Unit Analysis Activity

Unit Analysis Activity Page 1 of 3

Unit Analysis/Cancellation Activity

Review of Concepts In a fraction, common factors in the numerator and denominator divide to make 1: 3  1 3 27 9  3 3   Note: 27 and 18 have a common factor of 9 which divides out. 18 9  2 2

When multiplying fractions, common factors in the numerators and denominators can be removed before multiplying. 6 2  5 3

2 3 2  6 can be broken into factors of 2 and 3 5 3

2 3 2  The common factors of 3 in the numerator and denominator divide to make 1 and 5 3 are removed. This is sometimes called “canceling”.

4 Multiply the remaining numbers in the fraction (2x2) and (5x1) 5

Review Practice 3 32 2x3 x 2 y 3 1)  2)  8 9 y 2 8

Introduction to Unit Analysis In unit analysis, this same concept is applied to units of measurement in order to perform conversions and other calculations using units of measurement.

Example: Set up a conversion from minutes to days. minutes x hours = minutes The common units of hours in the numerator hours day day and denominator “cancel”. This leaves minutes/day. Unit Analysis Activity Page 2 of 3

The fraction minutes/hours in the example above is based on the fact that we know a relationship between minutes and hours (60 minutes = 1 hour). We also know a relationship for hours and days. Right now we are going to work only with units and not worry about the numbers.

Note about terminology: The word “per” means division. So “miles/hour” literally means miles divided by hours or mile/hour. The mathematical word for this is “ratio” which means a comparison by division.

Practice with Units 3) Arrange the following ratios in fractions so that you will get a result of cups/gallon. Use only the units below (no numbers).

fluid ounces (fl. oz) per gallon fluid ounces per cup

______x ______= cups gallon

4) Arrange the following ratios in fractions so that you will get a result of $/miles. miles/gallon $/gallon

______x ______= $(dollars) miles

Conversions with Numbers Once you have the problem set up with the units, you can put in the relevant numbers based on conversion factors (the ratios using two units referred to above).

Example: Convert 200 fluid ounces to gallons. [Note that unlike the examples above, 200 fluid ounces is not a ratio (it’s not “per” anything). When writing it as a fraction, the denominator is “1”.] Find your conversion factor: 128 fl. oz = 1 gallon

Step 1: Set up the problem using the units first: fl. oz. x gal. Since we have 200 fl. oz. it goes over one. 1 fl. oz.

Step 2: Now put in the numbers: 200 fl. oz. x 1 gal. Because fl. oz. appear in the numerator and denominator, 1 128 fl. oz. they will cancel leaving gallons. This tells you the problem is set up correctly.

Step 3: Multiply and simplify. 200 gal. = 1.563 gallons Remember that a fraction bar means division. 128 Unit Analysis Activity Page 3 of 3

Harry Potter and the Amazing Unit Analysis

Solve each problem using unit analysis. You must show the use of units (as shown in the previous page) in order to receive credit. Do your work on separate paper.

1. As Hagrid says, wizard money is very easy to understand. There are three coins: knuts, sickles and galleons. There are 29 knuts in one sickle and 17 sickles in one galleon. How many knuts in a galleon?

2. Ron has carefully horded every knut he’s found for 10 years and now he has three huge bags. He didn’t want to count every coin so he weighed the bags and found he had 75 pounds of knuts. One knut weighs 2 ounces. How much does he have in galleons? 16 oz. = 1 lb

3. Harry is practicing flying on his Firebolt. He does 10 laps around the Quidditch field in 18 minutes. One lap of the field is 700 meters (m). How fast is he going in kilometers (km) per hour? 1000 m = 1 km

4. One of the most important ingredients in Polyjuice Potion (used to make you look like someone else) is dried boomslang skin. As you know, boomslangs are very small which is why boomslang skin is so expensive. It takes 32 boomslangs to make 1 teaspoon (tsp.) of dried boomslang skin. The potion calls for ½ cup (c.) of skin. How many boomslangs have to give their lives for the recipe? 3 tsp. = 1 tablespoon (tbsp.) 16 tbsp. = 1 cup

5. The evil Barty Crouch Jr. had to drink at least 4 fl. oz. of Polyjuice Potion every hour in order to maintain his disguise as Mad Dog Moody. How many days would 5 gallons of potion last him? 128 fl. oz. = 1 gallon