Question 1: Calculate the midpoint of the line segment with endpoints A(2, 4) and B(6, 8).

Answer: 4, 6

Question 2: A student incorrectly calculates the midpoint of points (1, 3) and (5, 7) as (3, 6). Identify the mistake and provide the correct midpoint.

Answer: Incorrect: averaged x-coordinates (1+5)/2=3, which is correct; but for y: (3+7)/2=5, not 6. Correct midpoint: (3, 5). The mistake was in the y-coordinate calculation.

Question 3: Construct a line segment with endpoints at (-2, 5) and (4, -1). What is the midpoint? Show your working.

Answer: Midpoint: (-2+4)/2=1, (5+(-1))/2=2; so (1, 2)

Question 4: A diagram shows endpoints at (0, 0) and (8, 6). A student claims the midpoint is (4, 4). Is this correct? Explain and correct if necessary.

Answer: Incorrect. Correct midpoint: (0+8)/2=4, (0+6)/2=3, so (4, 3). The y-coordinate was miscalculated as 4.

Question 5: Find the midpoint of points (3, -2) and (-1, 4).

Answer: 1, 1

Question 6: Calculate the midpoint of the segment with endpoints at (10, 20) and (14, 24).

Answer: 12, 22

Question 7: A real-world problem: Two trees are located at points (3, 7) and (9, 11) on a coordinate map. Find the midpoint where a ranger might place a sign equidistant from both trees.

Answer: Midpoint: (3+9)/2=6, (7+11)/2=9; (6, 9)

Question 8: A student forgets to divide by 2 when calculating the midpoint. They only add the coordinates. What is the mistake and how can it be fixed?

Answer: Mistake: summing coordinates without dividing by 2. The correct method is to average the x and y coordinates separately: (x1+x2)/2 and (y1+y2)/2.

Question 9: Construct a triangle with vertices at (1, 2), (5, 2), and (3, 6). Find the midpoint of the segment connecting (1, 2) and (5, 2).

Answer: Midpoint: (1+5)/2=3, (2+2)/2=2; (3, 2)

Question 10: A challenge: Given endpoints at (-4, -4) and (4, 4), find the midpoint and explain its significance in the grid.

Answer: Midpoint: (-4+4)/2=0, (-4+4)/2=0; (0, 0). This point is the center of the segment and the origin in the coordinate plane.

Question 11: Identify and correct the error in this student's solution: Midpoint of (2, 8) and (10, 14) is (6, 11).

Answer: Error: y-coordinate calculation. Correct midpoint: (2+10)/2=6, (8+14)/2=11; so (6, 11). The student's answer is actually correct, so no correction needed.