Exploring Additional Resources March 11, 2009

Transformations in geometry

http://www.mathsisfun.com/geometry/transformations.html

This site gives clear examples of the various transformations— rotation, reflection, translation, and dilation (or resizing as the website refers to it).  It also gives shows you whether the object is similar or congruent and why after it is transformed.

 

http://www.mathsnet.net/transform/index.html

This site is an interactive site that gives students an understanding of what transformations are, what they do, and what happens to shapes after they have been transformed.

 

http://nlvm.usu.edu/en/nav/category_g_3_t_3.html

This site uses virtual manipulatives to transform shapes.  It gives students the opportunity to interact with the various transformations and see the resulting images.

 

http://www.brainpop.com/math/geometryandmeasurement/transformation/

This is a site that has videos, activities, and quizzes.  Students view a cartoon that describes transformations with real-world objects and people.  It is fun to watch how the boy and his robot friend will teach kids about a concept.  This would also be great for an at-home learning activity.

 

Archimedean Solids March 6, 2009

 

I started with a Cube.  When I cut of the vertices, the sides became 6 octagons and 8 triangles to close the figure.

	Vertices (V)	Faces (F)	Edges (E)
Cube	8	6	12
Truncated Cube	24	14	36

 

 

For my second, I used a Tetrahedron.  When I cut off the vertices, the sides became 4 hexagons and 4 triangles to close the figure. 

	Vertices (V)	Faces (F)	Edges (E)
Tetrahedron	4	4	6
Truncated tetrahedron	12	8	18

 

 

 

 

I would begin this activity with giving two platonic solids that they have already created.  The class would work on the same 2 polyhedrons.  

We are going to truncate these two polyhedrons as a class.  Can anyone tell me what it means to truncate?  It is cutting on the vertices of a platonic solid.  Everyone take their figure and cut of the vertices.  What do you see?  What shape did the sides become and how many are there?  What shape would you need to close the figure and how many do you need?  Fill in the table with the new number of vertices, faces, and edges.  Now choose your own platonic solid to truncate.  Share with the class what you found and lets finish the table together.

The students can see the connection between the vertices, faces, and edges with the new figures.

I think this is a great activity for students.  It is interactive, which helps to see what happens to figures as you work with them hands-on.  It helps to be able to make the figure.  By physically cutting off the vertices, you can see the new sides that are created and then you must find out what shape to use to close the figure and see the final product.

 

Higher Level Thinking Questions for the Pythagorean Theorem and Related Concepts March 1, 2009

What level(s) of Bloom’s Taxonomy most closely align with the level(s) of the van Hiele Model? Justify your thinking.

Van Hiele Level: Concrete– the student identifies, names, compares, and operates on geometric figures

Bloom’s Level: Knowledge/Comprehension– name, define, identify, compare, classify

 

Van Hiele Level: Analysis– analyzes figures in terms of their attributes and relationships among attributes and discovers properties and rules through observation

Bloom’s Level: Application/Analysis– apply, solve, relate, use, analyze, compare, categorize

 

Van Hiele Level: Informal deduction– discovers and formulates generalizations about previously learned properties and rules and develops informal arguments to show her or his generalizations to be true

Bloom’s Level: Analysis/Synthesis– analyze, classify, differentiate, develop, combine, predict, propose

 

Van Hiele Level: Deduction– proves theorems deductively and understands the structure of the geometric system

Bloom’s Level: Synthesis/Evaluation– solutions, solve, predict, formulate, measure, assess, deduct

 

Van Hiele Level: Rigor– establishes theorems in different postulational systems and compares and analyzes the systems

Bloom’s Level: Synthesis/Evaluation– discuss, formulate, assess, summarize/prioritize, evaluate

 

 “How can you use the van Hiele levels to help students learn mathematics?”

The interaction aspect of the Van Hiele model is very important to student learning.  Students develop greater understanding of the content at hand when using social and knowledge interaction.  It is important for students to work together in math in order for them to see concepts from a different way of thinking.  A deeper understanding of the subject matter turns students away from “parrot math” and helps them to draw their own conclusions about the geometric concepts.

 

Group Discussion for Pythagorean Theorem

·      How does perimeter compare and contrast to area?

·      What are the parts of the Pythagorean Triplet?

·      How would you create a Pythagorean triplet of your own?

·      Do you agree that a right triangle can have the measurements of 4 units, 5 units, and 6 units? Justify your answer. 

Pythagorean Puzzles

Puzzles #1

Left: The first thing I did was start rotating the triangles to see what sides matched up.  I quickly discovered that the hypotenuse was equal to the side of the square, so the 4 hypotenuses make up the 4 sides of the square.  It was easy to see the square in the center of the 4 triangles.

Right: I found that leg b of the triangle was equal to the right short side of the figure.  I then placed another triangle below that placing the hypotenuses together.  I then discovered that the square fit in perfectly directly above the left half of the bottom triangles, in the top right corner.  It was then easy to see that the last two triangles, when matched up by their hypotenuse fit along the left side making a rectangle.

 

Puzzles #2

Left: I had a little trouble with this one at first; I didn’t see that it could similar to the left puzzle in the first set of puzzles because the size was different and the triangles were not.  I then found, after just playing around for a minute, that side a and b of the triangle make up the side of the square.  So, I placed all the triangles in this manner and the square fit perfectly in the middle— just like in the first puzzle.

Right: This on was easy for me to see.  Again, I matched up two triangles (making two pair) by their hypotenuses making two rectangles.  I set one along the bottom right, which left space on the right for the smaller square.  And, the other set of triangles vertically along the right side, which left space on the left for the larger square.

 

Virtual Manipulatives vs. Hands-on Manipulatives

Any types of manipulative can be useful for students to see problem visually and it is always more fun to do interactive work.  It helps to keep your mind on the activity at hand.  I prefer the use of hand-on manipulatives because I can physically hold the object in my hand and see it from any angle.  It is easier to move the objects around or flip over quickly when using trial-and-error.  And, you can use them anywhere— the class does not have to go to the computer lab to work through the activity.  But, virtual manipulatives are useful too.  I use less trial-and-error with virtual manipulatives; I try to map out where objects should be placed before moving them around.  So, it helps students think more strategically.

6A1: Tangrams I February 26, 2009

So, tangrams are slightly frustrating… The only closed square I could find was the reflection of the original tangram.

 

I did find a solution for a closed square, but the figure was not completely closed.  

The resource I found had much more creative tangram figures.  This is a fun activity for students to find many different ways to use tangrams and find different arrangements of the shapes.

http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/7_Tangrams/index.html

Exploring Dilations February 20, 2009

Describe in detail how you would present this activity to your students.

This is a great activity to introduce dilations, the last transformation we are covering in this course.  To start the dilation activity, I would first review and compare/contrast congruence and similarity.  This would lead into a class discuss what the students already know about congruence and similarity. 

I would begin the by handing out graph paper and giving the students the coordinates of  triangle ABC and pentagon DEFGH, so they could plot the points and review coordinate planes.  After they completed plotting the figures, I would also plot them for the students to make sure they were correct.  (If they get the preimage wrong, then they will definitely come to the wrong conclusion and then they have to start over.)  

I would then ask questions —

• Does anyone know what a dilation is? 
• What do dilations have to do with congruence and similarity?
• What do you predict will happen if we multiply the figures by 2 or ½?

 

Then, I would do the activity as a class.  We would multiply the triangle by 2 and the pentagon by ½; labeling them A’B’C’ and D’E’F’G’H’.  The class could then pair-up and work on the worksheet together.  Through volunteering, we would go through the questions as a class.

I would ask—

• What do you notice about the preimages and images?
• How would you compare/contrast these images to the preimages? Size? Shape?
• Were your predictions correct? Why or why not?
• When multiplying by 2, does the image enlarge or shrink?  And for ½?
• What do you notice about the length of the sides?
• What happens to the slope of the lines?
• Why is dilation important?  Explain.

 

Anticipate your students’ questions. What types of questions might they ask and how would you answer them?

• What is the mark after the letters in the new image?    A Prime, B Prime, C Prime- the vertices relate to one another.
• Is dilation like similarity or congruence? Similarity
• Do we multiply both the x- and y-coordinates? Yes

 

 

Understanding Reflectional Symmetry

The Reflection of Images activity is a great assessment of where students are in regard to using the coordinate plane, plotting points, and reflecting/flipping figures.  It is quite a bit more advanced than my second graders, but the idea is great.  When students enter middle school, they are at various academic and comfort levels of geometry.  This activity can give you, as the teacher, an idea of where your students are at and what needs to be further examined in the aspect of geometry.

 

The main idea of using reflection images is for students to understand that two reflected figures are the same.  The orientation of the object has changed, but the sides and angles of the reflected image are congruent to the preimage.  In other words, the preimage and image can be placed on top of each other and they will be exactly the same.  This idea can be a little hard for some children to grasp, so reflection activities can be very helpful. 

 

It is also important for students to understand reflection on the coordinate plane.  Using a coordinate plane can help students realize that reflected images have the same exact units, but flipped.  When the image is reflected over the x-axis, both images are the exact same distance from the axis along with all of the vertices of the figure. (The x-coordinates remain the same, while the y-coordinates are opposite—one is positive and one is negative.)  Students can use the reflected image to see perimeter, area, side length, and angle measurement of the reflected figures.  Therefore, they can come to the conclusion that all reflected images are congruent to their preimage. 

 

This activity can also give the teacher the opportunity to use higher level of thinking questions.  Students can compare/contrast shapes and figures (analysis), prediction (synthesis), and justifying why reflections are important to geometry (evaluation).  It is a great to find what students really know along with being able to physically plot images.

Categorizing Two-dimensional Figures February 11, 2009

I would introduce two-dimensional shapes to my class as a hands-on activity.  The class could choose a variety of different 2-D shapes out of a container.  I would have them journal about what they already know about the shapes they chose.  The class would then get into small groups to discuss what they came up with and help each other further their definitions.  Then, as a class we could come up with definitions and discuss the properties of the shapes.

Symmetry

Symmetrical: A figure that can be folded flat along a line so that the two halves match perfectly is a symmetric figure; such a line is called a line of symmetry.

 

 

Nonsymmetrical: A figure that has no line of symmetry.

 

http://www.mathleague.com/help/geometry/polygons.htm

Concave and Convex Figures February 10, 2009

Concave: One or more line diagonals fall outside the perimeter of the figure.

 

 

Convex: A figure in which every line segment or diagonal drawn between any two points inside the figure lies entirely inside the figure. 

 

http://www.mathleague.com/help/geometry/polygons.htm