Dimensions, Perimeter, Area, Polygons, Apothem, Isometric Rhombuses, and Functions

Getting carried away with perimeter and area is so much fun! The first page of the pdf was just straight rectangles calculating perimeter in linear units and area in square units. There are so many ways to calculate perimeter (2L + 2W or 2(L+W)) or just use your pencil point to count each linear unit by ones (not very exciting but effective). I prefer, adding the length and the width and then doubling that sum.

For area, the obvious method is finding the product of the length times the width. But when children say, “I don’t yet know multiplication,” which of course they do, I remind them that it is just repeated addition. Adding horizontally or vertically by multiples of each dimension can be so much fun. So a 5x8 rectangle, you can either multiply 5x8=40 or add 5, 8 times or add 8, 5 times. If they do not know their math facts, then counting by 5s until they get 8 fingers will quickly yield the same result. Of course, there always is the tedious approach of touching each square unit with your pencil point in its center and count by ones.

The second page in the pdf, I cut off each of the four vertices with a square to create a dodecahedron. The exploration showed them that the perimeter from the original rectangle does not change at all and the area just subtracts the relevant number of cut-off units.

The third page in the pdf, shows them that the area of a rectangle simply looks at the original rectangle area gets cut in half; so base times height (length times width/2) divided by 2 is the area of a triangle. The perimeter is simply adding the three sides. I used whole number lengths for the third side (hypotenuse) since I did not want to add pythagorean theorem to this week’s lesson (we will cover it in a future lesson). 

The fourth page in the pdf, I use isometric dot paper to create rhombus like shapes with each rhombus equal to one square unit. The perimeter was each linear unit connecting two dots vertically or diagonally (horizontal segments are greater than one unit; actually approx. 73% greater or root 3 greater). If they connect the dots vertically and diagonally, they can see the individual rhombus units appear and find rectangle-like structures appear. This makes finding the area easier. Beware of half square unit triangles, two of which make a whole square unit rhombus.

The fifth page in the pdf., I created regular polygons (these have equal side lengths and equal angle measures) and introduced the concept of apothem (this is the perpendicular distance from the center of the regular polygon to each side). If you multiply the apothem by the perimeter of the figure (side length multiplied by the number of sides), and divide by 2, you get the area of the regular polygon. My focus was on first taking half of the perimeter (if it was even), then multiplying by the apothem. This teaches the concept of cross simplifying fractions before ACROSS multiplying. 

Pages 8 and 9 of the pdf are square dot and isometric dot paper so the children can make their own shapes and calculate area and perimeter.

For my advanced 5/6th grade class, page 10 of the pdf, I explored functions that would generate rectangles with equal perimeters and areas when the rectangles had one square unit removed from each vertex. For example, a 3x10 original rectangle with the vertices removed would have equal perimeter and area of 26 units and square units, respectively. Also, a 4x6 original rectangle with the vertices removed would have equal perimeter and area of 20 units and square units, respectively. You could never have a side length of 1 or 2 and the larger one of the dimensions became (approaching infinity), the smaller the other dimension became as it approached 2 (but would never be 2).

An apothem of length 2 would also generate equal perimeter and area. Finally, we delved into equal surface area and volume of a sphere and also a cylinder. For the sphere, we found that a radius of 3 would generate a surface area and volume of 36pi square units and cubic units, respectively. For the cylinder, we found the same function as between the length and width of a rectangle with equal perimeter and area. 

R = (2H)/(H-2)

So a cylinder radius and height of 4 units each or radius of 3 and height of 6, or vice versa, will generate equal surface area and volume in square units and cubic units, respectively.