Jenni's math blog

Transformations:

www.nctm.org A great activity on transformations of geometric figures is presented in “A Treasure Hunt: Reflecting, translating, and Rotating Points on a Coordinate Map.”  What a fun way to bring transformations to a real life situation.  Who doesn’t like a treasure hunt.

 http://illuminations.nctm.org/LessonDetail.aspx?id=U104 A six lesson unit on transformations.  The culminating activity is a paper quilt.  I would use many parts of this unit in teaching students patterns, transformations and how transformations are used daily in making quilt patterns

www.mathsnet.net/transform/index.html  This is a great interactive site that allows the students many options.  In working with transformations, students have the option to observe movements, learn more about transformations, explore through a manipulative template, and test you knowledge through constructing your own transformations.  This activity would work well in a differentiated lesson that could meet the readiness levels of all your students while still engaging them in a fun technology based activity.

Creating a truncated Tetrahedron

 The Tetrahedron started with 4 sides which were all congruent equilateral triangles.  I cut off each of the three vertices.  The original sides are now hexagons.  The new sides of the truncated figure are 3 triangles.

 Creating a truncated Cube

 The Cube started with 6 sides which were all congruent squares.  I cut off each of the 8 vertices.  The original sides are now octagons.  The new sides of the truncated figure are 8 triangles.

 Setting up the activity for students

 This would be a great follow up activity to discovering and creating 3-dimensional figures using nets.  During the previous activity, a discovery of the number of sides and vertices as well as the shape of the faces would have taken place.  This would lead into a discussion of non platonic solids and Archimedean solids. 

Students would be given a choice in which one of the 3-dimensional figures they created in the first activity they would like to truncate for this activity.  I would model how to cut the corners, being careful to make straight cuts, and not ‘pinch’ the corners during cutting.  I would have them describe the new sides and their shapes and how the number new sides relate to the number of vertices of the original figure.

 Would the students enjoy this?

I believe they would enjoy anything that deals with building and working with manipulative.  They may become frustrated when the corners do not cut quite straight.  I may decide to use Styrofoam figures instead and have students cut off the corners or even use cardstock and box cutters for more precise cutting.

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

• Level 0:  Knowledge and Comprehension- van Hiele and Blooms both have students identifying, and comparing information not applying.
• Level 1:  Application and Analysis – Students analyze the information they have learned.
• Level 2: Synthesis and evaluation – students develop arguments and judge the information.
• Level 3:  Evaluation – students make predictions and support their predictions with facts.
• Level 4 : no Blooms level this high – this level of van Hiele has students creating their own theorems where as the highest level of Blooms only has students evaluating and judging existing information.

1. Answer the question asked in the article: “How can you use the van Hiele levels to help students learn mathematics?”
   • The most important thing is for teachers to realized where students are by recognizing the students behavior at each level.  By knowing this, we can develop appropriate lessons so we are not talking below or above their level.
2. 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.
   • Evaluation: Can you select one tile to remove from the original figure that will change the area but not the perimeter of the figure? 
   • Synthesis: Could you rearrange the tiles of the original figure so that the perimeter and the area remain the same but the shape changes?
   • Analysis: What is the greatest perimeter you can form by adding only one more tile to the original figure.

Puzzle 1 -   The square took me a few tries to fill because I was not turning the square.  I was putting the square in straight and trying to move the triangles around it.  Once I realized I should turn the square at an angle, I easily fit the rest of the pieces in.   The Hexagon was easier because I saw right away that two of the triangles could be put together to form a rectangle to fill in the bottom right area this left another rectangle area and the square filled the rest.  This would be a useful tool in showing students that two figures can have the same area even if their shape is different.  I especially like this activity as a problem solving activity.

Puzzle 2 – I filled the first square the same way I filled the square in puzzle 1.  For the second square I first tried turning the large square, but then I realized that that would not leave any shape that could be filled with the remaining pieces.  I put the large square in the corner and noticed that I again had rectangular areas to fill.  I put two triangles together in each of the area to fill in the rectangular space and was left with a small square area.   What I did not do as I am sure many of my students wouldn’t do also is use the information given to solve.  It would have been much easier if I had used the labels to fill in the squares.  This would be a great lesson in using the information given as well as problem solving.   I know this activity was supposed to be used with the Pythagorean theorem  but I think I would use it differently.

I like both hands-on and virtual manipulative.  With tangrams I think it would be important to have students work with them hands-on first to really understand the concept of creating shapes, then the virtual manipulative could be used to reinforce more complex concepts.

The hypotenuse is labeled C and each of the legs are labeled A and B.  The area of the square formed by the tangrams on leg A is 4 and the area of the square formed by the tangrams on leg B is 4.  The area of the square formed by the tangrams on leg C is 8.  We could say that the area of square A plus the area of square B is equal to the are of square C, but this would not help us with every size of triangle without forming squares.  Since we know that area is found by multiplying the length by the width, we can say that the area of Square A was found by multiplying the sides A times A or A² and the area of square B would be B times B or B² and the area of square C would be C times C or C².  Since we already established that the area of square A plus the area of square B equals the area of square C.  We can then conclude that A² plus B² equals C².

Students will be able to see that it takes twice as many triangles to form the square for C as it does for A or B.  They will further see that the number of triangles for A plus the number of triangles for B equals the number of triangles for C.

This activity would be a good introduction to square roots in that students will be able to see that the area of the square is found by multiplying side by side, but that square root finds what times itself is being multiplied.  They will see that if the area of the square on A is 4 and 2 time 2 equals 4 so the leg must be 2 units.  Side B would be the same.  They would also be able to see that the area of square C is 8, we could use this to introduce how to find what times what would equal 8 and that this is the square root of 8.  Since the square root of 8 is 2.828 (rounded) they can learn about decimals too.

I would introduce this activity much like it was set up for this module.  Students would be given a set of tangrams to work with and we would work through the process together.  The questions related to the Pythagorean Theorem would be address as well though classroom discussion

I have to say that I was very excited in reading this activity.  Before Spring Break, I always have my students participate in what I call “The Great Cartoon Blow-up” where they are given a picture of a cartoon character and they draw it enlarged on grid paper.  We have always just changed the size of the grid to do this activity.  During the activity we review similarity, which was in a previous unit, but they just have fun doing the drawing that the lesson itself seems to be lost.

I plan to use this activity to recreate the cartoon project.  After our lesson on similar figures I will use the activity described in the book.  I will add additional shapes and have the students perform the dialations by muliplying and dividing by different numbers.

When we begin our pre-spring break project, rather than having students just use larger grids, I will have them mark the points of the pictures (I will have to come up with different pictures) and have them choose what scale they want to use and muliply by that number to plot their new points. 

I think the students would have questions regarding “how” this works.  We would discuss scale drawings and how they are just smaller exact replicas of the real thing.  I would prompt them by asking what we would need to muliply a figure by to get it to be twice, or three times, or 1/2 the original size.  Perhaps asking them what they think a blue print of a house was muliplied by to get the actual house built.

This was a simple but fun activity that could definately be used in the classroom.  I think I would use this activity in the beginning of a unit.  I would introduce reflections using mirrors and images.  I would then let the students work on this worksheet.  I would not have explained the process of reflections yet.  After students worked on this activity I we would discuss thier processes and we would come up with our own strategies for drawing reflections.  One thing I like about teaching reflections is the reinforcement of the x and y axis.

If I did this activity with my class I would definately give the students a choice in which shapes they would define.  I would present them with a list of shapes that we would be working on and have them pick 2-3 to complete.  I would have them write the defintion in their own words first, then look it up in their textbook or a dictionary.  I would encourage them to draw a picture of their shape and to label each of the parts.  We would then make a class list of the entries to share.