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Measurements

Problem Solving

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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To solve a problem:• Identify the given unit• Identify the needed unit

Problem: A person has a height of 2.0 meters. What is that height in inches? The given unit is the initial unit of height. given unit = meters (m)

The needed unit is the unit for the answer. needed unit = inches (in)

Given and Needed Units

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Learning Check

An injured person loses 0.30 pints of blood. How

many milliliters of blood would that be?

Identify the given and needed units in thisproblem.

Given unit = _______

Needed unit = _______

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Solution

An injured person loses 0.30 pints of blood. How many milliliters of blood would that be?

Identify the given and needed units in this problem

Given unit = pints Needed unit = milliliters

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• Write the given and needed units.

• Write a unit plan to convert the given unit to the needed unit.

• Write equalities and conversion factors that connect the units.

• Use conversion factors to cancel the given unit and provide the needed unit.

Unit 1 x Unit 2 = Unit 2Unit 1

Given x Conversion = Needed unit factor unit

Problem Setup

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Guide to Problem Solving

The steps in the guide to problem solving are useful in setting up a problem with conversion factors.

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Setting Up a Problem

How many minutes are 2.5 hours?Given unit = 2.5 hrNeeded unit = ? minUnit Plan= hr min

Set up problem to cancel hours (hrs).

Given Conversion Neededunit factor unit2.5 hr x 60 min = 150 min (2 SF) 1 hr

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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A rattlesnake is 2.44 m long. How many centimeters long is the snake?

1) 2440 cm

2) 244 cm

3) 24.4 cm

Learning Check

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A rattlesnake is 2.44 m long. How many centimeters long is the snake?

2) 244 cm

2.44 m x 100 cm = 244 cm

1 m

Solution

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• Often, two or more conversion factors are required to obtain the unit needed for the answer.

Unit 1 Unit 2 Unit 3

• Additional conversion factors are placed in the setup to cancel each preceding unit.Given unit x factor 1 x factor 2 = needed unitUnit 1 x Unit 2 x Unit 3 = Unit 3

Unit 1 Unit 2

Using Two or More Factors

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How many minutes are in 1.4 days?

Given unit: 1.4 days

Factor 1 Factor 2

Plan: days hr min

Set up problem:

1.4 days x 24 hr x 60 min = 2.0 x 103 min

1 day 1 hr

2 SF Exact Exact = 2 SF

Example: Problem Solving

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• Be sure to check your unit cancellation in the setup.• The units in the conversion factors must cancel to

give the correct unit for the answer.

What is wrong with the following setup?1.4 day x 1 day x 1 hr

24 hr 60 min

Units = day2/min is Not the unit needed

Units don’t cancel properly.

Check the Unit Cancellation

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Using the GPS

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A bucket contains 4.65 L of water. How manygallons of water is that?

Unit plan: L qt gallon

Equalities: 1 L = 1.057 qt

1 gal = 4 qt

Set Up Problem:

Learning Check

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Given: 4.65 L Needed: gallons

Plan: L qt gal

Equalities: 1 L = 1.057 qt 1 gal = 4 qt

Set Up Problem:

4.65 L x 1.057 qt x 1 gal = 1.23 gal

1 L 4 qt

3 SF 4 SF exact 3 SF

Solution

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If a ski pole is 3.0 feet in length, how long is the ski pole in mm?

Learning Check

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3.0 ft x 12 in x 2.54 cm x 10 mm =

1 ft 1 in. 1 cm

Calculator answer: 914.4 mm

Needed answer: 910 mm (2 SF rounded)

Check factor setup: Units cancel properly

Check needed unit: mm

Solution

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If your pace on a treadmill is 65 meters per minute,

how many minutes will it take for you to walk a

distance of 7500 feet?

Learning Check

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Given: 7500 ft 65 m/min Need: min

Plan: ft in cm m min

Equalities: 1 ft = 12 in 1 in = 2.54 cm 1 m = 100 cm

1 min = 65 m (walking pace)

Set Up Problem

7500 ft x 12 in x 2.54 cm x 1 m x 1 min 1 ft 1 in 100 cm 65 m

= 35 min final answer (2 SF)

Solution

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Percent Factor in a Problem

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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How many pounds (lb) of sugar are in 120 g of candy if the candy is 25% (by mass) sugar?

Learning Check

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How many pounds (lb) of sugar are in 120 g of candy if the candy is 25%(by mass) sugar?

% factor

120 g candy x 1 lb candy x 25 lb sugar

453.6 g candy 100 lb candy

= 0.066 lb sugar

Solution

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STANDARDS OF MEASUREMENT

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• Measuring is an important skill, especially in the field of science. However, in order to be useful, they must be standardized.

• Standard of measurement an exact quantity that people agree to use as basis of comparison.

• All measurements are compared to the standard. This way all measurements can be compared to each other. A meter in the U.S. is the same as a meter in France

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SI Base Units for Measurement Used in Science

Length

Volume

Mass

Measured in meters (m)Measured in meters (m)

Measured in milliliters (mL) Measured in milliliters (mL) or cubic centimeters (cmor cubic centimeters (cm33))

Measured in grams (g)Measured in grams (g)

Metric ruler:

Graduated cylinder:

Balance:

the distance from one point to another point in a straight line

the amount of space occupied by an object

the amount of matter in an object

Density the mass per volume of a material

Measured in grams per Measured in grams per millimeter (g/ml)millimeter (g/ml)

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Temperature

Measured in degrees Celsius (Measured in degrees Celsius (OOC)C)Thermometer:

the average kinetic energy of the particles in a substance

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Time

Electric current

Light (luminosity)

Amount of a substance

Measured in seconds (s)Measured in seconds (s)

Measured in amps (A)Measured in amps (A)

Measured in candelas (cd)Measured in candelas (cd)

Measured in moles (mol)Measured in moles (mol)

Stopwatch:

ammeter:

Light meter:

Avagadro’s number

an interval on a continuum measured from a point in the past to a point in the present or future; duration

a flow of electric charge

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Practicing in MeasuringVolume, Mass and Length

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Volume

• Graduated cylinders

Glass

Plastic

Clear fluid problems

Procedure for finding volume of irregular objects

• Beakers

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• You should be able to measure the volume of liquids in a graduated cylinder. How precisely you can measure volume depends on the size and type of graduated cylinder you use. Generally, you should be able to estimate between the etched or printed lines.

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On this 100 milliliter cylinder, the numbers are 10, 20, 30, etc., so there is a 10 milliliter increment between them. Since there are 10 divisions between consecutive numbers, each division represents one milliliter. Therefore, you should be able to estimate to tenths of a milliliter by reading between the lines.

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It is important to notice what each line or interval on the graduated cylinder represents.

Different kinds of graduated cylinders are set up differently. A 10 milliliter cylinder, for example, usually has one tenth of a milliliter for each graduation, but some have two-tenths milliliter for each graduation.

The way to check this is to count the divisions between consecutive numbers.

8.8 mL

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On some cylinders, there may only be five divisions between numbers. Or there may be ten divisions for a 2 milliliter increment. In these cases, each of the divisions represents 0.2 milliliters, rather than 0.1. You need to be aware of that when you're using the cylinders like these, and adjust your between-line-estimates accordingly.

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We need to see the rest of the graduated cylinder to know what increments these lines are in.

?

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A characteristic of liquids in glass containers is that they curve at the edges.

You measure the level at the horizontal center or inside part of the meniscus.

Read the bottom of the meniscus. 5.9 mL

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Reading the Graduated Cylinder• Your eye

should be level with the top of the liquid

• You should read to the bottom of the MENISCUS

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6.8 mL

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The visibility of the meniscus can be enhanced by using a card with a dark stripecard with a dark stripe on it,  placed behind the cylinder. 

Adjusting the placement of the card can give you either a white meniscus against a black background or a black meniscus against a white background.

~ 48 mL

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36.5 mL

12 mL

16 mL

21.63

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Reading the Graduated Cylinder

•What is this reading?

•18.0 mL

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Reading the Graduated Cylinder

•What is this reading?

•36.5 mL

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Reading the Graduated Cylinder

•What is this reading?

•42.9 mL

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Reading the Graduated Cylinder

•What is this reading?

•47.0 mL

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Reading the Graduated Cylinder

•What is this reading?

•61.2 mL

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Finding volumes of regular objects and irregular objects.

75 – 50 = 25 mL

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1) Measure the volume of liquid in a partially filled graduated cylinder,

2) Add the solid (making sure it is submerged) and note how the level of the liquid goes up,

3) Measure the combined volume of the solid and liquid. The difference between the initial and final volumes is the volume of the solid.

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Beakers with only one scale of numbers.

43 or 44 mL

~ 159 mL

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Beakers with two scales of numbers.

~ 2050 mL

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Mass

• Kinds of balances

• The standards

• The way to read the scales

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The same worldwide.

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Mass MeasurementThe triple beam balance is commonly used to measure mass in the biology lab. This device is named for its three long beams on which sliding bars called riders (or tares) are used to determine the mass of an object placed on its platform.    It is very important that the riders on the rear beams are in the notch for the whole number of grams and not in between notches. The front beam is a sliding scale graduated in grams. The rider on this beam can be positioned anywhere on the scale. Masses on a triple-beam balance can be read to tenths of a gram and estimated to hundredths of a gram.

~ 100.41 grams

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The picture shows the measurement of a mass in progress.    Without estimation, the mass of the object appears to be 373.3 grams (g).

~ 373.3 grams

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Accuracy and Precision

The usefulness of measurement is enhanced by knowledge of its level of certainty. 

Multiple measurements of the same property are like multiple shots at the same target. 

The pattern of the shots tells you something about the measurement and its ability to describe the 'true' value of the property being sought.  

The patterns depict possible outcomes of different experiments to measure the same property. 

Expt IV is of course the best, because it give very reproducible results (precise) and also results that are very close to the true value or bull's-eye (accurate). 

Experiment III is precise but not accurate.  It exhibits systematic error, which is very difficult to estimate at times.

http://www.brainpop.com/science/scientificinquriry/precisionandaccuracy/

A short video about this topic.