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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Geometry: Applying Parallel Lines, Transversals, and Special Angle Pairs with a Mini Flip Book

Pg 4. Over the past three years, I have approached these theorems/postulates in different ways.

First attempt: Completed a worksheet and summarized our findings as seen below.

Second attempt:

We completed two flip books with sentence strips that the students had to match the if statement to the then statement and to the diagram.

Third attempt:

Within a large flip book project, we wrote the postulates/theorems out along with each converse.

For the large flip book project, students were to google an image that they could illustrate parallel lines and a transversal on.

INTERESTING: Students brought up the interesting difference between parallel in the real world versus the result of a two dimensional image. For example, one student wanted to use an image of railroad tracks knowing that they were parallel in the real world. However, another student pointed out that when viewed on a two dimensional service the tracks appeared to intersect at the perspective point. Is the picture a valid option? After this discussion, I had students justify their chosen image. How did the idea of parallel apply?

Next, they labeled the angles and identified the special angle relationships.

This had some interesting results. I had the students make up some algebra problems, solve and check them, and then exchange with a partner. Some of the results the students would makeup would have infinite solutions, no solution, and result in negative angle measures.

Here are the postulates/theorems and converses.

 

 

 

 

Out of all that I teach in Geometry, this is one of the hardest concepts to get my students to understand. Proving lines paralell or angles congruent. It took me two years to see the pattern. Maybe next year I will be able to present it better.

 

I gave students two angle puzzles to complete. We discussed every detail and justified each strategy used. Next, students were to create their own angle puzzles for homework. The next day, I mixed up each puzzle and passed them out to the class. Great idea, but kind of overwhelming. Students could critique and troubleshoot each others puzzles, but some would use incorrect properties and their answers wouldn't work out. This activity requires quite a bit of time and would probably do better if the students work with a partner.

 

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Geometry: Parallel Lines, Transversals, and Special Angle Pairs

It has taken me about three years to grasp the point of this unit. The angle relationships are later used in Unit 6 Quadrilaterals and Unit 7 Properties of Two Dimensional Figures.

Pg 1.

Pg 2. Transversals and Special Angle Pairs

OR

Diagram of Lines and Transversals

 

Pg 3. Special Angle Pairs

OR

 

 

 

 

 

I have noticed in the past that students sometimes struggle with all the numbers assigned to the angles. A fellow teacher at a math and science symposium shared this star-dot labeling she has used in the past. She said that this is to be used once an exploratory is complete where a student builds an understanding of congruent and supplementary angles. This allows the students to quickly relate two symbols with location instead of dealing with the numbers.

 

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Having Trouble With Protractors, Angles, and Polygons?

TOOL TO USE: Anglegs!

I have had students struggle with how to use a protractor in the past. The main issue for students was "Where do I put the center?" and "I can't line up the center and the base at the same time!".

Anglegs can be used for angles, triangles, quadrilaterals, etc.

My students call them click sticks. :)

ANGLES:

Students can lay them flat.

Students can stand them up and rotate the extended side to watch the change in degree measure.

 

TRIANGLES:

Use Anglegs to discover the Triangle Angle Sum. (Can also discuss that the length of the side relates to the degree of the opposite angle.

QUADRILATERALS:

These are an excellent tool for students to use to discover and explore quadrilaterals. I, myself, made a discovery regarding trapezoids.

This is just a small look at the possibilities of Anglegs!

 

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Geometry: Angle Basics, Protractor, Compass, and Types of Angles

Pg 10. Compass: What, How, and Use

It was important to diagram (trace) and label the components of a compass. We took a few minutes and practiced constructing circles with the compass on scratch paper to discuss the essential task of a compass.

I felt that it was critical to have students write out the steps for each concept. By writing the notation and terminology, students used them when teaching and working with others. Even though arcs haven't been officially introduced, I still used the term and notation for the diagram. This was a good experience for them. They began to observe and identify arcs as they came up throughout the year.

The independent practice problems that they store behind the notes previously completed by glueing the notes down 'like a pocket'.

FAVORITE: One of my perfered methods of teaching is to introduce the concept, complete a guided example together, and finish with one independent practice problem that day. Then on a seperate day, I have them go back and do one more. When they finish four days later, students either have it down, or we complete another set every other day. This helps them retain the skill over a period of time. There are some days when my students are independently mastering a series of concepts. I enjoy those days.

 

Pg 11. I begin angles with a page called Angle Basics. This page focuses on the components of an angle. I feel that in geometry it is very important to know and continually recognize angles.

 

Pg 12. How To Measure an Angle

For this page, my students are given a regular plastic protractor an angle, and together they are to build a set of three to four step instructions on how to measure and angle.

Then we take an envelope, cut it half and use it as a pocket for their own personal protractor that I print off on a transparency.

EXCELLENT IDEA: Print reusable tools like rulers, protractors, and grids on transparencies. Templates can be found through google. I use the grids for transformations of shapes and functions, and I also discovered that they are a great tool for layering and comparing systems of equations (especially those with infinite solutions because students then understand when we say the two equations are the same line - one over the other) from a list of four to five equations.

Pg 13. Angle Construction

Each student constructed their own angle and completed a set of sentence stems.

I really like the idea of students individualizing their experience-creating diagrams, writing problems for given information, etc. This encourages students to collaborate with others about their findings and allows them to teach each other without cheating tendencies.

Next year I plan on inserting a page of Angle Construction using the compass to copy angles, bisect angles, etc.

 

Pg 14. Types of Angles

This page is a half fold consisting of five flaps.

 

BETTER IDEA:

I pulled this diagram from a SmartPal template book. Using a push brad and straw, we created a reusable took for types of angles. We used the straw to physically demonstrate each type of angle and then we defined them under the 'area' for that angle.

LESSON SEQUENCE:

1. Engaging: Read Sir Cumference and the Great Knights of Angleland to my high school students. (They actually enjoyed it!) As I read they were given post-its and asked to write down any words or phrases they thought were important to the world of Geometry. From here, I knew that they caught the cute geometry references along with the important vocabulary. They demonstrated a significant prior knowledge for angle types (which I should expect). They then posted and organized the class post-its into some form of organization. Discussion ensues.

2. Exploring and Explaining: We complete the foldable.

3. Evaluating: I would ask them questions that required them to demonstrate a certain angle measure and type using their foldable and hold up their journals when asked. (Surprise(Elaborating): Disscussion of the two angles that actually exist each time came up!)

It just so happened that I ended up with a unannounced evaluation that day. It couldn't have gone better. I will never forget that lesson.

 

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