Intermediate Test Prep (Grades7-8) Working Backwards

This lesson will investigate the proper use of unit labels when calculating 1-dimensional, 2-dimensional and 3-dimensional measurements.

1 - Dimensional Measurements
When you measure something in 1-dimension you are
measuring it's length.
For example, when you measure the length of your driveway, or how tall you are, or how far you can throw a ball, you are measuring lengths. When you measure the dimensions of your bedroom, or a piece of wood for a bookshelf you are still just measuring 1-dimension....it's length. When you measure the distance between trees in your front yard, or how far away two houses are from each other, or the distance between two towns you are still operating in the first dimension...length.
So as you can see, there are all kinds of measuring problems which involve only the first dimension.
We phrase many 1-dimensional measuring questions by asking:
"How long is..."
"How tall is..."
"How far is..."
"How wide is.."
Another important use for 1-dimensional measurement is in finding the perimeter of a shape (the distance around).
When labeling any 1-dimensional measurement the answer is always given in terms of plain units.
For example: The length of a standard sheet of notebook paper is 11 inches. The width of that sheet is 8 1/2 inches. The length of a yardstick in feet is 3 feet, in inches it is 36 inches.
All 1-dimensional measurements must always be labeled in the unit by which they were measured. Miles, kilometers, meters, yards, feet, centimeters, inches, millimeters are all examples of the types of units used when measuring the
length of an object.

2 - Dimensional Measurements
When you are asked to determine measurements in
2-dimensions, you are being asked to find the area of a shape.
When you find the area of a closed plane figure (that's a fancy way of saying a shape that doesn't have any breaks in it's boundary) you are actually calculating the number of square units that are contained within the boundaries of the shape.
A really great way to visualize this concept is to look at a checkerboard. A checkerboard is a large square which has been divided into 8 equal units down each side of the square. The resulting pattern, inside the boundaries of the square, is 64 smaller identical squares. Since the 64 squares fill up the entire area inside the boundary, we say that the area of the checkerboard is 64 square units. Each of these smaller squares has sides of length 1 unit. If the checkerboard is 8 inches by 8 inches, then each square has sides of 1 inch, or an area of
1 square inch. If the board is 48 cm by 48 cm, then each small square has sides of 6 cm, or an area of 36 square cm.
There are area formulas for many of the polygons we have in mathematics, and we will work with those in another lesson. What is important in this lesson is that after you have done the calculation to determine the numerical part of your answer you make sure to label your answer in square units!

3 Dimensional Measurements
When you measure using 3 dimensions you are calculating the "volume" of a shape. Volume, also called "capacity", is the amount of something that a container can hold. We will investigate volume problems in another lesson. Remember, when you measure in 1 dimension you are determining it's length. When you measure in 2 dimensions you are calculating the area of a shape.
When we were developing the concept of area we saw that what really is happening is that you are determining the number of smaller square units which fit inside the boundary of the shape. When we move into the third dimension we move from unit squares to unit cubes. So when you calculate the volume of a shape you are really finding how many unit cubes can be packed inside the shape. That is why that after you do the calculation you must label all volumes in "cubic units".
Let's look at a quick example. Suppose you want to find the volume of a rectangular box that has a length of 10 inches, a width of 8 inches and a height of 6 inches. The formula for finding the volume of a box (a rectangular prism) is
length x width x height
Plugging in the values we find the volume of our box would be:
10 x 8 x 6 = 480 cubic inches.

A Quick Recap
When measuring in 1 dimension (length) the answer is labeled in "units".
When measuring in 2-dimensions (area) the answer is labeled in "square units".
When measuring in 3-dimensions (volume) the answer is labeled in "cubic units".