How Big Is the Property?

Hectar is in dispute over the amount of property taxes he owes to the state for his farmland. Property taxes on farmland are set by the state government and are determined by the total area of the property, measured in acres. The state is using the aerial photo below to define the total size of the property.

The boundaries for the property can be described using coordinates on a grid overlaid on the aerial photo. If Hectar walks the property boundary, he arrives at coordinates (320, 0), (700, 0), (700, 340), (320, 340), (320, 520), (0, 520), (0, 180), (320, 180), (320, 0). All coordinates are defined in units of feet.

The state has a property tax rate of $326 per acre. An acre is equal to 43,560 square feet. As a result, the town is currently charging Hectar $2,444 in property taxes. Hectar believes he is paying too much in property taxes.

Help Hectar prepare a letter to the state government explaining how much he believes he should be paying the state for his property. Be sure to include all of your mathematical thinking.

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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.

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More Accessible Version

Hectar is in dispute over the amount of property taxes he owes to the state for his farmland. Property taxes on farmland are set by the state government and are determined by the total area of the property, measured in acres. The state is using the aerial photo below to define the total size of the property.

The boundaries for the property can be described using coordinates on a grid overlaid on the aerial photo. If Hectar walks the property boundary, he arrives at coordinates, (300, 0), (700, 0), (700, 300), (300, 300), (300, 500), (0, 500), (0, 200), (300, 200), (300, 0). All coordinates are defined in units of feet.

The state has a property tax rate of $300 per acre. An acre is equal to 43,560 square feet. As a result, the town is currently charging Hectar $2,500 in property taxes. Hectar believes he is paying too much in property taxes.

Help Hectar prepare a letter to the state government explaining how much he believes he should be paying the state for his property. Be sure to include your mathematical thinking.

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More Challenging Version

Hectar is in dispute over the amount of property taxes he owes to the state for his farmland. Property taxes on farmland are set by the state government and are determined by the total area of the property, measured in acres. The state is using the aerial photo below to define the total size of the property.

The boundaries for the property can be described using coordinates on a grid overlaid on the aerial photo. If Hectar walks the property boundary, he arrives at coordinates (320, 0), (700, 0), (700, 340), (320, 520), (0, 520), (0, 180), and back to (320, 0). All coordinates are defined in units of feet.

The state has a property tax rate of $326 per acre. An acre is equal to 43,560 square feet. As a result, the town is currently charging Hectar $2,444 in property taxes. Hectar believes he is paying too much in property taxes.

Help Hectar prepare a letter to the state government explaining how much he believes he should be paying the state for his property. Be sure to include your mathematical thinking.

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Plan

Underlying Mathematical Concepts

Possible Problem-Solving Strategies

Formal Mathematical Language and Symbolic Notation

Teacher Notes:

Possible misconceptions that students may encounter:

  • Interpret and plot coordinates as (y, x) instead of (x, y).
  • Invert the values to convert sq. ft. to acres (i.e. divide 43,560 by 238,000 giving a much smaller answer than desired).

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Possible Solutions

Task Solution

Hectar should be paying $1,793 in property taxes. The area of his property is about 5.5 acres.

The boundaries of Hectar’s property are shown on the coordinate grid:

 

Students can find the area of Hectar’s property by decomposing the property into 2 rectangles as shown and use the coordinates to find the dimensions and area of each rectangle.

 

The rectangle on the left is 340 ft. (520 – 180 = 340) by 320 ft. (320 – 0 = 320), so its area is 108,800 sq. ft.

The rectangle on the right is 340 ft. (340 – 0 = 340) by 380 ft. (700 – 320 = 380), so its area is 129,200 sq. ft.

Total area = 108,800 + 129,200 = 238,000 sq. ft.
Total Area ÷ sq. ft. per acre ≈ Total acres
238,000 sq. ft. ÷ 43,560 sq. ft. per acre ≈ 5.5 acres

Cost: 5.5 acres x $326 per acre = $1,793

Students could also find the area of the property by first finding the area of the rectangle that would surround the boundaries of the property. The extra area that isn’t part of the property would then be subtracted from that area.

The dimensions of the larger rectangle are 520 ft. by 700 ft.
Area of the larger rectangle: 520 x 700 = 364,000 sq ft.

The dimensions of the smaller extra rectangle are:
320 ft. (320 – 0 = 320) by 180 ft. (180 – 0 = 180)

Area of the smaller extra rectangle: 320 x 180 = 57,600 sq. ft.

The dimensions of the larger extra rectangle are:
380 ft. (700 – 320 = 380) by 180 ft. (520 – 340 = 180)

Area of the larger extra rectangle: 380 x 180 = 68,400 sq. ft.

Area of the property:
364,000 sq. ft. – 57,600 sq. ft. – 68,400 sq. ft. = 238,000 sq. ft.
Or: 364,000 sq. ft. – (57,600 sq. ft. + 68,400 sq. ft.) = 238,000 sq. ft.

238,000 sq. ft. ÷ 43,560 sq. ft. per acre ≈ 5.5 acres

Cost: 5.5 acres x $326 per acre = $1,793

More Accessible Solution

Hectar is paying too much in taxes. Hectar should be paying about $1,425 in property taxes, based on the area of his property being about 4.82 acres.

More Challenging Solution

Hectar is paying too much in taxes. Hectar should be paying $2,252.66 in property taxes, based on the area of his property being about 7 acres.

Note: Rounding his acreage to 7 might produce an acceptable tax rate of $2,282.