Free math worksheets

- Check the side lengths of each shape first. - Then check whether all four corners are square corners. - The correct shape must satisfy both instructions.

1. Shape a) has \(4\) equal sides and \(4\) square corners. 2. Shape b) has \(4\) equal sides, but its corners are not square corners. 3. Shape c) has \(4\) square corners, but not all \(4\) sides are equal. 4. Therefore, shape a) follows both instructions.

Shape a).

- Focus on the attribute about equal side lengths. - Remember that one category does not require square corners.

1. A quadrilateral with \(4\) equal sides is a rhombus. 2. A rhombus does not have to have square corners. 3. When it does not have \(4\) square corners, it is a non-square rhombus.

A rhombus.

- Focus on the corner attribute. - Recall which quadrilateral category is defined by four square corners.

1. A rectangle is a quadrilateral with \(4\) square corners. 2. The side lengths do not need to be known to guarantee the rectangle category.

Rectangle.

- Think about the meaning of the word quadrilateral. - Check whether a square has the attribute needed for that larger category.

1. A quadrilateral is any closed shape with \(4\) straight sides. 2. A square has \(4\) straight sides. 3. Therefore, a square is a quadrilateral, so Lena is not correct.

No. A square has \(4\) straight sides, so it is a quadrilateral.

- Start with the clue about the side lengths. - Then use the corner clue to decide whether the shape is a square.

1. Four equal sides make the quadrilateral a rhombus. 2. A square must also have \(4\) square corners. 3. Because the corners are not all square corners, the shape is a non-square rhombus.

It is a non-square rhombus.

- 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.

- First confirm that each choice has \(4\) sides. - Rule out any shape with \(4\) square corners. - Rule out any shape with \(4\) equal sides.

1. Shape a) has \(4\) square corners, so it is a rectangle and does not follow the instructions. 2. Shape b) has \(4\) equal sides, so it is a rhombus and does not follow the instructions. 3. Shape c) has \(4\) sides, but its sides are not all equal and its corners are not all square. 4. Therefore, shape c) follows all the instructions.

Shape c). Shape a) is a rectangle, and shape b) is a rhombus.

- Use the defining number of straight sides for a quadrilateral. - Test every choice, including shapes with more specific names.

1. A square has \(4\) straight sides, so A is a quadrilateral. 2. A triangle has \(3\) sides, so B is not. 3. A rectangle has \(4\) straight sides, so C is a quadrilateral. 4. Choice D directly describes a quadrilateral. 5. A pentagon has \(5\) sides, so E is not.

A, C, and D.

- Ask which clue separates rectangles from other quadrilaterals. - Test whether each clue could also describe a non-rectangle.

1. Every quadrilateral has \(4\) sides, so A is not enough. 2. Four equal sides guarantee a rhombus, but not necessarily a rectangle. 3. Four square corners guarantee a rectangle. 4. Choice C is correct.

C. It has \(4\) square corners.

- Keep the corner attributes that come with the word rectangle. - Combine those corners with the new information about all four sides.

1. A rectangle has \(4\) square corners. 2. The shape also has \(4\) equal sides. 3. A quadrilateral with both attributes is a square.

It is a square.

- Use the corner clue to identify the main category. - Compare all four side lengths. - A square needs both four square corners and four equal sides.

1. Four square corners make the quadrilateral a rectangle. 2. The side lengths are not all equal because \(6 \ne 3\). 3. Therefore, the shape is a rectangle, but not a square.

It is a rectangle, but not a square.

- 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.

- Focus on the corner rule that every shape in the category shares. - Remember that the category may include squares as well as non-squares.

1. Every quadrilateral with \(4\) square corners is a rectangle. 2. A square is included because it is a rectangle with all sides equal. 3. Non-square rectangles are also included. 4. The category is rectangle.

The category is rectangle.

- Focus on the one attribute shared by every shape in the category. - Remember that the category can include a more specific shape.

1. Four equal sides define a rhombus. 2. Squares are included because they also have \(4\) equal sides. 3. Non-square rhombuses are included as well. 4. The category is rhombus.

The category is 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.

- Use the four-equal-sides clue first. - A square is also a rectangle. - Use “not a rectangle” to rule out the square choice.

1. Four equal sides make the quadrilateral a rhombus. 2. Since it is not a rectangle, it does not have \(4\) square corners. 3. Therefore, it is a rhombus, but not a square.

Rhombus, but not a square.

- The side lengths are already the same in both descriptions. - Compare the corners instead.

1. A square has \(4\) square corners. 2. A non-square rhombus does not have all \(4\) corners square. 3. The distinguishing attribute is whether all four corners are square.

A square has \(4\) square corners; a non-square rhombus does not.

- Trace only the outside boundary. - Use the grid to compare a horizontal side with a vertical side. - Check the corner attribute before choosing the most specific category.

1. The inside seam is not part of the outside boundary. 2. The outside shape has \(4\) square corners, so it is a rectangle. 3. Its horizontal sides are \(4\,\text{units}\) long and its vertical sides are \(2\,\text{units}\) long, so all \(4\) sides are not equal. 4. Therefore, the most specific category is rectangle, but not a square.

Rectangle, but not a square.

- First confirm that each figure has four sides. - Rule out any figure with four square corners. - Rule out any figure with four equal sides.

1. Figure a) has four square corners, so it is a rectangle. 2. Figure b) has four equal sides, so it is a rhombus. 3. Figure c) has four sides, but its sides are not all equal and its corners are not all square. 4. Therefore, figure c) is the other quadrilateral.

Figure c).

- Look for four square corners first. - Then compare adjacent side lengths. - A non-square rectangle must have two different side lengths.

1. Figure a) has four square corners and two different side lengths, so it is a rectangle but not a square. 2. Figure b) has four equal sides and four square corners, so it is a square. 3. Figure c) does not have four square corners.

Figure a).

- Start with what would make a rhombus become a square. - Use the words “not a square” to rule out one corner condition.

1. A rhombus has four equal sides. 2. If all four corners were square, the rhombus would also be a square. 3. Because it is not a square, at least one corner is not square.

At least one corner is not square.

- Check equal sides and square corners separately. - Combine those two yes-or-no attributes for each figure. - Use the most specific phrase that fits.

1. Figure a) has four equal sides and four square corners, so it is a square. 2. Figure b) has four square corners but two different side lengths, so it is a rectangle but not a square. 3. Figure c) has four equal sides but not four square corners, so it is a rhombus but not a square. 4. Figure d) has neither four equal sides nor four square corners, so it is an other quadrilateral.

a) Square b) Rectangle but not a square c) Rhombus but not a square d) Other quadrilateral

- Do not use the direction the figure is facing as a classification rule. - Compare the lengths of adjacent sides on the peg grid. - Check whether the corners are square corners.

1. The figure has four square corners, so it is a rectangle. 2. One pair of opposite sides is longer than the other pair, so all four sides are not equal. 3. Therefore, it is a rectangle but not a square. Tilting the figure does not change its category.

Rectangle but not a square.

- Track the pair of sides that becomes longer and the pair that stays the same. - Use the square-corner clue before deciding the shape name. - Compare the attributes of the starting square with the attributes of the new shape.

1. The top and bottom sides become \(7\) units, while the left and right sides remain \(4\) units. The list is \(7, 4, 7, 4\). 2. Shape b) still has \(4\) square corners, so it is a rectangle. Its sides are not all equal, so it is not a square. Its most specific name is rectangle, but not a square. 3. The shape still has \(4\) sides and \(4\) square corners. The attribute “all sides equal” changed and is no longer true.

a) \(7, 4, 7, 4\) b) Rectangle, but not a square c) It still has \(4\) sides and \(4\) square corners, but its sides are no longer all equal.

- For shape a), compare all four side lengths on the grid. - For shape b), check the corners and compare adjacent side lengths. - A square must have both \(4\) equal sides and \(4\) square corners.

1. Shape a) has \(4\) equal sides, but its corners are not square corners. It is a rhombus, but not a square. 2. Shape b) has \(4\) square corners and two different side lengths. It is a rectangle, but not a square.

a) Rhombus, but not a square b) Rectangle, but not a square

- Separate what is given from what is not given. - Ask whether equal sides alone tell you anything about the corners. - Choose only statements that must be true.

1. Four equal sides guarantee that the shape is a rhombus, so A is true. 2. The shape is already stated to be a quadrilateral, so B is true. 3. The corner information is missing, so the shape may or may not be a square or rectangle. 4. C and D are not guaranteed.

A and B.

- For each clue, imagine a quadrilateral that has that attribute but is not a square. - Decide what additional information is missing. - Compare what each clue guarantees by itself.

1. Clue a) alone proves the shape is a rhombus, but it could be a non-square rhombus. 2. Clue b) alone proves the shape is a rectangle, but it could be a non-square rectangle. 3. Neither clue alone proves the shape is a square. Both clues together would.

a) No. b) No. Both clues together are needed to prove the quadrilateral is a square.

- For each part, begin with the side clue or the corner clue. - Then use the second clue to decide whether the shape can be a square. - Choose one exact phrase from the bank for each part.

1. a) Both defining attributes together make a square. 2. b) Four equal sides make a rhombus, and the corner clue shows it is not a square. 3. c) Four square corners make a rectangle, and the different side lengths show it is not a square.

a) Square b) Rhombus, but not a square c) Rectangle, but not a square

- Use the corner clue to identify the guaranteed broad category. - Consider one case in which all four sides are equal and one case in which they are not. - Check each phrase from the bank against the square-corner clue.

1. Four square corners guarantee that the shape is a rectangle. 2. If all \(4\) sides are equal, the shape is a square. 3. If the side lengths are not all equal, the shape is a rectangle, but not a square. 4. A rhombus that is not a square cannot have four square corners, so that description is not possible.

The possible descriptions are “square” and “rectangle, but not a square.”

- 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.

- Use the corner requirement for the rectangle category. - Remember that squares are included. - Reject descriptions that do not guarantee all four square corners.

1. A has \(4\) square corners, so it is a rectangle. 2. B has \(4\) square corners, so it is a square and also a rectangle. 3. C is a non-square rhombus, not a rectangle. 4. D does not guarantee a rectangle. 5. A and B belong in the bin.

A and B.

- Use the side-length requirement for every rhombus. - Remember that squares are included in the rhombus category. - Do not assume equal sides when the description does not say so.

1. A has \(4\) equal sides, so it is a square and also a rhombus. 2. B has \(4\) equal sides, so it is a non-square rhombus. 3. C is a non-square rectangle and does not have \(4\) equal sides. 4. D does not guarantee a rhombus. 5. A and B belong in the bin.

A and B.

- Use the corner clue for one category. - Use the side-length list for the other category. - Check whether both definitions are satisfied.

1. Four square corners make the shape a rectangle. 2. The side lengths are not all equal because \(4 \ne 6\). 3. Therefore, it is not a rhombus. 4. It is a rectangle only.

It is a rectangle only.

- Think about the side attribute required for every rhombus. - Use the category that does not fit. - Decide what must be false about the four side lengths.

1. A rectangle has \(4\) square corners. 2. If all \(4\) sides were equal, the rectangle would also be a rhombus and therefore a square. 3. Because the rhombus category does not fit, the side lengths cannot all be equal.

Its \(4\) side lengths are not all equal.

- Start with the clue about all four side lengths. - Use the words “exactly one” to rule out the other category. - State what must fail about the corners when the rectangle label does not fit.

1. Four equal sides make the shape a rhombus. 2. The problem says exactly one of the two labels fits, so rectangle does not fit. 3. A rectangle must have all \(4\) square corners. 4. Therefore, at least one corner of the rhombus is not square.

The rhombus label fits. At least one corner is not square.

- Identify the category guaranteed by the side lengths. - Think about the extra attribute required for a rectangle. - Consider two possible corner arrangements.

1. Four equal sides make the quadrilateral a rhombus. 2. If all \(4\) corners are square, the rhombus is also a rectangle and a square. 3. If the corners are not all square, the rhombus is not a rectangle. 4. The side information alone does not determine the answer.

No. It is a rectangle only if all \(4\) corners are square.

- Read each row as a pair of yes-or-no attributes. - Use equal sides to test the rhombus category. - Use square corners to test the rectangle category, then combine the results.

1. W has all equal sides and all square corners, so W is a square. 2. X has all square corners but not all equal sides, so X is a rectangle that is not a square. 3. Y has all equal sides but not all square corners, so Y is a rhombus that is not a square. 4. Z has neither special attribute, so Z belongs in the “other quadrilateral” category.

W: square X: rectangle that is not a square Y: rhombus that is not a square Z: other quadrilateral

- Separate side-length information from corner information. - Recall the defining corner attribute of a rectangle. - Decide whether the given clue guarantees that attribute.

1. A rectangle must have four square corners. 2. Equal opposite sides do not tell us that the corners are square. 3. Therefore, the given information is not enough to prove the quadrilateral is a rectangle.

No. You also need information that guarantees four square corners.

- Compare “one square corner” with the full rectangle requirement. - To disprove the statement, keep the given clue true while making the rectangle rule fail. - You only need one possible counterexample description.

1. A rectangle needs all four corners to be square. 2. A quadrilateral can have one square corner and another corner that is not square. 3. Such a quadrilateral is a counterexample, so Lila's statement is false.

No. A counterexample is any quadrilateral with one square corner but not all four corners square.

- Use each “not” statement separately. - Ask which defining attribute must fail for a rectangle. - Ask which defining attribute must fail for a rhombus.

1. If all four corners were square, the shape would be a rectangle, so A must be true. 2. If all four sides were equal, the shape would be a rhombus, so B must be true. 3. It is already a quadrilateral, so it has four straight sides; C and D are false.

A and B.

- Use the perimeter to find the equal side length. - Keep the rhombus side condition true. - Change the corner condition needed for a square.

1. A rhombus with perimeter \(20\,\text{units}\) has four equal sides of \(20 \div 4 = 5\,\text{units}\). 2. Choose a quadrilateral with side lengths \(5\,\text{units}, 5\,\text{units}, 5\,\text{units}, 5\,\text{units}\) and at least one corner that is not square. 3. The shape is a rhombus because all sides are equal, but it is not a square because not all corners are square. 4. This description is a counterexample to Owen's claim.

A rhombus with four sides of \(5\,\text{units}\) and at least one corner that is not square.

- Test the rule with a square. - Decide whether side lengths must be different for every rectangle. - Keep only the attributes shared by all rectangles.

1. The rule excludes squares because a square does not have two long sides and two short sides. 2. A square still has \(4\) square corners, so it is a rectangle. 3. A complete rule is: A rectangle is a quadrilateral with \(4\) square corners.

The rule leaves out squares. A complete rule is: “A rectangle is a quadrilateral with \(4\) square corners.”

- Find four equal side lengths that total \(24\,\text{units}\). - Find two different side lengths that repeat as opposite sides and have the same total perimeter. - Compare the categories of the two examples.

1. A square with side lengths \(6\,\text{units}, 6\,\text{units}, 6\,\text{units}, 6\,\text{units}\) has perimeter \(24\,\text{units}\). 2. A non-square rectangle with side lengths \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) also has perimeter \(24\,\text{units}\). 3. The same perimeter can belong to both categories. 4. Therefore, perimeter alone does not determine whether the rectangle is a square.

No. For example, \(6\,\text{units}, 6\,\text{units}, 6\,\text{units}, 6\,\text{units}\) makes a square, while \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) makes a rectangle that is not a square. Both have perimeter \(24\,\text{units}\).

- Find the shape that appears in both attribute groups. - Use the “exactly one square” clue before classifying the other two. - Then match the remaining corner and side attributes.

1. Q has both four square corners and four equal sides, so Q is the square. 2. P has four square corners but is not the square, so P is a rectangle but not a square. 3. R has four equal sides but is not the square, so R is a rhombus but not a square.

P: rectangle but not a square Q: square R: rhombus but not a square