Math helps the Good neighbourhood

Mathematics & Logics


This WebQuest is intended to be conducted through collaborative and cooperative pair work, using the Internet in order to complete the assigned research and tasks.

Students will determine the cost that neighbours will pay for building a fence that separates their yards. They will have to use the Pythagorean theorem, convert measurement and pricing units from SI to imperial and determine an equitable means for sharing the cost.


Mr James has just moved to Cyprus from England and has settled in beside his Italian neighbour, Ms Diana. Mr James needs to build a fence between their two yards to keep his dog out of his neighbour’s yard. Being on tight budgets, both of them want the job done well but neither want to pay for any extra costs. Both would like to have the job done quickly so they decided to choose pre-manufactured panels (available at select hardware stores). As their friendly neighbour, you offer to help them calculate the length and cost of this fence and determine how much each of them should pay. Each neighbour has emailed you their property line measurements but they each gave you different measurements for the fence line!


Your role is to research the cost of building the fence and make a recommendation for sharing the expense with Mr James and Ms Diana. Explain your calculations, choices and rationale. To calculate the cost of the fence per neighbour, consider:

  • Mr. James got a measurement of 110.21 feet for the length of the fence to be constructed. Ms. Diana believes the fence should be 32.5 metres long. These measurements are not equivalent. Fortunately, the surveyed property measurements will allow you to simply calculate the accurate fence line and save you a trip to measure it! Apply the Pythagorean Theorem to calculate the length of fence required to separate the two yards.
  • Convert between the SI units and Imperial units so that you can provide the measurements and costs in the Imperial system (for Mr. James) and the Metric system (for Ms. Diana).
  • Calculate the total cost of the materials for the fence; The original fence had been removed years ago, but the footings are still in place on the property line (4’ gaps between each footing). Research the type of pre-manufactured fence panel you would recommend and the cost of the materials needed. The only materials to be purchased are:
    • Fence panels (4 ft / panel; assume each kit includes hangers),
    • Posts, and screws (or nails)
  • Justify a reasonable cost for the labour of building the fence. Ms. Diana is elderly and is unable to help build the fence. She does, however, have 3 extra fence posts (10.16 cm square) she salvaged from the last fence and is happy to contribute them to the project. Mr. James is willing to build the fence using his tools. Determine a reasonable amount for the value of his labour.
  • Determine a fair way to share the total cost of the fence, considering the need, value of the posts and the labour. What proportion should each neighbour pay?



  1. Your first task is to determine the length of the fence to separate the two yards. To do this, you can use the Pythagorean theorem if you know the length of the two sides of the triangle that align to the fence length.
    • Can you find the height of the triangle? You know the two opposite lenghts for the front and back of Ms. Diana’s yard.
    • Can’t you find the lengh of the triangle? You know the depth of Ms. Diana’s yard.


  1. Once you have found the height and the length, use these values in the Pythagorean theorem to calculate the length of the fence.

  1. Since we have used the measurements provided for Ms. Diana’s yard (in metres) and we have to share the information with Mr. James, we will need to convert the length into feet. Which conversion factor will you use?

1 metre = 3.2808 ft                                     1 foot = 0.30480 m

  1. Was either of the measurements that Mr. James and Ms. Diana provided correctly? Which measurement was the longest? Is it possible to quickly compare these lengths even though they are in different units? Explain how.
  2. Choose a fence panel (available at a local hardware store). You may look online or consult advertisement flyers. Keep in mind the “tight budget” when making your selection. Calculate how much the building materials that need to be purchased will cost to complete the project.

  1. Does the length of fence work out to a whole number of panels? If not, what will you do to address the gap? Can you use a portion of the panel?
  2. Determine a reasonable credit for the posts that Ms. Diana has supplied. (How much is each pole worth?)
  3. How long do you estimate it will take to build the fence? What factors are you considering?
  4. Determine a cost for the labour to be done by Mr. James. How much do you think the time and effort Mr. James will put into building the fence is worth? Consider the following factors:
    • What is minimum wage? Should this be the agreed amount of compensation? Why or why not?
    • Does his expertise or loss of recreation time factor in?
    • What about the use of his tools or the clean-up of the area?
    • Are there any other factors that you feel should be considered?



  1. Justify the reasonableness of the amounts you feel each neighbour should pay to complete the project. Who needed the fence? How much should this factor in?


If you add all the expenses (the materials purchased, the materials and labour provided), what proportion will each person pay?


Throughout this WebQuest, the following mathematical processes are specifically addressed:

  • Communication: communicate in order to clarify, reinforce and modify ideas.
  • Connections: connect mathematical ideas to each other or to the real world.
  • Problem-Solving: develop and apply new mathematical knowledge through problem-solving. Reasoning: use reasoning skills to analyze a problem, reach a conclusion and justify or defend that conclusion.
  • Technology: utilize technology as a tool for learning, solving problems and presenting solutions.
  • Visualization: understand mathematical concepts and make connections among them.


This WebQuest is oriented to develop these students’ competencies:

  • Critical Thinking
  • Problem Solving
  • Managing Information
  • Communication


Evaluation of learning achievements

Practice with your pair.

  • One person asks the questions and the other person gives the answers. Then switch jobs.
  • How is it possible to compare the lengths provided in different units?
  • How does the fence length compare to the number of panels used to build the fence?
  • How did the measurements that were provided compare to the actual length of the fence?


Each student makes a presentation that answers these questions:

  • What fence panel do I recommend them to use and why?
  • How do I suggest we address any gaps between the fence length and the multiple panel length?
  • How did I arrive at an estimate for how long this project will take? What factors did I consider?
  • What were all the considerations I made to evaluate Mr James labour? Did I exclude any I didn’t think were relevant?
  • What makes my recommendation “fair”? Do my calculations support this?

In this section we will not dive very deep into the underlying educational theories about evaluation and testing: there’s too much out there than we could possibly cover in this small project report.

Instead, we want to concentrate on procedures that enable both students/pupils and their teachers to establish if the learning goals of the Webquest were achieved and, if so, to what extent. We recommend teachers make use of a combined evaluation procedure, that consists of:

  1. Statements by learners (after being asked to do so)
    • telling what they learned about the subject (knowledge-oriented self-evaluation): now (after going through the Webquest) I know that …
    • telling what he/she learned about herself/himself (formative evaluation, in this case, diagnostic self-evaluation): now (after going through the Webquest) I know about myself that I …
      This pair of basic statements add up to a so-called learner report, in which the pupil/student reflects on what the Webquest brought him/her in terms of acquired knowledge and new personal views and attitudes concerning the subject.

    For instance:

    • ‘I learned that in medieval times the hygiene of people was hardly a concern which helped to let epidemic diseases like the Plague cause so many casualties’ Or:
    • ‘I learned the facts and I know the earth is warming, but I cannot understand why people were so stupid to pollute the world and let it warm up so much.
    • ‘I learned from the information about diseases that this subject is more appealing to me than I would expect in advance: maybe I should consider a medical career’. Or:
      ‘The Webquests confirms what I thought already: I could not care less about the climate and global warming. In fact, I thought it was all a hoax and I still do!’

    This kind of assessment seems more subjective than it actually is: in his standard work on testing and evaluation (and much more), simply called Methodology (1974), Prof. A.D. de Groot described how consistent the student’s self-evaluations appeared to be: when asked again after 5 or 10 years, their evaluation would almost be the same. De Groot advised teachers to use the learner report as a start for joint evaluations, striving for consensus between teacher and student/pupil about the learning outcomes and their value for the learner, but also compared with the learning objectives as stated in the curriculum.

  2. The learning achievements are visible in the output produced by the students: it is physical evidence: reports, answers to questions asked in the Webquest, presentations, and performance during presentations (preferably recorded). The teacher completes an evaluation grid stating clearly what the learning outcomes for the student/pupil are. The categories in the grid can be modified by the teacher to cover more precisely the content of a Webquest.

    >We advise teachers to use the grid to start a joint evaluation discussion, aiming at consensus or at least understanding between the teacher and the student/pupil about the learning outcomes: were they achieved (as planned in the curriculum and communicated before the Webquest started) and to what extent? To communicate the learning goals clearly before any learning activity starts, is a transparency requirement that is widely acknowledged in the educational community. The history of making learning objectives explicit goes back to the evaluation ‘Bible’ by Bloom, Hastings and Madaus: ‘Handbook on formative and summative evaluation of student learning’ (1971), a standard work that also served as inspiration for the earlier mentioned Prof. De Groot.


The procedure also applies when students/pupils have worked together on a Webquest. The teacher will ask questions about individual contributions: ‘What did you find? What part did you write? How did you find the illustrations? Who made  the final presentation?’

All the evidence (of learning efforts and outcomes plus joint evaluations) is preferably stored in the learning portfolio of the student, or in any other suitable storage system (folders with written or printed documents, online collection of files, etcetera ).

Changes in personal points of view and feelings are harder to value and here the consensus between teacher and student/pupil about experiences during the learning process provides essential insights.

The grid below gives an example of how the evaluation of the learning process and achievements can be shaped: what kind of reactions to the Webquest does the teacher expect and how valuable are they? Is the teacher capable to explain the value or score allocated to answers or presentations given by pupils? Does the pupil/student understand the evaluation outcomes, and does he/she agree? If an agreement (consensus is not possible, it is still the teacher who decides how to value the student’s work.

Please note that the text in the grid addresses the pupil/student directly: this is important and it is in fact a prerequisite for using such an evaluation grid: it is specifically meant to enable a discussion of learning results between teacher and student and not to communicate learning achievements of learners to others who had no direct role in the Webquest.

Evaluation Grid

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Funded by
sCOOL-IT erasmus logo EN

The European Commission’s support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Talk To Us

t: +357 2466 40 40
f: +357 2465 00 90

Funded by
sCOOL-IT erasmus logo EN

The European Commission’s support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Talk To Us

t: +357 2466 40 40
f: +357 2465 00 90

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