I would use the following website when implementing math journals in my classroom.  This site offers great strategies in getting started with math journaling.  It also gives you some math journal prompts to get you started.  http://math.about.com/library/weekly/aa123001a.htm

The following site gives additional writing prompts.  http://futureofmath.misterteacher.com/Writing%20Prompts.pdf

Math journal writing about problem solving prompts:

http://www.readwritethink.org/lesson_images/lesson820/JournalPrompts.pdf

Scholastic article Math Journals Boost Real Learning

http://content.scholastic.com/browse/article.jsp?id=3583

Provides links regarding different types of journals:

http://www.sdcoe.k12.ca.us/score/actbank/sjournal.htm

The tetrahedron is composed of 4 equilateral triangles.  When truncated those triangles became 4 congruent hexagons and 4 new congruent, equilateral triangles were formed.

The second shape I started with was the Platonic Solid – Cube.  When truncated it became the Archimedean Solid – truncated cube.

The cube is composed of 6 congruent squares.  When truncated those 6 squares became 6 congruent octagons and 8 new congruent, equilateral triangles were formed.

Middle school students can perform this task just as I did.  I would enlarge the templates so that they do not have to work with such small figures, but overall, I think that they could manage just fine. 

Most students will enjoy this activity.  You can allow them to decorate their solids as they wish.  Some students might get frustrated folding and cutting straight lines, but I’m certain that other students will be willing to offer a helping hand to those in need.  Any time that we can get away from book work and incorporate manipulatives in our classrooms, we will be more likely to captivate the attention of our students. 

Level 0:  Concrete is most similar to Bloom’s knowledge.  The student can identify and compare geometric shapes.  Comparing shapes at this level is not necessarily at Bloom’s understanding level.  I think it is more of a visual comparison.

Level 1:  Analysis is similar to Bloom’s understanding level.  This is where the student could examine the shape and classify it by its properties.  Analysis could also be compared to Bloom’s apply level.  It’s at this point that students can apply formulas and compute areas.  They could also look at different shapes and compare their perimeters and area.

Level 2:  Informal deduction corresponds to Bloom’s analysis level.  Students begin to analyze and differentiate properties of shapes and apply them to one another.

Level 3:  Deduction must be at Bloom’s synthesis level.  Using deduction requires a thorough understanding of geometric principles and combining them in such a way to formulate theorems.

Level 4:  Rigor is at the highest level of thinking so I would compare it to Bloom’s evaluation level which is also is the top of the scale. 

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

I think van Hiele’s levels relate closely to Vygotsky’s zone of proximal learning.  Students can not learn something unless they are ready for that stage of learning.  van Hiele’s levels demonstrate this as well.  We can not ask students to perform at a level higher than they are currently on.  As teachers, we must evaluate our students and where they fall within this framework.  Activities must be presented to them at their levels.  As the previous article showed, one lesson may cover more than one level.  Students will simply think at different levels and be able to apply concepts in a variety of ways. 

3.  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.

Use the tiles to configure two different shapes with the same perimeter.  Do you think that their areas will be equivalent as well?  Test your prediction.  Compare the two areas? 

Construct a 3, 4, and 5 triangle.  Using each of the sides construct squares.  Evaluate a² + b² = c².  Now using the same 3, 4, and 5 triangle construct triangles from each of the sides.  Predict whether the same holds true?  Does a(triangle) + b(triangle) = c(triangle).  Discuss why or why not?

 I spent a lot of time working on the first figure.  After a many attempts, I found out that it would work if I rotated the square.  Once, I did this, then the other pieces fell into place.  The second one was fairly easy.  I placed the square in the upper right corner, and I could then see two rectangles that I needed to make using the 4 triangle pieces.

 Puzzle #2

The first one came to me easily.  I aligned the triangles around the four sides and rotated the square to fit in the middle.  The second figure was a little more difficult, but it didn’t take long to figure out that I needed to create two rectangle with the 4 triangle pieces leaving two square areas.  A small one and a larger one.

      This type of exercise would be useful to students because they could compare similar shapes and congruent shapes.  They will see how different shapes fit together to form new shapes.  Students can discover that the area of one particular shape might be equivalent to the area of a different shape.  This also allows the students to think outside of the box.  They will learn to look at the tans from different aspects by rotating and flipping them.  This exercise is also a great problem solving exercise. 

     I’m not sure that the hands-on manipulatives or the virtual manipulatives are better than the other one.  However, the virtual manipulatives might be easier to work with since they will stay in place and not move on you.  When demonstrating to the class, the virtual manipulatives would work better since you can link your computer screen to an overhead projector.  On the other hand, I don’t know how many math classrooms have more than a couple of computers in them.  It might be easier just to use the hands-on manipulatives.

     On the legs of the yellow, mid-size triangle, it took 4 triangular units to form each of the squares.  On the hypotenuse of the mid-size triangle it took 8 of the triangular pieces to form a perfect square.  Although using a different size right triangle, it still took twice as many pieces to form the square on the hypotenuse as it did on either leg.

     On the legs of the large, blue triangle, it takes 8 triangular pieces to form a square.  On the hypotenuse of the large triangle, I used all 7 pieces of the tangram to form a square.  Based upon data from the small and medium triangle, we can assume that it would take 16 small triangular pieces to form a square along the hypotenuse.

     In all right triangles, we notice that it takes the square of A + the square of B to = the square of C(hypotenuse).  Hence, A² + B² = C ².  Students have just discovered the Pythagorean Theorem!
     This is a great introduction to square roots.  This hands on activity walks the students step-by-step through the process of really understanding square roots. 
     I would present this activity as it was presented to us.  I’d have them follow the 6 steps.  Students would be placed in pairs and each pair could work together to unravel the Pythagorean Theorem.  We would use the Tangram Blackline Master, make copies, color and cut out the shapes.  I would use heavyweight paper so the students wouldn’t get as frustrated as I did when placing the pieces together.  They kept moving on me!

We would calculate the side lengths and compare the two.  Calculate the perimeter and compare the two.  We could discuss scale factor and how to determine it.  I would ask them to compare the original coordinates to the coordinates of the dilated figure.  Ask them what correlation they found between the two and how this correlation affects the dilated figure.  If you multiplied the coordinates by 2, then would the dilation enlarge or shrink?  If you multiplied the coordinates by 1/2, then would the dilation enlarge or shrink?  Why? 

The students might ask how to plot a dilated figure.  I would encourage them to investigate what happens when they multiply all coordinates by the same number.  Maybe, some of the more inquisitive students would ask about the relationship between the coordinates of the original figure and the dilated figure.  I would encourage them to investigate and discover the connection.

This particular activity will be a good introduction to the new concept of reflection.  I don’t think that it will be too difficult, but yet it will challenge the students, especially the figure in C.  As with many endeavors, there no better way to learn than by doing and that is just what this exercise is about.

After the students have completed this activity, they could check their work by folding their papers along the x-axis and y-axis.  Each reflection should “lay” on top of the original figure, preimage.

The students should be able to recognize that when images are reflected across the x-axis they appear to be upside down but with the same original characteristics.  When images are reflected across the y-axis they also have the same attributes but what once was on the left is now on the right and vice versa.

Concave- when drawing diagonals on a concave figure, at least one is outside of the shape

 Convex – when drawing diagonals on a convex figure, all of them are inside of the shape

  Visit this websit to see pictures:

http://www.mathsisfun.com/geometry/polygons.html

Polygon – a shape that is closed and has straight sides

Regular Polygon – a shape that has sides of the same length and equal angle measurements

Equilateral Triangle – triangle whose sides are of equal length

 Isosceles Triangle – triangle having two sides of equal length

 Scalene Triangle – triangle having no sides of equal length

 Right Triangle – has one angle measuring 90°

 Obtuse Triangle – has one angle measuring more than 90°

 Acute Triangle – has 3 angles all measuring less than 90°

  For pictures of  triangles visit site:

 http://www.mathsisfun.com/triangle.html

Parallelogram – is a quadrilateral having two pairs of parallel sides

Rhombus – is a parallelogram with 4 sides of equal length

Rectangle – is a parallelogram that has four right angles; opposite sides are of equal length

 Square – is a parallelogram with 4 sides of equal length and 4 right angles

Visit this site for pictures:

http://www.mathsisfun.com/quadrilaterals.html

How could it be that an egg is left in the basket and three people each take an egg?

What does it mean to say that each person takes an egg?

Could all three people take an egg leaving one in the basket?

Act it out! 

Must each person remove an egg from the basket?

Manholes

 Why is it better to have round manhole covers than square ones?

     I believe the manhole question is at the level of evaluation.  To determine a feasible explanation for round covers the student may have to appraise/assess why they would be round as opposed to square.  On the other hand, this question may be at the synthesis level.  Students would have to combine information they know regarding  squares and circles to develop reasoning why round would be better than square.  They would have to discuss their reasoning and logic behind round manhole covers instead of square ones. 

Name:  Angela Jackson

Concept/Subject: 

            Mathematical education does not focus on developing logical and lateral thinking skills enough.  It’s been my experience, that our math classes are just barely touching on these issues.  Talk about preparing our students for the real world, what better education could there be than to develop these skills? 

            I’m currently taking an on-line problem solving course.  Throughout the coursework I’ve tossed around the idea of incorporating similar, more appropriate problem sets in my math classroom when I have one.  I find the problems challenging, but also very fun.  Sometimes I even become obsessed with finding the solution.  They challenge me to think in ways that don’t come naturally.  Therefore, I believe that my students can also greatly benefit from such an experience.