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.