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A Fence for Gretchie and Ashby!

A lesson by Carol McNair

This math lesson uses the following benchmarks: Novice, Apprentice, Practitioner and Expert (see Below). The Student Samples are examples of work that falls into each category, with annotations explaining why.

Grade Level: 6-8

Task

My husband and I want to build a fence around our yard so that our dogs won't be able to run into the road. Below are some different types of fences that we can choose from. We have about $1600.00 budgeted to spend. All sections below are 8 feet long.
Scallop Fence
Scallop Gothic Fence
Pieces needed to build each section of the Scallop Gothic Fence: 2 end posts, 2 back rails, and 12 pickets


Shadow Board
Pieces needed to build each section of the Shadow Board Fence: 2 end posts, 3 thick back rails, 2 thin back rails, and 13 pickets
Shadow Fence

Gothic Fence
Gothic Picket Fence
Pieces needed to build each section of the Gothic Picket Fence: 2 end posts, 2 back rails, and 13 pickets

Your task has 3 parts:

Part 1: Given that each section costs approximately $50.00 determine the number of fence sections we would need to buy in order to create a large area in which our dogs can play.

Part 2: Choose the fence design you think would look best.

Part 3: Create a shopping list for the materials we would need in order to build each fence with the dimensions you determined in Part 1 of the problem.

Other things you may need to know:

  • The dimensions of our house are 35' x 30'.
  • Our house sits on our land in the following way (See diagram to the right.):
  • Your final product should contain a letter to me and Mr. McNair that tells us where to build the fence, how many sections we will need, how many of each type of building materials we will need, and the approximate cost. You should also explain how you made your determinations, so we can check your work before purchasing the materials.

    Context
    This task was given to students that were studying functions and algebra. I was hoping that students would apply this knowledge to this problem-solving situation. As we were actually building a fence for my dogs, one of whom had recently been hit by a car, this task had real meaning and relevance to my students.

    What This Task Accomplishes
    This task combines the opportunity to use function and algebra concepts with geometry and measurement concepts. The function and algebra part of the task comes into play as students realize that one section of the fence requires two end posts, but two sections require three end posts as the two sections share a post. The measurement aspect comes into play as students decide where to put the number of fence sections that our budget will allow. Students see that they can maximize the area for the dogs by using sides of our house in the perimeter.

    Time Required For Task
    One or two 45-minute periods.

    Interdisciplinary Links
    The types of fences presented in the task were taken from a brochure given to me by the fence company from which we purchased our fence. Students could investigate the many types of fences, their different purposes, and their histories. Students can also visit a fence company and learn more about how they build and market their fences. Students could also read and discuss the Robert Frost Poem "Good Fences Make Good Neighbors."

    Teaching Tips
    For students who need to develop stronger problem-solving skills, I adapted the task in the following manner:

    Your task has 3 parts:
    Part 1: Choose a fence design you think would look best.
    Part 2: Create a shopping list for the materials we would need in order to build each fence using 28 sections.
    Part 3: What dimensions could the fence with 28 sections be?

    Suggested Materials
    Some students required graph paper; others used calculators.

    Possible Solutions
    Given our budget of $1600, we could afford to buy 32 sections of fence. Students could place the fence however they wished as long as it stayed within the confinements presented in the task.

    Pieces needed to build 32 sections of the Scallop Gothic Fence: 33 end posts, 64 back rails, and 384 pickets.

    Pieces needed to build 32 sections of the Shadow Board Fence: 33 end posts, 96 thick back rails, 64 thin back rails, and 416 pickets.

    Pieces needed to build 32 sections of the Gothic Picket Fence: 33 end posts, 64 back rails, and 299 pickets.

    Benchmark Descriptors

    Novice
    The novice will show little or no work to support the solution and the solution will be incorrect. No math language will be used and diagrams will lack labels.

    Apprentice
    The apprentice will show little or no work. Some parts of the student's solution will be correct but many parts will be far from correct. Some math language will be used and diagrams will lack labels. Some parts of the student's solution will be unconnected or unclear.

    Practitioner
    The practitioner will find a correct solution and the approach used will be explained. Most work will be shown. Math language will be used and an algebraic rule will be used to solve the task.

    Expert
    The expert will use math language throughout to communicate the approach and reasoning used to solve the task. Correct answers will be achieved. Representations will be accurate and labeled. The student will find a rule to solve the problem and will show how the rule can be used for any number of fence sections.

    Author
    Carol McNair teaches grade 6 at the Camels Hump Middle School in Richmond, Vermont. She has a master's degree from the University of Vermont in curriculum and instruction. She has worked as the mathematics assessment consultant for the Vermont Department of Education's Portfolio Assessment Program and acts as a consultant to school districts and to publishing companies. She is also an editor for Exemplars.


    Student Samples
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