The Scales problem

 There is one way that I can see to measure each integer up to 40 using only 4 weights. They would have to be weights of 1, 3, 9, and 27. The equations below calculate how we can arrive at each desired weight (1-40) using the 4 different weights (1, 3, 9, 27).

But why? Lets have a look. We need 1 to weigh 1, no way around that. If we put 1 on the other scale, the one with the items being purchased, we can use this weight of 1 as negative 1. So we don’t need 2 because we can use 3 and negative 1 to make 2. This lets us avoid having a weight of 2. Instead, we use 3 and negative 1. So we need 3. Then we can calculate 1,2,3,4 no problem using 1 and -1 and 3. Next, instead of a weight of 5, we use our weights, 1 and 3, on the other scale so negative 1 and negative 3. Therefore, we can skip 5,6,7, and 8 and include just 9 because we can get to 5 with 9 combined with both negative 1 and negative 3.

To extend this problem more deeply we could look at a general solution for calculating the next heaviest necessary weight to continue weighing things that weigh more than 40. To do this, we need to look first at the weights that we have already determined are necessary: 1,3,9 and 27. Upon first glance, I see that this could be rewritten as 3­^0, 3^1, 3^2, 3^3. Looking at it this way, we can see that the next weight will need to be 3^4. 3^4 is 81.

1,3,9, 27 are our tools that we can use to calculate each integer from 1-40.

1 = 1

2= 3-1

3=3

4=3+1

5= 9-3-1

6=9-3

7=9+1-3

8=9-1

9=9

10=9+1

11=9+3-1

12=9+3

13=9+3+1

14=27-9-3-1

15=27-9-3

16=27-9-3+1

17=27-9-1

18=27-9

19=27-9+1

20=27-9+3-1

21=27-9+3

22=27-9+3+1

23=27-3-1

24=27-3

25=27-3+1

26=27-1

27=27

…

40=27+9+3+1

 To keep going we would need 81 as our next weight.

 

Entrance Slip for Monday, Nov. 23 article on textbooks

 

•How you respond to the examples given here -- as a teacher and as a former student

As a student, I agree language is super important, every text has its own viewpoint so teachers must interrogate texts, and pictures help to impact mathematical experiences. As a student and a teacher, this article made me really think about why math texts are written the way they are. After so much experience with math texts, I can see why math texts could benefit from a more personal touch, such as including personal pronouns. As a teacher, I like what the article mentions about structuring texts to be about 10 – 20% inclusive imperatives such as describe or explain and the rest of the imperatives are exclusive like calculate. Both types have a role to play and both are necessary for a complete understanding. I find oftentimes in math, people read a math text and think they understand what they read but then have trouble when it comes time to apply it to a question. Inclusive questions offer these students an opportunity for their understanding to be demonstrated outside of the typical, numerical setting, (which has validity as well,) and opens up thinking to further possibilities as well.

•What are your thoughts about the reasons for using/ not using textbooks, and the changing role of math textbooks in schools?

I think that textbooks are helpful for learning math and I wouldn’t want to see them removed from their role as a foundation of math class. This is because as a math student, I often learned more from my text than from anywhere else. In math classes and non-math classes without a textbook, I often felt lost or disconnected from the material or that I was being deprived of the opportunity to truly learn anything.

I think the reasons for using math texts are to provide thorough explanations of the material as a resource for students to learn from as well as questions for them to do to develop their own mathematical skills.

I think the reasons for not using math textbooks may be because of the cost of the books, which are often large. They may be seen as unnecessary now that material is available online, which is part of the changing role of math texts. Furthermore, online videos are being used more and more to replace texts as a way of providing students with a resource for mathematical explanations. I feel that videos can be good for showing alternative approaches to solving math problems as well as for slowly, clearly and methodically going through very complex or detail-oriented explanations. Videos also offer unique opportunities for powerfully demonstrating ideas with visuals. However, good videos can be tough to find and making good videos is a lot of work. Therefore, I would say stick with the textbooks for now but maybe down the road when more good videos are available, teachers may be able to compile them to be a sort of video textbook. However, I think it is important that students don’t have to go chasing down their own material.

Exit Slip Nov 16. Microteaching Reflection

I feel that our microteaching lesson on graphing a linear equation did not go very smoothly. We tried to cover too much material from too many different categories. To start the process, I suggested we do graphing a line with the equation y=mx+b and using desmos for a visual understanding. However, when we each went off individually to make the lesson plan and the slides, the topic ballooned to include functions, introducing new notation, and including other ideas like absolute values and algebra. I tried to stress to my groupmates the need to keep things simple, but met resistance over what a grade 10 student has already learned. With so many concepts, our lesson did not have any time left for the fun part, playing with desmos. Overall, my takeaway from this microteaching lesson is the difficulty of working with other teachers to try to teach a math lesson as a group. I think this is because everyone has their own unique way of understanding math, deciding what aspects of math are important, what to expect from students, and how to go about teaching math.

Next time, I will try to focus more on simplicity and staying focused on teaching one thing at a time. It is already difficult enough to grasp one mathematical concept at a time, so trying to grasp more than one at a time is almost impossible. Also, I will focus more and give more lesson time to the interactive, fun aspects of the lesson (such as desmos) as opposed to more time standing and delivering. This is because math is not learned by simply listening. It is learned kinesthetically, by doing it, which is what should be done by students in math class, not just listening.

 

Microteaching Group Project

Here are the links to our lesson plan and presentation slides.

 

https://docs.google.com/document/d/1ST2O2muK88lWVg1SH33gCe6wG3jpW_85/edit

https://docs.google.com/presentation/d/1hpTqnFjh0ofj-Sw_AmRU5kHEuvj5BaHSc36J5iUtKnM/edit#slide=id.ga992be6037_0_38

Tomato Soup Can

 

Q1. Lots of math problems could come out of this story. Here's one: given the size of the actual Campbell's Soup can (of normal size) and the height of the bike in the photo, what are the dimensions of the volunteer fire department's water tank? What is its volume? Does it hold enough water to put out an average house fire?

The average height of a bike is 31 inches (according to Google). The bike in the photo looks to be approximately one third of the height of the can. Of course, the can is on its side so 31x3=93 inches represents the diameter of the can. Therefore, the radius of the can is 46.5 inches, quite big. Radius is the most important variable when considering volume because it has the highest power so the fact that this water tank has a big radius means it probably can hold enough water to put out an average house fire.

According to Google, a house fire requires about 1 gallon of water for every 3 square feet of house. To be on the conservative side, which is where you want to be when it comes to emergency situations like house fires, lets say an average house on Honrby Island is 2000 square feet. This is an educated guess because this statistic is harder to find. So, a tank would need to hold 2000 divided by 3 gallons, which is 667 gallons.

The can looks to be about twice as high as it is wide so the height is four times the radius. So the radius is 46.5 inches, the height is 186 inches. Therefore, the volume is 5,470 gallons. While this number may seem precise, it is not in fact very accurate because of all the estimating that was done to arrive at it. Nonetheless, it appears that the tank is more than sufficient in terms of holding enough water to extinguish a house fire many times over.

Q2. Your task: work on this puzzle yourself, and let your 'teacher bird' and 'student bird' notice how you approach it, where you can use reasoning and where you need to research, where you get stuck and un-stuck.  

My teacher bird always seems to take a backseat to my student bird, as was the case with this problem. I approached it with a mixture of educated guessing, unreliable google research, and with consideration for the real-world context of the problem. This type of problem reminds me of the type of questions that come up while building things and planning/designing, which is always fun. I got stuck looking for the exact dimensions of a Campbell’s soup can because the only dimensions that I found were 51cm by 61cm but that can’t be right because it looks much taller than that ratio of width to height.

Q3. Then work on either: (a) extending this puzzle, or (b) coming up with your own puzzle for secondary math students based on a real-life observation you have made (and include a photo or graphic to support it).

I think it would be interesting to extend this puzzle to consider how much soup that would be if it was actually full of soup. 5, 470 gallons of soup, how long would that last if you ate a big bowl of soup every day for lunch. How many tomatoes would go into making 5,470 gallons of tomato soup? How much would that many tomatoes cost compared to how much would that much soup cost? What if we buy in bulk?

Entrance Slip Nov 9 Flow Video

Some things that jumped out to me from the Flow video were: the state of flow, the value of intrinsic motivation, and work for its own sake. The state of flow is something that I think we've all experienced. Watching the video, I thought of my own experiences with the state of flow and immediately I smiled thinking about snowboarding down a mountain on a day with perfect snow conditions and weather. 

While snowboarding, I often achieve a state of flow. Although it is not a work-related endeavor and therefore does not benefit society, snowboarding does allow for a flow state and meets all of the criteria. It is outside of the everyday, highly-involved, requires inner clarity in order to balance, you feel are succeeding as you do it, there is serenity being out in nature, timelessness from the high speeds, and it is intrinsically motivated.

The video also got me thinking about intrinsic motivation, which is very important. Extrinsic motivation can cause people to stop doing things that they enjoy if the extrinsic reward stops coming. Therefore, intrinsic is the better option. However, it is harder to fuel. In the classroom, one way that I will try to fuel student's intrinsic motivation to learn math is with daily reminders of the connection between math and money. I plan to teach math from a financial literacy perspective because that will motivate many students to learn math. It may be argued that money is an extrinsic reward and therefore using it to motivate is not increasing intrinsic motivation. To that argument, I would say that while money is itself an extrinsic reward, in this situation, I am not rewarding them with money so it is not an extrinsic reward in this situation. They are being rewarded with mathematical knowledge in exchange for their efforts, which will one day provide them with an extrinsic reward, money. It is an extrinsic source for intrinsic motivation.

Work for its own sake is the ultimate in intrinsic motivation. That is when you do something because you are interested in it and want to see it done and not for any reward or advantage from it. When I think of work for its own sake, I think of things that better the world, especially as it relates to nature and the many people who are happy to do things to help nature, such as gardening and tree planting.

 

BC Secondary Math Curriculum

 

 (1) Please write about two things that were new to you or surprised you from the curriculum orientation guide and/or glossary of new terms, and

I was gladdened to see “constructivism” in the glossary which is the connection between learning through experiential, inquiry-based, project-based and other forms of active learning because I value project-based learning as a fun and effective way of teaching STEM. This aligns with what we’ve been learning about the value of embodied learning as an effective tool for learning.

“Habits of mind” is an interesting term that I was not previously familiar with. Reading it got me thinking about how to teach different habits that help with problem solving and encourage an open mind towards thinking critically in new ways. Some habits of mind that I have already been encouraging in my own math teaching include assigning a letter (variable) to unknown numbers and the divide and conquer approach to problems which involves breaking large problems into smaller, more manageable problems.

 

 (2) Create your own schematic chart of possible pathways in the courses of the BC Math curriculum from Grade 8 - 12.

 

I tried to breakdown each year into about 9 units so each unit will be about 3 or 4 weeks of the school year.

Grade 8:

Quantities, averages, estimates

Ratios, and percent

Financial literacy and best buys

Independent and dependent variables

Linear relationships

Substitution

Fractions

Theoretical probability

Experimental probability

 

Grade 9:

Rational numbers

Squares, square roots, cubes, cube roots

Pythagorean Theorem

Surface area, volume, nets

Exponents

Polynomials

Two variable graphing

Spatial proportional reasoning

Statistics in society, central tendancy

Financial literacy, budgets, transactions

 

Grade 10:

Financial Literacy, gross pay, net pay

Arithmetic sequences

Prime factorization

Polynomial factoring

Functions, data and graphs

Linear function equations

Systems of linear equations

Trigonometric Ratios

Metric conversions

 

Grade 11:

Rational Functions

Forms of reasoning

Angle relationships

Modeling to scale

Graphing inequalities, quadratics

Graphing systems of equations

Optimization

Applying statistics

Financial literacy, interest, investments, loans, decisions

 

Grade 12:

Rates of change

Geometric sequences and series

Constructions, conics, fractals

Graphing polynomials, logarithms, exponents and sinusoids

Transformations

Logarithmic operations

Regression analysis

Combinatorics

Probability and expected value

Financial planning