This is an exciting and challenging exploration for pre-algebra, algebra for geometry students in grades 6 - 8. Depending on the grade level and the quality of the students the activities in this exploration can take anywhere between 3 to 7 regular class periods. This exploration was originally designed as a hands-on workshop for teachers and mathematics education students. The duration for this work-shop can be from 1/2 hour to 1 1/2 hours.
This exploration can be presented, on-line or onsite, without needing to refer to the original source material upon which the workshop is based. However, some presenters may find it useful to explore the manipulative, Dr. Super's Triangles, and scan the book, Dr. Super's Triangles: Algebra Explorations before doing this workshop with other teachers or students in a classroom.
If time is limited, reading at least the first activity in the book -- Week 1: Meet Fitz D. Pieces (Page 4-15) will be most helpful. It is useful to also look at the home project (page 58) and the exploration evaluation sheet (page 62) for the week 1: Meet Fitz D. Pieces activity.
Onsite or on-line individuals can explore the activity as individuals. When possible, it is best to pair participants. The interaction usually enhances the learning for both persons. To begin the workshop, distribute the following items to participants:
a. A copy of the single sheet handout. On one side, titled Puzzles for Dr.Super's Triangles, there are a rectangle and two triangle puzzles. On the other side, titled Fitz D Pieces, there is a triangle monster puzzle. For the best results, you should print and copy these two puzzles on the back and the front of sturdy card stock. But, the workshop can be done using ordinary paper as well. Alternatively you can distribute 2 separate pieces of paper, one with Dr. Super's Triangles Puzzles and the other with the Fitz D Pieces puzzle.
Note: Depending on your hardware/software configuration, you may or may not be able to print the front and back of the activity sheet. If you can not, for a limited time you can obtain one free by contacting Dr. Super.
b. An 8 piece subset of the Dr. Super's Triangles manipulative that has 1 large triangle, 3 medium triangles and 4 small triangle. Importantly, the 1 large triangle and the 4small triangles should be of the same color, while the 3 medium triangles are of a different color.
Note: Experience suggests that the activity will flow more smoothly if you have packaged the8 piece subset into small plastic bags before the presentation. Doing so makes distribution,management and storage of the materials easy and efficient.
Note: One set of Dr. Super's Triangles will make a total of 8 of these 8 piece demonstration subsets.
At this point, it is not necessary to give formal instructions to participants about what to do. Merely encourage them to explore with the pieces and the handout while you are waiting to start the presentation. This provides a few minutes of structured informal exploration time for participants to get acquainted with the manipulative pieces.
If you are doing this workshop on-line, you will want to maximize the viewing area. To do so on the browser you are using, hide the Toolbar, Status Bar, and all of the other Bars. [If you are not familiar with how this is done, the following generally works in Netscape. And, most other browsers will be similar. Select options from the Toolbar. Generally, you will see a "check mark" next to the choices of Show Toolbar, Show Location, Show Directory Buttons. Click on each of these. The "checkmark" will disappear and the bars will not be visible, giving you the maximum area in which to view the workshop.]
For an onsite workshop for teachers you can display the Washington Post METRO article titled Math Made Easy. You can also distribute a copy of this article to each of the participants if you wish.
The article shows classroom middle school teacher Sherry Gorrell of the Fairfax County Public Schools using Dr. Super's Triangles in her classroom. Mrs. Gorrell, a Presidential Award winner,was among a half a dozen teachers that pilot tested Dr. Super's Triangles before they were published.
Suggested points that could be made in reference to this article include:
a. Dr. Super's Triangles have been completely tested by teachers and students in real classroom settings.
b. As reported in the article, Dr. Super's Triangles promote an exciting hands-on approach to learning mathematics, an approach that has captured the attention of knowledgeable teachers as well as reporters and editors of a major daily paper.
c. In the activity today that is similar to what the students were engaged in Sherry's classroom the four main goals of the NCTM standards for middle school are addressed as described in the introduction. At the end of each Step the presenter can revisit how theses goals are addressed in that step.
If time permits, a nice ending for the presentation is to share the story of the Babylonian tablet and Dr. Super's Trigram Puzzle. Here is the story in capsule form. Embellish as appropriate.
Before developing Dr. Super's Triangles for Creative Publications, Dr. Behrouz Aghevli (pronounced Bay rooz' Ah gav' lee) actually discovered (invented) a Tangram like puzzle using 1 large, 3 medium and 5 small triangles in what is now the Dr. Super's Triangle set. The discovery of that puzzle, which he calls the Trigram puzzle, came when Dr. Aghevli was trying to divide an equilateral triangle into 5 congruent right triangles.
Dr. Aghevli was aware of and fascinated by the reported 4000 year old history of the traditional 7 piece Tangram puzzle. As the story goes, the Emperor Tan had a square mosaic tile in his possession. One day, while strolling in the garden, Tan dropped the tile and it broke into seven pieces. You guessed it!! Those seven pieces are the familiar and famous seven Tangram pieces. And, it was reported that Tan spent hours and hours trying to rearrange those seven pieces into the original square, hence the origin of the famous Tangram puzzle.
While FINDING the tale intriguing, Dr. Aghevli believes that the reported story of the origin of the Tangram puzzle, circulated in the early 1900s by several well known English puzzle writers, is "phoney baloney," as they say.
Anyway, early in their working relationship (during the summer of 1991), Dr. Aghevli showed his Trigram puzzle to his collaborator and co-developer of the Dr. Super's Triangle materials, Dr. Mark Spikell, a professor at George Mason University in Fairfax, VA. And, in a spirit of comedic interplay, when Dr. Spikell inquired about the origin of the Trigram puzzle, Dr. Aghevli wove a fairly tall tale about discovering the puzzle on an ancient, 4000 year old Persian tablet that had been in his family for centuries. Dr. Aghevli was, of course, "pulling Professor Spikell's leg," so to speak, and had no such tablet in his possession.
Unknown to Dr. Aghevli at the time, Professor Spikell never actually believed the tale and proceeded to humor his colleague by allowing him to think he believed the tale he had been told.
On Christmas day of the same year, Dr. Aghevli was showing a friend his newly developed Trigram puzzle. A few days later, the friend called Dr. Aghevli and said they had to meet for lunch to discuss a very urgent matter. He would not reveal the nature of the meeting.
At the meeting, the friend told Dr. Aghevli he had quite a surprise for him. He handed Dr. Aghevli a physical replica of an ancient tablet he had seen in a museum in Baghdad, Iraq, many years before. The friend had been so captivated by the actual tablet that he purchased a replica, to bring back to the United States.
Show the picture of the original tablet in the museum in Iraq. Direct the participants attention to the triangular shape at the top of the tablet.
Yes, you guessed it again. The triangles pictured look very much like the triangles in the Dr. Super's Triangles set.!!!!!!
You may point out that a large, a medium and a small triangle fit on the tablet almost perfectly.
It is truly an understatement to say that both Dr. Aghevli and Dr. Spikell were very surprised to learn of the existence of such a tablet containing the Dr. Super's Trigram Puzzle pieces. They were literally incredulous. And, Dr. Aghevli began the academic journey into the history of mathematics to track down the real truth and story of the tablet.
Here is what Dr. Aghevli learned.
Scholars in the Babylonian culture and language, informed Dr. Aghevli that the tablet is believed to be one of the first mathematical problem texts from the Old Babylonian era. The triangles pictured are not actually the 306090 triangles in Dr. Super's set, though they surely "appear" to be of those angle measurements. Actually, it turns out that the triangles are 345 right triangles, a fact apparently made clear when the text on the tablet is translated.
Interestingly, it is believed that this tablet represents one of the first known demonstrations of the very famous theorem attributed to the Greek mathematician, Pythagorus, a theorem learned (or memorized) by virtually all school children. The theorem states that in a right triangle, a2 + b2 = c2 , where a and b are the lengths of the legs and c is the length of the hypotenuse. And, if true, this Babylonian demonstration of Pythagorus' theorem would have preceded the work of Pythagorus by some 1300 years.
Through his research on this tablet, Dr. Aghevli learned a great deal about the contributions of the Babylonians in mathematics. And, as a result, he and Professor Spikell included in the book, Dr. Super's Triangles: Fraction Explorations, the interesting Babylonian accordion that is made with a folded rectangle the size of 60 small triangles. The accordion can be used to show all the important fractions: 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30 and 1/60.
Note: This is an ideal opportunity to introduce any number of topics from the history of mathematics. Teachers could use the story about the tablet to motivate a discussion of the valuable contributions to mathematics of various civilizations and cultures, not just the Babylonians.
A nice touch at the end of the presentation is to offer a few suggestions of what could actually happen in the classroom with pre-algebra, algebra and geometry students.
Certainly, one good way for students to begin their work with Dr. Super's Triangles would be to explore the very activity outlined in this presentation for teachers.
A good extension or follow up activity might be to invite students, perhaps in teams of 2-4, to design their own Triangle Monsters, using a larger number of Dr. Super's Triangles. Their monsters could be traced on paper, colored and displayed in the classroom, perhaps for parents night.
And, of course, students could be asked to find the area and the perimeter of their own Triangle Monster in terms of a and b.
Spending a few moments displaying or mentioning the three books available for Dr. Super's Triangles might be appropriate in some settings. Show the cover of the book, Dr. Super's Triangles: Algebra Explorations. The books contain a wealth of mathematical topics that can be presented to pre-algebra, algebra and geometry students and could be a valuable resource for students as well as teachers. Some of the topics covered in the books are:
A. Dr. Super's Triangles: Algebra Explorations -- writing word sentences about relationships; translating sentences into algebraic statements and equations; FINDING lengths of sides of geometric shapes in terms of a and b determining equations for areas of triangles and rectangles; finding areas and perimeters of triangles and rectangles; graphing linear equations, determining line slope and y intercepts; a proof of the Pythagorean Theorem.
B. Dr. Super's Triangles: Symmetry Explorations -- determine horizontal and vertical axes of symmetry; build and record 2, 4, and 8 piece mosaic designs; work in teams to find the total number of 8-piece mosaic designs; create mosaic designs with a given number of mirror images, axes of symmetry, etc.; explore mirror and rotational images; define 180 degree rotation and negative images; prove various transformational equalities.
C. Dr. Super's Triangles: Fraction Explorations -- construct a fractional pattern that shows given fractions; explain why 1/2, 4/8, 6/12, 9/18, and 12/24 are equivalent; complete factor tables for 1/8, 1/18 and 1/24; create repeating designs that cover given fractions; perform addition and subtraction of fractions; model multiplication and division of fractions; determine the greatest common fractional factor of two fractions; determine the least common multiple of 2 whole numbers; relate the least common multiple of whole numbers to the greatest common fractional factor of two fractions; make and use a Babylonian Accordion and table to do fractional arithmetic.
