MA.912.GR.7.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.GR.3.3 
• MA.912.GR.5.3
• MA.912.GR.5.4
• MA.912.GR.5.5 
• MA.912.GR.6

Terms from the K-12 Glossary

• Circle 
• Domain 
• Radius 
• Range of a Relation or Function

Vertical Alignment

Previous Benchmarks

• MA.8.GR.1.1
• MA.8.GR.1.2 
• MA.912.AR.3.8 
• MA.912.F.2

Next Benchmarks

• MA.912.GR.7

Purpose and Instructional Strategies

• Instruction includes various methods for students to graph the equation of a circle. 
  • For example, students can use the equation of a circle to identify and plot the center of the circle on a coordinate plane. Then students can identify and use the radius to determine at least 4 points on the circle and plot those points. Students can then use those 4 points to sketch the graph of the circle, or use a compass to graph the circle. 
• Instruction includes identifying and interpreting the domain and the range of the equation of a circle. Students should understand that these are the domain and the range of a relation, not a function. 
  • For example, the domain and the range can be determined by the coordinates of the center ($h$, $k$), and the radius, $r$. The domain is $h$ − $r$ ≤ $x$ ≤ $h$ + $r$ and the range is $k$ − $r$ ≤ $y$ ≤ $k$ + $r$. The domain can be interpreted as the points on the circle having a minimum $x$-value at $h$ − $r$ and a maximum $x$-value at $h$ + $r$. Likewise, range can be interpreted as the points on the circle having a minimum $y$-value at $k$ − $r$ and a maximum $y$-value at $k$ + $r$.  
• Within the Geometry course, it is not an expectation for students to master the concept of eccentricity as a key feature of circles even though it is mentioned within Clarification 1. For enrichment purposes, eccentricity can be included within instruction; the eccentricity of all circles is zero. 
• Problem types include graphing circles, determining key features of circles and interpreting key features of circles within a mathematical or real-world context.

Common Misconceptions or Errors

• Students may try to always plot the center of the circle at the origin, not the actual center of the circle. 
• Students may confuse the signs of ($h$, $k$) when determining the center of the circle. 
  • For example, given the equation ($x$ − 1)² + ($y$ + 2)² = 1, the center is (1, −2) and not (−1, 2). 
• Students may forget to take the square root of the $r$² term to determine the radius of the circle.

Instructional Tasks

• Nikita is trying to determine which sprinkler to buy for her backyard. One rotating sprinkler has a throwing radius of 32 feet, which costs $13.99, and the other rotating sprinkler has a throwing radius of 42 feet, which costs $16.99. Note that the sprinkler throwing radius refers to the radius of the spray when the sprinkler is being used. 
  • Part A. Write an equation that describes the region each sprinkler will cover if centered at the position ($h$, $k$). 
  • Part B. Nikita’s backyard is approximately a rectangle with dimensions 80 feet by 110 feet. Nikita would like to place her sprinklers so that she waters the majority of her backyard, doubling coverage with two or more sprinklers when necessary. Develop a pattern of sprinklers that would cover the backyard. 
  • Part C. Compare your sprinkler pattern and cost with a partner. Can you and your partner determine a better, and cheaper, solution?

Instructional Items

• A florist serving the Orlando area located at (9,8), and marked with an X on the coordinate plane shown where each unit is 10 miles. The florist has a 50-mile delivery radius. 
  
  
  
  • Part A. Write an equation that describes the delivery area. 
  • Part B. Does any of the florist’s delivery area include part of Seminole County? 

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