Geometry: Special Right Triangles

Pg 10. Rationalize Square Roots

Before we dived into Special Right Triangles, I though we'd go over rationalizing square roots. This went much smoother this year than in the past because my students were familiar and comfortable with square roots and simplifying them. Surprisingly, most of the students took this very well and rationalized square roots like it was nothing! (Still amazed!) It has been such a struggle in the past.

 

 

Pg 11. Special Right Triangles

This has always been my toughest area to teach to my students, until this year. When it was all said and done, I confessed to my students that this is one of the hardest concepts/lessons for me to teach. Then, they blew my mind. My students told me this was the easiest thing they've done so far and loved it. Throughout the rest of the year they chose to use the special right triangles method to solve most other problems over trigonometry!!

I think that part of reason for this is the method I used to teach this concept. I pulled several resources for notes, guided practice, and independent practice. I worked all problems myself to identify a pattern to the differing levelss of questioning. After this, I taught each question type as a lesson by itself.

For 45-45-90, I focused on finding the legs first and then the hypotenuse.

For 30-60-90, I began with the questions that required simple mathematical relationships in order to solve and then we worked up to the questions involving rationalizing.

We would work out an example with extensive explanations, then complete a student let guided practice. For the independent practice, I encouraged them to work as a team and I graded them immediately. I actively monitored for independent and collaborative work and I made sure to intervene when I saw attempted cheating.

Special Right Triangles

Cover:

Deriving the formulas:

Lesson examples:

Guided Practices: These were folded and tucked behind the above notes that were glued down like a pocket.

For each question, you'll notice a select number of homework problems. I assigned what my students thought was randomly selected problems. I would assign a small two to three problem practice and then the next day come back and do one to two more or the same. In the end, we completed the worksheet.

Honestly this whole journal page probably took us four or five days to complete. We looped through lesson, guided practice, and independent practice for each type of question. I felt that this was an important tool needed later for properties and measurement of two and three dimensional shapes.

Usually, the lesson using trigonometry to solve right triangles is introduced here. However, this lesson hit right before Christmas break. I looked through upcoming units and decided that I could make up the lessons and concept in Unit 8 Measurement of Two Dimensional Figures. It turned out great! Not sure if I will keep it this way next year or not. I guess it depends on timing.

This is a page from my second year journal.

Optional Pg. Right Triangle Solving Strategies

This foldable was used after the lessons on Trigonometry to summarize the strategies used to solve right triangles.

 

 

 

 

 

 

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Geometry: Triangle Congruency and Similar Triangles

Pg 13. Congruency and Similarity Relationships

This is a half page book fold with half page extensions on the inside.

We first wrote and diagramed all triangle congruence theorems.

We then briefed over similar and non congruent.

Finished with CPCTC and two proofs.

 

 

Pg 14. Proof Patterns of Congruent Triangle Proofs

The first year I taught, I noticed that students struggled with proofs. My third year, I decided to focus on building a pattern/strategy for proving congruent triangles.

1. Label what is given.
2. List the triangle congruency theorems/postulates we could use.

3. Start with the given.

4. Prove by listing the needed information (side, then angle, then side). This began to vary as the diagrams and proofs changed.

5. Prove statement.

 

Pg 15. Proof Book

This was a collection of about eight proofs. The students would complete one to two proofs each day as warmups. Any down time we had, the proof book would be used.

 

Pg 16. What Makes Triangles Congruent

This was originally a warm up quiz that turned into an impromptu lesson and journal page. As I graded their completion immediately, I noticed that many students struggled with number eight.

IMPROMPTU LESSON: I grabbed some patty paper (awesome stuff) and we traced the two triangles from the given statement and labeled the congruent segments. Once the triangles were separated, students saw the relationships and congruency.

We folded the paper in half (my go to impromptu journal fold) and glued the patty paper to the front.

Pg 17. Similar Triangles

 

Pg 18. Similar Postulates, Theorems, and Proofs

This was a new area for me this year so it's going to be a little rusty.

The statements on the front are actually students conclusions based on our warmup discussion while prepping (passing out papers, getting journals, etc.).

 

 

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Geometry: Euclidean and Non-Euclidean Geometry

For my third year, I began with an in depth discussion over the types of Geometry. We had a few debates and lots of illustrations and demonstrations. I couldn't believe how engaging these discussions were and how useful an old basketball could be!

Then we dove into the foundations of Euclidean geometry. Next year, I plan on teaching constructions as a continually developing concept through each unit. I love the vocabulary development that results from these experiences. Constructions became a hands-on collaborative activity for my classes.

I will try to post my Geometry Journal ideas and experiences in the chronological order that I use. I will be sharing differing foldables and information from about three different journals-year two experience, year three experience, and the 'creating an idea' journal.

Pg. 1. Foundations of Geometry: (This is the Unit 1 I have been creating for next year. Information has been pulled from the CSCOPE curriculum mostly. Some have been pulled from older journals.)

Unit Pocket (yes, it is backwards)

Pg. 2. Geometry - "To measure the earth" (These are foldables-mainly half folds, that are glued on a sheet of construction paper folded in half. I use the half fold book alot.)

 

 

My students came up with the real world example for the Non-Euclidean Geometry, and we celebrated with Pringles the next day!

Pg. 3. Structure of Euclidean Geometry

OR (the original from year two)

pg 1

pg 2

 

 

 

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