Draft+Page

==Hi guys, here is a Draft Page. Feel free to put anything here, ideas,questions, Just make sure to refer specifically what you are writing about. For example, From VDW.....make sure to include page number (if applicable).==

Big Ideas p.456

 * 1) "chance has no memory." Chance is independent, that is "the outcomes of prior trials have no impact on the next."
 * I like the example that many people have on tossing a coin -that after a run of several heads, their is a common assumption that the following toss will have a higher probability of being tails instead of 50%.

2. The Probability characterized along the continuum from impossible to certain (Impossible, least likely, likely, certain) (probability of an event expressed in number: decimal 0 to 1; percent 0%-100%) //(students may ask "why not negative, 1.5...or 110%) I think this is really important - maybe even worthy of a lesson!// 3. As number of trials increase, better estimate of probability increase...approaching close to theoretical probability.
 * I had an idea of compiling data from an event and recording it for each of my 5 classes and show the students how data changes from just looking at the results from one class to looking at the compiled data of all 5 and how this shows the way larger numbers more closely reflect the //theoretical probability//. I like this too - could we compile data from a game and use in our thrid lesson?

4. Independent vs. dependent in 2-event experiments ( I don't know if we'll actually get this far in our mini-unit) 5. From VDW (p.461) "Probability has 2 distinct types. ... the 1st type involves any specific event whose likelihood of occurrence is known (e.g. that fair die has 1/6 chance of producing each #)... Vs. the 2nd type which ,involves any event whose likeliness of occurrence isn't observable -but can be established through past events //-empirical data//. The examples used are a basketball player's likelihood of making free throws in a game (based on the player's previous record) or the chance of rain (based on how often it rained under equivalent conditions). The text brings up this "Although this (2nd type) of probability is less common in the school curriculum, it is the most applicable to most fields that use probability and therefore important to include in your teaching."

Math contents Connection (or prerequisite knowledge)

 * fraction and percents
 * ratio and proportion
 * data analysis ..

Implication for Instructions (to answer: why do I need to learn this?) p. 464

 * develop simulation approach to solve real-world problems -Yes in VDW (p.468) they discuss using //simulation// "as a technique used for answering real-world questions or making decisions in complex situations... many times simulations are conducted because it is too dangerous, complex, or expensive to manipulate the real situation". Immediately that show "Mythbusters" comes to mind that has been on the //Discovery// channel for the past few years -also using dummies in cars for safety tests in new cars. They also use the example on this page of when designing a rocket and testing systems for possible failure, using a model, even a computer model, helps give a more accurate estimate of possible failure because using a computer model for the event allows the possibility of conducting 1000's of trials which ties back in the higher the amount of trials brings the result closer and closer to //theoretical probability//.
 * More Intuitive
 * Educated guesses versus wild guesses
 * For "infinite" number of trials, the relative frequency (built from experiments) and the theoretical probability will be the __**same**__ (technically, it is not the same but very, very close to it). But the question for us to think is...can we produce "infinite" number of trials in real-world contexts..//.[ just a friendly reminder "infinite" = infinity is an idea; it is not a number (because if it is a number, what is it?]//
 * //"Informed citizens need to be numerate in data and chance and need to know how to decipher and make sense out of information that is presented in newspapers, medical reports, consumer reports and environmental studies." A Research Companion to Principles and Standards for School Mathematics, pp223. From same source, people are required to make decisions in their lives under conditions of uncertainty. An understanding of chance and risk, an ability to read and interpret graphs, and an ability to question situations involving data and chance are all essential skills for making these decisions.//

I think a vocabulary list will be a great addition. And I also think our lesson(s) seems to fit in the "close to the beginning" if not the "beginning" of a Probability Unit.
 * I agree with this too -it will be much easier to start at the beginning of a Unit. Also vocabulary will be key for reducing confusion and misconceptions. For example, VDW also points out on page 466 "Of special note is the word //or//, since its everyday usage is generally not the same as its strict logical use in mathematics. In mathematics, //or// includes the case of //both.//"

**__ Introduction lesson __** can be something like ....Give examples of real-world (locally) that you see probability is being used (implied, related)....students may answer Casino, Weather and etc
if weather is used....ask for example...what does it mean to have "60%" chance of snow" or casino is used....What the chance of going in a Casino and getting out winning after playing Blackjack (or slot machine)?
 * I think VDW makes a good suggestion that "to change early misconceptions, a good place to begin is with a focus on possible and not possible -then after playing around with that idea, introduce the idea of impossible, possible, and certain. VDW has some suggested activities p. 457 - I really like this idea, as we talked about yesterday. I think it is a good place to start: define with examples impossible and certain, and then place possible in the middle. I think this might even be good for one of our assessments. I AGREE, HERE. WE SHOULD ASSESS THEIR UNDERSTANDING OF THE COCEPT. IT IS CRUCIAL TO THEIR OVERALL UNDERSTANDING OF EVERYTHING WE WILL DO FROM THAT POINT ON. Either sort these things into impossible, possible or certain, or write examples from life of things that are impossible, possible or certain.
 * I think this flows naturally into the fact that probablities fall in the numerical range from 0 - 1. Do we want to spend any time on fractions, decimals and percentages? YES YES YES. THIS IS IMPORTANT CONCEPTUALLY AS WELL. THIS IS ALSO A GOOD OPPORTUNITY TO REVISIT THEIR WORK EARLIER IN THE YEAR ON THE CONNECTIONS BETWEEN FRACTIONS, DECIMALS AND PERCENTAGES. THEY NEED PRACTICE TO ACHIEVE FLUIDITY, PLUS I KNOW THEY WILL BE TESTED ON THIS IN THE MSP.If not, I think we should specify that our students are already competent or at least list these in our prerequisites for the lesson. Then I think we should develop an understand of a single event probability and how experimental and theoretical are similar and different. I AGREE HERE TOO. IF WE CAN MAKE IT WORK USING THE MEXICAN GAME, IT MIGHT HAVE MORE DIRECT RELEVANCE SINCE THERE IS A HIGHER NUMBER OF LATINO/LATINA STUDENTS IN OUR JR. HIGH THAN NATIVE STUDENTS. Maybe use one of the games from another culture to do this? Seems more interesting than rolling a die or flipping a coin. I'll read more carefully and see if there is a relatively simple game for a single event probablitity. (See my review below)
 * I also like the suggestion VDW makes (on page 466) on 2-event Probabilities... "Students like to explore data about themselves. Consider the context of birthdays of the entire seventh-grade class. Asking students which animal represents their Chinese birth year and which season they were born represents two independent events... (Not that this should be our introductory lesson, but I thought it was a good idea) It shows an example of this in Figure 22.10 which uses an area model for determining probabilities. VDW also points out that "The area approach is accessible to a range of learners, as it is less abstract than equations or tree diagrams." p. 466
 * From A Research Companion to Principles and Standards 5 ideas on teaching probability: 1. Begin teaching at young age. 2. Emphasize the value and importance of building the sample space for a probability experiment. 3. Make connections between probablity and statistics. 4. Introduce probability through data: start with statistics to get to probability. 5. Adopt a problem solving approach to probability. Children need a chance to investigate on their own.
 * Review of mulitcultural games article: 6 different games all linked to probability. They are Hubbub (Native American) a bowl with 6 rocks or beans painted to have two colors. Toss the beans in the bowl If you end up with 6 alike you get 3 points and another turn, 2nd six alike gets 9 points and another turn and a 3rd 6 alike gets 9 points and pass. 5 alike earns 2 points, 2nd 5 alike 4 points, and 3rd 5 alike 6 points and pass. Less than 5 alike 0 poionts and turn is over. Play to 50 points. Mancala: beans in egg carton - start with 4 beans in each depression. The idea to to get as many beans as possible to your home. You pick up beans and put one in each depression until you run out. Player with most beans in their home wins. Toma Todo - play with a 6 sided top from Mexico called a pirinola. Players all start with 10 beans and put one in the center. The fair spinning top says: put in 1, put in 2, take all, take 1, take 2 or all players put in 1. After take all, all players put 1 in pile to start again. Play until one player has all beans. There is also a Dreidl game that is similar only uses a dreidel (4 sided top) instead. Ashii is a Native American game with painted sticks that looks great, but the scoring is complicated (see article). Lu- lu is a Hawaiian game with 4 rocks, plain on back, and either 1, 2, 3 or 4 dots on the other side. Players throw all 4 rocks and get the score for the dots showing. First player to 50 wins. I like either Toma Todo or Lu-lu. What do you guys think? I think a game is great for engagement and group work.