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    How Many People’s Lives Can Be Helped with PET Bottle Caps That Have Been Collected?—Using Proportions to Determine the Number of PET Bottle Caps.  This lesson is part of Tokyo Gakugei University International Secondary School’s mathematics curriculum (functions) for first-year junior high school students.  The school’s effort links the mathematics curriculum with a non-profit organisation’s activities to collect PET bottle caps (hereafter “caps”) to fund vaccines for children across the globe.  The educational practice arose from the idea that even in the subject of mathematics, one should aim to encourage children to think about what it means to have a life that is truly rich and what they can do to realise such a life.

Lessons on Proportions
    In these lessons, students consider methods for determining the number of caps in the full bags of PET bottle caps they have collected.
    Caps are accepted by a recycling company at a rate of 10 yen per 400 units, and 20 yen will buy polio vaccine for 1 person.  Children like to know how many children’s lives they can save through their own efforts.
    Students think that if they divide the weight of an entire bag by the weight of one cap, they can determine the number of caps, and they conduct these weighings. However, there are caps that weigh 2 grams and caps that weigh 3 grams.  The students pursue a solution.  Various ideas are raised, with a group taking an average of 2.5 grams, and others measuring the weight of 10 and 30 caps to determine the weight of one cap. 
The next issue for students is determining which way of thinking is good.  They actually try testing with 100 caps.  They then realise by repeatedly selecting 10 caps at random and weighing them, that using the average weight is the best method.  Then, they turn their eyes to why this is the case.

Garbage Helps To Save Lives!
    Let’s look at comments written by students on their worksheets:
“I think it’s good that things that will become garbage can save lives.  It was interesting to be able to determine the number of caps not only by counting, but by weighing the caps.”
“It’s a small number, but I think it is good that we are able to save four people’s lives by collecting things that would be thrown away.”
“With this many caps we can only help four people….I thought that this was a small number and that it would be quicker just to give 100 yen.  However, it is wonderful that what would have been thrown away as garbage can be useful to people.”
    We can see that, even if the children are a bit confused by the fact that the nearly 4,000 caps in front of their eyes can pay for vaccines for only four people, their thinking does not stop at the idea of recycling PET bottles.  They express pure, positive feelings towards the possibility that garbage can be used to save people’s lives.   

Students weighing the caps

    Deepening their thinking, a student pointed out that “Because the price of PET bottle is 130-150 yen, making a 150 yen donation from the start will permit the purchase of more vaccines.”  Further, students pointed out aspects they thought showed inconsistency, as in: “I think that even if this looks at a glance to be eco-friendly, it isn’t at all if one thinks of the CO2 emissions resulting from shipping the caps.” 

Our Own Activities
    At this school, the activity of collecting caps was led by the student council.  This was the result of the idea raised to collect caps in addition to the “bell marks”* that the school had collected in the past, the teachers’ wish to have activities more closely connected with society, as well as students’ interest in recycling.
    There was one other piece of background to the students’ interest in recycling.  This was that they wanted the school to install a vending machine.  Three years ago at the school, there had been a vending machine that sold drinks in paper-based packaging.  However, these could not easily be recycled and the machine was removed.  Current students at the school do not know about this history, but teachers were opposed to the introduction of a vending machine based on this experience.  What caught the eye of the student council was that, in addition to recycling the PET bottles, they could participate in giving international assistance.  As a result of discussions with the teachers, the council’s appeal to the student body to maintain good manners when using the machine and promise to collect caps, the vending machine was eventually introduced.  Thus, to the students at this school, cap collection was truly an activity they “owned.”

A vending machine selling bottled drinks in the student lobby (right), a bottle collection box (centre), and an eco-cap collection box (left)
A student council poster resulting from much work (“Road to Installation of Vending Machine with Attached Deposit Capability”)

Student council poster

*Bell Mark collection activities aim at raising funds for schools and other educational or welfare-related organisations.  Participating groups must gather the Bell Marks that appear on the donor company’s products, and receive funds after sending them to the Bell Mark Foundation.

Addressing Real Issues
   Let’s look back one more time at these lessons.  Determining the number of units from the weight of the whole is content that is also covered in elementary school.  However, in these lessons, real occurrences connected with life in the society are turned into subject matter for mathematics.  Mathematical methods are used to address the issue, the results are compared with reality, and then students make judgments—with a focus on this complete cycle.  Such a cycle would include such things as determining the weight of one cap on one’s own, verifying by testing the weight of 100 caps, and becoming conscious of the assumption that all the caps were the same weight when dividing the weight of the whole by the weight of one cap.  The capacity to pursue such a cycle is indispensable for pursuing a rich life in its true meaning.  For the mathematics curriculum at this school, original, supplemental learning materials were developed that addressed various issues in actual society, and great effort was made to develop students’ capacity to pursue such a cycle.
  • Will bottlenose dolphins become extinct?
  • Let’s put a sensor near the bus door
  • Is it true that children’s physical abilities have decreased compared to children long ago?
  • How many strokes does it take to complete an 18-hole golf course?
  • Let’s examine the causes of traffic deaths
These questions topics are examples of ones that come up as issues for exploration in the supplemental learning materials.  While at a glance the topics seem to have no connection to mathematics, students will likely realise that the mathematics they are learning can be useful for solving the problems.

The Ability to Pursue a Rich and Fulfilling Life
    The teachers wish that when the students become adults that they will not pursue financial richness, but that each student, in her or his own way, will pursue the richness of fulfilment gained through striving to make more people happy.  Teachers aim to communicate through these lessons that mathematics ability is necessary for students to be able to do this.

Tokyo Gakugei University International Secondary
School Supplementary (TGUISS) Education Materials
[Mathematics, Seventh and Eighth grade]

Message on the inside cover of the supplemental education materials

Lesson Plan
Teaching Unit: Proportions and inverse proportions (Seventh grade)
Introduction and application of proportions – 1-2 hours

Learning Objectives
1) To be able to form the hypothesis that the weight and number of PET bottle caps are in a proportional relationship
2) To verify with a small number of caps whether or not the hypothesis will stand
3) To be able to determine the weight of one cap in order to estimate the number of caps overall

Instructional Methods

The teacher needs make the necessity arise for testing the weight of one PET bottle cap--in other words, for determining the constant proportion.  To do this, the teacher should prepare a scale that can only weigh items in units of 1 gram.  It is easier for students to recognise the need for this testing with a scale that can weigh items in units of 1 gram, because units of 1 gram must be rounded off.

Rubric for This Educational Practice*
Note: This rubric does not evaluate knowledge and understanding of such things as volume and geometric figures.
   Interest, Enthusiasm, Attitude  Mathematical Viewpoint and Way of Thinking

Mathematical Expression and Computation
 A  The student verifies the hypothesis that the weight of a cap and the number of caps are in a proportional relationship using a small number of units and tries to estimate the number of caps overall.  In addition, the student tries to compare the number of people who can be helped by not buying PET bottles at all and by donating the money the student would have spent.    
The student has a strong foundation and can formulate the hypothesis that the weight of a cap and the number of caps are in a proportional relationship.
The student can estimate the number of caps by determining the weight of one cap and is able to explain the method.
 B  The student tries to estimate the number of caps by determining the weight of one cap. The student can formulate the hypothesis that the weight of a cap and the number of caps are in a proportional relationship.
The student can estimate the number of caps by determining the weight of one cap.
C  The student cannot estimate the number of caps by determining the weight of one cap  The student cannot formulate the hypothesis that the weight of a cap and the number of caps are in a proportional relationship. The student can estimate the number of caps by determining the weight of one cap.

*A rubric (evaluation index) is a chart of evaluation standards with a scale that indicates children’s accomplishments in their study and a description in words of each of the levels.