Transcript FirstWeek
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 The Handshake Problem • If each student in this class shakes hands with every student, how many handshakes will there be? • 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 Handshake problem (the multiplicative way) • Each person shakes 19 hands → 20*19 • But there is multiple counting… how much? 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 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 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 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 Relating the two ways… • 1: ●●●●●●●●●●●●●●●●● ●● • 2: ●●●●●●●●●●●●●●●●● ●● • 3: ●●●●●●●●●●●●●●●●● ●● Or, with smaller numbers • • • • • • • 1: 2: 3: 4: 5: 6: 7: ●●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●●● (end of day 2) Even and odd numbers (geometric) ●● ●● … ●● ● ●● + ●● = … ●● ●● ● ●● ●● ●● ●● … 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 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 Chapter 1 Homework • pg 28 - 30: #18, 22, 29, 39; • pg 53 - 57: #5, 13, 36 Tuesday, 6/5 • Alphabitia • Creating a number system • Making a poster of your number system 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… 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. A new Alphabitia system • Here is a partial number system… – A=● B=●● C= ●●● – A0 = | AA = | ● ( similar to 'our' number 11) AB = | ● ● ( = 12) AC = | ● ● ● ( = 13) AD = | ● ● ● ● ( = 14) B0 = || use for 0 --> place holder D= ●●●● • One key idea here is new: place value 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. Other systems of different bases • Talk of other bases. What do you think these mean? baseplace value- • Exploration 2.8: – Part 1 (1, 2, 4, 5, 7, 8) – Part 3 (#2) – Part 4 (#1) • What does abcx equal in base 10? Other bases New vocabulary ● • • • • • place value base place holder units, longs, flats, cubes, super longs... expanded form: 275610 = 2*1000 + 7*100 + 5*10 +6*1 275610 = 2*103 + 7*102 + 5*101 + 6*100 34215 = 3*53 + 4*52 + 2*51 + 1*50 – What does 5035 mean? base 10 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: abcx = (a*x2 + b*x1 + c*x0 )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? • There are 10 kinds of people in the world… • There are 10 kinds of people in the world… those that understand base 2 and those who don’t.