Getting Started 
Misconception/Error The student has little or no understanding of the Triangle Inequality Theorem. 
Examples of Student Work at this Level The student:
 Attempts to use another theorem, such as the Pythagorean Theorem, to find the length of the third side of the triangle.
 Identifies the given lengths, 12 cm and 21 cm, as the upper and lower limits of the length of the remaining side of the triangle.
 Confuses restrictions on the measures of the sides with restrictions on the measures of the angles.
 Adds the given lengths and indicates that this is the upper limit while the lower limit is the smaller of the two given lengths (i.e., 12 < x < 33) or subtracts the given lengths and indicates that this is the lower limit while the upper limit is the larger of the two given lengths (i.e., 9 < x < 21).
 Adds the given lengths and indicates that this is the upper limit and provides no lower limit (i.e., x < 33).
 Subtracts the given lengths and indicates that this is the lower limit and provides no upper limit (i.e., x > 9).

Questions Eliciting Thinking What were you given in this problem? What are you asked to find?
Do you remember the Triangle Inequality Theorem? What does this theorem say?
How can you apply the Triangle Inequality Theorem to find the possible values of x? 
Instructional Implications Review the Triangle Inequality Theorem. Use manipulatives or a compass and straightedge to demonstrate why the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Assist the student in applying the theorem to the given problem. Be sure the student understands that the theorem can be used to find both the lower and upper limit on the length of the third side of the triangle. Ask the student to consider why the given lengths are subtracted to find the lower limit and summed to find the upper limit. Have the student explain why the length of the third side cannot equal either 9 cm or 33cm.
Give the student sets of three lengths and ask the student to determine which sets can represent the lengths of the sides of a triangle. Ask the student to justify each decision by appealing to the Triangle Inequality Theorem.
Provide additional opportunities to apply the Triangle Inequality Theorem to determine all possible lengths of a side of a triangle given the lengths of two sides. 
Moving Forward 
Misconception/Error The student does not understand that the third side of the triangle can have a real number length. 
Examples of Student Work at this Level The student correctly applies the Triangle Inequality Theorem to calculate 9 and 33 and understands that these values are related to the upper and lower limits on x. However, the student then subtracts 1 cm from one or both of these values or adds 1 cm to one or both of these values to find the limits on x.

Questions Eliciting Thinking What would happen if you tried to form a triangle out of the lengths 12, 21, and 50?
How much bigger than 9 cm does the third side need to be to form a triangle? How much less than 33 cm does the third side need to be to form a triangle ?
Does the length of a side of a triangle have to be a whole number? Can the length of the third side of this triangle be 20.4 cm? Can the length be irrational? 
Instructional Implications Use manipulatives to demonstrate what happens when the length of a side of a triangle equals the sum of the lengths of the remaining two sides. Explain to the student that typically lengths are not restricted to whole numbers but can be rational (or even irrational). Guide the student to understand that x can be any real number value that is greater than 9 and less than 33. Have the student consider if it is possible to represent the possible values of x (i.e., 9 < x < 33) using “less than or equal to” symbols in the context of this problem.
Provide additional opportunities to apply the Triangle Inequality Theorem to determine all possible lengths of a side of a triangle given the lengths of two sides. 
Almost There 
Misconception/Error The student’s justification is incomplete. 
Examples of Student Work at this Level The student correctly determines that the length of the third side of the triangle must be greater than 9 cm and less than 33 cm. However, when justifying this conclusion, the student:
 Describes how the limits were calculated rather than justifying the calculation.
 Cites the Triangle Inequality Theorem without any further explanation.
 Provides a nonmathematical explanation (that may or may not be true).
The student may also use notation incorrectly when conveying the possible values of x. For example, the student writes:
 9 < x > 33
 9 > x > 33
 9 – 33

Questions Eliciting Thinking How did you know to add and subtract 12 and 21? What theorem supports your calculations?
Why does adding 12 and 21 produce the upper limit of x?
Why does subtracting 12 from 21 produce the lower limit of x?
Can you read this this notation that you wrote (e.g., 9 > x > 33)? What does this actually mean? Is it possible for x to be less than nine and also greater than 33? 
Instructional Implications Provide feedback to the student concerning any omissions in his or her justification. Explain that a complete justification includes citing any relevant definitions, postulates, or theorems that support conclusions drawn. Provide additional opportunities for the student to apply the Triangle Inequality Theorem to solve problems. Ask the student to cite the theorem and to be explicit in explaining how it is applied.
Provide feedback to the student concerning any misuse of terminology or notation and allow the student to revise his or her work. 
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 correctly applies the Triangle Inequality Theorem to determine that the length of the third side of the triangle must be greater than 9 cm and less than 33 cm. The student represents the possible lengths of x using notation correctly. For example, the student writes 9 < x < 33. The student explains that the Triangle Inequality Theorem says that the sum of the lengths of any two sides of a triangle is greater than the length of the third side so that x must be greater than (21 – 12) cm and less than (21 + 12) cm.

Questions Eliciting Thinking What theorem are you using in your explanation?
What would happen if x = 9 cm or x = 33 cm?
Why does adding 12 and 21 produce the upper limit of x?
Why does subtracting 12 from 21 produce the lower limit of x? 
Instructional Implications Challenge the student to find two different values of x so that 12, 21, and x represent the lengths of the sides of a right triangle.
Ask the student to prove the Triangle Inequality Theorem. Consider implementing MFAS task Proving the Triangle Inequality Theorem (GCO.3.9). 