I thought this site could be very helpful for teachers who are not comfortable with using technology for instructing translations, reflections and rotations. The site provides a very detailed explanation of what the teacher should be discussing, guided practice, independent practice and how to close the lesson. I really thought the Transmographer site (http://www.shodor.org/interactivate/activities/Transmographer/ ) would be a great for teachers to use as a demonstration tool. I also think that students could then go back and use the site to further develop their understanding of the topics.

**2. MathsNet: Transformations** http://www.mathsnet.net/transform/index.html

This site intrigued me. It provides the student with the opportunity to observe the characteristics of the transformation, understand how the properties function, explore the characteristics in guided format and use the newly acquired knowledge to construct meaning. I would use this site as an inquiry site for the student to explore prior to the introduction of the unit.

**3. ****GradeAmathHelp.com **http://www.gradeamathhelp.com/transformation-geometry.html

This site provides a very basic definition for all of the transformation geometry terms. I can see this site being a resource for students who are absent when the topics are discussed in class. The definitions are very easy to understand and the illustrations are very easy for students to understand.

**4. ****Transformation Golf **http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/golftrans.html **Post the Shapes **http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/postshape.html

These two sites caught my attention because they allow students to use what they know about transformations to complete an activity. I have been to some sites that are supposed to reinforce a skill in a game like environment but the students spend more time playing than actually using the newly acquired skill. This appeared to be different. The student must understand the properties of the different types of transformations in order to complete the activity. This site may be too easy for some students but I still think they would enjoy it. This site would be great for a day when a lot of students are out of the classroom (sickness, pictures, field trips, early dismissal prior to a holiday, etc.).

**5. ****Pilot Math 7: Transformations **http://www.watchknow.org/Video.aspx?VideoID=16785

This is a short video that could be used as an introduction when starting the unit on transformation. I am not sure if junior high students would like it or not but I think that sixth grade students would enjoy it. I thought it was cute and to the point.

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**What shapes did you start out with? **

Platonic Solid |
Vertices |
Faces |
Edges |

Cube | 6 | 8 | 12 |

Tetrahedron | 4 | 4 | 6 |

**What were the shapes after you truncated them? **

Platonic Solid |
Vertices |
Faces |
Edges |

from Cube | 24 | 14 | 36 |

from Tetrahedron | 12 | 8 | 18 |

**How can you set up this type of activity for your students? **

I would provide the students with a variety of materials that could be used to create the solids. They would then be allowed to select the material that they would like to use. Once the material is selected the student would select which shape they would like to start with. I would have them complete a chart like the one above for their beginning shape. They would then be allowed to create their own truncated solid. Once they are finished they will complete the chart for the new shape. After everyone has finished their truncated shape we would have a class discussion. Since there are so many ways to create truncated solids the chances are high that everyone in the class will create a different one which would lead to a great discussion.

**Do you think this is an activity that students will enjoy? Why do you think this is the case?**

Students who enjoy puzzles would definitely like this activity. Anytime students can use their hands while learning they enjoy it. Students remember and learn more if they are actively involved. This activity gives the students a chance to prove to themselves that Eucler’s Formula works with all types of platonic solids.

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Bloom’s Taxonomy |
Van Hiele Model |
Justification |

Knowledge, Comprehension |
Level 0- Concrete |
Students know what the shapes are based on what they look like. |

Analysis |
Level 1- Analysis |
Students can identify the properties of shapes. |

Application, Analysis, Evaluation |
Level 2- Informal Deduction |
Students would be able to illustrate their understanding of a theorem through informal deduction |

Evaluation, Synthesis |
Level 3- Deduction |
Students are able to arrange statements in an order that develops a proof that is accurate |

Knowledge, Evaluation, Application |
Level 4- Rigor |
Students are able to complete proofs by providing supporting theorems and definitions |

Answer the question asked in the article: “How can you use the van Hiele levels to help students learn mathematics?”

**Helping students learn mathematics involves as much application as possible. Every year there are students in classes that are at different levels of understanding. According to the article it is advantageous to all students to work together on the same problem. I need to work harder on placing students of different levels in the same group. As an instructor I need to work harder on modeling all the levels of the van Hiele. I need to remember to use the correct terminology and appropriate tools when presenting concepts to my students. I must also remember to start at the lowest level and work my way through them regardless of where I think the students are located.**

Review the “Guiding Questions for Group Discussion.” Using the Questioning Cue Words from Module 4 < link to table from previous module>, develop additional questions that you could ask students if you were to use this lesson in your classroom. Use the Bloom’s Question worksheet you used in Module 4.

**How many units are needed to increase the perimeter to 16 units?****Give examples of figures with a perimeter of 16 units.****Demonstrate the largest figure that can be created but still have a perimeter of 16 units.****Create a figure that has the largest area with a perimeter of 16.****Argue if it is or is not possible to create a figure with an odd perimeter.**

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NOTE: All shapes were manipulated in Microsoft PowerPoint and it is harder than you think to get the lines to match up but I tried.

After comparing the three drawings I came up with the following table to show the areas of each square in each step.

Area of Square on Side A | Area of Square on Side B | Area of Square on Side C | |

Step 2 | 2 | 2 | 4 |

Step 3 | 4 | 4 | 8 |

Step 4 | 8 | 8 | 16 |

The sum of the areas of the squares on the legs (side A and side B) is equal to the square of the hypotenuse (side C).

After completing this activity I would hope that students would realize that the number of small triangles that is required to create the square on each side of the original triangle is equal to the area of the square on the hypotenuse side.

This activity allows students to use concrete examples to illustrate a complex concept. Once students realize that the sum of the areas of the squares on each leg is equal to the sum of the square on the hypotenuse. This can be demonstrated by showing the area of each square in terms of length times width. This would be a great introduction into squares and square roots. After completing all the steps students should understand that a square is a number times itself and that in inverse is called square root.

I really liked the way that this lesson was presented to us. I would have the students work in pairs or small groups. Once they are in their groups I would provide them with a handout (directions like we did for this assignment) and multiple sets of tangrams. I would then expect them to do all the steps we did. I used virtual manipulatives but I would want my students to trace the shapes on paper for each step and label all the sides. After the groups have completed steps 1-4, I would pull them back together and do steps 5 and 6 together. While doing steps 5 and 6 I would introduce the appropriate vocabulary.

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When I started this puzzle I thought about how students would enjoy doing this activity with actual pieces. I also thought about how many students no longer do puzzles at home. When I was growing up we always had a puzzle up on a card table in the living room. When I looked at the square and realized that the sides were equal to the long side (hypotenuse) of the triangle I knew that I would rotate each triangle so that hypotenuse was matching with the sides of the square. Once all the triangles were used, I just needed to rotate the blue square to fit in the middle. The other white shape (located on the right) seemed just as easy to me. I knew that the section labeled B matched that small leg of the triangle and that one of them would have to be rotated to fit there. Once that was in place I could tell that an inverted triangle needed to be placed on top of that triangle. Once these two shapes were in place the blue square fit into the top right area exactly. The remaining white space was a rectangle which I knew could be created by placing two triangles together.

**Puzzle #2**

When I started this puzzle I immediately thought of all the quilting patterns I have constructed in the past. The square with one square was exactly like the one in puzzle #1. I did it exactly the same way. The square with two square was also easy because there is a quilt pattern exactly like it. I knew that the two squares when placed side by side would complete the side of the square. I knew from the previous puzzle that two triangles could be place together to create a rectangle. I place the large square in the bottom right corner of the white square and the small square in the top left corner of the white square. This left 2 rectangles (one going vertically along the left side of the square under the small square and one going horizontally above the large square.)

**Preferred Manipulatives**

As a technology person I really do like the virtual manipulatives but I also know the value of students actually moving the pieces with their own hands. One of the advantages of virtual pieces is that students cannot lose them or drop them on the floor when they should to be working. Another advantage of virtual pieces is that the pieces are the exact size that they are meant to be for the activity. Physical pieces can be lost easily. If the students are responsible for creating their own pieces and they have poor fine motor skills, the resulting shapes may not be accurate. If the shapes are not cut correctly then the puzzles do not always work out correctly. An advantage of the physical pieces is that no additional supplies or skills are needed. When students use virtual manipulatives they must know how to move and rotate the shape using the mouse or the keyboard in order to complete the activity. Another advantage of the physical objects is they can be taken anywhere and virtual shapes can only be used wherever a computer is located. If I had to select one over the over I would have to say the physical objects win but I really do like the technology tools too.

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I would start this lesson by displaying graph paper background on the Smartboard. I would then have the students create a triangle. Three different students would determine the vertices. I would then create a copy of the triangle and paste it on the page. With the copy I would enlarge it using the corner and dragging it (a feature in Smartboad). We would talk about what I did and how it is the same and different from the original. I would place the new enlarged triangle in such a way that the coordinates can be determined. I would have the students write the new coordinates for each vertex. We would do this with a couple different shapes.

After I am sure the students understand the coordinate system I would give them the coordinates for two shapes and have them plot it on their own graph paper. We would then look for similarities and difference. The next problem would be for me to give them coordinates for a shape and provide them with some of the coordinates and ask them to determine which ones are missing. The final problem we would work together would be to provide the coordinates for the shape and ask them to create the new shape with only one coordinate provided. I would then provide some coordinates for them to graph and ask them to create the new shape based on the property provided. (I would only use whole and half numbers so that graphing would be easier, I have inclusion students in all of my classes and this would allow them to do the activity without getting lost in the conversions.) If students finished this I would then ask them to create two objects (different than what we had already used in class) so that someone else could determine what multiplier was used.

**Anticipate the questions you might ask students during this activity. What type of prompting questions would you ask during this activity?**

What do you notice about the two shapes I had you graph?

What similarities and differences do you notice?

What coordinates are missing? Explain

How do you know what coordinates are missing?

What did you do to determine the missing coordinates?

Can you add or subtract and create a similar shape?

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

*Does it matter where the shape is located on the graph paper?*

No because the paper is just for you to use as a reference, it has nothing to do with the shape.

*Is multiplication the only way to do this?*

No. Can anyone tell me another operation that could be used to make a dilation? Division. (Some of my students understand that division is the same as multiplying by a fraction but most do not.)

*Does the length of each side change?*

Yes the do. Can you tell me by how much they change? How do you know?

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I think that students would enjoy this activity. Anytime that a student actually gets to manipulate or create materials they develop a stronger understanding of the concept. I do feel that this activity is geared more for sixth graders than eighth graders. Students learn about the coordinate system in the late primary years and could easily build upon it to complete this activity.

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I think that vocabulary is very important in mathematics, especially in geometry. I like the idea of having the students create their own vocabulary journal. I do not see having my students do this activity online but I do see it as a paper and pencil activity. I would have them use a folder with the brass fasteners in the middle. This type of folder would allow the students to add extra pages as they need them. [Prior to the chapter I would go through and decide how many vocabulary words would be met and what letter they started with. This would allow me to know about how many pages are needed for each letter in the alphabet. (Since I do not see them needing a “Z” for example I would have them skip this letter.)] At the beginning of the class as we start the chapter I would let the students know that we will be developing personal vocabulary journals during the chapter. When we get to the first vocabulary word in the chapter I would illustrate how I would like to see them create their journal. I would want them to write a definition and have an example. Their definition can be either from the book or in their own words; however, I would encourage their own words. They can draw their example or tape a picture into their journal.

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It was hard for me to answer this question because I was torn between the logic problems and the higher level questions. I enjoy doing logic problems personally and I do not think that students get enough exposure to them during their schooling. I liked the fact that we given actual problems that could be used with students. I included the website on my blog so that I would have it for future reference. I also felt that the higher level questions activity was valuable because this is an area that I can always use additional practice in. The assignment forced me to create questions and I also viewed questions by my peers which were very good. Their questions gave me more examples to share with my fellow teachers.

**Do you feel that mathematical education today focuses enough on developing logical and lateral thinking skills? Why or why not?**

I do not think that mathematical education today allows teachers to develop logical and lateral thinking skills in our students. Teachers are more worried about Meeting and Exceeding standards so that our schools meet the requirements put forth by No Child Left Behind. Everything in my district revolves around the state testing that take place in the spring. Once the testing is over teachers do not feel so pressured that they will allow the students a little more time to explore concept development. The higher functioning students (“the gifted”) have opportunities to develop these skills through various team competitions. The average student is just left behind with very little exposure. I feel that we are doing our students an injustice by not giving them more opportunities to develop these skills.

**Describe how you will incorporate logical/lateral thinking activities in your classroom.**

** **I will share the websites that we used in this course with the mathematics teachers in my district so that they can use them with their students. I will continue to search for sites to share with my fellow teachers so that they will have an ample supply to share with their students.

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Puzzles — http://www.mycoted.com/Category:Puzzles

Logic Problems — http://www.folj.com/puzzles/

Lateral Thinking Problems — http://www.folj.com/lateral/

Blooms Taxonomy — http://officeport.com/edu/blooms.htm

Blooms Taxonomy with Technology – http://www.techlearning.com/article/8670

Free Blog — http://wordpress.com/

National Library of Virtual Manipulatives – http://nlvm.usu.edu/en/nav/topic_t_3.html

National Council of Teachers of Mathematics – http://www.nctm.org

Paper Models of Polyhedras – **http://www.korthalsaltes.com/**

Totally Tessellated: Templates – http://library.thinkquest.org/16661/templates/index.html

Math Word Wall (some word cards shown)- http://www.broward.k12.fl.us/STUDENTSUPPORT/ese/PDF/MathWordWall.pdf

Math Word Wall Words – http://t4.jordan.k12.ut.us/cbl/index.php?option=com_content&task=view&id=43

Best Practice Word Walls (Secondary) – http://www.markville.ss.yrdsb.edu.on.ca/mm/oise2005/bestpractice/wordwall.htm

**Interactivate: Translations, Reflections, and Rotations** http://www.shodor.org/interactivate/lessons/Translations/

**Transmographer site** http://www.shodor.org/interactivate/activities/Transmographer/

**MathsNet: Transformations** http://www.mathsnet.net/transform/index.html

**GradeAmathHelp.com (Transformations)**http://www.gradeamathhelp.com/transformation-geometry.html

**Transformation Golf **http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/golftrans.html

**Post the Shapes **http://www.mathsonline.co.uk/nonmembers/gamesroom/transform/postshape.html

**Pilot Math 7: Transformations (Video)**http://www.watchknow.org/Video.aspx?VideoID=16785

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