Three Fourth Vertices and a Claim About Squares
Ms. Kowalski's Grade 5 class is working on the coordinate plane and the quadrilateral hierarchy. The grid has the origin at the lower-left corner, and every grid line represents 1 unit.
Three vertices of a rectangle are given: A(2, 3), B(7, 3), and C(7, 6). Students must name a fourth vertex D that makes ABCD a rectangle and justify it.
- Tessa writes D(2, 6) and says: same x-coordinate as A, same y-coordinate as C.
- Ben writes D(7, 9) and says: I went up 3 more units from C.
- Diego writes D(2, 0) and says: I went down 3 units from A.
During the share-out, Ben adds that the answer cannot be a square, 'because a square has four equal sides and a rectangle has two long sides.' Ms. Kowalski wants the class to settle the vertex question first and then use the same figure to address Ben's claim, with coordinates rather than appearances doing the arguing.
1. Evaluate all three proposed vertices against the properties of a rectangle, justify the correct one with coordinates, and write one question that would push Ben to test his own answer.
Read the worked answer
Tessa, D(2, 6), is correct. AB runs horizontally from x = 2 to x = 7: length 5. BC runs vertically from y = 3 to y = 6: length 3, so the angle at B is a right angle. Completing the figure requires a horizontal side of length 5 from (7, 6) back to (2, 6) and a vertical side of length 3 from (2, 3) up to (2, 6). Opposite sides are equal and parallel, and because every side is horizontal or vertical, all four angles are right angles.
Ben, D(7, 9), fails. B(7, 3), C(7, 6), and D(7, 9) all lie on the line x = 7, so three vertices are collinear and the figure does not close into a quadrilateral at all.
Diego, D(2, 0), fails. DA is vertical with length 3 and AB is horizontal with length 5, but CD runs from (7, 6) to (2, 0), a length of √61, so opposite sides are unequal and there is no right angle at D.
Question for Ben: 'At your point D, which two sides meet, and what is the angle between them?' Plotting shows the three points stacked in a line.
2. Respond to Ben's claim that a square cannot be a rectangle. Use the attribute definitions, give an example of each category from the grid, and explain why the class cannot settle this by looking at the shapes.
Read the worked answer
Ben is treating the two categories as exclusive, but they are nested.
- A rectangle: a quadrilateral with four right angles. Nothing is said about side lengths.
- A square: a quadrilateral with four right angles and four equal sides - every property of a rectangle, plus one more.
So every square is a rectangle, and not every rectangle is a square. ABCD with sides 5 and 3 is a rectangle that is not a square, which is the counterexample to any claim that rectangles must be squares. A Venn diagram (quadrilaterals containing parallelograms containing rectangles containing squares) makes the nesting visible, and an attribute table with a 'four right angles' row and an added 'four equal sides' row shows where the extra condition enters.
Ben's 'two long sides' criterion is about appearance, not attributes, and appearance is exactly what fails here: a square rotated 45° on the grid looks like a diamond and has no horizontal or vertical sides, yet it still has four right angles and four equal sides. Ms. Kowalski should insist on side lengths and angles measured from coordinates, not on how a figure is oriented on the page.
3. Keeping A(2, 3) and B(7, 3) fixed, move exactly one vertex so that the figure becomes a square, give the new coordinates, and explain how the coordinate grid makes the equal-side check arithmetic rather than visual. Then state what this construction proves about Ben's claim.
Read the worked answer
Move C from (7, 6) to (7, 8) and set D at (2, 8).
Checks with coordinates:
- AB: (2, 3) to (7, 3) - horizontal, length 7 − 2 = 5.
- BC: (7, 3) to (7, 8) - vertical, length 8 − 3 = 5.
- CD: (7, 8) to (2, 8) - horizontal, length 5.
- DA: (2, 8) to (2, 3) - vertical, length 5.
All four sides are equal and each side is horizontal or vertical, so each pair of adjacent sides meets at a right angle. The figure is a square, and it therefore also has four right angles - it is a rectangle as well.
The grid turns 'equal sides' and 'right angles' into subtraction: same y-coordinate means a horizontal side, same x-coordinate means a vertical side, and length is the difference of the coordinates. No protractor or ruler is needed, and orientation on the page is irrelevant.
This construction is the direct answer to Ben: increasing the height from 3 to 5 makes the 5-by-3 rectangle into a 5-by-5 square, which is still a rectangle. Making a shape more special does not remove it from the broader category.