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.

    

Math Art: Artwork by Owen "OAT" Rohm + Reflection

Artwork by Owen "OAT" Rohm

Math Art PowerPoint

Reflection:

    After this project and participating in my classmates' presentations, I realize how many different activities can be done throughout the bc math curriculum.  In my experience, high school math was taught only through notes and videos.  I believe I would have enjoyed and would have been able to appreciate it more if it was presented through art.  

Group: Yi Wei Liu, Do Hoon Lee, Michelle Li

Letters from Future Students

Letters

Hello,

    I was in your grade 10 class this year.  Just in case you're my teacher again in grade 11 or 12, I wanted to let you know that I didn't enjoy the way your taught my class.  Teachers are always looking for feedback right?  

    I don't know why, but your classes were so long.  Maybe I was just not interested in the topics you were teaching.  I did not like the projects and discussion you made us do.  I didn't have anything to say and I don't see the point of these in a math class.  The projects were not very interesting and it took forever for my group to pick a topic.  I also didn't really understand what you were saying most of the time.  Maybe I'm just not good at math.  

Thanks

Hello,

    I was in your grade 12 math class last year and I was the one always sitting at the back.  I wanted to thank you for believing in me even when it took me forever to understanding anything in your class.  I hated math all my life, but because of your class, I hate it a little less now.  My favorite parts of your class were when you talked about how much you love math, when we went outside for class, and of course the fun videos you showed in class.  Though I didn't enjoy it all the time, your mini one-to-one sessions were really helpful!

    I'm in college now, but not in the sciences!  When I was writing a reflection for school about my learning, I thought about your class and all the non-math related things I learned from you.  You cared so much for your students and I could tell you always cared about me and were willing to support me.  You reminded me that any improvement is progress.  At the end of the year, I found the motivation to keep going and even did a little exploring on youtube for math videos!

Thank you so much!

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Reflection:

    I worry that even with the strategies I learned from teacher education, I won't be able to implement them effectively in my classroom.  I didn't like discussion even small groups when I was in high school, so I have reservation about using them in my classroom even though it can open up exploration and promote critical thinking.  I also worry that I won't be clear enough in my explanations, whether it be in a math concept or giving instructions on an assignment.  What I fear most is that because of my ineffective teaching, students will come to believe that they just aren't a "math person".

    I believe that caring student-teacher relationships are essential for good learning and teaching.  From my experience, students are more willing to pay attention and learn when they know the teacher is open, supportive, and caring.  My main goal in the classroom is not to get students to love math, but to explore what math is and get something out of it.  That may include being challenged by problem solving questions or learning to take on a challenge and find a path to the other side.  I want students to try.

Favorite and Least Favorite Math teachers

Favorite Math Teachers

    In grade 9, my math teacher gave mini quizzes at the start of almost every class to check our understanding from last class' material or from homework.  It was accounted for a very small percentage of our mark and was peer-marked.  The teacher also used humour throughout their lessons and showed relevant material through different media.  I found the mini quizzes to be a low-stress way to assess students.  

    In grade 12, my math teacher provided organized notes, similar to what one would expect in college.  They gave instruction on how to take effective notes that proved useful when I started post-secondary.  They formed caring relationships with students, held extra study sessions at school, and was always available for extra help.  I appreciated that they also encouraged peer to peer support and learning.

Least Favorite Math Teachers

    In grade 8, my math teacher was inconsistent with the material as well as attendance.  The flow and pacing of their class was often disrupted by tasks, irrelevant to students.  They gave regular timed drills which made me and other students anxious.  Because they were often absent, they were not available nor approachable.

    In post-secondary, one of my math teachers taught in a teacher's (or expert's) perspective instead of student's perspective (proofs were too "elegant" to be understood by a student who was just introduced to proofs).  In this class and most uni math classes, few assessments were provided by ways of weekly assignments, a midterm and a final.  I didn't learn much from this class and left feeling discouraged with a near failing mark.  Reflecting on this experience, I believe it's important to give appropriate level examples and start the class with a solid introduction and foundation to the material.

LOCKHART: A Mathematician's Lament

    I agree with Lockhart that once a subject is made mandatory, extinguishes the curiosity and joy for that subject in most students.  Reading is an example of this.  Just in this past week, I had two group discussions about reading for fun.  School was the main factor affecting our attitudes towards reading.  It was mandatory, we didn't get to choose the book, and it was not fun.  Over time, most people mentioned it's difficult for them to read because they want to.

    I somewhat disagree with Lockhart's point about mathematics being an art, like music and painting. Math, music, and painting can all teach us to think creatively and abstractly. However, unlike music or painting, math explicitly teaches skills we can use to solve real life problems.  Math demands somewhat more structure than some other forms of art.  What one learns from math is directly applicable to the world around us and can be related to most subjects, especially the sciences.

Math puzzle: The Locker Problem

  Assume the locker numbers start from 1 and increase by one as the students arrive at locker number 1000.  After student #1, all lockers are closed.  After student #2, all locker numbers divisible by 1 and 2 are open.  Those that aren't are closed.  After student #3, all locker numbers with factors of 1, 2 and 3 are closed.  The rest are open.  This pattern continues.

Observations:

    After student #1 has their turn, locker number 1 is left untouched for the rest of the students.  After student #2 has their turn, locker number 2 and any locker before that is left untouched as so on.  Looking at the diagram above, the number of times a locker changes state is equal to the number of distinct factors.  1 has one factor, 2 has two distinct factors, 4 has 3 distinct factors, 12 has 6 distinct factors.  Hence, those with an odd number of distinct factors will be closed and those with even ones will be open.

    Looking at the diagram above, from number 1 to 17, lockers 1, 4, 8, and 16 remain closed at the end of the 1000 students.  I hypothesize that every perfect square will be closed and the rest will be open if this pattern continues.  To check, we can take another square and use the factors approach mentioned above.

• Let's take 100.  It has factors 1, 2, 4, 5, 10, 20, 25, 50, 100- nine distinct factors.  Because it has an odd number of factors, it must be closed.
• Let's take 900 with 27 distinct factors.  This is odd, so the locker is closed.
• Let's take 50- not a perfect square.  It has 6 distinct factors, so it is open.

Conclusion:

    Locker numbers that are perfect squares are closed.  The remainder are open.  

    I used my knowledge of factors to find out the state of bigger numbers and simplification to determine the state based on odd or even number of factors.  I modelled the first few lockers to visually understand the problem, looking for patterns.

RE: Skemp Article Discussion

    I like Mariana, Alan, Yiru, and Rebecca's suggestion to slow down the pace and reduce the number of topics to address math anxiety.

    It makes me wonder, if so many students struggle with math in high school, it might be more beneficial to learn fewer topics over the same period of time so that students have plenty time to process and reflect on their learning. I also wonder if there is a reason why teachers are required to teach a set number of topics squished into one year when clearly, it might not be moving at an appropriate pace for students. For students who are needing a challenge, schools can offer advanced classes to support them.

SKEMP: Relational vs Instrumental Understanding

I agree with Skemp’s argument that relational understanding is important in math.  I've thought about this a lot after taking some courses in teaching and about how people learn. When tutoring my students, I regularly ask concept check questions to confirm the student's relational understanding. It is easy to breeze past a section of homework or topic without checking the students ability to show that they really understand.  Possibly, this is a result of the teacher’s assumptions on students’ ability.  I have done similar reflection on topics I studied in school. I computed without thinking about what I was really doing and followed a set of rules laid out by the teacher. Without really knowing what I was doing, there was no room for challenging the process or finding new ways to solve the problem. It limited my thinking. 

Skemp states that aiming for student relational understanding is a long process compared to instrumental understanding.  However, the former has long term benefits.  He relates instrumental understanding to learning a set of routes with instructions at each choice point determined “by the local situation”.  Instead, relational understanding builds conceptual structures (schemas) that “produce an unlimited number of plans''.  Students who learn through relational understanding become independent of external guidance for a route map.  There are long term benefits that help students get further while taking less time in total because they do less relearning.  This promotes life-long, self-sustained learning where students are aware of a range of possibilities, intrinsically motivated, and more satisfied with their learning.