For this Do Now question, my students came up with a plethora of relationships between angles 1 and angles 2. We had a long list of terms that showed the relationship between these, including supplementary, a linear pair, and adjacent. After going through this list, we talked about what are the most important ideas (or which ideas tell us more information!). We had a great conversation about this, and it seemed to clarify for students the hierarchy of concepts to know.
My students said things like, "It seems like it's good to know that the angles are next to each other but knowing that angles are congruent seems more helpful." Another student added on, "If you know that two angles are on a line then they add up to 180. If those angles are congruent though you could find the actual measurement of the other angle."
I also added in angle 3 right across from angle 1, and extended this conversation to include vertical angles and then all supplementary angles. Overall, a great way to get my students thinking about how angles relate to each other!
We will extend students knowledge of algebra properties to geometry new concepts relating to geometry. The images presented in Student Notes will help students to visualize and remember the key parts of each property, reflexive, symmetric and transitive. These are the key properties that will be used in future lessons to help students prove triangles congruent.
In the previous unit, students will have established a strong understanding for key properties like segment addition, angle addition, segment subtraction, angle subtraction and segment addition postulate. Students are provided with a space to draw pictures to represent each key term, and students can work in pairs to draw a diagram representing each key term. You may want to reinforce that we are working on formally naming each idea so that we can then use them to prove geometric concepts and eventually in a 2-column proof (MP #7).
Before jumping into this lesson, I first handed out a Unit Outline to help them keep organized and share this information with their families. I also purposefully put a picture in the right corner so it's easy for students to identify this sheet as their unit outline, and if possible, I copy this outline on colored paper.
Also, since this outline provides all of the homework assignments for the unit, it is easy for me to modify each assignment when needed for my special education students. I usually put a star next to the unit name for any modified unit outlines as to differentiate this outline in a subtle way. Here is an example of this modified outline.
To extend this unit, I also provide my students with a challenge homework assignment which can be worked on throughout the unit and I usually make due the day of the unit test. This assignment provides students with an extra challenge and focuses on word problems and applied questions that involve more complex topics in geometry and algebra.
On page 4 of Student Notes, you will have an opportunity to introduce students to a formal 2-column proof using basic geometry concepts. Two column proofs are just one way to develop student thinking about these concepts; paragraph proofs are also a great alternative and can allow for deeper engagement of student thought. This lesson includes a focus on pair-shares specifically to help engage students in this kind of discussion and teachers can use a paragraph proof instead of 2-column, if preferred. I typically ask students to just turn and talk to their partners and make observations about the proof. Then I will ask students some leading questions like the questions below. Possible students answers are below.
- How do we know what to prove?
(Look at the prove statement, use the diagrams)
- What are we told to help us get started and to prove this concept?
(Given statement, diagrams, prove statement too!)
- Try to explain in words to your partner why the statement is true? (Again, we can emphasis how important using formal language is in keeping our proofs concise, clear and readable by others)
- Why do we have 2 columns for this proof?
(Organize our work, make it easier to read, help to use formal vocab)
- Do we need to justify each step? Why or why not?
(Yes, this allows us to explain why we can use each step, if we don't include a justify step then we are not backing up our reasoning)
Then, I like to have at least 2 students explain how they would prove this Geometry Practice Proof in their own words. As a class, we can go through each step and add in the reasons and justify steps. Some possible student answers may focus on the formatting of a proof and the 2-column set-up. By using a structured format like a 2 column proof, we are helping to keep our work organized, legible and easy for others to read and follow. You may notice that students just want to write the statements without a justify or reason statement. It’s important to remind them that the goal of these ideas is to show why a concept is true, not just that it is true. We are interested in the logic and thought behind how we developed our proof.
I have provided the entire Geometry proof in the students notes already complete, however, you may want to remove certain parts, like the justify steps or even the statements. Once you have discussed the questions above with students and explained the proof, students can work in small groups to fill in the practice proof. You can ask a student to write their proof on the board, and again, reinforce how we use properties to prove these ideas in an organized format.
If time remains, teachers can have students work on the short Activity and then review the answers to ensure that students understood the key concepts from the lesson (MP 3). Students will be asked to finish activity and will be assigned questions in the textbook for homework.
Students will complete an Exit Ticket question that asks them to justify an algebra problem after each statement.