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).

- The app already guarantees the square-corner attribute. - Decide what must be different so the result is not a square. - Compare the two numbers in each pair.

1. A square has equal length and width. 2. A rectangle that is not a square needs two different side lengths. 3. Only \(2\) and \(6\) are different, so that pair works.

Use side lengths \(2\) and \(6\).

- 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.”

- 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.

- Keep the instruction that gives the shape \(4\) equal sides. - Ask which choice is still possible for a square. - Choose the condition that rules out all squares.

1. The selected instructions already guarantee a rhombus because the shape has \(4\) equal sides. 2. Choice B is also true for a square, so it does not prevent the result from being a square. 3. Choice C would make the result a square. 4. Choice A guarantees that the rhombus is not a square.

A. Make at least one corner not square.

- The current instructions already make one quadrilateral category. - Think about the extra side attribute that makes this category more specific.

1. The selected instructions guarantee a rectangle. 2. To make that rectangle a square, all \(4\) sides must be equal. 3. Add the instruction “all sides equal.”

Add “all sides equal.”

- 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.

- Identify the side-length feature that every square has. - Look for the choice that makes that feature impossible. - A condition already true for all rectangles does not rule out squares.

1. A square has all \(4\) sides equal. 2. If two adjacent sides have different lengths, the rectangle cannot be a square, so A works. 3. Choice B forces a square. Choice C is true for every rectangle, including squares, so it does not rule out a square.

A. Make two adjacent sides different lengths.

- Find the side opposite the question mark. - Opposite sides of a rectangle have equal lengths. - Use the label on the matching opposite side.

1. Opposite sides of a rectangle have equal lengths. 2. The side marked with a question mark is opposite the side of length \(2\) units. 3. Therefore, the missing side length must be \(2\) units.

The missing side length is \(2\) units.

- Compare all four side entries. - A square must have four equal sides. - In any four-square-corner shape, opposite sides must also match.

1. A square needs all \(4\) side lengths to be equal. 2. Three side lengths are \(4\), while one is \(5\). 3. The list also cannot close as a four-square-corner shape because opposite side lengths would not match. 4. Change the \(5\) to \(4\). Then the list is \(4,4,4,4\), which can make a square.

Change the \(5\) to \(4\).

- Recall both the side count and the closed-shape requirement. - Check whether every instruction uses the stated vertices to form a full boundary. - Eliminate any open path or wrong vertex count.

1. A hexagon is a closed polygon with \(6\) straight sides and \(6\) vertices. 2. Choice A creates a closed shape using all \(6\) vertices and \(6\) sides. 3. Choice B is open, and choice C has only \(5\) vertices. 4. Therefore, choice A will make a hexagon.

A. Place \(6\) vertices, connect each neighboring pair with straight sides, and close the shape.

- 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.

- In a four-square-corner shape, opposite sides must match. - In a side list written around the shape, compare the first with the third and the second with the fourth. - After finding a possible rectangle, check whether all four sides are equal.

1. In a rectangle, opposite sides must have equal lengths. In a list written around the shape, the first and third lengths must match, and the second and fourth lengths must match. 2. In A, the opposite pairs are \(7\) and \(7\), and \(3\) and \(3\). Because the two side lengths differ, this makes a rectangle that is not a square. 3. In B, the opposite pairs would be \(7\) and \(3\), so the sides cannot close to make a shape with four square corners. 4. In C, all four sides are equal, so the result would be a square.

A. List B cannot make a closed shape with four square corners, and list C makes a square.

- Check whether every instruction can be true at the same time. - Compare the two statements about the corners.

1. “All corners square” requires every corner to be square. 2. “At least one corner not square” requires one corner to fail that condition. 3. The two corner instructions contradict each other. 4. No shape can follow all the instructions.

No. The corner instructions contradict each other.

- Combine the rectangle corner rule with the equal-side instruction. - Identify the more specific shape that results. - Compare it with the “not a square” condition.

1. A rectangle with all \(4\) sides equal has \(4\) square corners and \(4\) equal sides. 2. Such a rectangle is a square. 3. Therefore, it cannot also be a rectangle that is not a square.

No. A rectangle with all four sides equal is a square.