Roberta's Blog

In your post briefly describe how you would introduce and conduct this activity with your class. Would you have them choose their own shapes to define or have a series of assignments where they define all two-dimensional shapes? Would you have them draw or construct the shapes?

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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What was the most valuable information/activity in this module? Why did you find it to be so valuable?

        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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These are sites that we have used in this course. If you have any others that you would recommend please post share.

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

Eggs in the Basket (http://www.folj.com/puzzles/)

It is difficult for me to tell you how I solved this problem because I have done it before in another situation. I do know that with students I would probably tell them if they are struggling that they should think outside of the box. Sometimes we tend to try and make situations harder to solve than they need to be.

Manhole Cover  (http://www.mycoted.com/Manhole_Covers)

The problem made me think more about shapes. At first I thought the circle due to the fact that it would be faster to take off and replace. Without corners the shape would fit easier, doesn’t require turning or aligning sides. I also thought about ease of moving, rolling. Once I looked at the answer I felt foolish for not even thinking about the lid falling into the hole.

I think the “Manhole Cover” question is at the analysis level. Analysis requires that the person uses problem solving skills to determine the answer or solution to the situation. I also see this problem as an application. In order to solve the problem you must apply information that is known about the shapes (circle and square). It can also be synthesis because a unique solution may be expressed in verbal form or with a physical object.

When presenting this problem to my students I would present it while we were in a geometry unit. I would make sure that we had already discussed the fact that the diagonal of a square is longer than the sides. Once I know that we have covered all of these concepts I would feel comfortable giving the problem to the students. I would give the students a few minutes to work on the problem individually and then I would allow them to partner up and discuss the problem. As the students are discussing the problem I would walk around to listen to their conversations. As I am listening to their conversations I would ask questions such as

1) Prove to me that the _________ is the best solution.

2)  Compare the traits/characteristics of the two shapes.

After an allotted amount of time I would ask for someone to share their response as long as they explain mathematically why their solution is correct. Once the solution has been given I would allow the other students to ask questions for their own clarification. If time allows I might ask them to 3) Create a list of other shapes that would and would not be appropriate for manhole covers.

1. Rotation

 Personal Definition – the object is moving around.  When teaching to students it is important that they understand that there is a set point in which the object is moving around. The point can be anywhere in/or the object or somewhere outside the object.

Definition from http://www.thefreedictionary.com/rotation – A transformation of a coordinate system in which the new axes have a specified angular displacement from their original position while the origin remains fixed.

Rotation

Image taken from http://mathworld.wolfram.com/Rotation.html

2. Translation

Personal Definition – slide an object with any rotation or flipping. When teaching students it is important they understand that the orientation of the object remains the same just the position of the object changes

Definition from http://www.thefreedictionary.com/translation – The changing of the coordinates of points to coordinates that are referred to new axes that are parallel to the old axes.

Translation

Image taken from http://mathworld.wolfram.com/Translation.html

3. Tangram

Personal Definition – seven puzzle pieces that are very specific in size and shape and when put together create a square.

Definition from http://www.thefreedictionary.com/ms –   A Chinese puzzle consisting of a square cut into five triangles, a square, and a rhomboid, to be reassembled into different figures

Tangrams

Images taken from http://mathworld.wolfram.com/Tangram.html

4. Concave

Personal Definition – a shape that has smooth outward edges

Definition from http://mathworld.wolfram.com/Concave.html– Curved like the inner surface of a sphere.

5. Convex

Personal Definition – a shape that has dips or indents

Definition from http://www.thefreedictionary.com/convex – Having a surface or boundary that curves or bulges outward, as the exterior of a sphere

ConcaveConvex

Images taken from http://mathworld.wolfram.com/Concave.html

6. Symmetrical

Personal Definition – a shape that can be divided in half and both halves match exactly. It can be divided vetically or horizontally.

Definition from http://www.thefreedictionary.com/ms –   having similarity in size, shape, and relative position of corresponding parts

AClipart - butterfly fantasy.  fotosearch - search  clipart, illustration,  drawings and vector  eps graphics images     B

A image taken from http://www.fotosearch.com/BDX238/bxp41131/ 

B image taken from http://gwydir.demon.co.uk/jo/symmetry/exsymm.htm

7. Nonsymmetrical

Personal Definition – a shape that cannot be divided so that each side matches the opposite side.

Definition from http://www.mathresources.com/products/mathresource/maa/nonsymmetric.html –   (of a figure or configuration) not identical with its own reflection in an axis of symmetry or center of symmetry

8. Regular Polygon

Personal Definition – a polygon in which all the sides and angles are congruent

Definition from http://www.mathopenref.com/polygonregular.html – A polygon that has all sides equal and all interior angles equal (This site has a great interactive object to demonstrate.)

 A     $\textstyle \parbox{5.84958pc}{\begin{center}\begin{xy} 0;<3pc,0pc>:<0pc,3pc>:: ... ... \ar@{-}c+(0.193096,1.52446);c+(0.974928,1.90097) \end{xy}heptagon\end{center}}$  A  $\textstyle \parbox{5.70633pc}{\begin{center}\begin{xy} 0;<3pc,0pc>:<0pc,3pc>:: ... ...11803) \ar@{-}c+(0.,1.11803);c+(0.951057,1.80902) \end{xy}pentagon\end{center}}$Photograph of a Stop Sign

A images from http://planetmath.org/encyclopedia/RegularPolygon.html

B image from http://geography.about.com/library/photos/blzz73.htm

9. Equilateral Triangle

Personal Definition – three sided polygon that has congruent sides and angles

Definition from http://en.wikipedia.org/wiki/Triangle –  In an equilateral triangle, all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

 EquilateralTriangle

 Image from http://mathworld.wolfram.com/EquilateralTriangle.html 

10. Obtuse triangle

Personal definition – a triangle with one angle greater than 90°

Definition from http://mathworld.wolfram.com/ObtuseTriangle.html – An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn’t be a triangle.)

ObtuseTriangle

Image from http://mathworld.wolfram.com/ObtuseTriangle.html

11. Quadrilateral

Personal definition – a four sided polygon

Definition from http://mathworld.wolfram.com/Quadrilateral.html – A quadrilateral, sometimes also known as a tetragon or quadrangle (Johnson 1929, p. 61) is a four-sided polygon. If not explicitly stated, all four polygon vertices are generally taken to lie in a plane.

 Quadrilateral

Images from http://mathworld.wolfram.com/Quadrilateral.html

12.  Trapezoid

Personal Definition – a four sided polygon (quadrilateral) with one side of parallel sides

Definition from http://www.mathopenref.com/trapezoid.html – A quadrilateral which has at least one pair of parallel sides. Every trapezoid has two bases, these are the parallel sides. The non-parallel sides are legs. Every trapezoid has two legs.

The Trapezoid

Image from http://id.mind.net/~zona/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html

13. Parallelogram

Personal Definition – a quadrilateral with opposite sides parallel and congruent

Definition from http://www.ies.co.jp/math/products/geo1/applets/para/para.html – A parallelogram is a quadrilateral that has two pairs of parallel sides.

A    B Parallelogram

A image from http://www.ies.co.jp/math/products/geo1/applets/para/para.html

B image from http://www.usefilm.com/image/1107115.html

14. Rhombus

Personal Definition – a four sides object with opposite sides are parallel and all sides are congruent

Definition from http://www.mathopenref.com/rhombus.html  – A quadrilateral with all four sides equal in length.

  B  File:Rhombus.svg

A image from http://www.dreamstime.com/search.php?srh_field=rhombus&s_st=new&s_sm=all&s_rsf=0&s_rst=7&s_mrg=1&s_ph=y&s_il=y&s_sl1=y&s_sl2=y&s_sl3=y&s_sl4=y&s_sl5=y&s_clc=y&s_clm=y&s_orp=y&s_ors=y&s_orl=y&s_orw=y&s_mrc1=y&s_mrc2=y&s_mrc3=y&s_mrc4=y&s_mrc5=y&pg=2

B image from http://en.wikipedia.org/wiki/Rhombus

15. Square

Personal Definition – four sided polygon with congruent sides, opposite sides parallel, and four right angles.

Definition from http://mathworld.wolfram.com/Square.html –  a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other

A Square  B IMG_2113.jpg 3/4 Inch Squares image by caroljt 

Image from http://mathworld.wolfram.com/Square.html

B image from http://media.photobucket.com/image/squares/caroljt/IMG_2113.jpg?=24

16. Pythagorean Theorem

Personal Definition – When the squares of the two legs of a right triangle equal the square of the hypthenuse.

Definition from http://mathworld.wolfram.com/PythagoreanTheorem.html  –  For a right triangle with legs a and b and hypotenuse c,  a^2+b^2=c^2.

PythagoreanTheoremFigure

Image taken from http://mathworld.wolfram.com/PythagoreanTheorem.html

17. Dilations

Personal Definition – When a shpae is enlarged or shrunk using a mulitplier

Definition from http://mathworld.wolfram.com/Dilation.html – A similarity transformation which transforms each line to a parallel line whose length is a fixed multiple of the length of the original line. The simplest dilation is therefore a translation, and any dilation that is not merely a translation is called a central dilation. Two triangles related by a central dilation are said to be perspective triangles because the lines joining corresponding vertices concur. A dilation corresponds to an expansion plus a translation.

Dilation

Image taken from http://mathworld.wolfram.com/Dilation.html

I want to tell you a little about myself. I come from a large family. I have 3 brothers and 3 sisters. All of us still live in Illinois so I am still able to see them a few times a year.

I have been married for over 25 years to the same person, he is my backbone. We have two wonderful grown sons who both live about 30 miles away. I finally got a daughter two years ago when my oldest son got married. Grandchildren are not a part of our lives right now but I am sure they will be in the upcoming years.

I teach in the same school district that I graduated from, it was one of my goals as a teenager. The name of the district where I teach is PORTA CUSD #202 in Petersburg, IL. It is a unit district in a rural farming community.

I am taking this course so that I can stay on top of what is going on in the geometry classroom and to help the teachers I work with stay attune to what is going on. Two things I would like to get out of this class are ways to help my fellow teachers strengthen the geometric abilities of their students and to find activities that those teachers can implement to involve transformations into their teaching.

Have a Great Day

Roberta

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  • Judy: Roberta, The Transformational Golf is a new site for me. I haven't seen this one and can't wait to share it with others. Many students struggle w
  • bkgeary: I like the idea of letting the student pick the material. Students never fail to amaze me with the great ideas that they come up with. I think that
  • joshbeals: I love puzzles too! I liked your comments about quilting patterns. It's awesome when we see math in the things we choose to do with our free time.