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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Geometry: Triangle Basics, Anglegs, and Special Segments of Triangles

Pg 1. Unit 4: Triangles

Pg 2. Classification of Triangles

Pg 3. Triangle Basics

ACTIVITY: I learned a great activity involving Anglegs for exploring types of triangles and the Triangle Angle Sum.

 

 

I use Anglegs for exploring polygons also. These are an excellent tool to provide students with a hands-on exploration and confirmation of understanding.

I had students draw their own triangle. We started with identifying the critical attributes. We then measured the length of the sides and degree of each angle. We talked about what the sum of the three angles should be and why would some of our measures vary. This led the discussion to the importance of precision. Lastly, we classified our own triangles and justified the classification orally. This will be written next year.

Pg 4. Special Segments of Triangles

PAPER FOLDING!!! Paper folding is a powerful tool in mathematics. I use it some in Algebra and mostly in Geometry. I'm looking for more, so if you know of some, tell me please!!!

We all started with a similar triangle (scalene acute triangle) in my first year teaching this lesson because I was unsure how students would react to the results of a right, isosceles, equilateral, and obtuse.

We labeled each vertex A, B, and C.

TIP: Have students use a ruler when creasing. This will help with accuracy.

Median: First locate the base, then fold in half and pinch to identify the midpoint. Using a straight edge, connect the opposite vertex to the midpoint.

Perpendicular Bisector: Fold the base in half again making sure that the base is lined up and crease. This fold and crease is important because it creates a corner/right angle. When unfolded, use a straight edge to trace the fold that shows a perpendicular line to the base through the midpoint.

Altitude: Students now know how to fold a right angle. The discussion approaches the strategy of how to fold a corner/right angle that will go through the vertex. This is the toughest one to fold and is usually not accurately done. I tell my students to do the best they can. Success is usually achieved when using a ruler.

Angle Bisector: If an angle bisector exists exactly half way between the two sides of an angle, then how do we fold? Students pick this on up quickly, but get the fold mixed up with the fold of a median. You fold the two sides on top of each other and crease.

OPTIONAL: Midsegment: I have students fold and pinch the midpoints of the other two bases and then use a straight edge to connect them for a midsegment.

Definitions: (Written on the back.) These can be done before or after each segment is constructed. I have done this both ways. Definitions done before allows students to determine how to fold the triangle to meet the definition. Definitions done after requires students to derive a definition based on the critical attributes used to create the segment.

NOTE TO SELF: However, after experiencing this again in a workshop, I would cut out all types of triangles and mark, with a dot, the base a student will use to view their triangle. After folding all special line segments, the presenter asked which participants only had a certain number of folds instead of all four/five different folds. These people would hold up their triangles and we would explain why that happened. This required us to connect the characteristics of the triangle to the type of triangle.

 

OR

Special Segments on Patty Paper

 

 

 

Patty Paper is easier for students to use to line up sides for creasing.

 

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