1 Dimensional Analysis Method Chapter 3.4 p84-89

2 Which is bigger? Pennies Quarters Nickels Answer = 5 The units are more important than the numbers!

3 What is Dimensional Analysis? Dimension – means Unit Analysis – means Problem Solving Solving problems using units! Use the units to find the solution to a problem!

4 What is Dimensional Analysis? Dimensional analysis is based on using “Conversion Factors” to convert one type of unit into another.

5 What is a conversion factor? A conversion factor is an equality written as a ratio. Every equality gives you two different conversion factors.

6 What is a conversion factor? As a ratio of two equivalent terms, all conversion factors equal 1. Since 1 dime = 10 cents, the ratio= 1

7 How are conversion factors used? When used correctly in a math problem, they allow use to convert from one type of unit to another! Notice the unit, dime, cancels and we are left with cents!

8 How are conversion factors used? When used correctly in a math problem, they allow use to convert from one type of unit to another! Since the conversion factor really equals 1, this means 4 dimes equals 40 cents. 1

9 How are conversion factors used? When used correctly in a math problem, they allow use to convert from one type of unit to another! The numerical value changes, but the actual size of the quantity measured stays the same!

10 Setup 1. Write down units of answer 2. Set up workspace 3. Write down given How many dimes are in 30 cents? 30 cents = dimes Step 2 Step 1 Step 3

11 Solution 4. Do I want to keep it? If you don’t, write the unit on the next line so it cancels (on bottom if it was on top). 5. What can I change it into? (What other unit is equal to the unit I don’t want?) –Write that unit on top and then include the numbers. –Our goal is to change to the target unit so look to see if you know an equality for this unit first. If you do, then you’ll be done after this step! 6. Repeat until the answer to #4 is yes! 30 cents = dimes cents dime1 10 Step 4 Step 5 Step 6 – cents cancel and we are left with dimes so “yes” go to step 7 This is an example of a single step problem! It uses 1 conversion factor!

12 Calculate 7. Everything on top is multiplied and everything on the bottom is divided. 8. Plug the numbers in the calculator and let it do all the work! 30 cents = dimes cents dime dimes Step 7 Step 8

13 More info – Step 4 Step 4 – Do I want to keep it? –The it refers only to the unit. Don’t recopy the number! –The units are part of the calculation, so to cancel a unit it must appear on the top and bottom of a divisor.

14 More info - Step 5 Step 5 -What can I change it into? –This really means “what do I know it equals”. In the example, we wrote dime on top because we know that 10 cents = 1 dime. We also know 100 cents = 1 dollar, but while we could write dollar on top, it wouldn’t help us solve the problem.

15 More info - Step 5 Step 5 -What can I change it into? –Always look to see if you know an equality between this unit and the target unit – if you do then you’ll be done after this step! –If not, ask what other unit (that gets you closer to the answer) is equivalent to the unit I don’t want?

16 More info – Step 5 continued –In this step, you are finishing what you started in step 4. You are actually completing what is termed a “conversion factor” (a ratio of two equivalent values). All conversion factors then equal seconds = 1 minute can be written: Both equal 1

17 Homework Use this technique to complete the Dimensional Analysis worksheet! Each problem is graded on the work – NOT the answer. You are learning a technique – show it if you want any points!

18 Multi-Step Problems

Some problems can’t be done with one conversion factor. –If we don’t know an equality between our given unit and our target unit, we’ll have to do the problem in more than one step. 19

Multi-Step Problems A good example of this is when we have a metric prefix unit and we are asked to change to a different prefix. How many dm are in 425 km? How many hg are in 14.3 mg? 20

Multi-Step Problems All of our equalities are between the base unit and the prefix unit. 1 km = 1000 m10 dm = 1m 1 hm = 100 m100cm = 1m 1 dam = 10m1000mm = 1m The only thing we can do with a prefix unit is change it into the base unit!!! 21

Metric Conversion Factors 1.Know the 4 metric prefixes you are supposed to know  kilo: 1 km = 1000 m  deci: 10 dm = 1 m  centi: 100 cm = 1 m  mili: 1000 mm = 1 m 22

Metric Conversion Factors 2.Remember, metric prefixes are interchangeable  If 1000 mm = 1 m, then 1000 mg = 1 g  If 1 km = 1000 m, then 1 ks = 1000 s 23

Metric Conversion Factors 3.You can change ANY prefix unit to the base unit  All prefixes are defined as equal to the base unit.  In 1 step, you can change any prefix to the base unit! 24

Metric Conversion Factors 4.AND you can change a base unit to ANY prefix unit  All prefixes are defined as equal to the base unit.  In 1 step, you can change a base unit into ANY prefix unit! 25

Metric Conversion Factors 5.It takes 2 steps to change one metric prefix into another type of prefix.  All prefixes are defined as equal to the base unit.  Therefore you must change the first prefix unit to the base unit and then change the base unit into the new prefix – TWO STEPS! 26

27 = 17.8 cm km cm m m km A pencil is 17.8 cm long. What is its length in km? Two step metric conversions (prefix unit to a prefix unit)

28 Other Multi-Step Problems = 2.5 wks s wks days 1 7 hours 1 24 hours min min s s How many seconds are in 2.5 weeks? How many years are in 1.35×10 7 seconds?

29 Complex Dimensional Analysis

30 1. Square and Cubic unit conversions = 3 cm 3 m3m3 cm m cm m cm m Or 3  m 3 How many m 3 are in 3 cm 3 ? cmcmcm

31 2. Complex Units (the hidden equality) What does 60 mi/hr (this is a complex unit) really mean? 60 miles = 1 hr or It is a bridge to convert time and distance units!

2.Complex Units a. Using as a given (At the start) The density of gold is 19.3 g/mL, what is it in cg/kL? = cg kL

2.Complex Units a. Using as a given (At the start) The density of gold is 19.3 g/mL, what is it in cg/kL? 19.3 g 1 mL = cg kL

34 2.Complex Units a. Using as a given (At the start) The density of gold is 19.3 g/mL, what is it in cg/kL? = 19.3 g = 1 mL cg kL 19.3 g 1 mL g cg mL L L kL  10 9

35 2.Complex Units b. Using as a conversion factor (In the middle) The density of gold is 19.3 g/mL. What is the mass if the volume is L? = L g L mL mL g  g = 1 mL

36 2.Complex Units b. Using as a conversion factor (In the middle) The density of gold is 19.3 g/mL. What is the mass if the volume is L? = L g L mL mL g  g = 1 mL