MATH - SCIENCE INTEGRATION

bullet1 CONVERSION FACTORS AND DIMENSIONAL ANALYSIS

bullet2 Converting from one unit of measurement to another.

Quick! Tell me how many seconds are in a year!
Most of us can't do that off the top of our heads.  So let's see how we can do it mathematically.


bullet3 Rules for solving problems

RULES FOR UNIT CONVERSIONS USING
DIMENSIONAL ANALYSIS

The best way to understand the rules for converting is to see them in action.  Let's start with an example problem and go through the rules as we solve the problem.  

You will want to have a pencil, a calculator, and a piece of paper ready to go.  



See also: CONVERSION CHART (Reference)

  • How much gift wrap do you need?
    Example Problem:
    A roll of Christmas gift wrap contains 2.5 meters of paper. What is the length of the paper in yards?


    Step 1:   Write down the known values and the unknown values.
    Known : length of paper in meters: 2.5 meters
          
    Unknown :   length of paper in yards

    Step 2:    Find the conversion factor that gives a ratio between meters and yards.  
    You can use the reference table (Conversion Chart) included with this lesson.

    Conversion factor    
    1 meter = 1.09 yards


    Step 3:   Write the conversion factor as a fraction with the unit you want to convert to on top and set the problem up as an equation.
    2.5 meters x  1.09 yards  =                           1 meter

    2.725 yards
    You can see that the unit meters will cancel.

    • Let's look at a problem in which we are converting to a larger unit.



     
    • Metric Mayhem
      Problem:   Convert 325 millimeters to meters.
      Step 1:   Write down the known values and the unknown values.
      Known :   length in mm = 325  
      Unknown : length in meters


      Step 2:    Find the conversion factor that gives a ratio between meters and millimeters.  
      You can use the reference table (Conversion Chart) included with this lesson.
      Conversion factor   (use reference)

      1 meter = 1000 mm


      Step 3:   Write the conversion factor as a fraction with the unit you want to convert to on top and set the problem up as an equation.
      325mm = 1 meter =
                      1000 mm

      .325 meters
      The unit millimeters will cancel.
      So far, the conversions solved have only required one conversion factor.  Let's solve a problem with several conversion factors.



       
      • Time Flies!

        Problem:   How many centuries are in 237,439 days?


        Step 1:   Write down the known values and the unknown values.
        Known : number of days = 237,439 days    
        Unknown : number of centuries


        Step 2:    Find the conversion factor that gives a ratio between days and centuries.
        You can use the reference table (Conversion Chart) included with this lesson.
        Conversion factor :  We do not have a conversion factor for centuries/days.  So we need to go from days to the nearest unit we know.    We know that there are 365 days in a year.  We know that there are 100 years in a century.


        Step 3:   Write the conversion factor as a fraction with the unit you want to convert to on top and set the problem up as an equation.
        Equation:
        237,439 days x 1year      x  1 century   =                        365 days       100 years      
        237,439   =  
          36500                                   
        6.51 centuries
        The units days and years will cancel.




         
        • You Can't Forget This!
          Conversion Factors and Dimensional Analysis are mathematical problems.  Here are a couple of things that you must remember:
          • Always remember to put the unit that you want to convert to on top!
          • All conversion factors must be labeled! (meter, liter, year, etc .)

          It's time to practice solving unit conversions using dimensional analysis. Try Basic Conversion Problems first to give you some practice.



           

bullet3 Practice conversions.

Practice makes perfect!
You would never play in a football game or take on a champion cross country runner without a lot of practice yourself.  You should approach math the same way.  Practice, practice, practice!


  • Basic Conversion Problems
    PRACTICE PROBLEMS

    DIRECTIONS:  Solve the following problems using dimensional analysis.  Show all work and don't forget to label all conversion factors (m, L, s, etc.).  If you need the conversion chart for reference, click on the link below.

    1.  Convert 2,378 feet into meters.

    2.  How many seconds are in 768 days?

    3.  A car has traveled 7.2 miles.  How many inches did the car travel?

    4.  How many centuries have you been alive?

    5.  If a truck has a mass of 23,456 kg, what is its mass in milligrams?

    6.  How many centiliters are in 3.7 gallons?

    7.  Write 2,465 milliseconds as seconds.

    8.  How many milligrams are in a 22 ounce steak?

    9.  How many hours are there in 6.5 centuries?

    10.  Convert 34,823 centimeters into miles.




    See also: Rules for solving problems
    See also: CONVERSION CHART (Reference)

     
  • Application Problems
    CAN YOU APPLY IT?
    DIRECTIONS: Look at the situation below and see if you can use your new conversion skills to solve the problems.

    PROBLEM #1:
    You have received an email from your cousin Madeline in France, who wants to make your family's famous Pistachio Ice Cream as a surprise for her parents anniversary party.

    Unfortunately, Madeline can not find her mother's recipe for the ice cream and wants you to email the recipe to her.  She also asks you to send it with metric measurements instead of American measurements. You need to do some converting!  









    See also: CONVERSION CHART (Reference)
     
    • Making Ice Cream
      The recipe below is in English American units.  Now get out your paper, pencil, calculator, and conversion chart and convert the measurements to metric.  Show your work using dimensional analysis and then rewrite the recipe using only metric measurements.  Remember that all fractions need to be changed to decimals since the metric system uses decimals.

      PISTACHIO ICE CREAM
      You will need the following items to pack the ice cream freezer:
      • 16 lbs of ice (crushed or chipped)
      • 4 lbs of rock salt
      Shell and blanch:
      • 4 oz pistachio nuts
      Pound them in a mortar with:
      • A few drops of rose water
      Add to them:
      • 1/4 cup sugar
      • 1/4 cup cream
      • 1 teaspoon almond extract
      • 1/4 teaspoon green vegetable coloring
      Stir these ingredients until the sugar is dissolved.  
      Heat, but do not boil:
      • 1 cup cream
      Add and stir until dissolved:
      • 3/4 cup sugar
      • 1/8 teaspoon salt
      Chill these ingredients.  Add the pistachio mixture and:
      • 2 cups whipping cream
      • 1 cup cream
      Put mixture into the canister for the ice cream churn.  Pack ice and salt around the canister.  Allow the pack to stand about 3 minutes before you start churning.  
      While waiting for the ice cream to harden, prepare the chocolate syrup topping.  
      TOPPING:
      Melt in a double boiler:
      • 2 oz unsweetened chocolate
      • 1 tablespoon butter
      Stir and blend well, then add:
      • 1/3 cup boiling water
      Stir well and add:
      • 1 cup sugar
      • 2 tablespoons corn syrup
      Let the sauce boil readily over direct heat for about 3 minutes.  Reduce heat and cook for about 2 minutes more without stirring.  Add just before serving:
      • 1 teaspoon vanilla extract
      The ice cream should be ready in about 20 minutes.  ENJOY!



       

bullet3 CONVERSION CHART (Reference)

CONVERSION CHART

BRITISH AMERICAN UNITS
 
Length:
Volume:
1mile (mi) = 1760 yards = 5280 feet
Liquid:
1 yard (yd) = 3 feet
16 fluid ounces (fl oz) = 1 pint
1 foot (ft) = 12 inches
2 cups = 1 pint
1 inch (in) = 1/12 foot = .083 feet
2 pints = 1 quart (qt)
 
4 quarts = 1 gallon (gal)
 
Dry:
 
2 pints = 1 quart (qt)
 
8 quarts = 1 peck (pk)
 
4 pecks = 1 bushel (bu)
Mass:
16 ounces (oz) = 1 pound (lb)
2000 pounds = 1 ton (t)
Time:
60 seconds (s) = 1 minute
60 minutes = 1 hour (hr)
24 hours = 1 day
7 days = 1 week
365 days = 1 year (yr)
10 years = 1 decade
100 years = 1 century
METRIC UNITS:
Length:
1 kilometer (km) = 1000 meters (m)
1 hectometer (hm) = 100 meters
1 dekameter (dam) = 10 meters
Base Unit for Length: 1 meter = 1m
1 decimeter (dm) = .10 meters (1m = 10 dm)
1 centimeter (cm) = .01 meters ( 1m = 100 cm)
1 millimeter (mm) = .001 meters (1m = 1000mm)
Volume:
1 kiloliter (kL) = 1000 liters (L)
1 hectoliter (hL) = 100 liters
1 dekaliter (daL) = 10 liters
Base Unit for Volume: 1 liter = 1L
1 deciliter (dL) = .10L (1 liter = 10 dL)
1 centiliter (cL) = .01L (1liter = 100cL)
1 milliliter (mL) = .001L (1liter = 1000mL)
Mass:
1 kilogram (kg) = 1000grams (g)
1 hectogram (hg) = 100 grams
1 dekagram (dag) = 10 grams
Base Unit for Mass: 1 gram = 1g
1 decigram (dg) = .10g (1g = 10dg)
1 centigram (cg) = .01cg (1g = 100cg)
1 milligram (mg) = .001mg (1g = 1000mg)
1 metric ton = 1000 kg
CONVERSIONS BETWEEN METRIC AND AMERICAN SYSTEMS
 
Length (Metric to American):
Length (American to Metric):
1km = 0.6mi
1mi = 1.6km
1m = 39.37in
1yd = 0.9m
1cm = 0.4in
1ft = 30cm
 
1in = 2.5cm
Volume (Metric to American):
Volume (American to Metric):
1L = 1.06qt
1qt = .95L
 
1 gal = 3.8L
Mass (Metric to American):
Mass (American to Metric):
1kg = 2.2lb
1lb = 0.45kg
1g = 0.04oz
1oz = 28.4g