Polygons and Algebraic Relationships

Lesson 11

Math

Unit 5

10th Grade

Lesson 11 of 15

Objective


Calculate and justify composite and irregular areas on the coordinate plane. 

Common Core Standards


Core Standards

  • G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Foundational Standards

  • 6.G.A.1
  • 6.G.A.3
  • 7.G.B.6

Criteria for Success


  1. Find the area of a figure on the coordinate plane by either:
    1. Identifying the area of a rectangle that “inscribes” the shape and subtracting the shape outside the figure (decomposition), or
    2. Identifying the lengths required by the formula to calculate.
  2. Find the perimeter of a figure on the coordinate plane by determining all the side lengths and finding the sum.
  3. Identify when an estimate of area is appropriate and devise a plan for a good estimate. 

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Anchor Problems


A family has a house on a plot of land and would like to build an addition. The map below shows the “footprint” of the original house, the proposed addition, and the family’s plot of land. Each grid line represents 10 linear feet. 

  1. The family’s plot of land is rectangular. What are the values of the missing coordinates, $$x$$ and $$y$$?
  2. What is the ratio of the proposed addition “footprint” area to the original house “footprint” area? 
  3. What is the area of land that will not have a building on it if the proposed addition is approved and built?
  4. Prove that the proposed addition “footprint” is a right triangle. 

Guiding Questions

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Target Task


Below is a map of an island superimposed on a coordinate plane. 

a) If each grid line represents 2 miles, what is the approximate area of the island in square miles?
b) If there are 1,500 people who live on this island, what is the population density?

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students must find the population density.
  • Create more problems similar to Anchor Problem #2 by adding grid overlay to images on GeoGebra. This video on SchoolTube provides guidance on how to do this.
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Lesson 10

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Lesson 12

Lesson Map

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Topic A: Distance on the Coordinate Plane

Topic B: Classify Polygons using Slope Criteria and Proportional Line Segments

Topic C: Area and Perimeter On and Off the Coordinate Plane

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