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Problem Solving for the 21st Century: Built for the Common Core

Herding Cats

Maru owns a free-roam cat rescue. Maru is working to enclose her roaming area with a cat-safe electric fence. Maru knows the cost of an electric fence installed is $0.89 per foot. The Meow Safe Fencing Company has provided an estimated price of $3,190 for fencing in the total property.

The boundaries for the property can be described using coordinates on a scaled grid overlaid on an aerial photo. Each unit is 30 feet. The boundary coordinates are (1, 0), (25, 0), (25, 22), (21, 22), (21, 16), (5, 16), (5, 22), and (1, 22).

Write a letter to the Meow Safe Fencing Company either accepting or rejecting their offer. Be sure to include all of your mathematical thinking.

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assessment
Domain:
Aligned Standard:
Mathematical Practices:
  • MP.2
  • MP.2

    Reason abstractly and quantitatively.

  • MP.3
  • MP.3

    Construct viable arguments and critique the reasoning of others.

  • MP.5
  • MP.5

    Use appropriate tools strategically.

  • MP.6
  • MP.6

    Attend to precision.

View all Grade 6 tasks

Plan

Underlying Mathematical Concepts

  • Graphing points on a coordinate grid
  • Distance on the coordinate plane
  • Perimeter
  • Scaling
  • Scale units
  • Addition
  • Rate

Possible Problem-Solving Strategies

  • Graph coordinate points
  • Create a scale drawing
  • Table
  • Calculate distance using coordinates
  • Determine cost based on unit cost (rate)

Formal Mathematical Language and Symbolic Notation

  • Coordinates
  • Coordinate plane
  • Perimeter
  • Distance
  • Vertices
  • Scale
  • Axis
  • Grid
  • Quadrant
  • Total/Sum
  • Length
  • Rate

Teacher Notes:

Possible misconceptions that students may encounter:

  • Determine perimeter and cost based on number of units without scaling by 30.
  • Interpreting and plot coordinates as (y, x) instead of (x, y).

Suggested Materials

Engagement Image:

Teachers may project the image below to launch this task in the classroom, to prepare students, to promote discussion, and to inspire engagement and problem solving.

Click image to enlarge

Task-Specific Evidence

This task requires students to determine how much it will cost to fence in an area based on a set of coordinate points and given scale. They will also need to write a letter to the fencing company to accept or reject the price the company plans on charging based on their findings.

Printer-Friendly Planning Sheet
Printer-Friendly Exemplars Rubric

Possible Solutions

  • Task Solution (active tab)
Task Solution

The cost of having the electric fence installed should cost $2,766.80, which is below what the Meow Safe Fencing Company has offered us. Students may decide The Meow Safe Fencing Company’s estimate is reasonable or not and should make an argument based on the information they consider. For example, other additional costs could be factored into their argument.

Possible Connections

  • The Meow Safe Fencing Company’s estimate is $423.20 more than the expected cost.
  • Determine the area of the roaming space (432 sq units or 388,800 sq ft).
  • The roaming area has a perimeter of 1,040 yards.
  • The Meow Safe Fencing Company charges $2.67 per yard of electric fencing.
  • Maru could maximize the area and spend less money on fencing if she made the roaming area a rectangle instead of an irregular shape.
  • Relate to a similar task and state a math link.
  • Cost for fencing can be stated algebraically: C = 0.89f.
    • C = total cost of fencing for a property
    • f = total feet needed for fencing
Printer-friendly Anchor Paper & Scoring Rationales

Assess

Printer-Friendly Exemplars Rubric

Novice

1

Apprentice

1
2

Practitioner

1
2

Expert

1

Novice

This student does not meet the standard.

Novice 1

Scoring Rationale

Problem Solving

Novice

The student confuses needing to find the perimeter of the enclosed space with their strategy of finding the area of the total yard. The student appears to divide the figure into rectangles and find the areas of these figures. This strategy will never lead to a correct solution for this question.

Reasoning & Proof

Novice

The student shows little correct reasoning in attempting to solve the task. No correct reasoning is present in finding the perimeter of the yard. Numerous calculations are present, but no reasoning is provided for what is being calculated.

Communication

Apprentice

The written explanation given for the work is limited and unclear. Work is difficult to understand and hard to follow. Significant interpretation is required to understand the student's strategy by the reader. The use of math language is limited.

Connections

Novice

No connections are present or attempted by the student.

Representation

Apprentice

Based on the drawing included, the student appears to correctly plot the coordinates given on a coordinate plane to create a diagram of the fenced area. The diagram is not labeled. The student then attempts to break the yard into rectangles to find the area of the total yard. The student does utilize area models for several calculations, although we are unsure what they are working to find.

Overall Achievement Level:

Novice

Apprentice

,
This student does not meet the standard.

Apprentice 1

Scoring Rationale

Problem Solving

Apprentice

The student correctly plots and connects the coordinates given to form a diagram of the fenced area. The student incorrectly calculates the perimeter of the polygon in unit lengths (records the bottom segment as 25 units instead of 24 units) to be 105 units instead of 104 units. The student also incorrectly uses 105 units as the perimeter, failing to multiply the measure by 30 feet/unit to get the perimeter in feet. The student multiplies the incorrect perimeter by $.89 (cost/foot) to get the incorrect cost of $93.45. These incorrect calculations still lead the student to the correct decision to reject the offer from the fence company.

Reasoning & Proof

Apprentice

The student demonstrates some understanding of the underlying concepts of the task, but fails to use the unit rate of 30 feet/unit length to find the perimeter in feet, which is needed to determine cost using the unit rate $.89 cost/foot. The student is inconsistent in finding the unit lengths of all sides of the figure. Errors in computation of the perimeter in unit lengths lead to subsequent errors in the solution process. The calculation 25un, 22un, 4un, 6un, 16un, 6un, 4un, 22un = 105 units x 0.89 = 93.45 is also an incorrect mathematical statement.

Communication

Practitioner

The student presents a sequenced response to communicate work. The explanations of the steps to the solution process are clear and easy to follow even though some of the calculations are incorrect. Math vocabulary is limited, yet appropriate.

Connections

Apprentice

The student attempts to make a connection but it lacks contextual relevance.

Representation

Apprentice

The student draws a diagram of the fenced area, but some of the unit lengths given for the sides are missing or incorrect. The diagrams units are not labeled.

Overall Achievement Level:

Apprentice

,
This student does not meet the standard.

Apprentice 2

Scoring Rationale

Problem Solving

Apprentice

The student’s strategy of creating a representation and finding the number of units, leading to the number of feet in the perimeter, is correct. However, the student fails to multiply the number of total feet by $0.89 to successfully solve the task. Despite failing to make the calculation of converting from total perimeter to a cost, the student does arrive at a correct answer.

Reasoning & Proof

Apprentice

The student provides some correct reasoning for several of the steps. However, the argument is incomplete and inaccurate because the student fails to multiply the total perimeter by the $0.89 per foot, leading to an incorrect cost for the fencing.

Communication

Practitioner

The student demonstrates a sense of purpose, clearly communicating this in the task statement at the beginning of the solution. The student’s overall approach is clear, organized and sequenced. Formal mathematical language, labels and symbolic notation are evident.

Connections

Practitioner

The student extends the task by finding the difference between the Meow Safe’s offer and their calculations.

Representation

Practitioner

There is an accurate and appropriate mathematical representation created in step 1 of the task. This representation is constructed to help portray their calculations and strategy for arriving at their final answer.

Overall Achievement Level:

Apprentice

Practitioner

,
This student meets the standard.

Practitioner 1

Scoring Rationale

Problem Solving

Practitioner

The student correctly plots the points given to create a diagram of the fenced area. The student determines the unit lengths of each line segment and converts these to feet by multiplying each value by 30 (feet per unit length). The student correctly determines the cost at $2776.80 by multiplying 3120 feet by $.89 (cost/foot). The student correctly determines that the offer from the fence company should be rejected.

Reasoning & Proof

Practitioner

The student’s argument is sequenced, logical and easy to follow. The student demonstrates correct reasoning of the underlying concepts of the task. The student determines the length in feet of the fence and then calculates the total cost by multiplying by the unit rate of $.89. Calculations are correct and support the solution given.

Communication

Practitioner

The student uses an organized and sequenced response to communicate work. The student explains each step to the solution process. Work is clear and easy to follow. Appropriate math language is used.

Connections

Expert

The student describes a second way the task can be solved (but does not actually show the calculations). The student also makes a connection between the mathematics in the task and how the strategy for solving the task could be extended to other cases, “when your coding."

Representation

Practitioner

The student plots the given points to create a diagram of the fenced area on a coordinate grid. The student correctly labels the length of each line segment in feet on the diagram.

Overall Achievement Level:

Practitioner

, ,
This student meets the standard.

Practitioner 2

Scoring Rationale

Problem Solving

Practitioner

The student correctly plots the points given to create a diagram of the fenced area. The student determines the unit lengths of each line segment and converts these to feet by multiplying each value by 30 (feet per unit length). The student correctly determines the cost at $2,776.80 by multiplying 3,120 feet by $.89 (cost/foot). The student correctly determines that the offer from the fence company should be rejected.

Reasoning & Proof

Practitioner

The student correctly plots points on a coordinate grid, finds the perimeter of the polygon created in unit lengths, and then converts units into feet in the diagram. Calculations for the total cost are present and correct. The student demonstrates an interesting strategy of finding half the total perimeter to enclose and then multiplying by 2 to find the other half of the perimeter.

Communication

Practitioner

The student uses an organized, sequenced, and labeled response to communicate work. The student identifies the task to be solved, explains each step in the solution process, and explicitly states the solution. Units of measure are accurately recorded. The student uses appropriate math language.

Connections

Practitioner

The student explores a mathematical phenomenon within their solution, “since this shape is symmetrical left to right, we can just find half the dimensions and multiply by 2.” This secondary strategy recognizes an interesting pattern in finding perimeter of symmetrical shapes.

Representation

Expert

The student creates two correct diagrams of the fenced area to help analyze the relationships between the different measurement for the perimeter of units versus feet.

Overall Achievement Level:

Practitioner

Expert

,
This student exceeds the standard.

Expert 1

Scoring Rationale

Problem Solving

Expert

The student provides an effective strategy and a second more efficient strategy to solve the task. The alternative strategy at the end is considered, which shows evidence of the student analyzing the situation and defining a more efficient strategy to solve the task, “but there is a different way…” The student then describes and diagrams this alternative strategy.

Reasoning & Proof

Expert

The student provides a systematic mathematical justification throughout. Because the student also supports their idea mathematically for an alternative strategy on how to solve the task more efficiently, “compensate for the 'dip' of 6 units at the top…” this student achieves Expert level work. Calculations are correct that support the solution.

Communication

Expert

The student uses an organized, sequenced, and labeled response to communicate their work. The student provides insight into the efficiency of their original strategy and defines a more efficient method for reaching the same perimeter. Formal math language is used throughout to communicate their ideas.

Connections

Expert

The student provides a deeper understanding of the mathematics in the task as they describe their alternative strategy to solve the task faster, “compensate for the 'dip' of 6 units at the top…” Experts articulate connections between various strategies for solving the task.

Representation

Expert

The student constructs a representation that clarifies their idea of using the perimeter of the rectangle “that this shape forms” plus the “the dip.” The construction of the two representations helps to clarify how they can each be used to solve the overall question.

Overall Achievement Level:

Expert

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