- Recall the defining attributes of the broad category. - Do not add attributes that would force a more specific quadrilateral.

1. A quadrilateral is a closed shape with \(4\) straight sides. 2. The instruction “make \(4\) straight sides and close the shape” gives exactly those attributes.

Choose “make \(4\) straight sides and close the shape.”

- Check each instruction against the attributes of a square. - Find the one numerical attribute that does not match.

1. A square is closed, has equal sides, and has square corners. 2. A square has \(4\) sides, not \(3\). 3. Replace “make \(3\) straight sides” with “make \(4\) straight sides.”

Change “make \(3\) straight sides” to “make \(4\) straight sides.”

- Use the two shown sides as adjacent sides of the completed rectangle. - Think about where the missing top and left sides would need to meet. - Check that the completed shape would have four square corners and two different side lengths.

1. Side \(AB\) is longer than side \(BC\), so a rectangle using those side lengths will not be a square. 2. The fourth vertex must line up vertically with \(A\) and horizontally with \(C\) to make four square corners. 3. Point \(P\) is in that location, so \(P\) completes the non-square rectangle.

\(P\)

- Find the two endpoints not already connected to each other. - A triangle must have a closed boundary with three sides.

1. The two drawn sides are \(AB\) and \(BC\). 2. To close a three-sided figure, join the open endpoints \(C\) and \(A\). 3. The completed closed shape has three straight sides, so it is a triangle.

Connect \(C\) to \(A\).

- Trace the outline you would need, starting at \(A\). - Avoid joining corners across the middle of the rectangle. - Does your completed path turn like a square corner each time?

1. Connect \(B\) to \(R\) to draw the right side. 2. Connect \(R\) to \(S\) for the top and \(S\) to \(A\) for the left side. 3. The four sides form a closed figure with four square corners.

Draw \(BR\), \(RS\), and \(SA\). They complete a rectangle with four square corners.

- Trace the three moves on the grid. - Look at where the final point lies compared with the starting point. - Once the shape closes, check both its side lengths and its corners.

1. The first three sides go right, up, and left, each for \(2\) grid squares. 2. The last point is directly above the start by \(2\) grid squares. 3. Draw a \(2\)-square vertical side downward. The four sides are equal and meet at four right angles, making a square.

Draw a line \(2\) grid squares straight down to the starting dot. All four sides are \(2\) squares long, and all four corners are right angles.

- Look at the corner of a sheet of paper. - Which choice could make a matching corner at \(A\)?

1. A horizontal and a vertical line meet at a right angle. 2. A line drawn diagonally up and right does not make a square corner with the horizontal line. 3. Draw the new side straight up from \(A\).

Straight up from \(A\), because a vertical side meets \(AB\) at a right angle.

- What stays the same when you move the corners of sticks joined at their ends? - Which corners must a square have? - Can the four sticks enclose a different shape without changing their lengths?

1. Join the sticks end to end with four right angles to make a square. 2. Join them again as a slanted rhombus, with two narrow corners and two wide corners. 3. Both shapes have four equal sides, but only the square has four right angles.

Make a square with four right angles. Then tilt the joints to make a rhombus with two narrow and two wide corners. Both use four equal sides.

- The result may be a square because squares are rectangles. - Identify the corner attribute required for every rectangle. - Choose the instruction set that guarantees it.

1. A rectangle is a quadrilateral with \(4\) square corners. 2. Choice A gives both the \(4\)-side and square-corner attributes. 3. Choice B could make a non-square rhombus, and C could make any quadrilateral. 4. Choice A is enough.

A.

- Keep \(AB\) as one side of the final figure. - The two new vertices must be the same distance from \(AB\). - Use the required height while keeping four square corners.

1. Segment \(AB\) is \(4\) grid units long. 2. One valid choice is to place one new vertex \(2\) grid units directly above \(A\) and the other \(2\) grid units directly above \(B\). 3. Connecting those two new vertices and the endpoints makes four square corners. 4. The resulting rectangle is \(4 \times 2\), so it is not a square.

For example, place one new vertex \(2\) grid units directly above \(A\) and one \(2\) grid units directly above \(B\).

- Match the length of each shown side. - The new sides must make square corners with the shown sides. - Look for the one grid point that is the same distance from both open endpoints in the needed directions.

1. The two shown sides are each \(3\) grid units long and meet at a square corner. 2. The missing corner must be \(3\) grid units directly above \(B\). 3. The same point is \(3\) grid units directly to the right of \(D\). 4. Connecting that point to \(B\) and \(D\) would make four equal sides and four square corners.

Place the missing corner \(3\) grid units directly above \(B\) and \(3\) grid units directly to the right of \(D\).

- The tick marks show that the two drawn sides match. What must be true about the missing ones? - Try each possible point and compare its distances from \(B\) and \(D\) on the grid. - A good choice must make all four sides equal, not just three.

1. From \(A\) to \(B\), move \(2\) squares right and \(1\) down. 2. Make the same move from \(D\): \(2\) right and \(1\) down reaches \(P\). 3. The opposite move from \(B\) to \(P\) matches the direction and length of \(A\) to \(D\). 4. All four sides match, so \(P\) completes a rhombus.

\(P\). From \(A\) to \(B\), from \(B\) to \(P\), from \(P\) to \(D\), and from \(D\) to \(A\), each side goes \(2\) grid squares in one direction and \(1\) in the other. All four sides are the same length.

- What must be true at the ends of the given side? - Can you change the height while keeping four right angles? - Which height would accidentally make a square?

1. One choice is to place a new vertex \(1\) square above \(A\) and another \(1\) square above \(B\). 2. Another choice is to place the two new vertices \(2\) squares above \(A\) and \(B\). 3. Connecting each pair of new points gives rectangles with four right angles. They are \(4\) by \(1\) and \(4\) by \(2\), neither a square.

For example: put both new vertices \(1\) square above the ends of \(AB\); for a second rectangle, put both \(2\) squares above the ends. Connect the vertices in each case.

- What can change when the top and bottom sides stay horizontal? - What extra rule do the corners of a rectangle follow? - Sketch one example with two vertical sides. Could another shape keep the horizontal sides without four square corners?

1. For the rectangle, draw a bottom side \(4\) squares long and an equally long top side directly above it, joined by vertical sides. 2. For the other quadrilateral, draw a bottom side \(4\) squares long and a top side \(2\) squares long centered above it; join their ends with slanted sides. 3. Both have parallel top and bottom sides. Only the rectangle has four right angles.

One example is a \(4\)-wide rectangle with vertical sides. Another is a quadrilateral with a \(4\)-square bottom, a centered \(2\)-square top, and two slanted sides. The first has four right angles; the second does not.

- Imagine folding the finished shape down a line through its highest and lowest corners. Which sides would need to match? - Once the shape closes, compare its corners with a square corner.

1. The first two moves make two matching slanted sides meeting at the top. 2. Move \(2\) squares left and \(1\) down, then \(2\) squares right and \(1\) down to return to the start. 3. All four sides use the same sideways and vertical steps, so their lengths are equal. 4. The corners are not right angles, so the shape is a rhombus but not a square.

Move \(2\) squares left and \(1\) down, then \(2\) squares right and \(1\) down. The four sides are equal, but the corners are not square corners.

- Where could the third point go so it does not lie on \(AB\)? - How can you close each triangle using the two ends of the given side?

1. One possible upper corner is \(1\) square right and \(2\) squares up from \(A\). 2. One possible lower corner is \(2\) squares right and \(3\) squares down from \(A\). 3. Join each chosen corner to both \(A\) and \(B\). Each finished shape has \(3\) straight sides and is closed.

For example, use a corner \(1\) square right and \(2\) up from \(A\), then a different corner \(2\) squares right and \(3\) down. Both make closed shapes with \(3\) straight sides when joined to \(A\) and \(B\).

- Make a closed shape with exactly four straight sides. - Try a slanted side rather than making every corner square. - Check that the four sides are not all the same length.

1. Start at a grid point. Draw \(4\) squares right, \(2\) squares up, then \(3\) squares left. Connect the last point back to the start with a straight line. 2. This makes a closed figure with \(4\) straight sides. 3. Its sides are not all equal, so it is not a rhombus. 4. Its corners are not all square, so it is not a rectangle or square.

One example: go \(4\) grid squares right, \(2\) up, \(3\) left, then draw a straight side back to the start. This makes a quadrilateral with unequal side lengths and at least one corner that is not square.

- For the square, split the perimeter equally among four sides. - For the non-square rectangle, choose two different side lengths that repeat as opposite sides. - Check that both lists total the required perimeter.

1. A square with side length \(5\,\text{units}\) has perimeter \(5 + 5 + 5 + 5 = 20\,\text{units}\). 2. A non-square rectangle with side lengths \(4\,\text{units}\) and \(6\,\text{units}\) has perimeter \(4 + 6 + 4 + 6 = 20\,\text{units}\). 3. These give two different rectangle categories with the same perimeter.

One valid pair is: square: \(5\,\text{units}, 5\,\text{units}, 5\,\text{units}, 5\,\text{units}\) non-square rectangle: \(4\,\text{units}, 6\,\text{units}, 4\,\text{units}, 6\,\text{units}\) Other answers possible.

- Keep the equal-side condition true in both descriptions. - Compare the corner condition for a square with the corner condition for a non-square rhombus. - State the attribute that changes while the four equal sides stay the same.

1. Both descriptions must keep four equal sides. 2. The square must have four square corners. 3. The non-square rhombus must have at least one corner that is not square. 4. The corner attribute distinguishes the two most-specific categories.

Square: four equal sides and four square corners. Non-square rhombus: four equal sides and at least one corner that is not square.

- Begin with two sides that meet like the corner of a grid square. - You still need two other corners. Where could you place one to change the angle without crossing any sides? - After closing the figure, check every corner, including those between two slanted sides.

1. One example starts at \(A\). Put \(B\) \(4\) squares to the right of \(A\), and \(C\) \(3\) squares above \(B\). 2. Put \(D\) \(3\) squares left and \(1\) square down from \(C\). Join \(A\), \(B\), \(C\), and \(D\) in that order, then close the shape. 3. The horizontal \(AB\) and vertical \(BC\) meet at a right angle at \(B\). At \(A\) and \(C\), the angles are smaller than a square corner. At \(D\), the angle is wider. 4. Thus the shape has exactly one right angle and is not a rectangle.

One example: from \(A\), go \(4\) squares right to \(B\); go \(3\) squares up to \(C\); go diagonally \(3\) squares left and \(1\) down to \(D\); then join \(D\) to \(A\). Only \(B\) is a right angle because a horizontal and vertical side meet there. Corners \(A\) and \(C\) are narrower than a square corner, and \(D\) is wider.

- Start by sketching any closed five-sided shape. - What happens if you move one of its corners toward the middle without adding a side? - Check whether any of your sides cross as that corner moves.

1. One example starts at a dot and makes five straight sides: \(4\) squares right, \(3\) up, diagonally \(2\) left and \(2\) down, diagonally \(2\) left and \(2\) up, then \(3\) down to close. 2. The corner after the first diagonal points inward. 3. The boundary is closed with five straight sides, so the figure is a pentagon.

One way: from the start, go \(4\) squares right, \(3\) up, diagonally \(2\) left and \(2\) down, diagonally \(2\) left and \(2\) up, then \(3\) down to close. The outline has \(5\) straight sides and one inward corner.

- How can you share the perimeter equally among four sides? - For a rectangle, which sides must have matching lengths? - Can equal side lengths go with different kinds of corners?

1. A square can have four sides of \(6\,\text{units}\) and four square corners, giving perimeter \(24\,\text{units}\). 2. A non-square rectangle can have sides \(4,8,4,8\) units and four square corners; its perimeter is also \(24\,\text{units}\). 3. A non-square rhombus can have four sides of \(6\,\text{units}\) and at least one non-square corner; its perimeter is also \(24\,\text{units}\). 4. The square has both equal sides and square corners; the rectangle has square corners but unequal adjacent sides; the rhombus has equal sides but not four square corners.

a) Four sides of \(6\,\text{units}\) with four square corners. b) For example, \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) with four square corners. c) Four sides of \(6\,\text{units}\) with at least one corner that is not square. Only the square has both four equal sides and four square corners. The non-square rectangle has the corners but not equal sides; the non-square rhombus has the equal sides but not the corners.

- Which listed shapes need all four sides to be the same length? - Think about putting the two equal pairs opposite each other. - Can you keep those sides but change the corners? - Check the requirements for each possible shape.

1. A square and a rhombus need four equal sides. The stick lengths are not all equal, so neither can be made. 2. For a non-square rectangle, place the two length-\(3\) sticks opposite each other and the two length-\(5\) sticks opposite each other, with four square corners. 3. For a different kind of quadrilateral, use the same opposite-side pairs but slant the corners so they are not square. 4. Thus the two possible kinds are a non-square rectangle and a quadrilateral that is neither a rectangle nor a rhombus.

Possible: rectangle but not a square; quadrilateral that is neither a rectangle nor a rhombus. Make the rectangle with opposite sides equal and four square corners. Make the other quadrilateral with equal opposite-side pairs but slanted, non-square corners.