Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 59183
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.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, quadrilateral, coordinates, special quadrilaterals, rectangle, diagonals
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_DiagonalsOfaRectangle_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Diagonals of a Rectangle worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to correctly identify the coordinates of vertex C in terms of the given coordinates.
Examples of Student Work at this Level
The student
Identifies the coordinates of vertex C as (b, a).
page 1 of 4 Introduces new variables to describe the coordinates of vertex C, e.g., C(c, d).
Questions Eliciting Thinking
Where would the point with coordinates (a, b) be located?
Can you describe the coordinates of vertex C in terms of the coordinates of vertices A and B?
Instructional Implications
Explain how to represent the coordinates of vertex C in terms of the coordinates of vertices A and B. Provide other examples of figures graphed on the coordinate plane
such as:
A square with one of its vertices is at the origin and sides that coincide with the axes.
A right triangle with the vertex of the right angle at the origin and sides that coincide with the axes.
An isosceles triangle with a base that coincides with the x-axis and positioned so that the y-axis is an axis of symmetry.
And ask the student to use the least number of variables to describe the coordinates of the vertices. Then introduce more challenging figures such as parallelograms,
rhombuses, isosceles trapezoids, and equilateral triangles.
Guide the student to develop an overall strategy for solving the problem presented in this task, i.e., (1) graph the vertices, (2) identify the coordinates of the fourth
vertex, (3) calculate the lengths of the diagonals using the distance formula, and (4) conclude that the diagonals are congruent. Ask the student to implement the strategy
and provide feedback.
If needed, review the distance formula and provide additional opportunities for the student to use the formula to calculate lengths in the coordinate plane.
If needed, provide feedback on the appropriate use of notation.
Moving Forward
Misconception/Error
The student does not use the distance formula correctly or at all to find the lengths of the diagonals.
Examples of Student Work at this Level
The student correctly identifies the coordinates of vertex C as (a, b) but, to prove the diagonals are congruent, the student:
Says to find the slope of each diagonal.
References the distance formula but then uses the midpoint formula.
Says to use the distance formula but provides no additional work.
Applies the distance formula incorrectly.
Questions Eliciting Thinking
What must be true of the diagonals in order for them to be congruent? How can you show this?
If you are asked to prove that the diagonals are congruent, is it sufficient to say, “just use the distance formula?”
Can you use the distance formula to calculate the lengths of the diagonals?
page 2 of 4 Instructional Implications
Review the distance formula and provide additional opportunities for the student to use the formula to calculate lengths in the coordinate plane.
Discuss with the student how to write a clear and complete proof. Show the student a model coordinate geometry proof of another statement and point out all of the
features that make it clear and convincing (e.g., steps are presented in a logical order, all work is labeled, computations are clearly presented, conclusions are explicitly
stated, and no extraneous work is left on the paper).
If needed, provide feedback on the appropriate use of notation.
Almost There
Misconception/Error
The student makes an algebraic error.
Examples of Student Work at this Level
The student rewrites the expression
as a + b.
The student may also not show work completely or label work appropriately.
Questions Eliciting Thinking
Is
really equal to a + b? Can you prove that?
You showed that you were using the distance formula to calculate the length of a diagonal. Can you tell me which length this is? Can you label that on your paper?
Instructional Implications
Provide a counterexample to demonstrate that
. For example, ask the student to evaluate each expression for a = 3 and b = 4 to show that the
expressions are not equivalent.
If needed, guide the student to show work completely, label all work appropriately, and use correct notation.
Consider implementing MFAS tasks Describe the Quadrilateral (G-GPE.2.4), Midpoints of Sides of a Quadrilateral (G-GPE.2.4), and Triangle Mid-Segment Proof (G-CO.3.10).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student states the coordinates of vertex C are (a, b). The student uses the distance formula to correctly calculate the length of each diagonal. The student then
concludes that since both diagonals have the same length,
, they are congruent.
Questions Eliciting Thinking
Would this proof be valid for a rectangle that is positioned differently from the one described on the worksheet?
How could you show that the diagonals bisect each other?
Instructional Implications
page 3 of 4 If needed, provide feedback to the student on any aspect of his or her work that might improve it (e.g., using the notation AB and CD rather than d to represent the
lengths of
and
, respectively, or enclosing ordered pairs in parentheses).
Challenge the student to prove geometric theorems using coordinate geometry. Consider implementing MFAS task Triangle Mid-Segment Proof (G-CO.3.10).
Give the student the coordinates of three of the vertices of a parallelogram ABCD, e.g., A(0, 0). B(b, 0), and D(a, c). Ask the student to identify the coordinates of vertex
C and prove statements such as:
The diagonals of a parallelogram bisect each other.
Opposites sides of a parallelogram are congruent.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Diagonals of a Rectangle worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
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.2.4:
Remarks/Examples:
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.
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