1. First Explorations 1. Handshake Problem (p. 3 #2) 2. Darts (p. 8 # 1) 3. Proofs with Numbers (p. 8 # 2) 4. relationships, graphs, words… Expl. 2.4, Expl. 2.5

2. The Handshake Problem If each student in this class shakes hands with every student, how many handshakes will there be?

3. Try several strategies. Would it help to solve a simpler problem? to draw a diagram? Find a pattern. Represent the pattern. Generalize the pattern. That is, how many handshakes would there be if there were n students? Explain your generalization. Does it work for this classroom? Use it to find the number of handshakes there would be in a room of 100 people. »End of day 1

4. Handshake problem (the multiplicative way) Each person shakes 19 hands → 20*19 But there is multiple counting… how much?

5. Handshake problem (the multiplicative way) Each person shakes 19 hands → 20*19 But there is multiple counting… how much? Each handshake is counted twice. So divide by 2 to get the actual number 20*19 / 2

6. The adding-up-consecutive - integers way How to add 19 + 18 + … + 2 + 1 ? How about 100 + 99 + … + 2 + 1? Or (n-1) + (n-2) + … + 2 + 1? –Avoiding “brute force” –Here is what Gauss did (in first grade!): 100 + 99 + 98 + …+ 3 + 2 + 1 1 + 2 + 3 + …+ 98 + 99 + 100

7. The adding-up-consecutive – integers way 100 + 99 + 98 + …+ 3 + 2 + 1 1 + 2 + 3 + …+ 98 + 99 + 100 Each ‘column’ add up to _101__. There are _100__ columns. So our answer is the product 100*101, right? Almost, adding up both columns doubles our answer, so divide by two.  100*101/2

8. Adding up consecutive integers - A little more formally, and generally: n-1 + n-2 + n-3 + …+ 3 + 2 + 1 =Ans 1 + 2 + 3 + …+ n-3 + n-2 + n-1= Ans ↓ n + n + n + … + n + n +n = Ans + Ans n*(n-1) = 2 * Ans So, Ans = n*(n-1)/2

9. Relating the two ways… 1: ●●●●●●●●●●●●●●●●●●● 2: ●●●●●●●●●●●●●●●●●●● 3: ●●●●●●●●●●●●●●●●●●● ….. 18: ●●●●●●●●●●●●●●●●●●● 19: ●●●●●●●●●●●●●●●●●●● 20: ●●●●●●●●●●●●●●●●●●●

10. Or, with smaller numbers 1: ●●●●●● 2: ●●●●●● 3: ●●●●●● 4: ●●●●●● 5: ●●●●●● 6: ●●●●●● 7: ●●●●●● (end of day 2)

11. Even and odd numbers (geometric) ● ● ● ● ● ● ● ● + ● ●= ● ● … … ● ● ● ● ● ● … ● ● ● ● ● ●

12. Even and odd numbers (algebraic) Even number must look like: 2 * n, for some integer n Odd number: 2*m + 1, for some integer m

13. Even and odd numbers (algebraic) Even 2 * n Odd 2*m + 1 So ( odd ) + ( odd ) looks like: (2*n + 1) + (2*m + 1) = 2*n + 2*m + 1 + 1 = 2*(n + m) + 2 = 2*(n+m+1) = even

14. Chapter 1 Homework pg 28 - 30: #18, 22, 29, 39; pg 53 - 57: #5, 13, 36

15. Tuesday, 6/5 Alphabitia Creating a number system Making a poster of your number system

16. Wed 6/6: Test driving the systems In groups: 5 minutes on each system… –Complete the Alphabitia table using the new system. –Find the sum of N + W in the new system. –Complete p.40, part 3, #2. 5 minutes as a tribe…

17. Test driving the systems In groups: 5 minutes on each system… –Complete the Alphabitia table using the new system. –Find the sum of N + W in the new system. –Complete p.40, part 3, #2. 5 minutes as a tribe… –Common advantages, common disadvantages. –Similar structures.

18. A new Alphabitia system Here is a partial number system… –A = ● B = ● ● C = ● ● ● D = ● ● ● ● –A0 = | AA = | ● ( similar to 'our' number 11) AB = | ● ● ( = 12) AC = | ● ● ● ( = 13) AD = | ● ● ● ● ( = 14) B0 = || use for 0 --> place holder One key idea here is new: place value

19. Working with Alphabitia and base 5. Complete the Alphabitia table using the new system. Complete the table in base 5. (A=1,...D=4) Find the sum of N + W in the new system. Complete p.40, part 3, #2.

20. Other systems of different bases Talk of other bases. What do you think these mean? base-place value- Exploration 2.8: –Part 1 (1, 2, 4, 5, 7, 8) –Part 3 (#2) –Part 4 (#1) What does abc x equal in base 10?

21. Other bases ● New vocabulary place value base place holder units, longs, flats, cubes, super longs... expanded form: 2756 10 = 2*1000 + 7*100 + 5*10 +6*1 2756 10 = 2*10 3 + 7*10 2 + 5*10 1 + 6*10 0 3421 5 = 3*5 3 + 4*5 2 + 2*5 1 + 1*5 0 base 10 – What does 503 5 mean?

22. Translating between bases (questions to ask) base x into base 10 -->expanded form –how many of each place value (units, longs, flats...)? –what is each place value worth? EX: abc x = (a*x 2 + b*x 1 + c*x 0 ) 10 from base 10 into other bases –how much is each place value worth? –how many of each place value do I need to 'use up' all the original base?

23. There are 10 kinds of people in the world…

24. There are 10 kinds of people in the world… those that understand base 2 and those who don’t.