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 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: 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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