Exploring additional resources: Pythagorean Theorem March 13, 2010

http://www.nsa.gov/academia/_files/collected_learning/middle_school/number-theory/how_many_jelly_beans.pdf

I was real excited when I read the little blurb about this lesson idea, but once I got into the lesson I found it was not exactly what I was looking for. Instead of “teaching” the theory, it is shown how to use and told to use it. I do really like the idea of the hands on learning with jelly beans (perfect for this time of year) so that is why I decided to go ahead and share it. I think I would use this lesson as a way to connect area and perimeter.

http://www.mathsisfun.com/pythagoras.html I could see using this as a quick review, for those students have been taught the theory before. I think the site does a nice job of just showing how  the pythagorean theorem works and why it is useful

http://www.arcytech.org/java/pythagoras/pyth_tip1.html This is a good tool to use for discovering the lengths of sides that work out so all are whole numbers. I could see it as an activity for students to find sides that work as whole. Or even this could be use to help in learning the theory. Charts can be done to see pattern. http://www.arcytech.org/java/pythagoras/pyth_tip5.html this site goes along with it. It is great way to physically see how a +b = c

http://www.ies.co.jp/math/java/geo/pythagoras.html This site I found to be similiar to the maniplatives we were using for this class. If I would use these I would use them to teach and reinforce the theory. I would hope to let the students do a lot of discovering with them

http://www.utm.edu/research/iep/p/pythagor.htm,  http://encyclopedia.kids.net.au/page/py/Pythagoras as we did our module on this theory I kept wondering who really was Pythagoras. I would use this two sites (and maybe some others) in a classroom to introduce the theory. I would do this so the students how a person to connect the theory with–make it seem more personable.

Well, I listed more than 4, but some are much better than others

7-B-2 Making my own Archimedean March 6, 2010

I have just found out how terrible I am at cutting 3-D shapes, well cutting the corners off I mean…

Starting shape #1: Cube

Truncated shape #1:  It was suppose to be a truncated cube made with octagons and triangles …but as I said before I am lacking in the cutting skills

 <–this was really the only good side…

Starting shape #2: Tetrahedron

Truncated shapes #2: This time I was attempting a truncated tetrahedron that consisted of triangles and hexagons…but I think my cutting skills were even worse than the first time!

I think I would use this as a way to teach types polyhedra and Archimedean. The first day we would make polyhedra from nets. Then create a chart to see faces, vertex, and edges. The second day I would introduce Archimedean and we would create those from our previously made polyhedra. The only problem I see with this is that I am struggling with the cutting…any advice? I think if I can figure out the cutting it would be an activity the students would enjoy. I am wondering if bigger shapes might make it easy, and possibly drawing the lines before I cut. I think the students will enjoy the part of creating and then creating something new from what was previously created.

7-B-2 Checking the formula March 6, 2010

Due to this taking up LOTS of ROOM I am putting it into 2 posts, this one is just checkinig the formula to see if it holds true…

Cuboctahedron:
Number of faces: 14
Number of edges: 24
Number of vertices: 12

14+12=24+2

Icosidodecahedron:
Number of faces: 32
Number of edges: 60
Number of vertices: 30

30+32=60+2

Truncated Tetrahedron:
Number of faces: 8
Number of edges: 18
Number of vertices: 12

8+12=18+2

Truncated Octahedron:
Number of faces: 14
Number of edges: 36
Number of vertices: 24

14+24=36+2

Truncated Cube:
Number of faces: 14
Number of edges: 36
Number of vertices: 24

12+24=36+2

Truncated Icosahedron:
Number of faces: 32
Number of edges: 90
Number of vertices: 60

32+60=90+2

Truncated Dodecahedron:
Number of faces: 32
Number of edges: 90
Number of vertices: 60

32+60=90+2

Rhombicuboctahedron:
Number of faces: 26
Number of edges: 48
Number of vertices: 24

26+24=48+2

Truncated Cuboctahedron:
Number of faces: 26
Number of edges: 72
Number of vertices: 48

26+48=72+2

Rhombicosidodecahedron:
Number of faces: 62
Number of edges: 120
Number of vertices: 60

62+60=120+2

Truncated Icosidodecahedron:
Number of faces: 62
Number of edges: 180
Number of vertices: 120

62+120=180+2

Snub Cube:
Number of faces: 38
Number of edges: 60
Number of vertices: 24

38+24=60+2

Snub Dodecahedron:
Number of faces: 92
Number of edges: 150
Number of vertices: 60

92+60=150+2

Higher Level Thinking Questions for the Pythagorean Theorem and Related Concepts March 1, 2010

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

Bloom’s               van Hiele       why

Knowledge             Concrete              Both models use this stage to build the understanding and learn the basic facts

Comprehension    Analysis              Both models have the students applying what they know to classifying or                                                                     identify

Application            Informal deduction         In both these models students are actually using application to do something–they take what they know to prove something else

Synthesis         Deduction        These are kind of similar because here the students are proving something

Evaluation       Rigor        I think these are probably the hardest to connect. When I think of evaluation I think of more creating something. Rigor makes me think more of doing long, hard proofs. Taking something and working with it instead of creating your own thing

 -I think I prefer Bloom’s over van Hiele, but this might be just because I am more familiar with it

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

I think the van Hiele levels can help students learn mathematics by helping them develop the background knowledge. I think with the model they focus on learning the concepts NOT being told the facts. Something I really like about this model is how it encourages students to figure it out on through their own questioning and discovering. I think I would use the model to encourage students to really find understanding in each topic

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.

If you add 2 tiles are there more than one to add them so that you have a different perimeter? different area?

(The questions asked if the perimeter can be odd) Explain why the perimeter cannot be an odd number?

Is it possible to add tiles and not change the area? perimeter? Prove how you know this.

Learning Activity 6-A-1 February 28, 2010

 

1) The relationship between the areas of the squares long each leg of the right triangle to the area of the square along the hypotenuses is that each leg’s square added together equaled the hypotenueses’ square. (See the attached powerpoint The Pythagorean Theorem with Tangrams)

2)What connection can students make between the numbers of shapes needed to create the sides of the triangle? I think the students will be able to see how if they add the smaller square’s triangles together they equal the amount in the larger square.

3) I think this is a great introduction to square roots because it takes an idea that is not very concrete to a student’s mind and gives them something concrete to learn about it.

4)How would you present this activity to your students?  I think we would do this as a class activity. Students would be in small groups and each assigned a one of the square building tasks. Then we would combine all our data to make generalizations.

6A3 Pythagorean Puzzles February 27, 2010

I think this whole website can be very useful to use with students; especially if I have access to classroom computers. Sometimes using technology students are automatically more interested. I think doing these two puzzles with students we  would do more of a matching up. Match up triangles’ sides with the sides of squares. To solve the first puzzle I saw that I can rotate the triangle and line it up with side C. Once I lined up all the C’s I could rotate the square and put it in the middle. For the second one on the first puzzle I started with the triangle again and lined up the B and the A. Then I rotated the triangles to make two rectangles. Then I was able to fill in the square. For the second puzzle I did the same process. The only difference was I was putting 2 triangles on each edge of the square and then filled the square in the middle. The second one was a little more difficult, but I ended up making the rectangles from two triangles to fill in the white space. I think the only difficulties was figuring out how it all tied together with the Pythagorean theorem.

There are many differences between virtual and hands-on manipulatives. Virtual can be more difficult I think; you have to give directions on how to move each object and hope that the technology is working. They are nice because you will never be missing any pieces. With the hands-on they are very dependable (except when pieces become missing).  I think either one are very valuable for teaching, I think the school’s budget largely effect the availability of either manipulatives. I think overall I would want to use hand-on manipulatives over virtual because I think they are easier to use.

Exploring Dilations February 20, 2010

I do not think this is an activity I would do with my present students, because they have not done much with graphing. In junior high I could see using this activity though. To present this with my students I think we would do it as a whole class. Going through each step together. After we make both figures I think we would start discussion. Where I would search out for the students to say how the two figures are similar and different. I would hope that some student might even say the shapes are similar.  I can imagine one saying it is the same shape but larger. When that is said we can talk about how all we did was multiple each point by 2.  I am sure this activity will require us to do more than just two figures. I would like to do some comparing and contrasting of previous things we have learned like congruence or translations. I wonder if some students will not make connects between the different points. I think some students will wonder “how this works” I think one way I will use to explain it is to ask them to think about multiplication and if you wanted a number to be twice a big you would multiple it by two or if you want to double your room size you would multiple it by 2. Then I would explain how by multiplying or points by 2 it was doing the same thing.

Learning Activity 5-B-1 February 20, 2010

It took me a little while to figure out this assignment. Once I did figure out what I was suppose to do I came to the conlcusion that this project can do well at teaching and reinforcing reflections. It seems so ovbioius that the shapes would be the same because you traced over them but for some students it will be beneficial to measure the sides and angles to prove they are the same. I think wehn I would ask students how the shape is the same a common response would be it is the same just in a different spot. It would be important to push the studnets to tell how they know it the same using size and shape.

Yay for snow… February 9, 2010

Even though this blog is mainly for Navigating through Geometry class I felt like making a post for the fun of it…this is because today was a snow day meaning that I could catch up on things I am behind with AND actually do some homework for this class before Friday night I know that most teachers are upset with snow days because they have to make them up in June, yes I will not enjoy that in the future…but as of right now my university does not require us to make up any snow days for our student teaching! Lucky me!

Categorizing Two Dimensional Figures February 9, 2010

As I was preparing my vocabulary list, searching online for photos and animations, I thought it would be fun for students to do the same thing, maybe even create a power point that is all about a certain 2-D figure (assigned one so that we would not have 24 squares). The students would then share their power points and at the end would we take each figure and decided how to separate them into categories using the diagram of Geodee’s. We could fill the chart out together or individually. One thing I think would be good is to have the actual shapes to look at while we fill out the chart.

Something else that would be good is to make posters for the figures that can be displayed in the room