Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 64722
Calculate Triangle Sides
Students are asked to determine the length of each side of a right triangle in the coordinate plane given the coordinates of its vertices.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, right triangle, Pythagorean Theorem, hypotenuse, diagonal, sides, legs, square root, Distance
Formula, coordinate grid
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_CalculateTriangleSides_Worksheet.docx
MFAS_CalculateTriangleSides_Worksheet.pdf
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 Calculate Triangle Sides worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to determine the lengths of the sides of a right triangle in the coordinate plane.
Examples of Student Work at this Level
The student:
Graphs the triangle incorrectly and is unable to calculate any lengths.
Counts unit lengths to determine the lengths of the legs but attempts a similar strategy to determine the length of the hypotenuse.
Uses some contrived formula or combination of operations rather than the Pythagorean Theorem.
Substitutes coordinates of vertices for either a or b in the equation
.
Finds the area of the triangle or the slopes of the sides rather than the lengths of the sides.
page 1 of 4 Attempts to use a proportion to determine the lengths of the sides.
Questions Eliciting Thinking
Can you explain how you calculated these lengths?
Did you estimate this length? Can you think of a way to calculate it instead?
What do the slopes you calculated tell you about the sides of the triangle? Are slope and length the same?
What kind of triangle is this? What theorem relates the lengths of the sides of a right triangle?
Instructional Implications
If needed, address graphing in the coordinate plane. Review the unit of measure for length and assist the student in counting unit lengths to find the lengths of vertical or
horizontal segments.
Review terminology related to right triangles. Be sure the student is able to identify the legs and hypotenuse of right triangles in a variety of orientations. Explain that the
longest side, the hypotenuse, is opposite the largest angle, the right angle.
Introduce the Pythagorean Theorem and provide opportunities for the student to apply the theorem to calculate unknown lengths in right triangles. Then introduce
segments in the coordinate plane. Assist the student in applying the Pythagorean Theorem to find the length of a segment or the distance between two points in the
coordinate plane.
Consider implementing the CPALMS Lesson Plan As the Crow Flies (ID 43471), a two-part lesson which guides the student to apply the Pythagorean Theorem to determine
the length of an unknown side in a right triangle, and then to apply the Pythagorean Theorem to determine the distance between two points in the coordinate plane.
Moving Forward
Misconception/Error
The student makes an error when applying the Pythagorean Theorem.
Examples of Student Work at this Level
The student uses the Pythagorean Theorem to write an equation of the form
but applies the theorem incorrectly or makes an error when solving the
resulting equation. For example, the student:
Substitutes the length of a leg for the length of the hypotenuse writing
Divides
by two or squares
.
to find the length of c.
Attempts to apply the distance formula but subtracts a y-value from an x-value, adds instead of subtracts coordinates, or neglects to square the differences.
Questions Eliciting Thinking
Can you explain what each variable in the formula you are using represents? How did you decide which value to substitute for each variable?
Do you think your answer is reasonable? What do you know about the lengths of each side of a right triangle? Which side must be the longest and why?
Is taking a square root the same as dividing by two? What is the difference between squaring and taking the square root of a number?
Instructional Implications
Review terminology related to right triangles. Be sure the student is able to identify the legs and hypotenuse of right triangles in a variety of orientations. Explain that the
longest side, the hypotenuse, is opposite the largest angle, the right angle. As needed, review the Pythagorean Theorem and how to correctly apply it. Be sure the
student understands the meaning of the variables in the equation
.
If needed, provide instruction on evaluating squares and square roots. Emphasize the inverse relationship between squaring and taking a square root. Model the use of the
square root symbol and be sure the student understands the distinction between evaluating square roots and dividing by two.
Address any errors related to the application of the distance formula. If not done so already, derive the distance formula from the Pythagorean Theorem. Provide
opportunities to use the distance formula to calculate lengths of segments or distances between two points in the coordinate plane.
Almost There
Misconception/Error
The student makes a minor error when calculating a length.
Examples of Student Work at this Level
The student makes a minor mathematical error in some step of the solution. The student:
Incorrectly finds the sum of 49 and 169.
Calculates the length of a leg incorrectly but all other work is correct.
page 2 of 4 Plots one vertex incorrectly.
Describes a length as negative (e.g., PQ = -14.8).
Writes a mathematically incorrect statement (e.g., 169 + 49 =
14.8 or
).
Rounds ZX incorrectly.
Questions Eliciting Thinking
You made an error in your work. Can you find and correct it?
Is it possible for a length to be negative?
Is this statement true: 169 + 49 =
(or
)? What might be a better way to show your work?
Instructional Implications
Provide feedback concerning any errors made and allow the student to revise his or her work. Explain that equations of the form
= p have two solutions although one or
both may not make sense in the context of the problem. If needed, assist the student in showing work in a manner that justifies strategies and answers. Remind the
student to label the units of measure.
Provide the student with additional practice using the Pythagorean Theorem to solve real-world and mathematical problems. Consider implementing the MFAS task New
Television (8.G.2.7) or How Far to School (8.G.2.7).
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 determines that XY = 13 and Y Z = 7. The student correctly applies the Pythagorean Theorem or the distance formula to determine that
(or an
appropriately rounded approximation).
Note: The student may have used the Distance Formula to correctly calculate all three lengths not recognizing that there is a more efficient way to find the lengths of the
legs of the triangle.
Questions Eliciting Thinking
Can you apply the Pythagorean Theorem to any triangle? Explain.
How can you prove this triangle is a right triangle?
Can you find the lengths of the legs without using the distance formula?
Instructional Implications
If needed, assist the student in recognizing that the distance formula is not needed to calculate the lengths of the legs of the triangle.
Introduce the concept of a Pythagorean Triple and challenge the student to find examples of triples.
Challenge the student to find the perimeter of a figure with no sides parallel to an axis. Give the student the coordinates of the vertices and guide the student to
decompose the shape into right triangles whose hypotenuses correspond to the sides of the figure.
page 3 of 4 ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Calculate Triangle Sides 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
MAFS.8.G.2.8:
Description
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
page 4 of 4