Getting Started 
Misconception/Error The student is unable to determine if the given lengths can be used to construct a triangle. 
Examples of Student Work at this Level The student:
 Indicates that he or she does not know how to determine if the given lengths form a triangle.Â
 Guesses or answers all questions incorrectly.

Questions Eliciting Thinking Will any three lengths form a triangle?
Did you think about trying to draw or construct triangles with these lengths?
Could you use a ruler and compass to draw triangles? Where would you begin drawing? 
Instructional Implications Guide the student to use a compass, straightedge, and ruler to draw triangles with sides of the given lengths. Begin with a set of lengths that will form a triangle. Explain that a good way to begin is by drawing a working line and â€śbuildingâ€ť the triangle on it. Show the student how to set the radius of the compass to a needed length and how to mark this length on the working line. Next, have the student set the compass to another given length and, with the compass point at one endpoint of the line, make a large sweeping arc. Then have the student set the compass to the remaining length and with the compass point at the other endpoint of the first side, make a sweeping arc that intersects the previous arc. Show the student how the point of intersection of the arcs is the given distances from the endpoints of the first side. Next, attempt to construct a triangle using a set of lengths that will not form a triangle. Explain to the student that if the arcs never intersect, the lengths are not long enough to meet. Model explaining that the sum of the lengths of any two sides of the triangle must be greater than the third length. A manipulative such as can also be effectively used to demonstrate this result.
Provide additional opportunities for the student to determine if three lengths can be used to form a triangle by attempting to construct the triangle. Then provide sets of lengths and ask the student to identify those that can be used to construct a triangle without actually constructing the triangle.
Consider implementing the CPALMS Lesson Plan Triangle Inequality Investigation (ID 40261), a handson lesson that teaches that only certain combinations of lengths will create triangles. 
Making Progress 
Misconception/Error The student is unable to write an accurate generalization statement. 
Examples of Student Work at this Level Although the student answers one through three correctly, his or her generalization statement contains some misconceptions. The student claims:
 The sum of the lengths of two sides must be â€śequal toâ€ť or greater than the length of the third side.
 The sum of the two shorter lengths must be greater than the longest length in order for the three side lengths to form a triangle.

Questions Eliciting Thinking Can the sum of two lengths be equal to the length of the third side? Can you confirm that statement by drawing a triangle with those conditions?
How does your statement, â€śThe sum of the shorter sides must be greater than the length of the third side,â€ť apply to an equilateral triangle? 
Instructional Implications Address the studentâ€™s specific misconception by providing a set of lengths that will serve as a counterexample and challenge the student to construct a triangle with that set of lengths. For example, provide the student with the set of lengths {3 in., 3 in., and 6 in.} to address the misconception that the sum of two lengths can equal the length of the third side. Ask the student to explain how a statement such as, â€śThe sum of the two shorter lengths must be greater than the longest length in order for the three side lengths to form a triangle,â€ť can be applied to sets of lengths that include two or three equal values. Assist the student in restating the generalization in a more precise way. Model explaining that the sum of any two lengths must be greater than the third length. Use mathematical terminology and notation to summarize this relationship. For example, explain that if a, b, and c represent any three lengths, then a triangle can be constructed from these lengths if and only if a + b > c, a + c > b, and b + c > a. 
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 5 cm, 8 cm, 12 cm and 12 in, 12 in, 12 in can be used to construct a triangle but that 3 ft, 6 ft, 10 ft cannot. The student explains that the length of each side of a triangle must be less than the sum of the lengths of the other two sides.
Note: This student made an error in his response to question #4 (repeated b + c > a rather than stating a + c > b). The student corrected this error when it was brought to his attention.

Questions Eliciting Thinking How can you be sure your statement is always true?
If a student said the sum of the two â€śshorterâ€ť sides must be greater than the third side, would he or she be correct? Explain. 
Instructional Implications Provide two lengths to the student such as 5 cm and 9 cm, and challenge the student to find the upper and lower limits on a third length so that the three lengths can form a triangle.
Consider implementing the MFAS tasks Drawing Triangles SAS, Drawing Triangles ASA, Drawing Triangles SSA, Drawing Triangles AAS, Drawing Triangles SSS, or Drawing Triangles AAA (7.G.1.2), if not done previously. 