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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

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

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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

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

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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

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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

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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

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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

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

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

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

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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

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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

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

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

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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…

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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.

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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

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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.

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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?

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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?

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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?

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There are 10 kinds of people in the world…

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

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