Draw an equilateral triangle. Divide each side in half and join the points to make four small equilateral triangles. Rearrange these four equilateral triangles into a parallelogram. Compare the perimeter of the original triangle with the perimeter of the parallelogram. Explain your results.

let the sides of the original be 2x

then each side of the 4 smaller trianges is x
Perimeter of original is 2x+2x+2x= 6x

Sum of all sides of the parallelogram = 6x

Not much to explain here, after you make the sketch just add up the sides to get the perimeter .

in how many ways can you arrange six things

To draw an equilateral triangle, follow these steps:

1. Take a ruler and draw a straight line segment of any desired length.
2. Place the compass at one end of the line segment and adjust its width to the length of the segment.
3. Keeping the compass width consistent, draw two arcs from the two endpoints of the line segment to create two intersecting points.
4. Use a ruler to connect the intersecting points with the endpoints of the line segment.

To divide each side of the equilateral triangle in half and create four smaller equilateral triangles, follow these steps:

1. Take a ruler and measure one of the sides of the equilateral triangle.
2. Divide that measurement by 2, and mark that distance from one endpoint of the side.
3. Use a ruler to connect the new point to the opposite endpoint of the side.
4. Repeat steps 2 and 3 for the other two sides of the equilateral triangle.

To rearrange these four equilateral triangles into a parallelogram, follow these steps:

1. Take one of the equilateral triangles and rotate it 180 degrees.
2. Place it adjacent to the original equilateral triangle, aligning their corresponding sides.
3. Repeat this process with the remaining three equilateral triangles, ensuring that their corresponding sides are aligned.

Now, let's compare the perimeter of the original equilateral triangle with the perimeter of the parallelogram:

The perimeter of the original equilateral triangle is the sum of the lengths of its three sides. Let's say the length of each side is 's'. Therefore, the perimeter of the original triangle is 3s.

On the other hand, the perimeter of the parallelogram is the sum of the lengths of its four sides. Since we rearranged the four equilateral triangles to form a parallelogram, each side of the parallelogram would still have a length equal to 's'. Hence, the perimeter of the parallelogram is 4s.

Comparing the two perimeters, we can see that the perimeter of the parallelogram (4s) is larger than the perimeter of the original equilateral triangle (3s). This is because when we rearranged the four equilateral triangles, we added extra sides to create the parallelogram. Consequently, the perimeter increases in the process.

In conclusion, the perimeter of the original equilateral triangle is less than the perimeter of the parallelogram formed by rearranging the four equilateral triangles.