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.