Standard #: MAFS.912.G-GPE.2.4 (Archived Standard)


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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).



Remarks


Geometry - Fluency Recommendations

Fluency with the use of coordinates to establish geometric results, calculate length and angle, and use geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields.

General Information

Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Geometry: Expressing Geometric Properties with Equations
Cluster: Use coordinates to prove simple geometric theorems algebraically. (Geometry - Major Cluster) -

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.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes

Test Item Specifications

    N/A

    Assessment Limits :
    Items may require the student to use slope or to find the distance
    between points.

    Items may require the student to prove properties of triangles,
    properties of quadrilaterals, properties of circles, and properties of
    regular polygons.

    Items may require the student to use coordinate geometry to provide
    steps to a proof of a geometric theorem.

    Calculator :

    Neutral

    Clarification :
    Students will use coordinate geometry to prove simple geometric
    theorems algebraically
    Stimulus Attributes :
    Items may be set in a real-world or mathematical context.
    Response Attributes :
    Items may require the student to determine if the algebraic proof is
    correct.


Sample Test Items (1)

Test Item # Question Difficulty Type
Sample Item 1

One diagonal of square EFGH is shown on the coordinate grid.

There are two highlights in the sentence to show which word or phrase may be incorrect. For each highlight, click the word of phrase that is correct.

N/A ETC: Editing Task Choice


Related Courses

Course Number1111 Course Title222
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1207310: Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1206300: Informal Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
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))
7912060: Access Informal Geometry (Specifically in versions: 2014 - 2015 (course terminated))
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 Resources

Formative Assessments

Name Description
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.

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.

Lesson Plans

Name Description
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.

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.

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.

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.

Problem-Solving Tasks

Name Description
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.

Student Resources

Problem-Solving Tasks

Name Description
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.



Parent Resources

Problem-Solving Tasks

Name Description
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.



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