Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 64705
Distance Between Two Points
Students are asked to find the distance between two points in the coordinate plane.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, Pythagorean Theorem, right triangle, distance, coordinates, coordinate plane, hypotenuse
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_DistanceBetweenTwoPoints_Worksheet.docx
MFAS_DistanceBetweenTwoPoints_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 Distance Between Two Points worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not use the Pythagorean Theorem (or the Distance Formula) to determine the distance between two points in the coordinate plane.
Examples of Student Work at this Level
The student:
Attempts to count the number of diagonal unit lengths between the two points.
page 1 of 5 Attempts to count the sum of the unit lengths that represent the horizontal and vertical distances between the two points.
Calculates slope instead of distance.
Questions Eliciting Thinking
What is one unit of distance in the coordinate plane? Is a diagonal of a square the same length as its side?
How would you calculate the distance between two points that have the same y-coordinate? How far is point K from (2, 6)?
Is slope the same as distance?
Can you apply the Pythagorean Theorem to determine the distance between the two points? How would you determine the lengths of the legs of the right triangle?
Instructional Implications
page 2 of 5 Review the properties of a right triangle as needed. Be sure the student is able to identify the legs and hypotenuse of any right triangle. Explain why the hypotenuse must
be the longest side of a right triangle in terms of the relative size of the right angle.
Assist the student in developing a thorough understanding of the Pythagorean Theorem before trying to apply it. Consider implementing the CPALMS Lesson Plan A
Hypotenuse Is a WHAT???? (ID 31918), which guides the student through the history and discovery of the Pythagorean Theorem or Pythagoras’ Theorem (ID 7726),
which introduces the Pythagorean Theorem and provides both visual and algebraic proofs for the theorem. Consider using the MFAS tasks Pythagorean Squares (8.G.2.6),
Explaining a Proof of the Pythagorean Theorem (8.G.2.6), and Converse of the Pythagorean Theorem (8.G.2.6) to assess the student’s level of understanding regarding
the Pythagorean Theorem. Also, address any specific misconceptions such as the confusion of slope with distance.
Then, 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 applies the Pythagorean Theorem incorrectly.
Examples of Student Work at this Level
The student uses the Pythagorean Theorem to write an equation of the form
but makes a significant error in the process of solving. The student may:
Write the formula incorrectly (e.g., 2a + 2b = 2c).
Perform the wrong operation during the process of solving (e.g., subtract instead of add).
Substitute the length of a leg for the hypotenuse.
Square the final value instead of taking its square root (e.g.,
or leave the answer as 225).
Confuse taking a square root with dividing by two.
Questions Eliciting Thinking
What formula are you trying to apply? Can you show me how you used this formula?
What variable in the formula represents the hypotenuse? What do you know about the hypotenuse?
How can you determine the lengths of the legs of the right triangle?
What does squared mean? Does the order of operations apply?
How do you square a number? Is squaring a number the same as multiplying by two?
What is the inverse of squaring? How do you take the square root of a number? Is taking the square root the same as dividing by two?
Instructional Implications
Review the properties of a right triangle as needed. Be sure the student is able to identify the legs and hypotenuse of any right triangle. Explain why the hypotenuse must
be the longest side of a right triangle in terms of the relative size of the right angle.
Focus instruction on the specific algebraic errors observed in student work. Consider implementing the CPALMS Lesson Plan Applying the Pythagorean Theorem (ID 48973)
and then the CPALMS Lesson Plan Origami Boats-Pythagorean Theorem in the Real World (ID 49055).
Provide instruction, as needed, on evaluating squares and square roots. Emphasize the inverse relationship between squares and square roots. Model the use of the square
root symbol and be sure the student understands the distinction between evaluating square roots and dividing.
Provide additional opportunities for the student to apply the Pythagorean Theorem to find distance between points in the coordinate plane. Consider using the MFAS task
Coordinate Plane Triangle (8.G.2.8) or Calculate Triangle Sides (8.G.2.8).
Almost There
Misconception/Error
The student makes minor errors when applying the Pythagorean Theorem.
Examples of Student Work at this Level
The student may:
Make a minor calculation error.
Miscount the length of a leg.
List a positive and negative answer (e.g.
Show an incorrect statement (e.g.,
).
).
Not show all work required to justify the answer.
Not assign units.
page 3 of 5 The student may apply the Distance Formula but does not know how it relates to the Pythagorean Theorem.
Questions Eliciting Thinking
I think you made an error in your work. Can you go back and review your work?
Can distance be negative?
Does c represent the same value as
? Explain.
How did you solve the equation
? If you take the square root of one side of an equation, should you take the square root of the other side as well? Why or
why not?
You did not show every step needed to justify your answer. Can you go back and fill in the missing steps?
It looks like you used the Distance Formula to find the distance between points J and K. Could you have also used the Pythagorean Theorem? How are these two
approaches related?
Instructional Implications
Provide feedback and allow the student to revise his or her work. Specify 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.
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 applies the Pythagorean Theorem and determines the distance between the two points is 15 units.
The student applies the Distance Formula and can explain how it relates to the Pythagorean Theorem.
Questions Eliciting Thinking
Why did you use the Pythagorean Theorem to solve this problem?
Is there another triangle you could draw to help you determine the answer?
Does the triangle have to be a right triangle in order to use the Pythagorean Theorem?
How can you be sure it is a right triangle?
It looks like you used the Distance Formula to find the distance between points J and K. Could you have also used the Pythagorean Theorem? How are these two
approaches related?
Instructional Implications
Provide the student with the coordinates of the four vertices of a parallelogram, rhombus, or trapezoid, and challenge the student to graph the polygon and then
determine the perimeter.
Introduce the student to the Distance Formula. Demonstrate the use of the Distance Formula to determine the distance between two points in the coordinate plane, and
then challenge the student to explain the relationship between the Distance Formula and the Pythagorean Theorem.
ACCOMMODATIONS & RECOMMENDATIONS
page 4 of 5 Special Materials Needed:
Distance Between Two Points 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 5 of 5