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.”

- Opposite sides of a rectangle match in length. - Keep the two given adjacent lengths different. - Recall the corner attribute every rectangle has.

1. Opposite sides of the rectangle have the same length. 2. Starting with adjacent sides of \(5\,\text{units}\) and \(2\,\text{units}\), the side lengths around the boundary are \(5\,\text{units}, 2\,\text{units}, 5\,\text{units}, 2\,\text{units}\). 3. A rectangle has four square corners. 4. Because adjacent side lengths differ, the rectangle is not a square.

\(5\,\text{units}, 2\,\text{units}, 5\,\text{units}, 2\,\text{units}\), with four square corners.

- 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.

\(P\)

- 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.

- Recall the exact side count for a hexagon. - Check whether each choice guarantees a closed polygon. - A guarantee must rule out shapes with the wrong number of sides.

1. A hexagon is a closed polygon with exactly \(6\) straight sides. 2. Choice A states those defining attributes directly. 3. Choice B is open, and choice C has only \(5\) sides. 4. Therefore, choice A guarantees a hexagon.

A. Make a closed polygon with exactly \(6\) straight sides.

- A square needs four equal sides. - Use the given side length for every side. - Recall the corner attribute of a square.

1. A square has four equal sides, so every side is \(3\,\text{units}\) long. 2. A square also has four square corners.

\(3\,\text{units}, 3\,\text{units}, 3\,\text{units}, 3\,\text{units}\), with four square corners.

- Keep the defining side attribute of a rhombus. - Compare that with the extra corner attribute of a square. - State the corner condition that prevents the rhombus from being a square.

1. A rhombus has four equal sides. 2. A square has four equal sides and four square corners. 3. Therefore, a rhombus that is not a square must keep the four equal sides and have at least one corner that is not square.

Four equal sides and at least one corner that is not square.

- 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 \(D\).

- 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.

- 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.

- Keep the defining four-sided condition for a quadrilateral. - Think about which side attribute would make the shape a rhombus. - Think about which corner attribute would make the shape a rectangle.

1. The figure must remain a quadrilateral, so it has \(4\) straight sides and is closed. 2. To keep it from being a rhombus, its \(4\) sides should not all be equal. 3. To keep it from being a rectangle, its \(4\) corners should not all be square. 4. Those two exclusions also keep it from being a square.

One valid description is a closed quadrilateral whose sides are not all equal and whose corners are not all 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 same four vertices except for the one named in the instruction. - After the imagined move, count the outside sides. - Use the broad category determined by that side count.

1. Moving one vertex changes some side lengths and corner sizes. 2. The new figure still has four straight sides and remains closed. 3. Therefore, it is still a quadrilateral, although it is no longer a square.

The new figure is still a quadrilateral.

- 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.

- Keep the same total perimeter while changing the defining attributes. - Separate the equal-side condition from the square-corner condition. - Check each perimeter and each category from your descriptions.

1. For a square, four sides of \(6\,\text{units}\) give perimeter \(24\,\text{units}\), and all four corners are square. 2. For a non-square rectangle, side lengths \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) give perimeter \(24\,\text{units}\), and all four corners are square. 3. For a non-square rhombus, four sides of \(6\,\text{units}\) give perimeter \(24\,\text{units}\), and at least one corner is not square. 4. The square has both four equal sides and four square corners; the non-square rectangle has four square corners but not four equal sides; the non-square rhombus has four equal sides but not four square corners.

a) Four sides of \(6\,\text{units}\) with four square corners. b) \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) with four square corners. c) Four sides of \(6\,\text{units}\) with at least one corner that is not square.

- First identify which categories require all four sides to be equal. - Then ask whether the two equal pairs can be used as opposite sides of a rectangle. - A quadrilateral does not need four square corners unless it is in the rectangle category. - Check every listed category, not just the first one that works.

1. A square and a rhombus both require four equal sides. The four sticks are not all equal, so neither category is possible. 2. A rectangle but not a square is possible: place the two sides of length \(3\,\text{units}\) opposite each other, place the two sides of length \(5\,\text{units}\) opposite each other, and make four square corners. 3. An other quadrilateral is also possible: use the same four side lengths in a closed four-sided shape but make at least one corner not square. 4. Therefore, the possible choices are a rectangle but not a square and an other quadrilateral.

Possible: rectangle but not a square; other quadrilateral. One rectangle plan is to put equal-length sides opposite each other and make four square corners. One other-quadrilateral plan is to use all four sticks in a closed four-sided shape with at least one corner that is not square.