10 Rats, 1000 Bottles of Wine

Assumptions:
1. Rats can and will drink a lot
2. This particular poison is also lethal to rats
3. Rats can be used more than once if needed and still alive
4. The poison is very strong and will not dilute when mixed with non-poisonous liquids

Phase 1: Picking the Best Half of Bottles
Pick 500 bottles to open and pour a tiny amount of each into a small vessel.
Allow the rat to drink from this and wait for it to possibly die (sorry rats :( ).


Phase 2: Narrowing down the Bottles
If the rat died:
    -One of the 500 bottles is poisoned so the other 500 can chill
    -Take 250 from the original 500 and mix it into a different small vessel
    -Allow a different rat to drink from this and wait for it to possibly die

If the rat did not die:
    -The original 500 are safe for human consumption and one of the other is poisoned
    -Open 250 of the unopened bottles, mix and let another rat taste test
    -While waiting for rat to possibly die, allow servants to begin decantering the wine so that it will be ready for when the guests arrive (serve snacks that are thirsty quenching so that guests will not be too thirsty hence buying some time just in case, this wine is the better of the wines so guests can savor it while still sober).

Phase 3: Repeat
Keep this halving process until left with a choice of 2 bottles and thus can determine which is poisoned as shown below.

Phase 4: Safely discard the poison and enjoy the party

Method:

After discussing with others in the class, it was determined that mixing wines was allowed and possibly necessary to figuring out this problem. Being able to mix wines helps to figure out which bottles are safe since you can eliminate many bottles at once. Hopefully the rats metabolize the poison quickly so that it does not take extremely long to test. Testing should not take too long if you can avoid the poison until the end. This process will become problematic if you choose the poisoned bottle in every mixture, which will also mean all the rats will die. If you are lucky and avoid the poison until the end, only 1 rat has to die. Being able to combine quantities and then ruling out entire groups systematically rather than testing one by one is a much more efficient way of sorting and determining something. In this case, it also saves the lives of at least 9990 rats. 

Image Source: https://shawetcanada.files.wordpress.com/2017/06/ratatouille9.jpg?quality=80&strip=all&w=605

Unit Plan Draft

Here is a link to my draft unit plan (lessons included).

Also, a picture for your entertainment.

Image Source: https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQXrWYA0ZSgXLK_Gm2_00AhN-U-Gm-C6PQJkQoHgFxR5ms_PU11

Un-fair Math Fair at West Point Grey

This was a very fun and exciting experience. I was extremely impressed with the projects the students developed. Many of them were ready to share about their calculations and how they determined if their game was fair/unfair. It was refreshing to see the young students excited about math and eager to tell others about the projects they've done. While talking to some the students, they expressed how this project was much more enjoyable than taking a test and I think it was a wonderful way to introduce the students to the subject and apply their knowledge to a real life situation where they were able to work with others and communicate with people outside their class.

I am curious to see how they will discuss their findings. While walking around I noticed that many of the kids who were playing the games would listen to the explanations of the students and then choose the winning combinations which could skew the results when calculating the experimental probability. 

Math Textbooks

How you respond to the examples given here -- as a teacher and as a former studentThe first example of modeling a person's height given their leg length gives a real world application of when linear relations are useful. Unfortunately the book does not mention anything about how this relation was discovered, why it is important, who would need to know this information etc. As a student this was a common theme in my math classes. Although it is nice to see math being used for a purpose, the purpose needs to be clear and appeal to the student's interest and get them thinking rather than just having an application for the sake of applying math. By getting the students involved in the application process would also help solidify the idea for them. If they had a choice in what they could explore, discuss with their peers what types of data they would need and determine how the data is related to each other would be better use of their time and application. Having a more natural setting rather than a forced problem that is set up at the end is much more meaningful. As a teacher, I would rather implement "word problems" as projects that students can fully experience for themselves rather than just doing the same calculations as before but with words attached.

What are your thoughts about the reasons for using/ not using textbooks, and the changing role of math textbooks in schools?
I am not entirely convinced by the exclusive/inclusive imperatives suggested by Rotman. Although the authors can write a textbook however they see fit, teachers have some autonomy in how they present the material in the textbook to the students. For example, the teacher can make exclusive imperatives inclusive by having students work in groups to calculate, copy and write things down. Having multiple students perform a calculation in order to see if they both arrive at the same answer, compare the processes they took to arrive at the answer and then discuss what their answer means with each other. Furthermore, if students are working in groups they can each work independently to come up with a way to represent their understanding of a topic and then share their representation with others. 

Another confusing part of the article for me was near the end when the author mentioned the way mathematics textbooks function in school settings. I don't believe that depersonalizing mathematics for communication is the most effective way of communicating the content. Letting the students have a personal connection to mathematics would allow them to become interested and want to learn about it and communicate math with others. I'm not sure if a textbook will inspire students to learn more about math but it can help make the communication and example creating process easier for teachers. 

Also, textbooks seem to take out the rich culture and history behind the math making it seem like a very dry and boring subject that is purely about calculating answers when is much more than that. During word problems, some textbooks use names from different cultures and include women which is a good start but it is very surface level and does not offer substance to learning.

The Scales Problem

What are the values of the four weights?
At first, I thought maybe the weights would be in base two since computers use base two to describe all numbers. However, after doing some trial and error it did not seem possible to get 4 numbers to sum all numbers 1 to 40.

I then switched to equations by letting each number be a variable a, b , c, d. Assuming that the weights could be added to both sides of the scale and that one of the weights had to be 1 gram, I proceeded to try and solve the equations. After doing a few equations for 40 - 34, I noticed either we needed one of the weights to be 3 grams or the difference between two of the weights had to be 3. Since it was easier to assign one of the variables to 3, I decided to continue my calculations with two variables and two known values. Eventually, I got to a point where the two other variable needed to sum to 36.

I began to narrow down the possibilities by trying to find values that are not possible for each pair that sum to 36. One pair that I struggled to find any discrepancies was (9, 27). I confirm that this was a solution, I found representations for each value 1 to 40 using 1, 3, 9 and 27 which ended up working.

Are there several correct solutions?
I'm not sure if there are several correct solutions but I did notice that this solution had values that were all powers of 3.

How could you extend this puzzle to help your students understand the mathematics more deeply?
We could use an actual balance scale to physically see what is happening and how the scale would balance. We could also try to see if we could do this same exercise except with more weights for a larger range (1-100 grams perhaps?) and see if we could use powers of 3 or if the 40 grams was just a special case. We could see if there are other solutions if we allow the number of weights to increase and then discuss why merchants would want as few weights as possible.

Cleaned up version of my rough work:

Pro-D Day - CUEBC

I attended the CUEBC (Computer Using Educators of BC) pro-d day event and it was very interesting. I was always very curious as to what the teachers did during pro-d days and was excited to find out. This event was a little overwhelming because there were so many different sessions to choose from and it was difficult to pick just one. I ended up learning about different apps that Microsoft offers for teachers and classrooms such as FlipGrid, OneNote, Skype in the Classroom and premade STEM lesson plans. Although this conference was designed for all teachers, it was interesting to see what technology is available for classroom use and it got me thinking on how I could potentially use and adapt some of these technologies to fit a high school math classroom. I also attended a session that described a special education version of the game Minecraft that could be used in a class that could be adapted depending on the subject and grade level. For the last session, I attended a session on self-regulation. This was a very interesting session but I found it difficult to see how to transfer many of the suggestions offered to a math class. The presenter was a business teacher and that environment allowed for more self paced learning where students can more easily determine there plan and pace for the class than a math class were there is more pressure to finish content on time and to be more or less in sync with the other math classes. However, the online exit slips and self evaluation forms could be useful if the students all have access to technology because it will force them to reflect and answer the questions rather than loosing the paper version and skipping the long answer/reflection portions.

Elliot Eisner and The Three Curricula

Two or three 'stops' you have in this article

1. "One of the first things a student learns - and the lesson is taught throughout his or her school career - is to provide the teacher with what the teacher wants or expects."
This quote reminded me of the reading about grades for our inquiry class. By telling students what is going to be on the test, we are conditioning them to pay attention to these topics and revisit them later when studying. This type of school culture is persistent throughout an entire lifetime. Children try to find out what they parents want and expect, they then try to give their teachers what they are looking for and often ask other students or former students such as older siblings what that particular teacher's tests are like and what they need to do in order to get a certain grade. As the reading points out, students spend an awful lot of their lives in school exposed to this culture and it continues throughout their lives. Whether they go on to post secondary and try to do the assignments so as to please the professor or if they go into the workforce and do whatever is necessary to please their boss and maintain their employment. Students are stuck in this very survivalist culture of trying to make it to the next course with a particular grade, or trying to get into university.

2. Implicit Curriculum
One thing that really stood out to me was the message that is conveyed immediately to the students by the physical set up of the classroom. If the desks are separate and in neat rows, the students know that it is an individualized learning environment with little opportunities to talk with others. On the other hand, if the desks are grouped together, students know to expect more group work and collaboration. This was less surprising to me than how the scheduling of classes affects how the subject is seen by a student. While I was in high school, our blocks were rotated throughout the year so that each subject was experienced at different times of the days. It was different in elementary school where art was typically reserved for afternoons. Many times while I volunteered at a kindergarten class the students were working on art or play in the afternoons while they had reading and math lessons in the mornings. This type of scheduling definitely contributed to the status of art and reading in the classroom. When a lesson was not finished on time, the math class or reading class would carry into the art class cutting it short. Certain students were unable to begin their artwork until after they completed their math/reading assignments. However, if an art activity was not finished it was seen as acceptable to just leave it for another day or the student could finish if they were able to complete the math/reading activity early. I was surprised that none of the students questioned why the math/reading was prioritized over art. Sure they were disappointed and complained but they never argued that art was as important as math/reading. It is amazing how early these subtle messages take affect.

3. Null Curriculum
As a high school student I always wondered how the duration of the subjects was determined (e.g why we needed 5 years of ELA, 4 years of math, and no mandatory art/music/dance/film studies etc.). What was so special about ELA, math and science and so dispensable about art and music?Having this conveys the message that ELA and math and science are superior subjects to art and music. Only students who are interested in pursuing art or music will immerse themselves in those subjects or enroll in special programs so as to get into a post secondary program for these areas. However, this implies that students other students will not benefit from having these types of experiences or classes throughout their schooling. Likewise, students entering art or music programs will see no need for math and science and determine it as a hurdle to jump over rather than something that could assist their studies and improve their learning. 

Another aspect of the null curriculum that struck me was the omission of certain topics within a subject. For example, conics was a section that was taken out of the curriculum while I was in high school yet it was something I encountered during my university studies. Why was this topic removed and what was put in its place (if anything)? What about conics did the curriculum decision makers feel was so unimportant that it was removed entirely?

Ways that this might expand our ideas about what is meant by 'curriculum'. How does the mandated BC Provincial Curriculum connect with Eisner's ideas?

Before this reading I equated curriculum with the explicit curriculum but I realize there is much more to curriculum then simply a list of requirements. Curriculum also encompasses what is left out, how the material is presented and how the environment is set up for the students among other things.

BC is helping to connect with Eisner's ideas by providing more opportunities for other types processes for students such as increasing collaboration, increasing ways of knowing such as Indigenous ways of knowing, using different modes of instruction, different modes of assessing. 
It also focuses on decreasing the null curriculum by adding financial literacy to the math curriculum. Students will leave high school with some basic understanding of finance such as knowing how they will be taxed and how to calculate their hours when completing a time sheet.