# MA.912.GR.7.3

Graph and solve mathematical and real-world problems that are modeled with an equation of a circle. Determine and interpret key features in terms of the context.

### Clarifications

Clarification 1: Key features are limited to domain, range, eccentricity, center and radius.

Clarification 2: Instruction includes representing the domain and range with inequality notation, interval notation or set-builder notation.

Clarification 3: Within the Geometry course, notations for domain and range are limited to inequality and set-builder.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Circle
• Domain
• Range of a Relation or Function

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students solved problems involving the Pythagorean Theorem. In Algebra 1, students rearranged formulas to highlight a quantity of interest, worked with quadratic equations and functions and performed single transformations on functions. In Geometry, this background provides a solid foundation for graphing the equation of a circle and determining and interpreting key features. In later courses, students will expand these skills to graph equations for other circles, parabolas, hyperbolas and ellipses and to determine and interpret their key features.
• 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.

• 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

Instructional Item 1
• 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?

Instructional Item 2
• The equation of a circle is given.
$x$2 + $y$2 − 6$x$ + 8$y$ + 5 = 0
• Part A. Determine the center and the radius of the circle.
• Part B. What is the ordered pair that contains the maximum $y$-value of the circle?
• Part C. Sketch the graph of the circle on the coordinate plane.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1202340: Precalculus Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.7.AP.3: Given an equation of a circle, identify center and radius, and graph the circle.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plan

Space Equations:

In this lesson, students model the orbit of a satellite and the trajectory of a missile with a system of equations. They solve the equations both graphically and algebraically.

Type: Lesson Plan

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.