M1 Class Summary for Jan 15, 2017

Class Summary Jan 15, 2017

Instructor: Fatima Al-Raisi

Attendees: Suhaib, Qadir, Eman 

Topics and Concepts: 

Counting:

A. Counting the number of items (numbers) in a list

1-list of numbers

2-starting with 1 or 0

3-with 1 as increment

4-starting with arbitrary numbers and with larger increments

5-including/excluding end points 

Strategies:

* for list of numbers with 1 as increment, convert the list to one that starts from 1 and considering the last number

* add 1 for lists that start from 0

* for general lists: (last - first) / increment + 1

* for more complicated lists some strategies are: split the list (even and odd), or count by exclusion (e.g., how many two digits numbers are divisible by 5? Consider all multiples of 5 until 100, there are 20 of them exclude 5 because it's a one-digit number and 100 because it's a 3-digit number)

B. Counting the number of steps/choice in a multi-step/choice procedure/decision 

1-multiplication rule

2-with or without repetition (introduction, simple examples, did not define combinations nCr)

examples: 

-number of ways Ali can can create outfit combinations from 4 shirts (different colors) and 3 pants 

(different colors)

-number of ways different ice-cream flavor/topping combinations can be made when buying a two-scoop ice-cream with a topping from a place with n flavors and m toppings given that the two scoops have to be different flavors 

-number of ways morning-afternoon-evening activity combinations can be made to create a 3-day camping program from a given list of 3 possible morning activities, 4 possible afternoon activities, and 3 evening activities 

-Question given to students as homework: what if we want each day to be different? Note: define different (two days with the same morning and afternoon activities but different evening activities are still different). To simplify the task, students were asked to find the number of different possible 3-day camping program such that each days is entirely different (all three activities are different).

Notes: as students were copying the list of activities for homework, instructor hinted that they only need to note down the number of activities in each category since they are only required to come up with the total number of options not with the actual program.

C. Counting and geometric concepts 

Examples:

-count the number of students standing (evenly spaced) along each side of a square shape classrooms 

-common (wrong) answer: number of students (assuming divisible by 4) divided by 4

-ideas: note these are discrete objects (not points in a line) --> common mistake of double counting in corners, still need to place 4 more students

-generalizing from square to any regular polygons 

-some students were able to generalize and come up with a rule for any number of students

-teacher refined generalizations by placing constraints on the # students in such examples 

General notes: to engage students

1. ask for their input in creating questions (shirt colors, camping activities, ice cream flavors)

2. ask them to work on paper even though the class is small since verbally answering discourages slower students from working out the problem

3. for more challenging problems they can discuss the answer

4. try to ask for alternative ways to solve the problem and compare answers 

Homework assigned from Ch4 Counting: 7-11, 15, 18, 19, 21, 22

Optional: 16 (read and try)