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This task can be implemented individually, with small groups, or with the whole class.

The teacher asks the student to complete the problem on the Distance Between Two Points worksheet.

The teacher asks follow-up questions, as needed.

TASK RUBRIC

Getting Started

Misconception/Error

Note: It is possible that some students may know to use the Distance Formula, therefore it was included in the following rubric. However knowing to use this strategy is not required.

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.

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

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 Pythagorasâ€™ 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, Explaining a Proof of the Pythagorean Theorem, and Converse of the Pythagorean TheoremÂ to assess the studentâ€™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., 2aÂ + 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 TriangleÂ or Calculate Triangle Sides.

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.Â

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.Â

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.

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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

Accommodations:

Note: It is possible that some students may know to use the Distance Formula, therefore it was included in the rubric. However knowing to use this strategy is not required.

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