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Standard 3 : Use coordinate geometry to solve problems or prove relationships.
Cluster Standards
This cluster includes the following benchmarks.
Visit the specific benchmark webpage to find related instructional resources.
- MA.912.GR.3.1 : Determine the weighted average of two or more points on a line.
- MA.912.GR.3.2 : Given a mathematical context, use coordinate geometry to classify or justify definitions, properties and theorems involving circles, triangles or quadrilaterals.
- MA.912.GR.3.3 : Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles and quadrilaterals.
- MA.912.GR.3.4 : Use coordinate geometry to solve mathematical and real-world problems on the coordinate plane involving perimeter or area of polygons.
Cluster Information
Number:
MA.912.GR.3
Title:
Use coordinate geometry to solve problems or prove relationships.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Geometric Reasoning
Cluster Access Points
This cluster includes the following Access Points.
- MA.912.GR.3.AP.1 : Select the weighted average of two or more points on a line.
- MA.912.GR.3.AP.2 : Use coordinate geometry to classify definitions, properties and theorems involving circles, triangles, or quadrilaterals.
- MA.912.GR.3.AP.3 : Use coordinate geometry to solve mathematical geometric problems involving lines, triangles and quadrilaterals.
- MA.912.GR.3.AP.4 : Solve mathematical and/or real-world problems on the coordinate plane involving perimeter or area of a three- or four-sided polygon.
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.
- 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.
- 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.
- Triangle Midsegment Proof: Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.
- Pentagon’s Perimeter: Students are asked to find the perimeter of a pentagon given in the coordinate plane.
Lesson Plans
- 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.
- Basic Definitions in Geometry: A set of basic definitions in geometry (line segment, ray, angle, perpendicular lines, and parallel lines) is addressed. The notation used in naming each defined term is also emphasized.
- Triangle Mid-Segment Theorem: The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.
- 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.
- To Be or Not to Be a Parallelogram: Students apply parallelogram properties and theorems to solve real world problems. The acronym, P.I.E.S. is introduced to support a problem solving strategy, which involves drawing a Picture, highlighting important Information, Estimating and/or writing equation, and Solving problem.
- The Seven Circles Water Fountain: Students will apply concepts related to circles, angles, area, and circumference to a design situation.
- 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.
- Musical Chairs with Words and a Ball: This lesson introduces students to concepts and skills that they will use throughout the year. Students will learn that the terms point, and line are considered "undefined." Students will play musical chairs while learning to develop precise definitions of circle, angle, parallel line, and perpendicular line, using counterexamples at different classroom stations. Students will identify models, use notation, and make sketches of these terms.
- 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.
- Triangle Medians: This lesson will have students exploring different types of triangles and their medians. Students will construct mid-points and medians to determine that the medians meet at a point.
- 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 Parallelograms Algebraically: This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram.
- 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.
- Observing the Centroid: Students will construct the medians of a triangle then investigate the intersections of the medians.
- Determination of the Optimal Point: Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.
- 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.
- Ellipse Elements and Equations: Students will write the equation of an ellipse given foci and directrices using graphic and analytic methods.
- The Centroid: Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.
- Congruence vs. Similarity: Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.
- 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.
- Sage and Scribe - Points, Lines, and Planes: Students will practice using precise definitions while they draw images of Points, Lines, and Planes. Students will work in pairs taking turns describing an image while their partner attempts to accurately draw the image.
- Let's Prove the Pythagorean Theorem: Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.
- Partition Me: Students will learn how to partition a segment. Turn your class into a partitioning party; just BYOGP (Bring your own graph paper).
- Fundamental Property of Reflections: This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.
- Intersecting Medians and the Resulting Ratios: This lesson leads students to discover empirically that the distance from each vertex to the intersection of the medians of a triangle is two-thirds of the total length of each median.
- 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.
- Concurrent Points Are Optimal: Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.
Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.
A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.
Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.
A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.
The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students.
- Determination of the Optimal Point: Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.
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.
- Using Geometry and Computers to make Art with CNC Machining: <p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>
- 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>
Perspectives Video: Teaching Ideas
- Exploring the Hat Polygon: Unlock an effective teaching strategy for engaging students with the newly discovered hat polygon in this Teacher Perspectives video for educators.
- Finding the Area of a Polygon on the Coordinate Plane: Unlock an effective teaching strategy for finding the area of polygons on the coordinate plane using nontraditional methods in this Teacher Perspectives video for educators.
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.
- Mt. Whitney to Death Valley: This task engages students in an open-ended modeling task that uses similarity of right triangles.
- 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.