Tangram Legend and Project Objectives: 1.To simplify radical expressions 2.To complete the Tangram Project
Tangram Legend Once upon a time, while the cherry blossoms were in full, fragrant bloom, a young Chinese artisan named Tan crafted his finest, square, ceramic tile, bejeweled with a glaze so exquisite that nations would war for centuries for just a fleeting glimpse of such fundamental perfection. Or so thought Tan.
Tangram Legend Thus, it was with a well-founded sense of pride that Tan traveled to the Emperor’s palace with the intention of showing His Majesty this supreme example of ceramic beauty, imagining that when His eyes finally settled on that magnificent glassine surface… the entire course of His Glorious Empire would irrevocably, but positively change forever.
Tangram Legend Although Tan was master of his work-worn hands, crafting innumerable pieces of rectangular genius, he possessed the distinct inability to control his own feet. Such is often the way of the world. And so, upon mounting the dais which held the Emperor’s throne, Tan’s feet betrayed him, and his precious tile tumbled terribly to break on the floor.
Tangram Legend As is well-known by now, Tan’s tile miraculously broke into seven geometric fragments. Needless to say, the Emperor was unimpressed. So with salty down-cast eyes, obsequious embarrassment, and obsessive regret, Tan stooped to gather his shattered masterpiece, vowing to restore it to its former glory, a task he tried in vain to complete for the rest of his life.
Example 1 1.Classify the seven pieces 2.List properties 3.Do the impossible: Reassemble Tan’s precious square square
Totally Radical Awesome! Matt & Kim Video “Yea Yeah” 1.First tryFirst try 2.Second trySecond try 3.Last tryLast try 4.After the last tryAfter the last try
Example 2 Write the first 15 terms of the following sequence: 1, 4, 9, 16, … Perfect Squares These numbers are called the Perfect Squares. Their square roots are integers.
Real Numbers real numbers The real numbers (are there unreal numbers?) can be divided into two infinite sets: Rational and Irrational. rational numbersAll rational numbers can be written as a ratio of integers. (Ex: 4, 2/3, -6/25, etc.)
Real Numbers real numbers The real numbers (are there unreal numbers?) can be divided into two infinite sets: Rational and Irrational. irrational numbersNo irrational numbers can be written as a ratio of integers. Their decimal expansion is never ending without pattern. (Ex: π,, e, etc.)
Exact vs. Approximate exact approximation So an exact answer for the irrational # would be simply. In terms of decimal expansion, you could start writing them down today, 1.4142…, and like Tan, you’d never complete your task by the time you died. Moreover, you’d never complete your task by the time the universe died, if it ever will. Thus, any decimal expansion for an irrational number will be an approximation.
Example 3 Use your calculator to approximate each of the following. 1. 2. 3. 4. 5. 6.
Adding Square Roots Adding or subtracting square roots is like combining like terms. 1.You can only add or subtract square roots if they have the same radicand. 2.Keep the square roots the same, just add or subtract the coefficients.
Example 4 Use your calculator to approximate each of the following. 1. 2. 3. 4. 5.
Multiplying Square Roots To multiply square roots, just multiply the coefficients and the radicands separately. Squares and square roots are inverses, so they annihilate each other.
Tangram Project In this project, you will be finding the area and perimeter of various composite Tangrams. The tangram bunny at the right is a sample of one you won’t be doing.
Tangram Project You will be responsible for completing each page of your portfolio. Start by arranging the seven tangram shapes within each outline, and then tracing off the perimeter of each shape within that outline. Now with the combined genius of you and your group mates, you’ll have to do some adding, subtracting, multiplication, and simplification to complete each problem.
Tangram Project Here are a group of folks of a geometric persuasion that you may rely upon for help:
Tangram Project Here’s what I will be looking for when I grade this project: 1.Tangram pieces traced off inside each shape 2.The measurements of the tangram pieces written along the perimeter of each shape 3.Perimeter calculation 4.Area calculation
Tangram Project Think of this portfolio as practice, where you’re learning how to position the tangram pieces and to find the perimeter and area of each shape. You may work together; you can even get help from me. On the day the project is due, you will have to complete a new shape in class all on your own. Here’s how your grade will be computed: 25%75% 6-page Tangram PortfolioSingle In-Class Problem