to give them an idea of the topics that we will be working on, helps to reactivate prior knowledge, and will also serve as a great review at the end of the unit, when I ask the students look over their initial responses and make corrections. (MP1)
Students are sitting in their cooperative learning groups. In this section I lead a discussion (not a lecture!) about the three undefined terms: point, line, and plane. I ask the students to tell me what these terms mean, and it becomes clear very quickly why these terms are undefined, as they are difficult to explain. I ask the students to discuss in their groups:
- Does a point have size? Can you measure a point?
- Does a line have length? Can you measure a line? How many points are on a line?
- Does a plane have dimensions? Does it have thickness? How many points does it contain? How many lines?
I then ask the groups to share out their conclusions, and we compare and contrast their answers. I have found that students are really intrigued by the concept of infinity and comparisons of infinity - does a plane contain more points than a line? Are there different sized infinities?
In the discussion, I make a point of representing each of these terms on the board, as a diagram and symbolically. I also incorporate vocabulary that the students need to know: collinear, coplanar, intersecting, horizontal and vertical, and I write these terms on the board as we encounter them. These words, as well as the undefined terms, will go on our word wall for this unit (MP1).
Next I pose questions to the groups about the intersection of lines and of planes. I have found that visualizing planes is difficult for a lot of students. Referencing the classroom walls, ceiling, and floor can be helpful when discussing intersections of planes:
- Can two lines intersect? How many points of intersection can there be?
- Can two lines not intersect? What is that called?
- Can a line and a plane intersect? How can they intersect?
- Can two planes intersect, and how? Do they have to intersect?
Groups will share out their answers to these questions.
We move from the undefined terms to our first defined terms: line segment and ray. Again, in the discussion I am careful to represent these terms both as a diagram and using their symbols. I also ask the groups questions similar to those I asked with regard to lines and planes:
- How many points are contained in a line segment? a ray?
- How many endpoints are there in a line segment? a ray?
- Can we measure a line segment? a ray?
- How can line segments and rays intersect?
During this discussion I introduce the term congruent, which I define as "same size, same shape." I explain that congruent segments are equal in length, and emphasize the difference between the symbols for congruent and equal, and the symbols for the name of a line segment and the length of that line segment. Again, I emphasize that we need to know how to represent geometric concepts both as a diagram and using symbols.
This discussion is truly that - a discussion, a conversation between me and the students, and, hopefully, between the students themselves. I do not expect the students to take notes. Instead, I look at this as an opportunity for everyone in the room to interact. It is a chance for me to get to know the students and to attach names to faces, it is a chance for the students to get to know my style of teaching and my expectations of them, and it is an opportunity for the students to interact with each other. Communication plays a large role in my classroom, and I like having the students communicate with me and with each other from the very start.
Applying Our Knowledge
I distribute a handout entitled Segments and Distance. In this handout, I have the students write out and apply their first postulate. I know many textbooks call it the Segment Addition Postulate, but I prefer to teach "The whole is equal to the sum of its parts." I know this may seem old-fashioned, but I read a lot and find references to this sentence constantly - in literature, in newspapers and magazines, all over. Therefore, I've decided that this sentence, The whole is equal to the sum of its parts*, will have much more meaning and significance in the students' lives than the term Segment Addition Postulate, and so I teach it instead. I do explain all this to my students, and have a couple examples of the phrase used on the Internet that I display on the white board for them.
The handout then asks the students to apply this postulate to four algebraic problems. The equations that they create are both first and second degree, and I do this intentionally, as I would like my students to be able to recall and apply their algebra skills right from the start of this course. I encourage the students to work together and to compare answers within their groups (MP3); as they work and discuss, I travel from group to group, listening to their discussions, and assessing the students' understanding of the concept and the ease with which they set up and solve the equations. As I circulate, I am careful to ask over and over again, "Does your answer make sense?" (MP 1) This is such an important question, and is one that I want my students to learn to ask themselves each and every time they do a problem.
Once everyone has completed the first page, we move on to the definition of a midpoint. I have the students write out the definition of a midpoint (A midpoint divides a line segment into 2 congruent segments.), we discuss the various numerical relationships that are created by the midpoint, and then the students again apply it to algebraic problems.
* Try googling "The whole is equal to the sum of its parts" and "The whole is greater than the sum of its parts." There is a song by a group called Ruby entitled "The Whole is Equal to the Sum of Its Parts." (Warning, however: the lyrics could be considered a little off-color!) Recently I also found an economics article and one on social media in which these phrases are prominent.
With 5 minutes left in the class period, I ask that the students put away their worksheets and finish whatever remains of them for homework.
Then I give each student an index card. I ask them to write on the card their name (using the name that they'd like to be called in class), their parents' names, and their parents' phone numbers and email addresses. I keep these cards handy, and will use them when I feel parental contact is needed.
Next I ask the students to write a little on their card about their experiences with math, and any special needs or preferences that they might have. I have included some question prompts that I display on my Smartboard.
Reading the students' comments is always interesting. A lot of them will be fairly superficial, but there are usually a few students who are up-front about difficulties and this helps me to begin to identify those who will need a little extra attention.