Geometry: Transformations on the Coordinate Plane

Pg 12. Transformations

 

I pulled the information from the given website and typed this assignment.

Pg 13. Translations

Pg 14. Reflections

 

Pg 15. Rotational Symmetry

Pg 16. Rotations

During my third year, I implemented a new strategy for transformations.

Materials needed:

• paper coordinate grid
• coordinate grid on a transparency
• push pin
• piece of cardboard

Plot the original image on the paper grid.

 

Layer materials:

1. cardboard
2. paper
3. transparency

Using a dry erase marker, copy/trace the original image; then using the transparency, complete a selected transformation. For translations, slide the transparency. For reflections, flip the transparency about the line of reflection. For rotations, turn the transparency in the desired direction and degree. Use the push pin to poke holes through the transparency, paper, and cardboard to mark the resulting image coordinates. Remove the transparency, and replicate the pre-image.

Pg 17. Dilations

Pg 18. Transformations Booklet

OBJECTIVE: This is another one of those 'create your own' projects. Students are given a grid and asked to plot a triangle of their own. I always use points that create a scalene triangle that does not have horizontal or vertical sides. I love watching students throughout this project because they see other's results and begin to connect patterns.

One year, I completed each section after each lesson and practice for the four transformations. The next year, we completed the entire booklet after all four lessons and practices were done. I'm not sure which had a better result. I think that the sequence depends on the level the students are performing at.

We then wrote a translation statement and completed a table using notation. I think students graphed first and then we completed the table. Graphing is easier for students because it it visual and kinesthetic.

We completed dilations after reflections and rotations.

I do remember stating at the beginning, that if you feel daring, then plot across the axes; however, if you are not sure, then keep it in a single quadrant.

 

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Geometry: Finding Distance, and Applying Coordinate Geometry to Quadrilaterals

Pg 10. Finding the Distance

 

 

This is a foldable that I wrote as a template.

Pg 11. Rectangle Analysis

FAVORITE: This is an individual project that we paced ourselves through as a class.

GEOGEBRA: We used Geogebra to graph a rectangle. This required students to be able to enter in two sets of parallel lines perpendicular to adjacent sides. Each student was required to create their own individual rectangle (no two projects were alike). This required some thinking and reworking. Once graphed, printed, and pasted, students had to justify that the shape is a rectangle. On page one, they listed the equations, intersections/ordered pairs, slopes, y-intercepts, and the parallel/perpendicular relationships.

We discussed the other critical attributes needed to justify the rectangle; they came up with midsegments and lengths of each side.

I tried my best to focus on notation, and I wish that we had written a final summary justifying the rectangle. Writing is an important component on math.

This kind of activity allows students to collaborate without copying another's work. They see multiple rectangles and help others troubleshoot.

 

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Geometry: Midpoint Formula and Distance Formula

Pg 5. Discovering Midpoint Formula

 

Pg 6. Midpoint Formula

 

Pg 7. Midpoint Formula: Working It Backwards

 

Pg 8. Where does the Distance Formula Come From?

 

Pg 9. Distance Formula

 

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Geometry: Angle Relationships

Pg 15. Angle Addition Postulate

The bottom half is a manipulative used to physically demonstrate the Angle Addition Postulate. The yellow rectangle was glued down 'like a pocket' for angles 1 and 2.

Pg 16. Angle Relationships

 

 

I use the layered flip book quite a bit.

 

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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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