Lesson Number 7 A GAME TO LEAD TO THE FUNDAMENTAL THEOREM OF ARITHMETIC. Prepare a lesson for the class on the topic described below. You can use the ELMO projector, the whiteboard, a power point mini presentation, handouts, etc. An average student led lesson might be about 15 minutes. Become an expert on the topic by practicing it yourself with many examples over many days or weeks. Teach it to friends, parents, grandparents, or children. You are teaching essential course material that will be on the tests. I have provided a few examples that you can demonstrate for the class. You need to make up more examples to present to the class or to have them use for in- class practice. You should call on volunteers. I consider a lesson that is very student centered to be the most successful. Therefore, seek to be a guide on the side rather than the sage on the stage. Plan to have the class actively involved. The more they talk and the less you talk the better! For some lessons, the students may need only one example before they are ready to come to the board to present your remaining examples. Prepare homework problems for the class to practice before the next class, approximately 10 – 15 minutes worth. You should give these problems to the class by a handout. Use a GVSU lab to print the copies. At the top of the homework, state the title of your lesson, your name, and a brief summary of your lesson. Send an electronic copy to firstname.lastname@example.org with “Lesson 7 homework” in the subject line. During the next class, plan to address the homework problems you assigned. You might do this by reading the answers or by asking classmates to read their answers or come to the board to show their work. Address the homework problems you assigned based on the needs/questions from the class. (For example, students might not have any questions and a simple check that everyone got the same answers may be sufficient. Use your judgment, keeping in mind that the material you taught will be on tests.) YOUR TOPIC: A GAME TO LEAD TO THE FUNDAMENTAL THEOREM OF ARITHMETIC. The Fundamental Theorem of Arithmetic says that every composite number can be uniquely expressed as the product of prime numbers. But…. DON’T TELL the theorem to the class just yet. Start with the game. The theorem WILL BE AN OBSERVATION THAT CAN BE SEEN BY THEM ON THEIR OWN AT THE CONCLUSION OF THE GAME PLAYING. THAT IS, THE THEOREM’S MEANING CAN BE SEEN BY THE STUDENTS AS OPPOSED TO BEING TOLD BY A TEACHER. Begin your lesson by demonstrating how the game is played with two examples (directions below). Be prepared for a third example if the class asks for another example. If they are satisfied in understanding the game after two examples, continue. It is a two-player game. Think of a way to do this so that each student can play the game with someone who is not in their group. (If the class has done “Clock Buddies”, you might use that.) Game directions: 1. Player A writes a number. 2. Player B factors the number into two factors. (The number 1 is never allowed as a factor in this game (else it could be played forever without end!!) 3. Player A uses what player B found to write the number using three factors. 4. Player B uses what player A found to write the number using four factors. Etc. 5. The game ends only when all of the factors are prime numbers. (Another reason why “1” cannot be used.) 6. If one wants the game to have a ‘winner’, it could be the player who was able to play last.* Example: Three different teams might decompose 1250 in the following way: Jason and Elise: J 1250 E 125 x 10 J 5 x 25 x 10 E 5 x 5 x 5 x 10 J 5x5x5x2x5 Boscar and Omar: B 1250 O 50 x 25 B 5 x 10 x 25 O 5 x 2 x 5 x 25 B 5x2x5x5x5 Emma and Spencer: E 1250 S 2 x 625 E 2 x 5 x 125 S 2 x 5 x 5 x 25 E 2x5x5x5x5 Each pair used different steps but all three found 1250 = 2 x 5 x 5 x 5 or 2 x 5 . It is the unique prime factorization of 1250. And that is what The Fundamental Theorem of Arithmetic is all about. 4 *Note: Since you and I know about the Fundamental Theorem of Arithmetic, we know that the number of factors in the last row of the game will be the same for every game being played. Thus, once a start number is selected, the winner (last person to play) is already predetermined. However, this is an observation that can also be made in the summary phase of the lesson. Ask the teams to each play four games and walk around to see when most of the class has finished. Game 1: 210 Andrea (Student A) writes 210. Game 2: 1000 Benny (Student B) writes 1000. Game 3: 1540 Andrea writes 1540. Game 4: 882 Benny writes 882. If they take turns being Player A, I think these numbers will allow them to each “win” twice. Please play each game to double check me on that. (Benny wins Game 1. Andrea wins Game 2. Andrea wins Game 3. Benny wins Game 4.) While the class is playing the game, divide the front board into two parts and back whiteboard into two parts and write each game number in one of the four parts, at the top of the whiteboard. After sufficient game playing time, ask each pair of players to select and show one (or more) of their complete game so that each number has at least 4 examples At this point, ask the students what they observe about the game. Give them at least one full minute to observe the boards. We’re hoping they notice that everyone’s last line has the exact same numbers. This fact illustrates the Fundamental Theorem of Arithmetic and now is a good time to write this theorem on the board. Give them time to write it down. I am hoping someone notices that ‘winning’ this game is simply a matter of luck and not of skill. However, a student (including an elementary child) could pre-find numbers that would insure a ‘win’ . Lastly, tell students that when they are asked to find the prime factorization of any number, they do not need to take as many steps as we did in playing the game. For example, for 1000 one could factor it as 10 x 10 x 10 and then see that each 10 has a 2 x 5 and quickly write the prime factorization as 2 x 5 . 3 3 Lastly, show an example of finding the prime factorization of a number more quickly than by the game directions. You can use 840. Perhaps you could start by writing 10 x 84. Then 2 x 5 x 2 x 42. Next 2 x 5 x 2 x 6 x 7. Since 6 is not prime, we’re not done, but almost. We write 840 as 2 x 5 x 2 x 2 x 3 x 7. If everyone puts the primes in order and uses exponents when able, we would all get the same final result. That is, the prime factorization of 840 is 2 x 3 x 5 x 7. 3 Some other numbers the class can factor could be: 720, 945, and 1024. Let the class have time to work on these numbers. Call on volunteers to bring their work to the board. This time, we’re not playing the game, but working alone. Final, important, note: The theorem is not one of several ‘theorems of arithmetic’, rather, it is THE Fundamental Theorem of Arithmetic, a testament to its importance.