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: Pythagorean Theorem

Pg 6. Right Triangle Basics (Optional - Used this my second year, but not my third.)

 

Pg 7. Pythagorean Theorem

LEFT: We took a square piece of paper and folded it until the entire square is made up of small right triangles. We then selected a right triangle and the three surrounding squares to demonstrate the Pythagorean Theorem.

RIGHT: Another time, we selected, drew, and then cut out the diagram to demonstrate the Pythagorean Theorem.

I have seen other diagrams using all the pieces to make a square, but I have yet to understand them or see the purpose. Any suggestions for enlightening resources?

Pg 8. Pythagorean Theorem and It's Converse

This next year, I will use Anglegs and worksheet to introduce and classify triangles.

My students came with excellent prior knowledge of the Pythagorean Theorem.

Pg 9. Multistep Pythagorean Theorem

 

 

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Geometry: Preparing for Right Triangles by Reviewing Squares, Square Roots, and Rules of Divisibility

Pg 1. Unit 5 Right Triangles

This has been a difficult unit for me to teach my first two year. My area of sttruggle has mainly been simplifying and rationalizing square roots and approaching methods used to solve right triangles.

Pg 2. Perfect Squares & Square Roots

After two years of struggling through teaching Pythagorean theorem, Special Right Triangles and square roots, I decided to start with the basics at the beginning. Squares and square roots.

After finding the squares by hand, I gave the students the square roots. I require squares through 25 to be memorized. I give several quizzes.

 

Pg 3. Rules of Divisibility

This page was just my go to half page fold for notes.

 

I printed out a list of primes from 1 to 1000 for students to use as a quick reference when simplifying radicals.

However, I ran across a layered flip book through pinterest for the rules of divisibility. The idea is from a blog called Growing in Fifth Grade.

Students discover the rules of Divisibility instead of being told and memorizing.

Pg 4. Simplifying Square Roots/Radicals

 

 

I wanted my students to be fluent in simplifying and using square roots. We ended up with an off day and decided to simplify all square roots from 1 to 200. My students loved this. We listed the square roots from 1 to 200 on my white boards around my room. I past out post-its and we set to work simplifying. Students had to show the steps and I modeled what I expected to see. They quickly learned that they could answer the perfect squares and prime number square roots using their journal as a reference. Then I watched as some students struggled and other students taught them how to simplify. We discovered patterns and they began to name these patterns and develop a number sense for simplifying square roots. I guess square roots were no longer scary, because after this, only one or two students struggled with them.

Pg 5. Patterns of Square Roots

 

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