Math Puzzle: Giant Soup Can of Hornby Island

 A geometric puzzle with real-life connections:

On Hornby Island, BC, local artists were commissioned to paint the volunteer fire department's water tanks in a dozen different locations around the island. Artist Pix Sutherland noticed that this tank was in exactly the same proportions as a Campbell's soup can and painted this (controversial) tribute to Andy Warhol's famous 1960s soup can pop art paintings.

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?

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.  

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).

Final Reflection

    I had a lot of fun in this course learning from class discussions, activities, and my peers.  I particularly enjoyed working on puzzles and non-curricular activities that could be incorporated into lessons as enrichment.  While unit planning, I have been sprinkle in some puzzles that I could put up on the whiteboard for students who like an extra challenge.  I found these activities and puzzles interesting, working with my peers and looking back on the semester, they are the most memorable.  Therefore, I believe students will remember learning in math class when things they find interesting are done in class.  

    I enjoyed learning through microteaching.  It was a gentle introduction to teaching in front of a group of people who I just met for a short amount of time on something that I knew a lot about.  The non-curricular teaching was interesting, where I saw natural teachers teaching another topic other than math.

JOHN MASON: Thinking about Proofs

    Mason (2001) mentions a study that "found recently that most students (their study was with nearly 2500 children aged 14-15 in 90 schools) base their confidence in the truth for a finite number of cases (an empirical perspective of proof)".  Mason mentions that checking on a finite number of cases has use in gaining confidence before taking on a proof by English standards.  By French standards, this process is the proof, though it may not always be formal.  

    I think finding a proof involving all cases, by induction, is satisfying, knowing that there are no exceptions.  However, to prove to myself, I find that a finite number of cases is enough to convince me.  I often fall into, what Mason mentions in their article, the '"just because" or "it just is", or "X said so" thinking' when it comes to math.  The way I was taught math in high school, I think, had an impact on how I think about mathematics.  However, I am trying to be more curious and inquisitive about mathematics and dig deeper into proofs, as well as history and background.

    Mason brings up a good point that I relate to, saying that when one tries to convince friends and associates, who in turn question and cast doubt on my explanations, I "respond 'in flight', by augmenting, elaborating or offering further examples to help them ‘see what you are trying to say’".  I often give up because I don't think I have sufficient knowledge in math.  However, if it is something I am confident in, Mason says that "you try to express your reasoning, your  way  of seeing,  so that it stands alone without the need for you to be present.".  I aspire to be able to explain in a way where there is no need for interpreting, elaborating, or augmenting and my reasoning leaves the person who receives the information thinking about what I said.

Opening up Closed Quesitons

1. If you take (5 > 3), and do this (5+2 >3+2), is it still true?

    Instead, use ideas from open middle math to generate an example.  

    I think this would open up the question, but I don't think it is necessary to show as an example for students.  Perhaps, students can try open middle math to investigate on their own if the multiplication rule of inequalities is true for every set of numbers.

2. Solve these question: 2x < 5 + 2x and 5x < 5x - 6.

    Instead: come up with two linear inequalities: one that has no solution, and one where the solution is all real numbers.

    I think opening up this question is a lot better than the original. As an way of assessing students' understanding, the open question assesses not only students' ability to recognized "no solution" and "solution is all real numbers", but also their knowledge about how to construct a linear inequality.

3. What happens when the variables add up to zero?

    Instead: add "give an example".

    I think getting students to come up with their own examples gives them opportunity to show and apply their knowledge.

Puzzle: Market Vendor

A market vendor sells dried cooking herbs in whole-number amounts from 1 to 40 grams. The vendor has an old-fashioned two pan weigh scale, and has exactly four weights of different amounts that allows them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight.

I assume that x must be 1 so that if y, a, and b were even or odd, I could easily make them one larger or smaller.

I will skip a 2g weight, since it can be represented by 3-1.  So, let weight y=3.

So far, we have x=1, y=3.

Since 4 can be represented by x+y, we will skip this number.

5 can be represented by 9-1-3, so let a = 9.

We continue...

• 6=9-3
• 7=9-3+1
• 8=9-1
• 9=9
• 10=9+1
• 11=9+3-1
• 12=9+3
• 13=9+3+1

14 can be represented by introducing b=27.  So, 14=27-9-3-1

We now have the only combination of four numbers that can be used to measure weights of 1-14g.

We also see that 27+9+3+1 = 40, so good sign.

We continue to see if this holds true...

• 15=27-9-3
• 16=28-9-3+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
• ... (we can stop here since we have proved number 1-13 using only x, y, and a and since we can represent 27 with just the 27g weight)
• example: 35 = 27 + 8 (8 was proved to be 8=9-1, so we can write 35 = 27+9-1)

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Some observations

• numbers 1-4 can be representation by x=1 and y=3
• numbers 5-13 can be represented by x=1, y=3, and a=9
• the numbers (1-4) created from x and y can be added to a to create a wider range of numbers (5-13)
• numbers 14-40 can be represented by x=1, y=3, a=9, and b=27
• the numbers (1-4) created from x and y and numbers (5-13) from x, y, and a can be added to b to create a wider range of numbers (14-40)
• the four weights found are ascending powers of base 3 (exponent=0, 1, 2, 3)

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To extend this problem, I would question whether this pattern of ascending powers of base 3 can be translated to herb amounts greater than 40g and whether this problem would use base 3 as well with greater powers.

Assignment 3: Unit plan

EDCP 342 Assign 3: Unit Plan- Michelle Li

Wagner & Herbel-Eisenmann on math textbooks

    I am surprised at the implications made by the author about how language affects students' positions in relation to mathematics.  The articles states that the absences of first person pronouns affects nature of mathematical activity" and "also distances the author from the reader, setting up a formal relationship between them".  I can see how this can contribute to students having a hard time relating themselves to the math content, that may already seem so static and theoretical on paper, rather than dynamic and practical in the real world.  

    The author also suggest that the linguistic choices in textbooks affects students' positions in relation to their experiences of the world.  As one flips through a textbook from front to back, it assumes the reader is progressing with it, though, of course, this progression is different for every student.  The author poses an interesting question: "Would the reader think that his or her everyday experiences matter less than their mathematical experiences?".  

    I think textbook use in classrooms can be beneficial to both the teacher and students.  It can offer a different perspective and give students another way to approach mathematical concepts and problems.  However, I think teachers and students should be flexible in their thinking and teachers should not rely on the textbook content and progression as a sole guidance to drive the course content.

Sketches: Truncated Cube and Icosahedron

 

Dave Hewitt’s secondary school algebra teaching

    In the number line demonstration we watched in class, I liked the idea of adding sound and movement to show the numbers as they increased when moved to the right.  Dave Hewitt didn't write on the board, making it very simple, letting the students imagine the numbers through sound and movement.  I think it makes it more engaging, maybe because students are used to seeing math visually and statically on paper or on a projected presentation.  

    I like that the whole class was encouraged to say the answer as a group, instead of the teacher telling the students or the teacher asking one student.  I find that students are more willing to learn when it is from their own peers.  Perhaps hearing the voices of their classmates would help students remember information in a more meaningful way while connecting it to a group activity they did in class.

    I don't think Hewitt mentioned to the students what they were learning.  Hewitt simply presented a pattern to the students by showing them with a ruler and the wall.  Students slowly caught on and soon, the whole class understood without any explanation from the teacher.

    I can see myself incorporating all three points mentioned above in my classroom.  Math can get boring when it is practiced and learned in the same way, especially when students are stick in their seats for the whole period.  Learning math in various ways that use movement and sound can help attract students' attention.

Arbitrary and necessary

    Hewitt describes something as being arbitrary is was given through external sources such as teachers, books, the internet, etc.  To be learned, it must be memorized and when a person is challenged by the question "why?" to that arbitrary knowledge, they are stumped.  

    Necessary aspects of math are things that can be figured out on your own, unlike the arbitrary aspects.  Using prior knowledge, new information and concepts can be worked out to be true because of the process one goes through.

    I think it is problematic if students start seeing all math concepts and terms as arbitrary.  When students stop questioning and stop being curious about the "why", math becomes abstract and meaningless.  Students can't make personal connections to the material they are learning and this may negatively affect their understanding and interest.  I believe it is important to constant question students' thinking process when looking at math concepts and terms to encourage inquiry.  

FLOW in Math

    I experience a state of flow mostly through learning music.  I can sit with a new piece of music that is challenging, but enjoyable, working through the tough sections and I wouldn't notice that I spent a few hours sitting with my instrument.  However, this doesn't happen every time I play music, only when I feel like learning something new.  

    Recently in one of my math courses, we completed a high school math test.  Class time that usually went by very slowly, went by quickly because I enjoyed doing math that was familiar but evoked thought because I hadn't encountered it in a long time.  

    I believe a state of flow can be achieve in secondary math classes.  New information must be presented in a way that is approachable to students and that invites students to be creative in their thinking.  In other words, inquiry-based learning environments can be a good tool for this.  When all the answers are given or when full instructions or procedures of how to solve a problem are given, it does not encourage students to think further about the questions.  Using various activities and opportunities for group work can help with exposing students to new perspectives that they haven't seen before, therefore challenging students' thinking.

EISNER: Three curricula all schools teach

     Eisner points out that values "are expressed in the kinds of illustrations that textbooks contain, in the language that is employed, and in emphasis that is given to the characters that constitute the stories that are read".  I find this interesting and I connect it to something I learned in psychology in terms of schemas.  Growing up, to me, curriculum meant the topics and content taught in classrooms.  New information is funneled into the minds of the students and that is the end of it.  However, in psychology class, I learned that there are two types of learning related to new information.  There is assimilation and accommodation.  New information either gets categorized into existing schema and possibly changes it or new schemas are created.

    The BC curriculum is split into different components that relate to each other.  The curriculum is based around the core competencies and describe the intellectual, personal, and social and emotional proficiencies students need for deeper learning.  Content is only one part of the curriculum and describes what students will know.  In any good learning environment, there will be more that just content learned.  Eisner's point about the values learned in schools can be seen in the big ideas that students will understand and the curriculum competencies that show what students will do and how they apply the skills learned. 

Battleground Schools Reflection

   I am surprised that the reading mentions that "there is no shame" in being "incapable of doing and understanding mathematics" and that it comes with positive social valuation.  The article mentions that during the progressivist reform, the reputation of math was that it taught students procedures to get to an answer but provided no explanation about the "why".  Therefore, many people thought of math to be useless in real life.

    John Dewey believe that there is a "split between knowing and doing, or abstract and applied knowledge".  He states that students "must engage in doing mathematics as part of a reflective inquiry" in order to become more knowledgeable.  This was during the early 1900s and I am surprised that the idea of inquiry and curiosity in mathematics for school mathematics was examined.  I feel that the current curriculum has only begun to incorporate more inquiry and project-based approaches in math classes.  Therefore, it has taken a long time to integrate these practices into classrooms. 

Oct. 21, 2021 Pro-d-day + Reflection

    After doing the TPI activity in our 342 class, I noticed that I did not see strong connections between mathematics and social issues.  I was glad to see presentations on social issues in the BCAMT conference.  

    The presentation I watched was "Challenged Accepted!  Use Project Based Learning To Explore Social Justice" by Carl Oliver.  Carl shared various activities that could be integrated into math classes and presented different frameworks that made sense to me once they were brought up.  

    The general outline of creating a project-based learning environment encompassing social justice is situation, exploration, and presentation.  I found it interesting that Carl shared Gutierrez's framework of rehumanizing math through connecting students to math by ways of history and culture.  It engages students and encourages them to think about themselves participating in making math in the real world.  Next is exploration by means of describing accurately a situation, making a prediction, designing something, or creating a product.  Lastly is presentation where students are given choice and autonomy to inquire and be creative.  

    I am surprised that these types of projects, that I have only seen arts, English, or social studies classes, can be so applicable in a math context.  I am excited to learn more about it and potentially use these ideas in my classroom.

Probability Microteaching Reflection + Feedback

Feedback from peers

Reflection:

While doing the lessons plan, I found this teaching activity to be a bit confusing at times because I wasn't sure how much content we could fit into 15 minutes.  The BC curriculum has many big ideas and curricular competencies and just focusing on one or two was a bit of a challenge for me.  

For the presentation itself, it did not feel rushed- something that I was anticipating.  For myself, I could have spoken louder and take more time between thoughts and explanations to check if the students understood.  I could do this by asking more concept check questions.

Microteaching: Workplace 10 Experimental Probability

 Lesson Plan

Group: Alan, Do Hoon, Michelle

Rhythm Microteaching Reflection

Rhythm microteaching peer feedback

Reflection:

    I appreciate the feedback I received from my peers.  It helped me realize my strengths and weaknesses when teaching with a time constraint.  From experience, I know I have difficulty estimating how much time an activity will take and choosing a topic that I could fit into 10 minutes for this microteaching was a challenge.  With more practice, I believe I will be able to better gauge the appropriate amount of material as well as the appropriate difficulty of the material.

TPI Results + Reflection

 

Talk about any surprises in your results

    I am surprised by my social reform results.  I do believe in preparing students to be engaged in and participate in society.  I was thinking in terms of teaching in math and I couldn't see any connections between the two.

Write about any interesting questions the TPI results raise for you as a new teacher.

    I wonder about my results for social reform and how I can integrate that into my math teachable subject.  I will need to do research on what social reform is referring to and find ways I could address those topics in math class.

Microteaching: Rhythm in Music

Lesson Plan

Powerpoint

Math Puzzle: Dishes Problem

Solving by algebra

Define variables and gather information given:

• Let number of guests = g
• Dishes of rice 
• R = x/2 (take the floor)
• Dishes of broth
• B = x/3 (take the floor)
• Dishes of meat 
• M = x/4 (take the floor)
• Total number of dishes 
• T = 65

T = R + B + T

65 = (x/2) + (x/3) + (x/4)

65 = (6x/12) + (4x/12) + (3x/12)

65 = 13x/12

780 = 13x

60 =  x

Since x is a whole number, we do not need to worry about taking the floor of R, B, M, and their sum.

Check: 

• R = 60/2 = 30
• B = 60/3 = 20
• M = 60/4 = 15
• T = R + B + M = 30 + 20 + 15 = 65
• Therefore, there are 60 guests if there are 65 dishes.

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Reflection

How could you solve this puzzle without algebra (or at least, without the algebra we are used to)?

    Without algebra, I would use a trial and error system.  I would see how many dishes there would be if there were 12 people, for example, since 2, 3, 4 can all go into 12 equal.  

If there are 12 people, there would be 6 dishes of rice, 4 dishes of broth, and 3 dishes of meat.  Therefore, there would be 13 dishes for 12 people.  

    I observe that the number of dishes is a number slightly bigger than the number of people.  I can use the number of dishes, 13, and double the number of people until I get close to 60 dishes:

If there are 24 people, there will be 26 dishes. 

If there are 48 people, there will be 52 dishes.  

13 dishes are missing to get to 60, and we know that if there are 12 people, there will be 13 dishes.

So we can add that to the 52 dishes for 48 people above.
So for 60 dishes (13 + 52), there will be 60 people (48 + 12). 

 

    This method is similar to how Babylonians would "multiply" numbers by doubling numbers and repeated addition. 

Does it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)

    Yes, I believe it does.  Similarly to movies and other media, when there is representation of one's own culture, especially a minority group, it an evoke a connection to the problem.  For this dishes problem that came from China, I paint a picture in my mind that is reflective of my experiences.  I imagine a big seafood restaurant where a banquet is going on.  The chefs in the kitchen are preparing multiple dishes at a time and ringing them up nonstop.  It's almost silly to think that a math problem can make me reminisce about my past travels to China, but I think it makes the problem more meaningful to me.

Do the word problem or puzzle story and imagery matter? Do they make a difference to our enjoyment in solving it?

    I find word problems accompanied with a story and imagery enjoyable.  I can see it in context, whether it is realistic or not.  I can imagine the world through mathematics, and I find it easier to think about.  I believe it is similar to students learning about fractions by looking at a pie.  Another example is using apples in place of just numbers to teach addition or subtraction (I have 2 apples, Jin has 5 apples.  How many apples are there in total?).  There is movement in the problem which can make it easy to imagine when I can use physical objects to visual the problem.