Shared Resources – resources and websites from other classmates.

These are resources and websites on topics that other classmates researched.

Transformations:

Below are a list of different resources that could be used with a Transformation Unit. This year I have used a combination of a math series called Digits, Task cards created by my district, and some ideas from my old math series.

Book: Mathematics Station Activities fro Common Core State Standards (gr.8) pgs.124-129

This has wonderful hands on labs that allow the students to explore transformations. I would use this for either assessment of understanding or further exploration of the concept.

Websites:

www.learnzillion.com– Wonderful tutorials and power points to help with explanations. Great to use as a review tool or for those struggling students.

www.softschools.com/math/geometry/transformations/quiz– Great basic quiz to check for basic understanding or pre-test on transformations.

www.illustrativemathematics.org/standards/k8– Higher order of thinking tasks for students on transformations. (Aligned with the Common Core)

http://jmathpage.com/middleschoolmath/jmsmcoordinatestransformations.html– Great games and virtual manipulatives for students or teacher to use in classroom.

www.epd86.org/epweblinks/8-math-Geometry.html#1 – Has a variety of websites and links on Transformations for a teacher to use inside the classroom.

Software

GeoGebra: Is a great virtual tool for students to use and see different types of Transformations.

Pythagorean Theorem:

1.  www. nlvm.usu.edu    The National Library of Virtual Manipulatives  The two puzzle activity found in 6-8 Grade Geometry allow students to explore Pythagorean and spacial relationships.

2.  http://www.StudyIsland.com Study Island (custom of PA)                                                Study Island gives lessons, rigorous practice, and remediation on all eligible content in the PA math curriculum.

3.  http://www.ixl.com/math/geometry/pythagorean-theorem                                                         This sight gives real world application problems.  Its is a good homework and skill building sight.

4.  http://www.wikihow.com/Use-The-Pythagorean-Theorem                                                  

Non-interactive sight shows 4 applications of the P.T.

A.  On Right Triangles

B.  On Parts of the Distance Formula                                                                                                 C.  On Non-Right Triangles                                                                                                                    D.  On Vector Applications

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8-B-2 Exploring Additional Resources

I am really excited about implementing journals into my classroom.  Since I don’t have much experience in it, my concern is being able to pull the information out of students.  Here are some sites I found for prompts to get started.

math.about.com/library/weekly/aa123001a.htm

This is a nice Q & A about journals and how you can apply it to the classroom.

thecornerstoneforteachers.com/free-resources/math/mathjournals

http://www.scholastic.com/teachers/lesson-plan/math-journals-boost-real-learning

http://www.glencoe.com/sec/teachingtoday/subject/int_writing_math.phtml

Believe it or not – I have found some fun ideas for math journals on Pinterest.

pinterest.com/mctalia/mathjournals/

A book to share is Writing to Learn Mathematics, by Joan Countryman.  I used this in another course.  It is great resource on the different strategies that you can take in regards to journaling.

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7-B-2 Archimedean Solids

Archimedean Solids

For the first shape, I started out with the Platonic solid – Tetrahedron.

Kris ' bday '13 (50) 008

When truncated, it became the Archimedean solid – Truncated Tetrahedron.

Kris ' bday '13 (50) 011

For the second shape, I went with the Platonic Solid – Octahedron.

Kris ' bday '13 (50) 017

When truncated, it became the Archimedean Solid – Truncated Octahedron

Kris ' bday '13 (50) 015

This would be a nice continuation of the Learning Activity 7-B-1.  After going over nets, vertices, faces, edges, and Euler’s Formula, Archimedean Solids are a natural progression when introducing the process of truncating a Platonic Solid.  If you used the same shapes from the previous activity, they can compare and contrast the nets of Platonic and Archimedean Solids.  After constucting the solids, again compare and contrast the 3-dimensional objects.

I do think the students will enjoy this activity.  The ones that enjoy the art side of mathematics will be especially drawn to it.  Some may get frustrated cutting, folding, and assembling, but with peer help they should be able at acheive this ok.  For the students with learning disabilities, they can stick with the basic solids like the Cube and Tetrahedron.  They also might benefit by working with a partner. The gifted/talented students can can assemble a more detailed Archimedean Solid like the Dodecahedron or the Icosahedron.

 

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6-C-2: Higher Level Thinking for the Pythagorean Theorem

6-C-2: Higher Level Thinking for the Pythagorean Theorem

Level 0: Concrete, identifies, names, compares, and operates on geometric figures. This level is most comparable to Bloom’s Knowledge and Comprehension level.  Students know the basics about geometric figures.   They can define the appropriate material at the time, but cannot use ideas associated with the subject to relate them other ideas.   They can work on assigned problems and example what they did.   

Level 1: Analysis, analyzes figures in terms of their attributes and relationships among attributes and discovers properties and rules through observation. This is closely comparable to Bloom’s Application Level.  Students can recognize what methods to use and use the method to solve problems.  They can apply ideas and concepts to new situations.

Level 2: 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. This level is comparable to Bloom’s Analysis level.  Students can explain why the solution process works.

Level 3:  Deduction, proves theorems deductively and understands the structure of the geometric system.  This level is comparable to Bloom’s Synthesis level.   Students recognize new problems and develop new tools to solve them. They can create their own plans, models, and/or hypotheses for constructing solutions to problems.

Level 4: Rigor, establishes theorems in different postulational systems and compares and analyzes the systems.  This level is comparable to the evaluation level.  Students can create a variety of ways to solve the problem and then, based on established criteria, select the solution best suited for the problem.    Students will have the ability to appreciate the value of ideas and concepts using appropriate criteria.

 

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

Since the Van Hiele model is such a natural process in mathematical maturity, the teacher will need to recognize the level the student is currently at.  The teacher will need to pay attention to questioning that will help the student understand and make a natural progression from one level to the next level.  The teacher will need to understand that not every student will be at the same level, but can use Bloom’s questioning to fulfill all levels so all students can be successful.

 

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

Analysis:

Illustrate the ways to increase the area, but the perimeter stays the same.

Compare the different perimeters you can have with an area of 16 units².  Compare and contrast the area and perimeter.

Synthesis:

Create another figure with the same area and perimeter as the original figure.

Create an object with an area of 12 units².  What is the greatest possible and least possible perimeter for an object of 12 units²

Evaluation:

Access if you can arrange the tiles to form an odd-numbered perimeter.  Explain if it’s possible or not possible.

 

Posted in Geometry Problems, Reflection | 1 Comment

6-A-1: Tangrams Part 1

Tangrams Part 1

Step 1:

First I traced the small triangle.  The relationship between the angles and sides are identified.

tans 001

Step 2:

When using the small red triangle, I formed perfect squares along each side of the red triangle using small red triangles.  For each of the two legs it took 2 small red triangles to form a perfect square, and for the hypotenuse it took 4 small red triangles.  If you add the triangles formed on the legs (2+2=4 red triangles), it is equal to the number of triangles formed on the hypotenuse (4 red triangles).

tangrams 007

Step 3:

I then used the medium yellow triangle.  I again formed perfect squares on each side of the yellow triangle using small red triangles.  For each of the two legs it took 4 small red triangles, and for the hypotenuse it took 8 small red triangles. If you add the triangles formed on the legs (4+4=8 red triangles), it is equal to the trangles formed on the hypotenuse (8 red triangles).

tangrams 008

Step 4:

I repeated this again using the large blue triangle.  I formed perfect squares on each side of the blue triangle using small red triangles.  For each of the legs it took 8 small red triangles, and for the hypotenuse it took 16 small red triangles.  If you add the triangles formed on the legs (8+8=16 red triangles), it is equal to the triangles formed on the hypotenuse (16 red triangles).

tangrams 006

  • “Discuss the relationship between the areas of the squares along each leg of the right triangle to the area of the square along the hypotenuse.” (from Step 5 from “The Pythagorean Theorem with Tangrams” – linked above)

All the pictures above show the relationship between the squares of the legs and the square of the hypotenuse.  The area of the square from leg ‘a’ + the area of the square from leg ‘b’ = the area of the square from the hypotenuse ‘c’.  If you use the area formula A = l × w, the area for the square with side length ‘a’ is a × a, or .  The area for the square with side length ‘b’ is b × b, or.  The area for the square with the side length ‘c’ (hypotenuse) is c × c, or .  So they would discover the formula a² + b² = c², which is Pythagorean Theorem.

  • What connection can students make between the numbers of shapes needed to create the sides of the triangle?

The students can experiment with the various shapes and make a connection between them.  They will make the connection that 1 medium yellow triangle is the same as 2 small red triangles.  Or 1 large blue triangle is the same as 2 medium yellow triangles (or 4 small red triangles).  Experimenting with this, they can create other models that would represent the same relationship and compare it to the models above working with only small red triangles.

  • In your own words explain why this activity is a good introduction to square roots and rational numbers.

This would be a good activity to lead into square roots and rational numbers.  After going over finding squares, the natural step is to explain square roots.  Especially when finding a missing ‘leg’ variable for the Pythagorean Theorem, they will get the connection to finding the square root and how value is associated to a rational number.  It would be a good time to review the basics of rational numbers.

  • How would you present this activity to your students?

I think this is a wonderful activity to do with students.  So often, the Pythagorean Theorem is taught and the students memorize the forumla but they don’t really know what the fomula means or how it was found.  I would first do this activity like the steps above and have them compare their findings with a partner. (Did the partner come up with the solution?  Was the same, or different when comparing with a partner?) After the developing the squares with the small triangle, have them predict what would happen with the medium and the large triange before continuing.  (Were the predictions true?)  I would have them experiment and challenge them to use different shapes to form the squares.  (What can we conclude about the different shapes in relation to the small red squares?)  I would also have them journal their findings.

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6-A-3 Pythagorean Puzzles

For the first puzzle, I noticed the length of the white square was ‘c’ and the length of the hypotenuse of the red triangle was ‘c’.  So I rotated each triangle from the previous one to line up side ‘c’ of the triangle to line up with side ‘c’ of the figure. What was left was a white area in the middle that was the exact shape for the blue square to line up with now gaps or overlaps. For the shape on the right, I noticed the veritical side ‘b’ on the white figure, so I lined up the red triangle with the shorter leg ‘b’ with the side ‘b’ in the figure.  I then place the 2nd red triangle in the figure to line up the hypotunuse of the first triangle to form a rectangle.  The length of the bottom of the figure is ‘a+b’, and since I already had a length of ‘a’ lined up on the bottom, I needed to rotate a 3rd red triangle around to line up side ‘b’ with the bottom of the figure. The 4th red triangle, I lined up the hypotenuse with the 3rd to form another rectangle.   What was left was a white square where I knew one side was ‘a-b’, so the blue square fit perfectly.

For the second puzzle, the figure on the left,  I used the same concept as the first figure in the previous puzzle.  I noticed the sides were ‘a+b’, so I used side ‘a’ of one triangle and side ‘b’ of another triangle lined them up with a side of the white square, keeping the right angle of the trangle lined up with the right angle of the white figure.  I rotated the remaining triangles so they lined up with the figure and the right angles lined up with the figure.  What was left was the square shape in the middle, so I rotated the green square it fit.  For the figure on the right, the sides were a+b, so I first put the large blue square in the upper left corner.  Then I took the 2 red triangles and placed them together to form a rectangle and lined it up with an exposed side of the large blue square.  I then took the last 2 red triangles, formed a rectange and lined it up with the other exposed side of the large blue square.  The space that was left  needed a square with the dimentions of ‘b’, which gave the small blue square, with side length ‘b’, a place in the figure.

I didn’t find any of the puzzles to be too difficult because I was looking at the relationships given to me in the pieces and the figures.  If I didn’t have the sides labeled and completing this at random, I would find it more difficult.

I would prefer the hands-on method because I like the idea of physically moving the pieces around, but vitual manipulatives are easier to manipulate since the stay in place better and won’t move around on you.  It really depends on the situation which you have in your classroom.  Depending on what kind of technology you have available would determine which would be better to demonstrate to the class.  If you don’t have many computers available, the hands-on approach would work too.

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Geometry 5-D-1: Exploring Dilations

Individually, I would have the students plot the original triangle on their graph, then multiply the coordinates by 2, plot the new points.   In partners, use the worksheet to talk about the similarities and differences of what they just did. Then I, or a student, could demonstrate what we should be seeing on an overhead for the class.   I would ask questions to compare how the coordinating points relate to the origin on the coordinate graph.  Then do the same thing with the pentagon, individually plot the points for the original figure, multiply each point by ½, then with a partner finish the questions on the worksheet.  As a class, have a different student or group present the graphing on an overhead. Again, look at how it correlates to the origin.

The following questions would be good classroom discussion and/or journal entries.  I would ask questions to compare the original coordinates to the coordinates to the dilated figures.  Did the shapes change?  How are they the same/different?     What would happen if we multiplied the original figure by 3, or ¼, will your figures enlarge or shrink?  Explain your reasoning.  How does if effect the relationship to the origin?  What do you notice about the distance from the origin to the 2 coordinating points of the figures?

At first, students might assume side length and angle measure will change in dilation.  But as you start comparing, you can explore the idea of similarity.  The students can cut out the figures to compare, side lengths change but the angle measure stays the same.  This is also a good way to confirm that no matter what size of triangle, pentagon, or any polygon, the angle sum of that particular figure will always be the same.  Students might also ask how small or large you can dilate a figure.  They can experiment with numbers with a partner until they come up with an answer, infinity both ways,  and be able to explain their reasoning.

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