In the previous lesson the units were selected so that they all matched. That is, time was in hours and speed was miles per hour. We did not have time in minutes in one place and in hours in another place. This must always be the case before you start solving any
problem. All your lengths must be in the same unit. All your times must be in the same unit. All your weights and temperatures must also be in the same unit. Never try to work a problem with both meters and feet, or pounds and tons, or days and hours.
Use only one unit for time, length, weight, and temperature.
Pick a unit for each type of measurement (time, length, weight, etc...) that makes it easy for you. Work the problem with those units. Then, after finding your answer, convert your units to the units the problem requires.
Now let’s look at an example. How far does a car travel if several drivers drive 50mph for 3 whole days? First, we notice that the speed uses hours, but the driving time is in days.
Hours must be changed to days or days to hours before we calculate. You know that 3^50 can’t be right!
Unlike variables, units only merge if they match.
Let’s change 3 days to 3*24 hours for 72 total hours. Now we can substitute in our formula d=rt. D=50*72=3600 miles. Since our problem was based on miles then our answer is in miles.
If we change hours to days we will still get the same answer. 50mph ^ 24 hours = 1200 miles per day times 3 days gives us 3600 miles. So you see, it does not matter what unit you use, but it must be only one unit before you calculate.
What is the area of a rectangle with a width of 6 inches and height of 2 feet?
A = 6in*24in = 144in
2A = .5ft*2ft = 1ft
21 square foot = 144 square inches
Both units give correct answers!
69) 2d shapes
For the next several lessons we will not introduce new math concepts, instead we will apply what we already know to common visual and word problems.
Two dimensional shapes are surfaces with length and width, or height and width, both are correct. What they lack is depth, also called height. (Now we wonder why kids get confused!)
The area of any surface is measured in square units, such as square inches, square centimeters, square miles, etc... Notice that “square” is represented with an exponent of 2. This corresponds with 2D, so it is easy to remember. The previous examples could be abbreviated: in2, cm2, and mi2. This works for all surfaces like paper, floors, land, etc...
This even works for curved surfaces like balloons and cylinders. And don’t be fooled by 3D surfaces like cereal boxes. Just “unfold” them so they lay flat and total up all the parts.
area = &r2 circumference
(perimeter) 2&r
radius
base length
height width
area of triangle = 1/2bh area of rectangle = lw
There are three basic shapes to learn, because they can be cut and/or combined to make up more complex shapes. (Of course, we are not including reeeeally complex curves and angles which require calculus and trig.) The shapes are circles, rectangles, and triangles.
Notice that the area of a triangle is derived from a rectangle. The base of a triangle is the same as the length of its containing rectangle and the height of a triangle is the same as the width of its containing rectangle. Now look at the left and right halves of the triangle and then of the rectangle. See how the triangle is half? It always works that way.
You may have memorized formulas for the areas of squares and parallelograms when you were younger, but these are just special rectangles with the same formula. Trapezoids
are just combinations of a rectangle and 1 or 2 triangles, but if you want a formula, it is:
vertical height^.5^(top+bottom)
One more thing just to be sure I cover my bases. You can have kids memorize formulas for the perimeters of different shapes, but except for the circle, it is superfluous. Just add up all the sides and you are done.
70) 3d shapes
3D shapes are able to enclose volume. They can hold air or water or solids. They have length, width, and height. Looking at them from a different point of view you can also say they have length, width, and depth. Either way, there are 3 dimensions to be multiplied which will give you a 3 exponent on your unit: mm3, km3, ft3.
There are two categories of shapes that concern us.
Those shapes that have the same outline on top directly over the bottom (vertical, with no skewing or slanting, as in the top diagram), and those that come to a point on top (cones or pyramids as in the bottom diagram).
The first category is called “right solids.” They are made by drawing a base, then lifting it exactly vertical (right angle) to a height. Their volume is found by multiplying the area of the base ^ height. This works for rectangles, triangles, other polygons, and circles.
Pyramids have a polygon for a base and cones have a circle for a base. Their formulas are the same. 1/3 ^ base area ^ height. So if you have an ice cream cone that exactly fits in a tin can, find the volume of the tin can and divide by 3.
A sphere (perfectly round ball) is our last object and it is 2/3 the volume of the cylinder into which it exactly fits. So its formula is 4/3&r3. This comes from 2/3^2r^&r2. That is translated from 2/3 ^ height(which is the diameter) ^ area of circular base.
base
base
height
height
base