§ Yesterday when we were completing our factor chart and playing the factor game, I heard a lot of interesting and intelligent comments about what you were noticing. Today we are going to continue to play and think about the factor game. Our goal for today is to continue to practice finding factors of different numbers as well as to incorporate some specific mathematical language about what we are seeing and learning about as we play our game.
§ Intro: Today we are going to pay specific attention to analyzing the first move in the factor game. As I told you yesterday, the more that I’ve played this game, just like anything else, the better I’ve become. So, today we are going to analyze the game by asking our self some questions about the best first move.
§ Model: So, if I take 26 as my first move, how many points does my opponent get? Remind students that my opponent will get all of the factors of my number. 1, 2, and 13 Great – so let’s see how many points they would receive by adding their numbers together. (16) So is that a good move or a bad first move? Ask Ms. F. It’s a good move because you will score more than your opponent. Interesting. So, what I make 18 my first move? Is that a good move or a bad move? How many points do you get? How many points does your opponent get?
§ Show students an organizer and put what we have discovered into the organizer.
§ Guided: do the number 6 together- what happens?
§ Independent: Assign students to groups (higher level to larger numbers, lower level to smaller numbers) If finished early they should continue to find out about the other first moves even if it’s not a part of their assignment. Ask students to fill in the chart when they are finished as a group.
§ Discussion: Let’s analyze the chart – What first moves give the other player exactly one point? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. (color on a chart) Ask if anyone knows what these numbers are called prime - Right a prime number is a number that has exactly two factors – itself and 1 – (students take notes as does teacher)
§ All other numbers except for the number 1 are called composite numbers. Name some composite numbers. What is the fewest number that a composite number can have? Three (like 4, 9, 25)
§ How many factors does the number 1 have? One Explain that the number one is neither prime nor composite. It is the only number with exactly one factor.
§ Look at the table again, were all of the prime numbers good first moves? Were all composites good first moves? Why or why not?
§ What was the best first move (twenty-nine) What was the worst first move? 24 or 30 Tell students that ancient Greeks had a special way of classifying numbers – they called one group abundant, one group perfect and one group deficient.
§ Bad first moves are abundant numbers because the sum of the proper factors is greater than the number
§ Good first moves are called deficient because the number is larger than the sum of the factors
§ Moves that are neither good nor bad are called “perfect” because the sum of the factors is equal to the number.
Students complete worksheet on Erathosthenes Sieve to determine all prime numbers less than 100. Students walk through the steps eliminating numbers - what is remaining is prime. Have discussion with students about why certain numbers were crossed out and how we know that only prime numbers are left.