## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to: use the Pythagorean theorem to derive the equation of a circle; and translate between the geometric features of circles and their equations.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP), Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution-NonCommercial-NoDerivs 3.0
- Media Format:
- Downloadable docs, Text/HTML

## Standards

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Use coordinates to prove simple geometric theorems algebraically

Standard: Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Use coordinates to prove simple geometric theorems algebraically

Standard: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Use coordinates to prove simple geometric theorems algebraically

Standard: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Use coordinates to prove simple geometric theorems algebraically

Standard: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Creating EquationsCluster: Create equations that describe numbers or relationship

Standard: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Creating EquationsCluster: Create equations that describe numbers or relationship

Standard: Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Creating EquationsCluster: Create equations that describe numbers or relationship

Standard: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Creating EquationsCluster: Create equations that describe numbers or relationship

Standard: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Translate between the geometric description and the equation for a conic section

Standard: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Translate between the geometric description and the equation for a conic section

Standard: Derive the equation of a parabola given a focus and directrix.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Expressing Geometric Properties with EquationsCluster: Translate between the geometric description and the equation for a conic section

Standard: (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Use coordinates to prove simple geometric theorems algebraically

Indicator: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Use coordinates to prove simple geometric theorems algebraically

Indicator: Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Use coordinates to prove simple geometric theorems algebraically

Indicator: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ‰ö_3) lies on the circle centered at the origin and containing the point (0, 2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Use coordinates to prove simple geometric theorems algebraically

Indicator: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Creating Equations

Standard: Create equations that describe numbers or relationship

Indicator: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Creating Equations

Standard: Create equations that describe numbers or relationship

Indicator: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Creating Equations

Standard: Create equations that describe numbers or relationship

Indicator: Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Translate between the geometric description and the equation for a conic section

Indicator: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Creating Equations

Standard: Create equations that describe numbers or relationship

Indicator: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Translate between the geometric description and the equation for a conic section

Indicator: (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Expressing Geometric Properties with Equations

Standard: Translate between the geometric description and the equation for a conic section

Indicator: Derive the equation of a parabola given a focus and directrix.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

## Evaluations

# EQuIP Rubric

Average Score (3 Points Possible)ELA | Math |

Alignment to the Rigor of the CCSS | N/A |

Key Shifts in the CCSS | N/A |

Instructional Supports | N/A |

Assessment | N/A |

Overall Rating for the Lesson/Unit | N |

Alignment to the Rigor of the CCSS | N/A |

Key Shifts in the CCSS | N/A |

Instructional Supports | N/A |

Assessment | N/A |

Overall Rating for the Lesson/Unit | N |

# Tags (9)

- Mathematics
- Geometry and measures
- CCSS
- Circles
- Common Core Math
- Common Core PD
- Equations
- Geometry
- Pythagorean Theorem

## Comments