Notes

Formulas are like picky eaters. They only want specific types of units to eat. Take the area of a rectangle for an example.
A
=
l●w
The formula in this case will only “eat” units that are associated with distance. You would not say a rectangle is three gallons long, but you might say it is three feet long.
Take a look at the speed formula as well.
speed
=
distance
time
This formula would need units that measure distance in the numerator, and units that measure time in the denominator.
The units for the result of the formulas will depend on what units you “feed” it. Formulas are what they eat!
•For the area formula, if the units of feet (ft) were used for the length and width, then the area would be measured in square feet. •Since x ● x = x2, then it follows that ft ● ft = ft2.
•The same goes for all units of measure, square meters (m2), square inches (in2), or square centimeters (cm2).
•For the speed formula, if the unit of meters was used for the distance and the unit of seconds was used for the time, then the speed would be measured in meters per second. •Division requires the use of the word “per” in the unit. Miles divided by hours equals miles per hour.
1 gallon is equal to 128 fluid ounces.
5 gallons●
128 fluid onces
1 gallon
=
640 fluid ounces
The paint-sprayer can handle 100 fluid ounces at a time, since Daniel has 640 fluid ounces (5 gallons) he would be able to fill it up 6 times.
640
100
=
6.4
1 fluid ounce is equal to 0.0078125 gallons.
100 fluid ounces
0.0078125 gallons
1 fluid ounce
=
0.78125 gallons
The paint-sprayer can handle 0.78125 gallons.
5
0.78125
=
6.4
The fraction of fluid ounces per gallon is referred to as a conversion ratio. This ratio can be derived from the conversion table linked above. On the table, it lists that 1 gallon = 128 fluid ounces. Daniel is converting from gallons to fluid ounces, so he correctly places the gallon information in the denominator of the ratio.
The ratio is set up this way so that the units will cancel out, leaving the units required.
Observe that the unit of gallon is in both the numerator and denominator. The unit of gallon can be canceled out. Occasionally, there will not be a direct conversion ratio. Some situations will require the conversion of a unit multiple times in order to get it in the unit desired.
If you recall, Daniel’s speed on the road was 88 feet per second. In order to make sure he was going the speed limit, multiple conversions must happen. His distance will need to be converted to miles, and his time will need to be converted to hours.
First, write his speed as a ratio:
88 feet
___________
1 second
Next, start lining up the conversion ratios.
•Feet will convert to miles.
•Seconds will convert to minutes.
•Minutes will convert to hours.
Observe that, in order to cancel out the unit of seconds in the denominator, seconds had to be placed in the numerator when converting to minutes.
Combine and simplify. Cancel out units when they occur in the numerator and denominator.Students are making and selling wax candles as a fund-raiser. They need to buy blocks of wax to melt and pour into the candle molds. Each candle mold has a volume of 25 fluid ounces of wax. Each block will melt into 10 cups of wax. How many blocks of wax should be purchased to make 1,000 candles?
The given information:
•Each candle will be made from 25 fluid ounces of wax.
•Each block of wax melts into 10 cups of wax.
•The students want to make 1,000 candles.
Unfortunately, there isn’t a direct conversion from fluid ounces to blocks of wax, so we will need a couple conversions to get there. The conversions are found within the question.
You can solve a problem like this by doing conversions one step at a time, but after practice, you probably will be able to do multiple conversions at once.Choosing the scale for a graph is similar to taking a photograph. If you zoom in too close, you might lose an important part of the picture. If you zoom out too far, it’s hard to see the details. Take a look at these three graphs.
Believe it or not, these graphs are of the same data! They all represent the table below.
If they all model the same data, why do they look so different? Take a very close look at the axes and see if you notice a difference.
•Graph A has too much room on the x-axis. It goes all the way up to 25, while the data only goes up to 5.
•Graph C has too much room on the y-axis. It goes all the way up to 25 while the data only goes up to 5.
•Graph B has just enough room to see the data points. This spreads the data across the coordinate plane.
Though graph B shows the data nice and clear, is it the best way to show the data? Selecting Units
•When selecting units, be mindful of the situation you are modeling.
•Make sure the unit is not too large or too small to represent the measurement.
Units in Formulas
•Use units that appropriately model the formula. Use units that match with the quantities measured by the formulas.
•Remember that units follow algebraic operations. (1 ft • 1 ft = 1 ft2)
•Division of units results in the use of “per” in the resulting unit rate. •Miles divided by hours results in miles per hour.
•Dollars divided by gallons results in dollars per gallon.
Converting Units
•Use the conversion as a ratio to assist in converting units.
•Any unit present in the numerator and the denominator can be canceled out.
Conversion table for measurement units
Distance
1 inch = 2.54 centimeters
1 meter = 39.37 inches
1 mile = 5,280 feet
1 mile = 1,760 yards
1 mile = 1.609 kilometers
1 kilometer = 0.62 mile
Mass / Weight
1 pound = 16 ounces
1 pound = 0.454 kilograms
1 kilogram = 2.2 pounds
1 ton = 2,000 pounds
Volume
1 cup = 8 fluid ounces
1 pint = 2 cups
1 quart = 2 pints
1 gallon = 4 quarts
1 gallon = 3.785 liters
1 gallon = 128 fluid ounces
1 liter = 0.264 gallons
Scales of Graphs
A graph is most useful when the units on the x-axis and the y-axis have the proper scales. The scale includes the low and high values and the increments (or steps) for each axis. The origin is the point (0, 0) on a graph. Use the scales to help interpret the axes and what the points represent.