Geometry: Quadrilaterals

This is one of my favorite units out of geometry now, but it wasn't the first time I taught. My secret... I will reveal in a later post. ;)

Pg 1. Unit 6 Quadrilateral

Pg 2. Map of Quadrilaterals

This is my original organization and understanding of quadrilaterals. Some students like the Venn Diagram feel of this graphic organizer. This notes page does not include the trapezium and also leads to the misconception that legs of trapezoids always extend outwardly. I added that small trapezoid image to remind me about that misconception (and that trapezoids should also be seen 'upside down').

Pg 3. Properties of Quadrilaterals

This page is used to define and list the attributes of quadrilaterals. This foldable is a collaboration of my second year project from my students.

 

There has been some controversy over whether or not the complex are actual polygons or not. Some of my resources are

• http://www.mathsisfun.com/geometry/polygons.html
• http://www.math.com/school/subject3/lessons/S3U2L1GL.html
  
• http://mathworld.wolfram.com/Polygon.html
  
• http://www.mathopenref.com/polygon.html

I always try to use multiple resources along with different curriculums and textbooks. When get to a gray area, I let my students know, provide them with the references, and tell them to keep an open mind because math is up for interpretation. (And I am not always right. :))

Pg 4. Flow Chart of Quadrilaterals

This page was a result of a tutoring session with a student at Sylvan. This student wanted to look at my class journal to compare my information to the information from her teacher. After viewing my Map of Quadrilaterals page, she disagreed with my graphic organizer and drew up the following flowchart on a scratch sheet of paper. My students really like the flow and relationships that this graphic organizer gives. I used this graphic organizer to visually notate the critical attributes of each quadrilateral. As we went shape by shape, I encouraged students to reference our previous notes page.

 

During my second year of teaching, I had students write a paper analyzing the relationships of quadrilaterals or depicting the relationships through fiction. My favorite result was the student that wrote about the rhombus and the rectangle that stayed out late one night, and nine months later, there was a square.

Pg 5. Properties of Trapezoids

This is an important page as far as properties go. I didn't know how to go about this journal page, so I went with my go to half page fold. I need to find another foldable or way to help organize and emphasize each property. Suggestions?

 

 

NEW: I viewed these in an amazing geometry workshop and I can't wait to try them out next year.

Who am I? Puzzles:

Read the lines one at a time to students. One teacher suggested giving students whiteboards to draw their guesses as each line is given. If a student makes a guess, then they are required justify their guess using attributes and definitions.

 

 

 

Posted with Blogsy

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.

 

 

 

 

 

 

Posted with Blogsy

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

 

 

Posted with Blogsy

iPads in the Classroom

These are my resources from an iPad training.

Resources:

• Algebra Touch *****
• Khan Academy *****
  
• TI-Nspire ******
• Educreations Interactive Whiteboard ****
• Youtube ****
• Solarwalk 3D Solar System Model ****
• Splashtop 2 - Remote Desktop *****
• Remind 101 ***
• Notability *
• iTunes U ***
• TeacherKit *
• Flashcards*
• ShowMe Interactive Whiteboard ***
• Minds of Modern Mathematics ****
• Elevated Math ***
• Math Ref ***
• Mathinnation ***
• Video Calculus *
• Math Stripes **
• Touch Physics *
• Math *
• Motion Math: Hungry Fish *
• Pre-Calculus & Calculus by WAGmob *
• Google Earth **
• How to Make Origami *
• Free Graphing Calculator **
• Skitch *
• iMathematics! *
• Kno Textbooks *
• inClass *
• Sketchpad Explorer
• Geometry Pad
• Geometry Constructions Tutor (Lite)
• iSpy - X
• Algebra Champ
• Doceri **
• MathBoard ***

Pending: (Apps I have not used, but shared by other teachers.)

• Quizzlet
• Easy bib (English)
• Books app (English personal)
• Guided Access on the iPad
• qr codes
• class dojo
• prezi (scroll all the way down to education)
• today's meet
• CamDraw
• MathAcademy
• BrainPop
• MeasureLength
• ButterflyMath
• MathOpen
• MathZombie
• MathPentagon
• Farmer'sMath

Resource:

• TCEA
• http://www.ck12.org/student/

Apple TV: Used to project your iPad activities.

 

Posted with Blogsy

How to make a Geoboard!

I was going through my workshop journal from a journaling workshop several years ago and I ran across the Geoboard page. This is a great idea, but I haven't used Geoboards in my classroom. This is very easy and the idea can be used elsewhere.

Materials needed:

• print a grid of desired measure onto cardstock (I was given 1 in. squares)
• push brads
• string

Decide first if you want this to be a removable tool from the journal or not. If you decide on removable, I would suggest making a pocket to store the Geoboard in.

Next, push the brads through each point of intersection or your desired scaling on the grid paper. Then glue a second piece of cardstock to the back to secure the brads.

I made a separate small pocket for the string.

My Geoboard in the picture is just a sample from a workshop. I have no experiences to reference for it, but I think it is a great idea.

 

Posted with Blogsy

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

 

Posted with Blogsy

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.).

 

 

Posted with Blogsy