The Scales Problem

 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 allow them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight. What are the values of the four weights? Are there several correct solutions? How could you extend this puzzle to help your students understand the mathematics more deeply?

Solution content:

1) Formulation of an extended problem of generating a range of positive integer value
2) A special case of two pan weight scale
2a) Attempt using base-two binary number system
2b) Attempt using base-three number system with modification
2c) A construction procedure to find alternative solution

Group Microteaching Lesson Plan

 

The Giant Soup Can Problem

 

Write the given information: 

Look at the photo of the fallen water tank with the bike. We choose the dimension of the water tank (height or diameter) that would produce the minimum error. The dimension chosen should be distorted the least, and is parallel to a bike dimension (for example, height). This dimension and bike dimension should be close to each other on the camera line of sight. 

Suppose we choose the diameter. From photo we measure and obtain D/(h_b) which is 
(Diameter of the tank)/(Height of the bike) value using a ruler. Then, following computations estimate diameter, height and volume of water tank.

We can find, or construct and agree on, what are acceptable definitions of average house fire and required water amount (volume) to put out the fire. Let (V_ f) be required water volume to put out average house fire. Then,

Expanding the idea: Let us work on the same water tank with student birds in class. Given the photo of the fallen and little bit distorted tank with a bike, how do we estimate the dimensions and volume of the water tank?

Some students would want to organize the information we already have first.

 What we have:
1) The water tanks have the exact proportion of the Campbell’s soup can
2) The photo showing the fallen distorted tank and a bike

A student asks what is proportion? Another student answers, “It is diameter to length ratio”. When a key moment arrives, teacher steps in and reminds that we need clear statement written in completion not to make confusion. The variable of interest is volume of cylindrical water tank. If V is volume of water tank, v is the volume of soup can, then

It sounds like teacher forgets something. A student in the class has parents working in sheet metal business. This student points out that we are forgetting the thickness of the shell material used in construction of water tank and soup can. Someone recommends that we add assumption.

 Assumption 1: Thicknesses of the materials are negligible compared to the volume.

 What is next? Students shout out we need that proportion. Some students argue that we do not have a road map. We have not got a plan yet. Then one says we are planning now. Someone quietly listening now suggests that it is all good because we are in an adventure. We argue, point out weak point, but everything is peaceful. Finally a constructive idea is reached.

 We would copy, cut and match the water tank photo and the actual picture of the soup can. Another debate erupts on “what to match and why?” The tank in the photo looks distorted. A student asks if we can match the size of the letters because these letters seem to be the most intact in the photo. Another student agrees but adds assumption.

Assumption 2: The proportionality includes letters on the label. That is,

Students use school computer to do some image works. They use zoom in and out while keeping the proportionality. They compare, and finally they manage to get the letters on the tank and soup can nearly the same size. Therefore, the original soup can picture serves as original water tank alongside the damaged fallen water tank with the bike.

If the assumption (2) is true, this picture surprises the teacher. The water tank in the photo is distorted significantly. The answer quietly calculated by the teacher before seems to be not realistic now. This would become another key point that ‘looking from different points of view’ is beneficial. Teacher decides to discuss this after the project.

Students take measurements on the final picture and compute the ratio H/L and D/L. Now we need more information. We need the actual length L of the bike. Students arrive at a mid-point conclusion that we need the length of the bike to get water tank dimensions. Therefore, the class adds a new assumption.

Assumption 3: We have the length L of the bike.

Students then do the computations:

We do not even need the actual dimensions of the soup can. We also have a student whose parents work in some engineering design jobs. This student remembers that our assumptions may be incorrect and we need to inspect. When the student points that out, the student is not very sure what exactly that is. Teacher catches another key moment to explain what is in the student mind.

Sensitivity analysis:
If a value in the computation is actually different from what we assumed, then the final result can change. What is the effect of a certain amount of variation in data on the final result?

Before the teacher bird opens her mouth to confess her prior solution and its weak points, student birds demand for explanation why we need so-called sensitivity. Instead of answering directly, teacher asks if they want their firefighting water tank to be smaller than the necessary size? This sparks another round of discussion for importance of responsible reporting of computational results in real life. This class project produces some important points:

• Assumptions must be made after reasonable amount of unbiased inspection
• Statements (including reporting) should be produced in completion to avoid misunderstanding
• When seeking a solution (in this case minimum error estimate), past experience and different points of views or methods should be applied
• Need of experiments, exploration, and learning of skills and reasoning required by the problem at hand (in this problem, knowing that two images need to be at the same state and actually making it happen)
•  Possible weak points (for example due to assumptions) of each method should be discussed
  
• Sensitivity analysis on all assumed values should be done
• Responsible reporting

Additionally, if we do not have a means to manipulate the pictures (advantage of technology), we would still need the actual dimensions (letter size also) of the soup can. Finally everyone agrees the importance of sharing of diverse views, methods and knowledge; and need to continue learning to gain greater maturity (experience). It is almost 7 pm. It is decided that all birds need a rest. 

1

Flow

 From my experience, when people see high interest, high value matching their world view, truth that worth knowing, challenge that matches and tests their skills in doing something they are into it. They are willing to spend time and energy into it. According to TedTalk, it is not that income and living standard, but it is something meaningful and worth doing for the person. It is ecstasy or a state of the person’s mind that transports the person into alternative reality. It is a completely engaging process in which the person sees creating a new thing. According to Peter Liljedahl’s presentation at SFU (Engaging students, understanding flow, 2015), there are nine items involved in the state of ‘flow’ in which a person is completely engaged and delighted. These are:

1. There is clear goal every step

2. There is immediate feedback on one’s action

3. There is balance between challenge and skill

4. Attention is focused on one’s action

5. Distractions are excluded from consciousness

6. There is no worries of failure

7. Self-consciousness disappears

8. The sense of time becomes distorted

9. The activity becomes satisfying in its own right

In teaching math, teachers can take part in dialogue with students on the world we live in, the society humans have built out of natural resources and the planet, the villages, towns, trades, transportations, necessary skills needed to contribute to work and earn, and sports, arts and so on. We will see that math can be found in almost every component. That way, teachers can establish interests in students. 

Students have their family and friend circles or communities. They grow up in certain neighbourhoods in which there are also works. Therefore, students may already have some interests. Or students can be interested in some other works they found in movies, the Internet or from someone. Teachers need to be well rounded, mature, or able to learn the subject matter of student interest in relatively shorter time. 

Teachers then ask the topics of interest of high value of the students. After getting a list of topics of interests, students are asked to rate their interest on a scale to each of the topics. That creates an interest level (1-5) table like this.

Teachers can declare that there would be a certain number of projects (3 or 4) to be undertaken in the semester. So the class would use the table and vote for the project topics. On each project teachers carefully plan the start point, possible process and ending so that the level of challenge is appropriate to the class, and design the project question based on the students’ inputs. Teachers also explain to the class about the concept of ‘flow’ ahead. Students will participate in keeping the project process within their challenge boundary without losing goal in each step. Students participation is important to let them keep themselves in ‘flow’.

Or the class can be divided into two or more subgroups working on different projects of interest to the sub-groups. This sub-group method could make the project and level of challenge more suitable to each sub-group. 

It is also important for teachers to discuss with the students that we as humans living on the planet earth (which is not our design and construction) can face any event in our life. Some events we have gone through in our life can be not our interest but we have to face it. So it is necessary to raise the issue that students need to be able to solve the problems of not only their interest, but also other problems.

Geometric/Numerical Puzzle

Rotational symmetry of order n means the object can be rotated about the point of symmetry by 360/n degrees and the result is the same as original in geometry. Circle has a rotational symmetry of infinite order with respect to its centre. That is, rotation of the circle about its centre by any angle does not change its geometry. Two diametrically opposite points on the circumference divide the circumference into two arcs of equal lengths. Thirty equally spaced points on the circumference make 30 equal arcs out of the circumference. From the point (7) one travels 15 arcs (half the circumference) to point (22) which is half way around. Therefore points (7) and (22) are diametrically opposite to each other.

The process to the solution is based on these two facts about the circle:
1) Circle has rotational symmetry of infinite order.
2) The same angles at the centre correspond to the arcs of the same length.

An extended puzzle: A farmer has fencing material of length 600 meters. The farmer mistakenly assume that circumference of a circle of radius r is 6r. He starts building the circular fence with radius 100 meters, thinking that he would get a circular enclosure. After laying just over 300 meters of fencing material, he realizes that he is wrong. He manages to close the fence by laying the last part straight. Does the farmer get the maximum enclosed area?

 The key to this question is to know what region on the flat plane has maximum area to perimeter ratio. The circle has the maximum area to perimeter ratio. The farmer finishes the enclosed area of the following shape.

What he gets is not circular region. If he correctly calculates the radius for the circle with 600 m perimeter, then he would get

With this radius he would get a circular area larger than the one in above diagram.

Regarding the solution: In our life when we face with problems some possibilities are:
1) There is no solution.
2) There is one solution.
3) There are finitely many solutions.
4) There are infinitely many solutions.
5) We know there is a solution. However, we are unable to compute exact solution due to our insufficient computing ability (capacity) or due to resource limitation. We can find approximate solution(s).
6) We can simplify the problem to a simpler form. However we cannot get better for now.
7) We do not know if the problem is solvable (need more knowledge).

The number of solutions (including zero) can be due to the constraints. For example, in the problem: x+y = 5 with both x and y are real integers of at least three, there is no solution. But if we can take x and y any positive integers of at least two, then there are two solutions:
(x,y) = (2,3),(3,2). If x and y can be any integers, then there are infinitely many solutions. Students should be allowed to experience a variety of solution-situations.

A puzzle (problem) truly geometric: A puzzle which can be solved by knowing the properties of the geometric figures involved. 

The new BC curriculum & secondary math course pathways structure

• Two things that were new to me or surprised me from the curriculum orientation guide and/or glossary of new terms
  
  First thing that I feel surprised is that “thinking” is listed as one of the core competencies. When I was in school, there was not an emphasis on training how students think. Even in subject like mathematics, I did not see any implementations that were dedicated toward training students to think critically and creatively. Normally math classes were taught in the form of a lecture, which was a one way communication from teacher to students. Students mimicked how teacher solve problems and learn the technique for solving problems. There were rarely any debates between teachers and students on concepts, instead, student would trust their teacher entirely and follow what the teacher did. There were not much creative nor critical thinking. I see the importance in teaching students to think but thinking itself is a very abstract object which is hard to grasp as well as to measure. I glad to see that this is properly addressed in nowadays curriculum and I would like to see how we can follow up with this.
  
  The second thing I read which was not seen during my school time was the use of inquiry-based approaches. As I mentioned, in a mathematics classroom, the method of delivering knowledge was through lecture. Usually after learning in class, we practiced our skills by the assigned homework and used the way we were taught in class. And the traditional way for assessment were tests and exams. Talking from my own experience, I had never done a mathematical project nor an open-ended small research problem in secondary school. All of what we learned is for the purpose of passing exams but not for applying them in the real life situations. I have heard a saying that “education is the kindling of the flame not a filling of a vessel”. It is very important for educators to elicit the interest and curiosity from students so that they can take ownerships of their learning. Inquiry-based approach is one good way to actually implement creative and critical thinking skills which is lacked in traditional approach.
  
  
• My own schematic chart of possible pathways in the courses of the BC Math curriculum from Grade 8 - 12.
  
   
  
  
  

Micro-Teaching lesson plan with self assessment