The Math is Fun site has a nice applet on the front page showing that the area of the two squares adjacent to the legs of a right triangle are equal in area to to the square adjacent to the hypotenuse, and it explains it in writing, too, helping visual and verbal linguistic learners understand the concept behind Pythagoras’ Theorem.  This page walks students through the visual proof, into the definition, through an algebraic proof, and adds two additional proofs that students can do using paper cutouts of the squares.  This would make an excellent introductory lesson for the students, utilizing concrete and abstract activities to gain a greater understanding of Pythagoras’ Theorem.

In the NCTM Illuminations site, students have the opportunity to watch an applet demonstrating Pythagoras’ theorem.   This applet offers a different proof of the theorem, which is attributed to Bhaskara, I a mathematician who lived in the 12^th century.   This would give students an opportunity to look at the theorem from a different perspective, and teach them that there is more than one way to offer proof that .

The University of Minnesota has an excellent site that includes real-world examples of problem solving using Pythagoras’ Theorem.   It provides “examples and contexts in which this invaluable theorem influences and can be involved in our everyday affairs.”  One problem involves a hot air balloon ride where three friends are in a ticklish situation.  They are in the air at right angles to each other (according to the diagram).  One is running low on fuel.  The other needs to decide if he has enough fuel to save his friend.  Students will use the information in this puzzle to determine a solution, which can then be checked.  The other puzzle on this page involves builders using the theorem to determine if their building has square corners.  This is a standard use for the theorem, put into action every day by builders.  This puzzle gives students enough information to solve, and then they can check their solutions on another page.  Both of these examples offer a terrific way to apply the theorem to real-world situations and would make wonderful follow-up lessons.

A third site that offers hands-on experience for middle school students who are studying the Pythagorean Theorem is this Pythagorean Theorem Webquest. This page offers a web quest for pre-algebra students in seventh or eighth grade.  Students can create their own scenario for the problem which involves traveling the shortest distance to your home. Students use the information on the website to find out about Pythagoras, then move into demonstrating proofs of their problem, giving them practice in applying the theorem.  The problem will appeal to students as it is directly applicable to their lives.

While there are many websites available on the Internet. The above set provides a sample progression.  When students can follow the progression at their own pace, learning is deeper, and the resources help the student understand the information that is available at their fingertips.

In my study of these figures, I chose to create the Platonic solids of tetrahedron and an octahedron.  When truncated, these shapes become the Archimedean solid of truncated tetrahedron and a truncated octahedron.  As I created the figures from paper “nets”, I realized how important it is for students to create such figures.  This hands on experience helps the student understand the figures more deeply.  As you cut and fold, the number of edges, faces and vertices comes to life.

In presenting this to my students, I would offer them a variety of nets with which to work.  Students would be requested to select the figure of their choice, and a truncated version of that same shape.  The data from each figure would be collected on a class chart, detailing the name of the figure, and the number of edges, faces and vertices on each.  The information would then become the focus of a brain storming session, looking for patterns among the data.

The beauty of the figures would be celebrated by creating a public display.  Hanging the figures using thread or fishing string would help others see the shapes clearly.  Using brightly colored paper or patterned paper would highlight the figures even more.  The display would not only please its audience, but also create pride in the students, and increase their interest in Platonic and Archimedean solids.

In Blooms, the second level is comprehension, earmarked by being able to summarize information.  In van Hiele, the second level is Analysis, where students are able to determine relationships between ideas. These correlate because you have to be able to analyze what is important before you can comprehend it. The next level in Bloom’s is Comprehension.  In this level, students can work with a new, abstract idea by connecting it with past knowledge.  In van Heiles next level, Informal Deduction, students have a working, common sense knowledge of geometry.  They may not be able to formalize their learning, demonstrating their understanding with common rather than specific terminology. They are, in effect, using what they know to understand more complex ideas.

Bloom’s describes a level called “Analysis”, in which students are able to break down the components of a concept or idea and show how the parts of the idea are related.  This would correspond to van Hiele’s level of Deduction, where students are proving ideas deductively. Finally, Bloom’s lists the evaluation level, where students demonstrate an ability to make informed evaluations based on self-developed criteria. The top van Hiele level is Rigor, where students are able to maximize their geometric understanding by comparing and analyzing the systems used to describe the processes used. The correlation here is that students are understanding the parts of the concept and are able to analyze and justify their decisions.

Teachers of geometry would want to be highly familiar with both systems in order to guide their students to a high level of competence and facility with geometric concepts.  They can use the van Hiele levels in helping students of different ability succeed.  Keys to this are encouraging students to work together in problem solving; designing questions to help students understand the next level; and most importantly, design educational tasks that move students through the levels.  Lessons plans should include the concrete stage, move into determining relationships, to having a working knowledge of geometry, to being able to express their ideas in formal geometric language, to studying geometry with rigor, and finally seeking to apply and prove theorem.

It is important for a teacher to have predetermined questions to encourage student growth.  After reading the Perimeter and Area Using the van Hiele Model, by Carol E. Malloy, the following  questions were developed that will assist students in moving to the higher levels of van Hiele:  Can you compare the different strategies for adding to the perimeter?  What is your analysis of the affect of adding a block to a corner?  To a side?  What criteria would you use to evaluate your process through this activity?  (could be asked before or after the activity.)  How can you verify your response?

The students will determine that the small triangle in the tangram set is one square unit for the purposes of this activity.  The area of the smallest triangle is 1/16 the area of the whole tangram square.  Students will build squares adjacent to each side of the three different size triangles in the tangram set.  (see photos- in each photo the white triangle is the central focus, with the adjacent squares in color.)  Students will then determine the area in units of each square. (see table)  Students will discover that in each case, the legs of the small square, when squared and added to the square of the square adjacent to the other leg, will equal the square adjacent to the hypotenuse.  This is expressed by a² + b² = c².

The number of shapes needed to create the squares that lie adjacent to the sides of the triangle are as follows:

This is an excellent introduction to square roots and rational numbers because it is hands on, and can be completed with both real and e-manipulatives, making it accessible to all students.  The numbers are small and even, removing much of the mathematics anxiety that can go along with understanding higher level math.  Students will discover the relationship between the three squares by manipulating and counting the units from each side, then compute to discover that the area of the two “leg squares” add up to the area of the “hypotenuse square”.  They will realize that this means that there is also a relationship between the sides of the right triangle.  A discussion of rational and irrational numbers would be appropriate at this point in the lesson, as the sides of the triangles may be irrational, such as √2.

I would present this activity to my students by having them follow the above listed steps.  Prior to this lesson the students would create their own set of tangrams.  This will engage them and help them internalize the relationship between the seven pieces of the puzzle.  Students would require a working knowledge of many other geometric terms.  This is particularly necessary as students write to explain each step.

Both puzzles were solved in a similar manner. Playing with the pieces is one good way to solve spatial problems.  This is a benefit of using a computer program.  In using an online resource in place of real manipulatives is a great benefit in mathematics.  The manipulatives do not have to be purchased by the school, or stored in the school building.  They do not have to be acquired in advance.  Students can access the manipulatives in moments both within the school setting, at home or in the public library.  These aspects of virtual manipulatives make them preferable to use in the classroom in my opinion.

In the first problem, there are two tasks.  The first task was to arrange four right triangles and one square within the outline of a square, without overlapping or leaving gaps.  The inclination is to place the square within the frame with edges parallel to those of the frame.  As the triangles are placed within the frame, it becomes quickly apparent that each triangle lies with its hypotenuse adjacent to one side of the frame, with the square nestled in the center of the triangles.  The hypotenuse is labeled c, and is adjacent to and congruent with the side of the first frame.  In the second frame, the square was placed first, quickly followed by the long, narrow right triangles.  With the configuration of the frame of this puzzle, one can determine that one leg of the triangle is a, and the other leg, b.  When you compare the two frames and the triangles that are embedded in them, it becomes apparent through study that a² + b² = c².   In the second puzzle, the sides a, b and c are again identified through their congruence with adjacent sides of squares.  Again, solving the puzzle proves that the square frames are congruent and the three square puzzle pieces are also congruent.

Solving these puzzles, along with other Pythagorean Theorem activities will give students the opportunity to understand how the theorem was developed and to see how it works across multiple examples.  This activity gives students another way to confirm that the Pythagorean Theorem works.

Resources:

http://nlvm.usu.edu/en/nav/vlibrary.html

http://louisville.edu/provost/wroffice/math72.html

In geometry, a figure is called similar if it has the same angle measurement as the preimage and the distance measurements are proportional to the preimage.  There are many types of similarity, such as geometric dilation, where the new figure’s sides are reduced proportionately to those of the originals, and expansion, where the sides are proportionately reduced from those of the preimage.  In teaching a fifth grade enrichment math class this transformation, the most important aspect of the lesson would be the guided portion where the students would be working along with the teacher in completing both dilations and expansions of the original figures.

Questions I might ask the students would be ranging through Bloom’s taxonomy.  I would ask knowledge and comprehension questions to check for basic understanding of the process.  Moving forward, the questions would be analysis and synthesis, having students do comparisons of the figures they create and consider predictions of the area of the new figures.  Devising a plan to estimate the area based on the area of the preimage would be a terrific synthesis question for the students, and would have them putting into action many geometry concepts. Prompting questions would be asked during independent work time, as misconceptions come to light in the drawings.  Students would ask knowledge questions to ensure that they understood how to make the dilations and expansions to make the new similar figure.

Resources:

http://mathworld.wolfram.com/Similarity.html

http://en.wikipedia.org/wiki/Similarity_(geometry)

https://whites-geometry-wiki.wikispaces.com/file/view/GIR276111474_455bfef3c3.jpg/305431/GIR276111474_455bfef3c3.jpg

Writing to explain is key to clearly communicating the steps involved in this activity.  It is very important that students be able to use correct terminology to describe the thinking process and the movements of the preimage (original figure).  In the Every Day Math program, up to grade three, the language used in describing transformations is flip, slide and turn.  In fourth grade, both common and specific terminology is used with the inclusion of the terms translation, reflection, and rotation.  In fifth grade and beyond, students are expected to use the correct terminology in describing such movements.  The idea is to have a progression toward using the mathematical terminology for the movements.  In the NCTM publications, it is recommended that students be permitted to use the common terms in conversation, and be familiar with use of the mathematical language for use when specificity is necessary.

Resources:  http://www.mathsisfun.com/geometry/symmetry.html

Concave – a closed figure with one vertex within the outline of the shape.  One memory key is con”cave”, where one corner is dented inward.

Convex – a closed figure with all vertices inside the outline of the figure.  No vertices are “dented inward”.

Polygon – closed figures with many sides – three or more sides – that lies on a plane.  Has width and length.

Regular polygon – all sides and vertices are equal or congrugent.

Quadrilateral – a closed figure with four sides.  

Kite – a quadrilateral with two congruent sides that are shorter than the other two congruent sides.

Trapezoid – a quadrilateral with two sides that are parallel – the other two sides are not parallel

Parallelogram – a quadrilateral with each pair of sides parallel to each other.

Rhombus – a quadrilateral with all four sides parallel and pairs conguent. Has symmetry in two directions.

Square – a quadrilateral with all four sides congruent and all four vertices right angles.

Rectangle – a quadrilateral with opposite sides congruent and all four vertices right angles.

Triangle – a three sides polygon of many types – can be equilateral, isosceles, right, or scalene.

Isosceles – a triangle with two congruent sides and one that is not congruent.

With my students, I would conduct this activity in a similar way.  I would present the students with the vocabulary terms from their textbook, have them select several terms, then define and illustrate the terms in their own vocabulary book using the Internet and other resources to investigate each term.  Students would be responsible for knowing the formula for finding the area of the polygon. Selecting the terms that are important to them will help the students to remember those key terms.  Students will present or share their booklets in order to learn all the terms in the unit.  Some students may wish to create the figures in a more tactile manner.  Pipe cleaners would be available for those students.

Many curriculum today are powered by an imposed schedule where the driver is the number of outcomes covered prior to state testing.  Teachers need to control their own schedules based on what their students need, rather than meeting artificial deadlines.  Time for presenting lateral and logical thinking has got to be found.  In order to maintain the United States’ lead in innovative thinking, students must have practice in applying logic and lateral thinking within the school day.

A great way to do this is to seek puzzle resources on the web. Some sites that fit the bill are www.folj.com, www.mycoted.com, and www.lateralpuzzle.com.  Presenting these to a classroom can be simple.  Put the puzzle on presentation software; either PowerPoint, ActivStudio, or your districts preferred program.  Find ten minutes in the day. Capture the students’ attention.  Give them paper.  Have them think individually about the problem, and write or draw a response.  Have them pair with their neighbor in discussing the logic of their response, then have the students share in some way.  The result will be an engaging activity that will help all students move forward in their lateral and logical thinking.

The point is, lateral thinking problems are not solved by knowing formulas, or having quick facility with basic facts. They are an activity which finds itself in Bloom’s Application level, where the thinker would determine the varying attributes of a round and a square manhole cover, then determine how those attributes relate to the manhole itself. Having said that, an argument can be made that the question could also be on the Analysis level, as the thinker has to analyze the factors involved, and use the information to come up with a response.

Some advanced elementary students and many middle school students are ready to solve a problem such as Manhole Covers. Out of 27 fifth graders, probably three quarters of them would come up with some solution. When I do the problem with my students, I will supply them with paper and have them do a think-pair-share activity with lots of paper and pencils so they can draw as they think. Toward the end of the class, the students would share their solutions with the entire group so they could learn from each other. Following the solution to Manhole covers, I would ask the students to look at other lateral thinking problems to enhance their ability to think outside the box.

An example of a slightly different puzzle is the “eggs in a basket” question. In this question, there are six eggs in a basket. Six people each take one, yet there is one egg left in the basket. This one was not as obvious to me, and the answer really requires thinking outside the box. **(Spoiler alert)** I did not come up with the correct answer, which is that the last person to take an egg takes it in the basket. This did not make as much sense to me. In the “manhole problem”, it is identified as both a logic problem as well as a lateral-thinking problem. For me, logic may be more understandable.

No matter what the puzzle type, giving your students one unique problem a day will help them expand their thinking and questioning techniques, as well as giving them a life-long interest in finding out-of-the-box solutions!