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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.
Standard 2 : Use coordinates to prove simple geometric theorems algebraically. (Geometry - Major Cluster)Archived
Cluster Standards
This cluster includes the following benchmarks.
Visit the specific benchmark webpage to find related instructional resources.
- MAFS.912.G-GPE.2.4 : 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).
- MAFS.912.G-GPE.2.5 : 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).
- MAFS.912.G-GPE.2.6 : Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- MAFS.912.G-GPE.2.7 : Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★
Cluster Information
Number:
MAFS.912.G-GPE.2
Title:
Use coordinates to prove simple geometric theorems algebraically. (Geometry - Major Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
912
Domain-Subdomain
Geometry: Expressing Geometric Properties with Equations
Cluster Access Points
This cluster includes the following Access Points.
- MAFS.912.G-GPE.2.AP.4a : Use coordinates to prove simple geometric theorems algebraically.
- MAFS.912.G-GPE.2.AP.5a : Using slope, prove lines are parallel or perpendicular.
- MAFS.912.G-GPE.2.AP.5b : Find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point.
- MAFS.912.G-GPE.2.AP.6a : Given two points, find the point on the line segment between the two points that divides the segment into a given ratio.
- MAFS.912.G-GPE.2.AP.7a : Use the distance formula to calculate perimeter and area of polygons plotted on a coordinate plane.
Cluster Resources
Vetted resources educators can use to teach the concepts and skills in this topic.
Original Student Tutorial
- High Tech Seesaw: Learn how to find the point on a directed line segment that partitions it into a given ratio in this interactive tutorial.
Formative Assessments
- Partitioning a Segment: Students are asked to find the coordinates of a point which partitions a segment in a given ratio.
- Centroid Coordinates: Students are asked to find the coordinates of the centroid when given the ratio of a directed segment.
- Proving Slope Criterion for Perpendicular Lines - 2: Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.
- Proving Slope Criterion for Perpendicular Lines - 1: Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.
- Proving Slope Criterion for Parallel Lines - Two: Students are asked to prove that two lines with equal slopes are parallel.
- Proving Slope Criterion for Parallel Lines - One: Students are asked to prove that two parallel lines have equal slopes.
- Midpoints of Sides of a Quadrilateral: Students are asked to prove that the quadrilateral formed by connecting the midpoints of the sides of a given quadrilateral is a parallelogram.
- Type of Triangle: Students are given the coordinates of three vertices of a triangle and are asked to use algebra to determine whether the triangle is scalene, isosceles, or equilateral.
- Writing Equations for Parallel Lines: Students are asked to identify the slope of a line parallel to a given line and write an equation for the line given a point.
- Diagonals of a Rectangle: Students are given the coordinates of three of the four vertices of a rectangle and are asked to determine the coordinates of the fourth vertex and show the diagonals of the rectangle are congruent.
- Perimeter and Area of an Obtuse Triangle: Students are asked to find the perimeter and area of an obtuse triangle given in the coordinate plane.
- Perimeter and Area of a Right Triangle: Students are asked to find the perimeter and the area of a right triangle given in the coordinate plane.
- Perimeter and Area of a Rectangle: Students are asked to find the perimeter and the area of a rectangle given in the coordinate plane.
- Writing Equations for Perpendicular Lines: Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given a point.
- Describe the Quadrilateral: Students are given the coordinates of the vertices of a quadrilateral and are asked to determine whether the quadrilateral could also be a parallelogram, rhombus, rectangle, square, or trapezoid.
- Pentagon’s Perimeter: Students are asked to find the perimeter of a pentagon given in the coordinate plane.
- Finding Equations of Parallel and Perpendicular Lines: This lesson is intended to help you assess how well students are able to understand the relationship between the slopes of parallel and perpendicular lines and, in particular, to help identify students who find it difficult to:
- Find, from their equations, lines that are parallel and perpendicular.
- Identify and use intercepts.
Lesson Plans
- Keeping Triangles in Balance: Discovering Triangle Centroid is Concurrent Medians: In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the centroid is a point of concurrency for the medians of a triangle and recognize its associated usage with the center of gravity or barycenter. This set of instructional materials provides the teacher with hands-on activities using technology as well as paper-and-pencil methods.
- Proof of Quadrilaterals in Coordinate Plane: This lesson is designed to instruct students on how to identify special quadrilaterals in the coordinate plane using their knowledge of distance formula and the definitions and properties of parallelograms, rectangles, rhombuses, and squares. Task cards, with and without solution-encoded QR codes, are provided for cooperative group practice. The students will need to download a free "QR Code Reader" app onto their SmartPhones if you choose to use the cards with QR codes.
- Airplanes in Radar's Range: For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.
- Partition Point For The Queen: Students will locate a point that partitions a line segment into a given ratio. Students will use a variety of methods; the activities range from informal student definitions and sketches to tasks using number lines and the coordinate plane.
- Who Am I?: Quadrilaterals: Students will use formulas they know (distance, midpoint, and slope) to classify quadrilaterals.
- Pondering Points Proves Puzzling Polygons: In a 55 minute class, students use whiteboards, Think-Pair-Share questioning, listen to a quadrilateral song, and work individually and in groups to learn about and gain fluency in using the distance and slope formulas to prove specific polygon types.
- Geometree Thievery: This geometry lesson focuses on partitioning a segment on a coordinate grid in a non-traditional and interesting format. Students will complete a series of problems to determine which farmers are telling the truth about their harvested "Geometrees."
- Proving quadrilaterals algebrically using slope and distance formula: Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then justify their proofs to their classmates.
- Quadrilaterals and Coordinates: In this lesson, students will use coordinates to algebraically prove that quadrilaterals are rectangles, parallelograms, and trapezoids. A through introduction to writing coordinate proofs is provided as well as plenty of practice.
- My Geometry Classroom: Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.
- Partitioning a Segment: In this lesson, students find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- Proving Quadrilaterals: This lesson provides a series of assignments for students at the Getting Started, Moving Forward, and Almost There levels of understanding for the Mathematics Formative Assessment System (MFAS) Task Describe the Quadrilateral (CPALMS Resource ID#59180). The assignments are designed to "move" students from a lower level of understanding toward a complete understanding of writing a coordinate proof involving quadrilaterals.
- What's the Point?: Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.
Note: This is not an introductory lesson for this standard.
- Polygon...Prove it: While this is an introductory lesson on the standard, students will enjoy it, as they play "Speed Geo-Dating" during the Independent practice portion. Students will use algebra and coordinates to prove rectangles, rhombus, and squares. Properties of diagonals are not used in this lesson.
- Partition Me: Students will learn how to partition a segment. Turn your class into a partitioning party; just BYOGP (Bring your own graph paper).
- Just Plane Ol' Area!: Students will construct various figures on coordinate planes and calculate the perimeter and area. Use of the Pythagorean theorem will be required.
- Going the Distance: This lesson uses the Pythagorean Theorem to derive several iterations of the Distance Formula. The Distance Formula is then used to calculate the distance between two points on both directional maps and the Cartesian coordinate plane. Vocabulary relating to vectors is also introduced.
- Graphing Equations on the Cartesian Plane: Slope: The lesson teaches students about an important characteristic of lines: their slope. Slope can be determined either in graphical or algebraic form. Slope can also be described as positive, negative, zero, or undefined. Students get an explanation of when and how these different types of slopes occur. Finally, students learn how slope relates to parallel and perpendicular lines. When two lines are parallel, they have the same slope and when they are perpendicular their slopes are negative reciprocals of one another.
Prerequisite knowledge: Students must know how to graph points on the Cartesian plane. They must be familiar with the x- and y-axis on the plane in both the positive and negative directions.
- When Will We Ever Meet?: Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine whether lines are parallel, perpendicular, or neither by looking at the graph and the equation.
- Forget Waldo - Where is 'the orthocenter'?: Starting with a set of three points, students will practice finding equations of lines and the lines that are perpendicular to them. The students will repeat this process three times - using different colors for differentiating one line from the next. The big finale brings all the work together and the students realize this activity leads to finding the orthocenter of a triangle.
- Investigating Lines With Our Minds!: Discover the relationships between the slopes of parallel and perpendicular lines. Students write the equations of lines parallel and/or perpendicular to a given line through a given point. Directions for using graph paper or x-y coordinate pegboards are given.
Perspectives Video: Professional/Enthusiasts
- Amping Up Violin Tuning with Math: Kyle Dunn, a Tallahassee-based luthier and owner of Stringfest, discusses how math is related to music.
Download the CPALMS Perspectives video student note taking guide.
- Cataloging Cats with Cartesian Coordinates: <p>This researcher knows where your cat lives! Watch how he uses coordinates and the distance formula to plot the location of hundreds of thousands of cats on a map.</p>
Problem-Solving Tasks
- A Midpoint Miracle: This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.
- Unit Squares and Triangles: This problem solving task asks students to find the area of a triangle by using unit squares and line segments.
- Triangles inscribed in a circle: This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.
Tutorials
- Parallel Lines: Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.
- Perpendicular Lines: Perpendicular lines have slopes which are negative reciprocals of each other, but why?
Video/Audio/Animations
- Parallel Lines 2: This video shows how to determine which lines are parallel from a set of three different equations.
- Parallel Lines: This video illustrates how to determine if the graphs of a given set of equations are parallel.
- Perpendicular Lines 2: This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.
Worksheet
- Midpoints of the Sides of a Quadrilateral: The students will construct a quadrilateral on graph paper, determine the midpoints of each of the four sides, then connect the midpoints of adjacent sides. The question then is the following: what are the properties of the resulting quadrilateral? Students need to justify their conclusions.