- Focus on the corner attribute of a square. - Compare it with the defining corner attribute of a rectangle.

1. Every square has four square corners. 2. A quadrilateral with four square corners is a rectangle.

Yes. A square is also a rectangle because it has four square corners.

- Count the sides that every rhombus has. - Name the broad category defined by that count.

1. A rhombus is a closed polygon with four sides and four equal side lengths. 2. Every four-sided polygon is a quadrilateral.

Quadrilateral.

- The direction a shape faces does not determine its category. - Check the side lengths to decide whether it could be a square. - Check the corners to decide whether it is a rectangle too.

1. Each side moves \(2\) squares sideways and \(2\) squares up or down. The four sides are equal and the corners are right angles, so the tilted figure is a square. 2. Any quadrilateral with four right angles is a rectangle. 3. The figure is both a square and a rectangle.

Yes to both. It has four equal sides and four right angles, so it is a square and also a rectangle.

- Start with the outside side count. - Check the corners, then compare neighboring side lengths.

1. The figure has \(4\) straight sides, so it is a quadrilateral. 2. It has \(4\) right angles, so it is a rectangle. 3. Its sides are not all equal, so it is neither a rhombus nor a square.

Quadrilateral and rectangle.

- Check whether all four sides are equal. - Think of a more specific category that uses that side attribute.

1. The four sides have equal lengths, so the figure is a rhombus. 2. A rhombus is also a quadrilateral, so that label can stay. 3. The corners are not right angles, so “rectangle” and “square” do not fit.

Add “rhombus.” The figure has four equal sides and also belongs to the quadrilateral category.

- Check the square against each category one at a time. - A shape can belong to a smaller category and a larger category at the same time.

1. The shape is a square. 2. Its \(4\) square corners make it a rectangle. 3. Its \(4\) equal sides make it a rhombus. 4. Its \(4\) sides make it a quadrilateral. 5. It belongs in all four categories.

It should receive all four cards: square, rectangle, rhombus, and quadrilateral.

- Keep the category named in the problem. - Ask which larger category contains every rectangle. - Use “not a square” to decide whether all four sides can be equal.

1. The shape must receive the rectangle card. 2. Every rectangle has \(4\) sides, so it must also receive the quadrilateral card. 3. A rectangle that is not a square cannot have \(4\) equal sides, so the rhombus card does not fit. 4. The square card does not fit because the shape is stated not to be a square.

The cards that must fit are rectangle and quadrilateral.

- Use both the side clue and the corner clue. - Choose the name that includes all the given attributes, not only one of them.

1. Four equal sides make the shape a rhombus. 2. Four square corners make the shape a rectangle. 3. A quadrilateral with both attributes is a square. 4. Square is the most specific name.

The most specific name is square.

- Keep the category name already given for this shape. - Ask which larger category contains every rhombus. - Use this shape's corner clue to rule out two categories.

1. The shape is a rhombus because that category is given. 2. It is also a quadrilateral because every rhombus has \(4\) sides. 3. It is not a rectangle or square because it does not have \(4\) square corners.

The names that fit are rhombus and quadrilateral.

- Use the side-length attribute that defines a rhombus. - Check whether a square has that attribute. - Do not assume that one category name removes another valid name.

1. A rhombus is a quadrilateral with \(4\) equal sides. 2. A square has \(4\) equal sides. 3. Square corners do not prevent a shape from being a rhombus. 4. Therefore, Diego is not correct.

No. A square has \(4\) equal sides, so it is also a rhombus.

- Think about whether squares are included in the rectangle category. - Check the corner and side attributes of a square.

1. A rectangle is defined by its \(4\) square corners. 2. Some rectangles also have \(4\) equal sides. 3. Those rectangles are squares. 4. Therefore, Mina is not correct.

No. A square is a rectangle with \(4\) equal sides.

- Write the defining attribute contributed by each category. - Combine the corner and side conditions.

1. Being a rectangle gives the shape four square corners. 2. Being a rhombus gives the shape four equal sides. 3. A quadrilateral with both attributes is a square.

Square.

- Think of a quadrilateral that satisfies both defining attributes. - Check both its side lengths and its corners.

1. A square has four square corners, so it is a rectangle. 2. A square has four equal sides, so it is a rhombus. 3. Therefore, the rectangle and rhombus categories overlap at squares.

Yes. A square is both a rectangle and a rhombus.

- Classify each figure from what the geoboard shows before comparing them. - Check the side lengths and the corners of each figure separately. - Keep only the category names that appear in both classifications.

1. Figure a) has four equal sides and four square corners, so it is a square, rectangle, rhombus, and quadrilateral. 2. Figure b) has four square corners but two different side lengths, so it is a rectangle and quadrilateral, but not a square or rhombus. 3. The category names shared by both figures are rectangle and quadrilateral.

Rectangle and quadrilateral.

- Classify each figure from what the geoboard shows before comparing them. - Check whether all four sides are equal and whether all four corners are square. - Keep only the category names that appear in both classifications.

1. Figure a) has four equal sides and four square corners, so it is a square, rectangle, rhombus, and quadrilateral. 2. Figure b) has four equal sides but does not have four square corners, so it is a rhombus and quadrilateral, but not a square or rectangle. 3. The category names shared by both figures are rhombus and quadrilateral.

Rhombus and quadrilateral.

- Name all the categories that fit a) and then b). - Keep only the names that belong to both lists.

1. Figure a) is a square, so it belongs to all four categories. 2. Figure b) has four straight sides but is not a rectangle or a rhombus. 3. Both are quadrilaterals. That is the only category from the list they share.

Quadrilateral only. Both are closed figures with four straight sides.

- What makes a rectangle different from some other four-sided shapes? - What do you know about the lengths of a rhombus's sides? - Test both rules on each figure before you choose.

1. To be both a rectangle and a rhombus, a figure must have four right angles and four equal sides. 2. Figure a) has four equal sides and four right angles, so it fits both names. 3. Figure b) has four equal sides but does not have four right angles. It is a rhombus, but not a rectangle. 4. Figure c) has four right angles but unequal side lengths. It is a rectangle, but not a rhombus.

Figure a) only. It has four equal sides and four right angles. Figure b) does not have four right angles; figure c) does not have four equal sides.

- Compare the required corner attributes first. - Then find a side-length feature that some rectangles have but squares do not. - Use one example type to show why the reverse statement fails.

1. Every square has \(4\) square corners, so every square meets the rectangle rule. 2. A rectangle can have two different side lengths. 3. A rectangle with two different side lengths does not have \(4\) equal sides, so it is not a square. 4. Therefore, all squares are rectangles, but some rectangles are not squares.

Squares have the \(4\) square corners required for rectangles. Some rectangles have two different side lengths, so they are not squares.

- Start with the most specific category marked yes. - Check which larger or overlapping categories every square must belong to. - Correct every entry that conflicts with those required memberships.

1. A square has \(4\) square corners, so every square is a rectangle. 2. A square has \(4\) equal sides, so every square is a rhombus. 3. If square is yes, rectangle and rhombus must also be yes. 4. Therefore, both no entries must be changed to yes.

Change rectangle and rhombus from no to yes.

- For “always,” look for a defining attribute that every shape in the first category has. - For “sometimes,” find one example that works and one that does not. - Use squares as the overlap case when appropriate.

1. Every square has four equal sides, so a) is always true. 2. Some rhombuses are squares and therefore rectangles, while non-square rhombuses are not rectangles, so b) is sometimes true. 3. Some rectangles are squares, while rectangles with two different side lengths are not squares, so c) is sometimes true.

a) Always b) Sometimes c) Sometimes

- Start with the category marked yes and identify its larger category. - Ask whether a square could have “rhombus: no.” - Use the overlap relationships, not the drawing's appearance.

1. Because rectangle is yes, the shape has four sides and is therefore a quadrilateral. 2. A square is both a rectangle and a rhombus. 3. Since rhombus is no, the shape cannot be a square.

Square: no Quadrilateral: yes

- Use the category that the shape is explicitly not in. - Ask whether every square must belong to that category. - Then decide whether the square label is possible.

1. A square has four square corners, so every square is a rectangle. 2. The mystery shape is not a rectangle. 3. Therefore, it cannot be a square; it is a non-square rhombus.

No. Every square is a rectangle, and this shape is not a rectangle.

- Treat each category independently instead of choosing only the most specific name. - Check equal sides and square corners for each figure. - Remember that a square also belongs to larger categories. - Orientation and concavity do not change whether a closed four-sided figure is a quadrilateral.

1. Figure a) has four equal sides and four square corners, so it is a square, rectangle, rhombus, and quadrilateral. 2. Figure b) has four square corners but two different side lengths, so it is a rectangle and quadrilateral. 3. Figure c) has four equal sides but not four square corners, so it is a rhombus and quadrilateral. 4. Figure d) is a four-sided figure without four equal sides or four square corners, so it is a quadrilateral only. 5. Figure e) is a rotated square, so rotation does not change its four category memberships: square, rectangle, rhombus, and quadrilateral. 6. Figure f) is concave but still has four straight sides, so it is a quadrilateral only.

a) Square, rectangle, rhombus, quadrilateral b) Rectangle, quadrilateral c) Rhombus, quadrilateral d) Quadrilateral e) Square, rectangle, rhombus, quadrilateral f) Quadrilateral

- Test the first claim with a rectangle that has two different side lengths. - Test the second claim with the most special rectangle. - Use one counterexample for each statement.

1. A rectangle that is not a square does not have all \(4\) sides equal, so it is not a rhombus. This disproves Mia's statement. 2. A square has \(4\) square corners, so it is a rectangle, and \(4\) equal sides, so it is a rhombus. 3. The square disproves Leo's statement. 4. Some rectangles are rhombuses, and some are not.

Both are wrong. A non-square rectangle is not a rhombus, but a square is both a rectangle and a rhombus.

- Test whether every rectangle must have the side attribute of a rhombus. - Test whether every rhombus must have the corner attribute of a rectangle. - Use the square to identify where the two categories overlap.

1. A rectangle that is not a square does not have all \(4\) sides equal, so it is not a rhombus. 2. A rhombus that is not a square does not have all \(4\) square corners, so it is not a rectangle. 3. A square has both attributes, so it belongs to both categories. 4. Therefore, neither category contains all of the other; the categories overlap at squares.

No. Some rectangles are not rhombuses, and some rhombuses are not rectangles. Squares belong to both categories.

- Consider the four main cases: a quadrilateral that is neither a rectangle nor a rhombus, a non-square rectangle, a non-square rhombus, and a square. - Count the category cards each case receives. - Check whether any case gives exactly three cards.

1. A quadrilateral that is neither a rectangle nor a rhombus belongs only to the quadrilateral category. 2. A non-square rectangle belongs to the rectangle and quadrilateral categories, and a non-square rhombus belongs to the rhombus and quadrilateral categories. 3. A square belongs to all four categories because it is a square, rectangle, rhombus, and quadrilateral. 4. Therefore, no quadrilateral in these categories belongs to exactly three of the four.

No. The possible membership counts are \(1\), \(2\), or \(4\), but not \(3\).