How can we convert units?

 Every measurement needs to have a value (number) and a unit (label).  Without units, we have no way of knowing what the actual measurement is  Sometimes the units that something is measured in, need to be converted into a comparable unit for a calculation  So how do we convert our units into new units?

 When we are converting from one metric unit to another, all we need to do it move the decimal point  Convert the following: k h da _ d c m dm = _________ hm s = _________ ms g = _________ kg

 Not every type of conversion that you will encounter will be a metric conversion where you can just move the decimal  Dimensional Analysis (Factor-Label Method) is the process that we can use to mathematically convert units from one unit system to another

 Before we can look at examples of dimensional analysis, let’s review some basic math principles:  What happens when you divide a number by itself?  What happens when you divide a unit by itself?  In both cases, you get the number 1.  Dimensional analysis involves multiplication and division using conversion factors.  Conversion factors : two numbers with their units that are equivalent to each other  i.e. 1 foot = 12 inches, 12 eggs = 1 dozen

Conversion factors can be written as ratios because both values equal each other Because they equal each other, if we divide the quantities they would be equal to one. or For Example: 12 inches = 1 foot Written as an “equality” or “ratio” it looks like: = 1 When a value is multiplied by a conversion factor the units behave like numbers do when you multiply fractions: If you have the same units in both the numerator and the denominator, they cancel!

How many feet are in 60 inches? Solve using dimensional analysis. All dimensional analysis problems are set up the same way. They follow this same pattern: What units you have x What units you want = What units you want What units you have The number & units you start with The conversion factor (The equality that looks like a fraction) The units you want to end with

Write this conversion factor as a ratio, making sure that the number on the bottom of the ratio has units that match the units of your starting units so that they will cancel 60 inches You need a conversion factor. Something that will change inches into feet: 12 inches = 1 foot x = 5 feet Do the math: 1. Multiply all of the numerators first: 60 x 1 = Multiply all of the denominators: 12 x 1 = Divide the product of the numerators by the product of the denominators: 60 ÷ 12 = 5

Using this format, the vertical lines mean “multiply” and the horizontal bars mean “divide.” The previous problem can also be written to look like this: 60 inches 1 foot= 5 feet 12 inches

 Let’s practice setting up dimensional analysis problems using nonsense units: 1. How many bleeps are in 12 cams? 2. How many nerds are in 6 tongs? 3. How many yips are in 15 cams? (Hint: Use 2 conversion factors!) Conversion Factors: 3 bops = 5 yips 20 nerds = 8 cams 2 cams = 1 bleep 2 nerds = 3 tongs 1 bop = 5 cams 12 cams x1 bleep 2 cams 6 tongs x2 nerds 3 tongs 15 cams x1 bop 5 cams = 6 cams = 4 nerds = 5 yips x 5 yips 3 bops

Units of Length 12 inches = 1 foot 3 feet = 1 yard 5280 feet = 1 mile 1 inch = 2.54 centimeters 1 foot = meters 1 mile = kilometers 1 mile = 1609 meters Units of Mass 16 ounces = 1 pound 2000 pounds = 1 ton 1 ounce = grams 1 pound = kilograms Units of Volume 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon 16 fluid ounces = 1 pint 1 gallon = 3.79 liters 1 fluid ounce = 29.6 milliliters Units of Time 1 hour = 60 minutes 1 minute = 60 seconds 1 hour = 3600 seconds

______ 2.54 cm 1 in Now let’s practice conversions with real units: 1. How many centimeters is 8.72 in? applicable conversion factors: equality: or 8.72 in x= 2.54 cm = 1 in ________2.54 cm 1 in Again, the units must cancel. () ______ 22.1 cm 2.54 cm 1 in

2. How many feet is inches? applicable conversion factors: equality: or in x= 1 ft = 12 in ______1 ft 12 in Again, the units must cancel. () ____ 3.28 ft 1 ft 12 in ______ 1 ft 12 in

3. Convert 65 meters/second into miles per hour. (2 part units!) 1. Convert your distance from meters to miles: 2. Convert your seconds into hours: 3. Divide your miles by hours: equalities:1 mile = 1609 meters 3600 s = 1 hour 65 meters x1 mile 1609 meters = miles 1 second x1 hour 3600 seconds = hrs miles hrs. = 145 mi/hr

6 minutes  seconds (6 minutes) 1 ( ) seconds minute 360 seconds 60 1 = (6)(60 seconds) (1)(1) = Step 1 – Read the question and determine what information it provides you with (starting point & final destination) Step 2 – Write down your starting point and your final destination 1 minute = 60 seconds Step 3 – Determine how you will get from your starting point to your final destination (list any “connections” or conversion factors) Step 4 – Create a fraction by placing your starting point over one Step 5 – Multiply between fractions Step 6 – Write in the bottom unit of the new fraction (this is the same as the top unit of your previous fraction) Step 7 – Write the appropriate conversion factor into the fraction. Your bottom unit will guide you. Step 8 – Determine if this top unit is the desired unit (your final destination). In this case the answer is YES, so we move on to step 9 Step 9 – Cancel all diagonal units. Once this is done, your final destination should be the only unit left – in this case seconds Step 10 – Multiply the top of the fractions; multiply the bottom of the fractions; divide the product of the top by the product of the bottom Starting Point Final Destination