January 20, 2009 Hope you enjoyed the long weekend!

January 20, 2009 Sec 1.3: Patterns (Exploration 1.1) Sec 1.4: Representations (Exploration 1.4) Sec 1.7: Connections Sec 2.3: Numeration

1.3 - Patterns Ex: Find the next term in the sequence. Describe the pattern you found (use a sentence). a) 1, 2, 4, 8, 16, … b) 25, 5, 1, 0.2, 0.04, … c) 1, 3, 6, 10, 15, … d)1, 0, -1, 0, 1, 0, -1, …

1.3 (cont’d) A little math history… Carl Friedrich Gauss ( )

Exploration 1.1 (handshakes) Goal: determine how many handshakes will occur if each student in this class shakes hands with every student. Work on this alone for a few minutes. Can you apply something we’ve discussed today?

Exploration 1.1 (handshakes) Discuss your approach to the problem with others at your table (the answer is not so important – it’s the solution that you should discuss). What were some of the ideas you had not thought of?

Exploration 1.1 (handshakes) For homework, answer the following two questions: 1.How many handshakes will there be in our class (24 people)? 2.How many handshakes will there be in a group of n people? Explain what you did to solve, and why you did it.

1.4 - Representations Diagrams Graphs Tables Sketches Equations Words Etc.

Exploration 1.4 (Darts) Purpose: think about solution strategies as opposed to trying random things. If you have 4 darts and the dart board shown above, which of the scores below are possible, assuming that every dart hits the board? (Explain your reasoning)

Exploration 1.4 (cont’d) What did you notice? Part 4a asks you to list all scoring outcomes, e.g. 7, 7, 7, 7 or 1, 3, 5, 7 etc. With your table, discuss how you could do this systematically.

Exploration 1.4 (cont’d) What if you only had 3 darts? What scores would then be possible? Answering this and explaining why will be part of your homework for next time.

2.3 - Numeration Have you ever thought about what it was like for people to invent counting? Abstract notion of “number” Representing discrete amounts and measures

2.3 (cont’d) As societies became more complex, so did number systems. Tally systems Roman (widely used for a long time – still has some usage) Egyptian (earliest known system) Babylonian (very advanced for its time) Mayan (probably first to develop concept of zero as we know it) Hindu-Arabic (system we use)

2.3 (cont’d) Roman system ValueSymbolValueSymbol 1I100C 5V500D 10X1000M 50L

2.3 (cont’d) Egyptian system 1 =(staff)10,000 = (pointing finger) 10 = (heel bone) 100,000 = (fish or tadpole) 100 =(scroll)1,000,000= (astonished person) 1000 = (lotus flower)

2.3 (cont’d) Babylonian system ValueSymbol 1  10  Positional system The number 60 Symbol for zero: (developed later)

2.3 (cont’d) Mayan system The number 20 Used the concept of zero, but only for place holders Used three symbols: 150 Wrote their numbers vertically: is = 10, is = 14

2.3 (Mayan system cont’d) Mayans left a vertical gap to represent place value: is one 20, and 0 ones = 20. is two 20’s + 6 = 46

2.3 (cont’d) What similarities/differences do you see in these systems? Advantages/disadvantages of each system?

Homework Read text section 1.7 Write up the handshakes exploration. Write up the darts exploration. Do text assignment posted on web pg Due Thursday, 1/22 at the beginning of class.