Exploring Additional Resources

One of the topics reviewed during the course was transformations.  Below is a list of useful websites devoted to the topic.

http://www.nctm.org

This site is useful because it has lessons on identifying, visualizing and  composing transformations.  I would use this site to have students explore the interactive applets and answer the reflection questions associated with the lessons.  It’s a great way to infuse technology into the lesson.

http://illuminations.nctm.org/ActivityDetail.aspx?id=193

This site is useful because it allows students to investigate the result of performing a specific transformation.  Like the site listed above, there is an interactive applet that allows students to transform a figure eight different ways.  It’s an ideal site for lower-level students to explore.

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

This site is useful because it allows students to investigate the three main transformations (reflections, rotations, translations).  The site provides definitions and visuals for all three transformations, as well as interactive activities and review questions at the bottom of each page.  It is a great source for an introductory lesson.

http://www.worsleyschool.net/science/files/transform/mainpage.html

This site has interactive graphs that allow students to explore transformations and experiment with different initial conditions.  Students can manipulate points within the graph to make predictions and draw conclusions.  There are also fun activities, quizzes and test associated with each type of transformation.  I would use this site to asses student’s understanding of transformations.

 

 

 

 

 

 

 

Additional Resources Shared By My Peers

http://www.shodor.org/interactivate/lessons/GeometryTessellation/

http://www.mathopenref.com/polygoninteriorangles.html

http://www.onlinemathlearning.com/index.html

http://discoveryeducation.com

7-B-2

The two platonic solids I started out with were a cube and a tetrahedron.

  

 

 

 

 

 

As a result of truncating the cube, there were 8 equilateral triangles and 6 regular octagons.  14 Faces + 24 Vertices – 2 = 36 Edges

 

 

 

 

 

 

 

As a result of truncating the tetrahedron, there were 4 equilateral triangles and 4 regular hexagons.  8 Faces +  12 Vertices  – 2 = 18 Edges

 

 

 

 

 

 

 

I think my students would enjoy creating the Archimedean Solids;  anything that’s hands-on is usually enjoyable.  Truncating the solids, on the other hand, would be a bit more challenging, for some.  Since both grade levels would have prior knowledge of 2-D & 3-D figures, I would definitely have them create all of the Archimedean Solids, so they could analyze the characteristics of each.  Additionally,  I would have them truncate the cube, only, so they can describe the transformation and make predictions about the faces, edges and vertices.  Since many of my students are avid soccer fans, I would use the dodecahedron as a model during my anticipatory set, and display all of the solids on the bulletin board, after the lesson is completed.

6-C-2

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

Level 0 in the Van Hiele Model aligns with Bloom’s Knowledge and Comprehension category, because both levels require identifying, comparing and naming.  Students are merely required to recall  information.

Level 1 in the Van Hiele Model aligns with Bloom’s Application and Analysis categories, because both levels require students to analyze and differentiate to determine relationships amongst attributes.  Students are required to make discoveries.

Level 2 in the Van Hiele Model aligns with Bloom’s Analysis and Synthesis  categories, because both levels require students to analyze and research to formulate arguments.   Students are required to make predictions and formulate arguments, based on their previous discoveries.

Level 3 in the Van Hiele Model aligns with Bloom’s Synthesis category, because both levels require students to test proofs and formulate an understanding of geometric ideas.

Level 4 in the Van Hiele Model aligns with Bloom’s Evaluation category, because both levels require students to assess, argue and conclude.  Students are required to create and compare their own theorems, and implement a sophisticated level of thought.  

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

I can use the Van Hiele Model to help students learn mathematics, because the Van Hiele Model clearly indicates what students should master at each level; thus, I can implement the appropriate activities according to students level of development.  Since the model provides the opportunity for students to engage in discovery-based learning, students will gain a better understanding of geometric concepts, and a foundation is laid for the next level of learning and thinking.  The model can definitely be used to ensure that students progress beyond a basic, superficial understanding of geometric thinking.

Develop additional questions that you could ask students if you were to use this lesson in your classroom.

Knowledge: Define area & perimeter.

Comprehension: Predict what would happen to the perimeter if you were to place tiles in the corners.

Application: Is it always true that the corner tiles add no units to the perimeter?  Develop a rationale.

Analysis: Compare and contrast the area & perimeter when tiles are placed in the corner, as opposed to any other place.  What do you notice?

Synthesis: Construct a different design that has a perimeter of 16.  Does your design have fewer or more tiles?  Explain why.

Evaluation: Assess the discoveries you made as a result of engaging in this activity.  Were some more beneficial that other?  Decide which discover was the best.  Justify your answer.

 

6-A-1

When I followed the directions using the small triangles, I created the visuals displayed below.  As on can see, the area of the square along Leg A is A², or two small triangles, and the area of the square along Leg B is B², or two small triangles.  This relates to the area of the square along the hypotenuse, because the area of that square is C², or four small triangles.

Even when I changed the size of the triangle in the center, all of the images clearly demonstrate that A² + B²  = C².  Therefore, students should be able to make a connection that the larger square will always have twice as many triangles as the smaller squares.  Or, in other words, the sum of the two smaller squares is equal to the larger square. For example:


Small Triangle:  A² (two triangles) + B² (two triangles)  = C² (four triangles)

Medium Triangle:  A² (two triangles) + B² (two triangles)  = C² (four triangles or eight small triangles)

Larger Triangle: A² (two triangles) + B² (two triangles)  = C² (four triangles)

This activity would be a good introduction to square roots and rational numbers because the lesson could be extended to have students find the value of the missing side (hypotenuse).  As a result, they would become familiar with triangles that have legs that are rational (the small & large triangle), and diagonals that are irrational, requiring them to find the square root and un-square the radical.  Since my students aren’t taught the Pythagorean Theorem in the 6th grade, if I were to present this lesson to my students, I wouldn’t change a thing.  I think it’s a great hands-on, introductory activity for students to learn about the theorem.

6-A-3

The Pythagorean Theorem Puzzles would be very beneficial to my students because, not only do they provide students with a visual of how and why the theorem works, but they also incorporate technology and transformations (rotations),  and they require students to engage in critical thinking.  I like how the puzzles diagram the dimensions of each side; thus, meaningful discussions could be generated about the theorem, in addition to using the activity to make a connection to finding the area of an irregular polygons.  Furthermore, most of my students wouldn’t necessarily consider the activity a lesson in geometry.  Rather, they would merely consider it a game or puzzle; therefore, I think they would exert more effort and thinking in an attempt to “win” and complete the puzzle.

When I solved puzzle #1, I found the puzzle on the right to be easier.  For example, when I analyzed the white area, prior to rotating the figures, I knew that the square couldn’t fit in the lower right-hand corner.  Thus, once I inserted the two triangles, everything else fell into place rather easily, because there was only one option left for placing the square.  To the contrary, however, the puzzle on the right took a little bit more time; it was more difficult than the other.  For example, since the square could fit into any one of the four corners, I spent more time deciding where to place the square.  I realized that, when I placed the square in one of the four corners, the triangles didn’t extend the full length of the square box.  Thus, I quickly realized that I needed to somehow rotate the square and place it somewhere other than one of  the corners.  This process took trial and error.

Fortunately, puzzle #2 didn’t give me any trouble at all.  For example, the puzzle on the right was a carbon copy of  the previous puzzle, and the one on the left was also pretty much the same.  I enjoyed using the virtual manipulative; however, as a teacher, I prefer having my students use hands-on manipulatives.  For instance, hands-on manipulatives allow students to create multiple examples at one time.  Therefore, students can visualize more than one example/option at a time.  In addition to this, students are given hands-on manipulatives on their standardized test.  Thus, I think it’s best that they have more exposure and experience with the hands-on manipulatives, than the virtual manipulatives.  Besides, hands-on manipulatives are more accessible; every teacher doesn’t have easy access to technology for an entire class.

Virtual manipulatives, on the other hand, break up the monotony of having students sit at a desk for the entire class, and they do a create job of incorporating technology into the lesson.  Unlike hands-on manipulatives, you can often alter the shape, size and color of virtual manipulatives, in addition to having access to a wider array of manipulative options.  However, as mentioned above, without access to technology, all of the advantages associated with virtual manipulatives are a mute point, although another added benefit is that they extend the class-time, since there’s no clean-up process.

Exploring Dilations

If I were to use this activity with my students, I would take all of the questions from the activity sheet and infuse them into various components of my lesson.  For example, I would provide students with a coordinate grid, and have them plot the pre-images and images for the triangle and pentagon. I think this would be a great question to start the class prior to beginning a full lesson on dilations; therefore, I would use this question as the Problem of the Day.

Once the P.O.D is complete, I would have a student re-plot the images on my drop-down coordinate grid so the entire class can see the answer and check their work.  Subsequently, I would engage students in a class discussion about Question #3.  Based on student’s answers to Question #3, I would then ask the folowing questions:

Teacher: What do you notice about the size of the images?

Student Response: One is bigger than the other.

Teacher: Why do you think one is bigger than the other?

Student Respone: Because we had to multiply each coordinate by two.

Teacher:  What do you think would happen if we multiplied each coordinate by .2?

Student Response:  The image would probably get smaller.

Teacher: Why? 

Student Response: Because we’re not multiplying by a whole number.

Teacher:  Take a look at the angles, the distance, and orientation of the figures.  Have they changed?

Student Response: The angles and the orientation are the same, but the distance is not the same.

Student Question: Why does it look like the images have been translated?

Teacher Response: Because with a diliation, the distance is not the same, but the points are enlarged or contracted on the same path of motion.

Student Question: So, is a dilation a transformation?

Teacher Response: Yes, a dilation is a type of transformation, but the dilated image is either smaller or larger than the original.

Post-Discussion Classwork/Reflection

Students will complete Question #5 in class to re-inforce their understanding of the lesson.  Subsequently, they will answer Question#4 in their journals.

Learning Activity 5-B-1

I enjoyed engaging in the refelcti0nal symmetry activity; it was very similar to the reflection activities my students complete in class.  I particularly liked the open-ended question at the bottom of the page.  For example, usually my students complete exercises that require them to reflect over the X and the Y axis.  However, I liked how this activity had students reflect over the Y-Axis first, then engage in higher-order thinking to consider what would happen if they reflected over the  X-axis.

A nice extension of this lesson would require students to identify the new coordinates after completing their reflections.  It would be a nice way to assess student’s understanding of reading coordinates.  Additionally, assuming they write the new coordinates correctly, they could compare and contrast the coordinates to discover the rules for reflecting over the X & Y axis.  This is a lesson I would definitely incorporate into my class.  However, I would modify it even further, and have students write the open-ended response in the journals, prior to reflecting the images over the X-axis.

Learning Activity: 4-A-1

Parallelogram:  A parallelogram is quadrilateral that is characterized by two pair of opposite sides that are equal, and two pair of opposite angles that are equal.

 

 

 

 

Rectangle: A rectangle is a parallelogram with four right angles.


 

 

 

Regular Polygon: A regular polygon is a polygon that has congruent sides and angles.

 

 

 

 

Square:  A square is a regular polygon with four 90 degree angles.

 

 

 

 

 

 

Symmetrical: A characteristic of an object or image that can be folded in half to create two identical sides. 

 

 

 

 

Acute Triangle:  An acute triangle is a triangle that has three acute angles that are less than 90 degrees. 


 

 

 

 

 

 

Obtuse Triangle:  An obtuse triangle is a triangle that has one obtuse angle that’s greater than 90 degrees, but less than 180 degrees. 

 

 

 

 

 

 

Right Triangle:  A right triangle is a triangle that has one right angle that’s exactly 90 degrees. 

 

 

 


 

 

If I were to introduce this activity to my class, I would have them create individual Polygon Dictionaries as a unit project.  For example, at the end of the unit, I would have students select six, two-dimensional shapes.  Once their shapes were selected, I would incorporate the Frayer Model and have them define the shapes in their own words, provide an image of the shapes, as well as examples, non-examples, and characteristics.

Students would type the name of each polygon, and all of the criteria outlined above.  They would arrange their dictionary in alphabetical order, one polygon per page, and eventually bind it together in a plastic cover.  Students would be permitted to draw their own images, or they could find one on-line or in a magazine.  Lastly, the outside of their dictionary must be decorated with a “polygon theme”.  For example, they could tessellate the cover, draw a “character” made out of polygons, or merely draw a variety of polygons and spell out the word dictionary.

Since I teach 6th & 7th grade, I would require my 6th grade to focus on 2-D shapes, while my 7th grade focused on 3-D shapes.  Student’s grades would be based on a geometry rubric, and they would also be required to write a reflection, in addition to submitting their dictionary.  The reflection would require them to answer the following questions: Think about the polygons you selected for your dictionary.  Where do you see these polygons the most in the real-world?  What purpose do they serve?  What can others learn by utilizing your dictionary?

 

3-C-2: Logic & Reasoning

The most valuable information in this module is Bloom’s Taxonomy.  As an educator, understanding the taxonomy is very  important because in addition to educating our students, we must know how to inspire them to think critically, make conjectures, and demonstrate their understanding.  More importantly, we have to know how to assess our student’s understanding, or lack thereof.  If our students aren’t able to analyze and synthesize the mathematics that’s taught within the classroom, then the education is useless because they’re unable to apply it to the real-world experiences they will encounter outside of the classroom.  Therefore, I believe the biggest value of Bloom’s Taxonomy is that it assists educators with enhancing our student’s intellectual growth.

 

 

 

 

 

Mathematics education today focuses more on standardized testing than logical and lateral thinking skills.  Therefore, unfortunately, our instruction is dictated by the previous year’s test results, not on whether or not students know how to engage in complex reasoning.  However, in an attempt to incorporate more logical/lateral thinking activities within my classroom, I plan to post a Problem of the Week on my classroom bulletin board.  Those students who answer the problems correctly will receive extra credit.  I also plan to infuse logical/lateral thinking problems into student’s journal writing.

3-B-2: Solving Lateral Thinking Problems

Eggs in a Basket Response

I think there is one egg left in the basket because one of the eggs was returned to the basket.   For example, six people took an egg out of the basket, one at a time, and they kept their egg.  However, one of the six people removed the egg from the basket, then returned it to the basket; thus, one egg was left in the basket.


 

 

 

 

 

Manhole Covers Response

Manhole covers are round because the manholes are round, and round covers are easier to remove and replace.  For example, round covers don’t have angles, so they can be replaced easier than a square cover with four angles.   Similarly, since circular covers don’t have angles, they can be rotated or reflected for removal purposes.  Squares, on the other hand, could only be reflected open.

 

 

 

 

As I think about the manhole covers question, I think it’s a Level 6 evaluation question on Bloom’s Taxonomy.   Level 6 comes to mind because the question requires one to make judgements and defend a position.   Nonetheless, the question can also fall under Levels 4 (Analysis)  and 5 (Synthesis).  For example, at Level 5, one is required to predict and imagine.  Therefore, in attempting to answer the question, one must predict why the covers are round, as well as imagine what would happen if the covers were square.  Similarly, at Level 4, one  would have to compare and contrast the properties of circles and squares.  Clearly, the properties associated with these figures have something to do with the fact that manhole covers are circular and not square.

If I were to present this question to my students, I would first make sure my students know what manhole covers are.  Therefore, I would print-out a few images from the internet, then proceed to ask them what they notice about the shape of all the various manholes.  Subsequently, I would have them discuss the question within their cooperative learning groups.  I would encourage them to create a manhole by cutting a circle out of a piece a paper, then I would provide them with square and circular manipulatives to use as their manhole covers.  The higher-order thinking questions I’d pose include the following:

Create a manhole by cutting a circle out of a piece of paper.  Use your diagram and the manipulatives  to explain what would happen if the manhole cover was a square. (Synthesis)

Analyze your “manhole” and both “covers”.  Which cover would be easiest to remove & replace?  Justify your answer using your prior knowledge of square and circles. (Evaluation)

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