Getting Started

The student does not demonstrate an understanding of the Pythagorean Theorem or the need to use it to solve the problem.

The student’s work contains no evidence of the use of the Pythagorean Theorem. For example, the student:

• Estimates the missing length based on the given dimensions.
  
  
• Uses a contrived formula or performs a combination of various operations using the known values.
  
  
• Uses the formula for the area of a triangle (A=bh).
  

Note: The student may or may not make calculation errors.

How did you determine your answer? Can you think of a way to find an exact answer rather than estimating?

How would you describe the triangle in the diagram? What theorems do you know that involve the lengths of the sides of right triangles?

Provide the student with basic instruction on the Pythagorean Theorem. Be sure to review the parts of a right triangle (e.g., vertices, right angle, acute angles, hypotenuse, and legs). When initially introducing the Pythagorean Theorem, emphasize that it only applies to right triangles. Be very explicit about what the theorem states describing it verbally and with mathematical symbols. Caution the student to be careful not to confuse the legs and hypotenuse when applying the theorem. Give the student the opportunity to find missing lengths in right triangles in both real-world and mathematical problems. Include problems in which the length of the hypotenuse is unknown, the length of a leg is unknown, the unknown lengths are integers, the unknown lengths are rational or irrational numbers, and those in which diagrams must be sketched and labeled. Guide the student to show work completely and in an organized manner.

If needed, review finding and approximating square roots.

Moving Forward

The student makes errors in applying the Pythagorean Theorem.

The student recognizes the need to use the Pythagorean Theorem but makes significant errors in its application. For example, the student:

• Does not substitute the correct values for the lengths of the legs or the hypotenuse.
  
  
  
  
• Reverses the operation of addition/subtraction when solving for the unknown variable.
• Finds the square of a length but neglects to take its square root or reports the square as the actual length.
  

Do you think your answer is reasonable? Why or why not?

How did you decide which value to substitute for each variable? Which dimension would represent the hypotenuse (or legs)?

Can you tell me what you know about the Pythagorean Theorem? How did you apply it?

What is the difference between and c? How do you find the square root of a number?

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.

Provide instruction, as needed, on evaluating squares and square roots. Emphasize the inverse relationship between squares and square roots. Use the square root symbol and be sure the student understands the distinction between evaluating square roots and dividing.

Give the student the opportunity to find missing lengths in right triangles in both real-world and mathematical problems. Include problems in which the length of the hypotenuse is unknown, the length of a leg is unknown, the unknown lengths are integers, the unknown lengths are rational or irrational numbers, and those in which diagrams must be sketched and labeled. Guide the student to show work completely and in an organized manner.

Almost There

The student makes minor errors when applying the Pythagorean Theorem or does not completely justify each step.

The student:

• Makes a calculation error while applying correct procedures.
• Does not assign units to the answer or assigns incorrect units (e.g., square or cubic units).
  
  
• Rounds the answer for the hypotenuse incorrectly.
• Uses = rather than when using a rational number to estimate an irrational answer.
• Gives both a positive and negative value (e.g., ).
• Writes an incorrect statement (e.g., or 625 - 100 =  = 23).
  
  
• Does not show all the steps needed to justify his or her answer.

I think you made an error. Can you go back and review your work? How can you correct this error?

What units were given in the problem? What units should you include with your answer?

What is the difference between = and ? How did you decide to round your answer?

Why did you give both a positive and negative value (i.e., cm) as your answer? Can distance be negative?

How does  ? If you take the square root of one side of an equation, should you take the square root of the other side as well?

You did not show every step needed to justify your answer. Can you go back and fill in the missing steps?

Provide feedback to the student regarding any errors made and allow the student to revise his or her work. Encourage the student to carefully label diagrams. Remind the student to show work neatly and completely to avoid careless errors.

Encourage the student to check for the reasonableness of the answer within the given context and to label the units of measure.

Got It

The student provides complete and correct responses to all components of the task.

The student applies the Pythagorean Theorem and shows work to justify the height of the pyramid as approximately 22.9 cm (or an appropriately rounded value).

Can you apply the Pythagorean Theorem to any triangle? Explain.

Can you use the Pythagorean Theorem to determine lengths of two sides if you are only given the length of the hypotenuse? Explain.

Edit the worksheet to give the student additional practice applying the Pythagorean Theorem. For example:

• Provide the pyramid height and slant height and ask for the perimeter of the base; or,
• Give only the height and length of the edge of the base of the pyramid and ask for the slant height of the pyramid.

Introduce the student to using the Pythagorean Theorem to solve problems in the coordinate plane.