Pre ctivity Using Tables for Conversions PrePArAtion · 2013-08-07 · Pre-Activity PrePArAtion Using Tables for Conversions Section 4.2 • Understand that a measurement may be expressed

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Pre-Activity

PrePArAtionUsing Tables for Conversions

Section 4.2

• Understand that a measurement may be expressed in different units

• Apply the Methodology for Solving a Proportion to calculate conversions between units

• Use unit analysis for validation

new terms to LeArn

metric

unit

unit analysis

Previously used

LeArning objectives

terminoLogy

Mississippi Mud Cake2 cups all-purpose flour2 cups sugar 1 teaspoon baking soda 1/2 teaspoon salt 1/2 pound butter 1/3 cup cocoa 1 cup water 1/2 cup buttermilk 2 eggs, lightly beaten 1 teaspoon vanilla 8 ounces mini marshmallows2 cups chocolate icing

Immigrants to the United States face many challenges. One is certainly our quirky resistance to convert to the metric system of measurement. The following scenario illustrates the point.

At a social gathering of a neighborhood watch group, two moms were trading recipes. Maria, a recent emigrant from Chile, wanted Sherri’s recipe for Mississippi Mud Cake. Sherri supplied the recipe at right from her mother’s favorite recipe file:

Unfortunately, Maria’s experience with baking was limited to using only metric weights and measures. Undaunted, she accepted the recipe and vowed to make the necessary conversions so that she could make a delicious family treat.

A little history on international measurement from www.bipm.org:The Convention of the Metre (Convention du Mètre) is a treaty which gives authority to the General Conference on Weights and Measures (CGPM), the International Committee for Weights and Measures (CIPM) and the International Bureau of Weights and Measures (BIPM) to act in matters of world metrology, particularly concerning the demand for measurement standards of ever increasing accuracy, range and diversity, and the need to demonstrate equivalence between national measurement standards. The Convention was signed in Paris in 1875 by representatives of seventeen nations. As well as founding the BIPM and laying down the way in which the activities of the BIPM should be financed and managed, the Metre Convention established a permanent organizational structure for member governments to act in common accord on all matters relating to units of measurement. The Convention, modified slightly in 1921, remains the basis of international agreement on units of measurement. There are now fifty-one Member States, including all the major industrialized countries.

�� Chapter � — Tables and S�mple Stat�st�cs

buiLding mAthemAticAL LAnguAge

English Units

In the United States, we use measurements of inches, feet, yards, and miles to show distances. To measure weight we use ounces, pounds and tons; and to measure volume we use ounces, cups, pints, quarts and gallons. These units are part of the English Measurement System and have been used for hundreds of years.

The following chart lists some common English measurement conversions.

Length Weight Volume/Capacity

12 inches (in) = 1 foot (ft) 16 ounces (oz) = 1 pound (lb) 8 fluid ounces = 1 cup (c)

3 feet = 1 yard (yd) 2000 pounds = 1 ton (T) 2 cups = 1 pint (pt)

5280 feet = 1 mile (mi) 2 pints = 1 quart (qt)

4 quarts = 1 gallon (gal)

Metric Units

The metric system of measurement was developed as a system that was simple to convert between units of the same measure and between units of volume, weight, and length. The metric system is the scientific standard of measurement used throughout the world. The following chart illustrates the connection between the metric system and place value.

Place Name Place Value Metric Name/Prefix

thousandth 0.001 milli–

hundredth 0.01 centi–

tenth 0.1 deci–

one 1 meter, gram, liter

ten 10 deka–

hundred 100 hecto–

thousand 1000 kilo–

Use the following equivalents to convert between metric units.

Length Weight Volume

1 meter (m) = 1000 millimeters (mm) 1 gram (g) = 1000 milligrams (mg) 1 liter (L) = 1000 milliliters (mL)

1 meter = 100 centimeters (cm) 1 kilogram (kg) = 1000 grams 1 liter = 10 deciliters

1 kilometer (km) = 1000 meters (m) 1 metric ton (t) = 1000 kilograms

1 centimeter = 10 millimeters

Length and volume are fundamentally related in the metric system:

1 cubic centimeter (cm3) = 1 milliliter

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Converting between Systems

Sometimes it is necessary to change from English to metric units or metric to English units. The following table lists some commonly used conversions between metric and English units.

Metric to English English to Metric

Length 1 centimeter ≈ 0.3937 inches 1 inch = 2.54 centimeters

1 meter ≈ 3.28 feet 1 foot = 0.3048 meters

1 kilometer ≈ 0.62 miles 1 mile ≈ 1.6 kilometers

Weight 1 kilogram ≈ 2.2 pounds 1 pound ≈ 0.45 kilograms

1 gram ≈ 0.035 ounces 1 ounce ≈ 28.3 grams

Volume 1 liter ≈ 4.23 cups 1 cup ≈ 236.6 milliliters

1 liter ≈ 1.057 quarts 1 quart ≈ 0.946 liters

Note: Use the internet to find other conversion tables and even conversion calculators. Sites like www.infoplease.com and www.factmonster.com have tables set up so that you can easily calculate between units as well as explore many more measurements than presented here. Enter “unit conversions” in the search space provided by your internet search engine and see thousands of hits on the topic.

Applying Proportions

Complications arise when we need to calculate measurements that have different units. It is necessary to convert between units for the same reason we find common denominators when working with fractions: we need to compare or calculate with “like” sizes.

Many conversions can be done mentally, and almost automatically. We can quickly convert 15 yards to 45 feet, or 1.5 pounds to 24 ounces, or six cups to three pints. We quickly multiply or divide by a known equivalent: 1 yard = 3 feet; 1 pound = 16 ounces; 1 cup = ½ pint, to name a few.

Sometimes comparing or changing units is not easy enough that we can convert units mentally. When that is the case, we can use proportions to change from one unit to another. The methodology on the following page can be used for converting between any two measurements.

Unit Analysis (also known as “Dimensional Analysis”)

Unit analysis is a way to validate the correct conversion from one unit to another. For example, let’s convert 6 feet into yards:We can perform unit analysis to determine that we have started and finished with the correct units:

ft ydft

yd:= , validating that our answer should be in yards.

While this is a relatively simple example, the more complicated the conversion problem, the more useful unit analysis becomes. While none of the problems in this section will be this complex, the conversion from 55 miles per hour to meters per second is shown below. You can see how unit analysis allows us to make sure that our calculation follows accurate conversions from beginning to end:

55 mile hour

5280 feet mile

1 meter feet

1 hour60 mi1 1 3 28

: : :. nnute

1 minute60 seconds

meterss

::

: :=

55 52803 28 60 60

24 59.

. . eecond

6 ft 1 yd ft

yd:3

2=


Steps in the Methodology Example 1 Example 2

Step 1

Locate the conversion in a table

Obtain the appropriate conversion table and the conversion needed.

Look under volume for the conversion:1 cup = 8 ounces

Step 2

Incorporate the table values and the units for conversion into a model (set up a proportion)

Set up a proportion using the information from the table and the measurement you need to convert.

Represent the unknown measurement with a variable.

Set up using a proportion:

12 1

8

c c ozx

=

Step 3

Test that units align and perform a unit analysis to determine the correct units for the answer

Make sure the units match in the proportion.

cupounces

cupounces

=

Changing cups (c) to ounces (oz); cups divide out, leaving ounces as the unit for the answer:

c oz c

c ozc

: :

:

=

=

x

x

Step 4

Calculate the conversion

Solve the proportion for the conversion.

Cross multiply and solve:12

c 8oz 1c: := x

12

c 8oz

1c

:

= x

12

4

: 8oz

Answer: oz

=

=

x

x

methodoLogies

Calculating a Conversion

Example 1: A recipe instructs you to use a half a cup of butter in the recipe but the wrapper only shows the butter divided into ounces. How many ounces do you need?

Example 2: A recipe for beer cheese calls for ¾ cup of beer. How many ounces is that?

►

►Try It!


Model 1A: English Measures (length)

Which is longer: 17 inches or 1.3 feet?

Step 1 Conversion factor Working in feet: or Working in inches:

Convert to feet (ft)12 in = 1 ft

Convert to inches (in)1 ft = 12 in

Step 2 Proportion 17 121

in inftx

=1 3 1

12.

ft ftinx

=

Step 3 Unit Analysis inft

inft

=

in ftin

ft:=

ftin

ftin

=

ft inft

in:=

Step 4 Calculate 17 1 12

17 112

1 4

in ft in

in ftin

ft

: :

:

=

=

x

x

x. .

17 inches ≈ 1.4 feet

1 3 12 1

1 3 121

15 6

.

.

.

ft in ft

ft inft

in

: :

:

=

=

=

x

x

x1.3 feet = 15.6 inches

Step 5 Validate 1 4 112

1 4 121

16 8

.

.

.

ft ftin

ft inft

in 17in

x

x

x

=

=:

. .

15 6 121

15 6 112

1 3

.

.

.

in inft

in ftin

ft 1.3ft

x

x

x

=

=

=

:

.

Answer: 17 inches ≈ 1.4 feet, longer than 1.3 feet, so 17 inches is

longer than 1.3 feet.

1.3 feet = 15.6 inches, shorter than 17 inches, so 17 inches is

longer than 1.3 feet.

modeLs

Steps in the Methodology Example 1 Example 2

Step 5

Validate your calculation by converting back to the original unit and comparing answers

For comparison problems worked in one unit, rework the problem in the other unit and compare answers.

8 14 8

1

4 18

12

12

12

oz coz oz

cozc

ozcoz c

oz

c

c c

=

=

=

=

=

=

x

x

x

:

��0 Chapter � — Tables and S�mple Stat�st�cs

Model 1B: English Measures (weight)

Which is heavier: 14.3 ounces or 0.81 pounds?

Step 1 Conversion factor Working in pounds: or Working in ounces:

Convert to pounds (lb)16 oz = 1 lb

Convert to ounces (oz)1 lb = 16 oz

Step 2 Proportion 14 3 161

.

oz ozlbx

=0 81 1

16.

lb lbozx

=

Step 3 Unit analysis ozlb

ozlb

= ,

oz lboz

lb:=

lboz

lboz

= ,

lb ozlb

oz:=

Step 4 Calculate 16 14 3 1

14 3 116

0 9

oz oz lb

oz lboz

lb

: :

:

x

x

x

=

=

.

.

..

14.3 ounces ≈ 0.9 pounds

0 81 16 1

0 81 161

13

.

.

lb oz lb

lb ozlb

oz

: :

:

=

=

x

x

x .

0.81 pounds ≈ 13 ounces


0 9 161

14 4

.

.

.

lb lboz

lb ozlb

oz 14.3oz

x

x

x

=

=:

. .

13 161

13 116

0 81

oz ozlb

oz lboz

oz 0.81oz

x

x

x

=

=

=

:

. .

Answer: 14.3 ounces ≈ 0.9 pounds, heavier than 0.81 pounds, so 14.3 ounces is heavier than

0.81 pounds.

0.81 pounds ≈ 13 ounces, lighter than 14.3 ounces, so 14.3 ounces is heavier than

0.81 pounds.

Model 1C: English Measures (volume)

Which is more: 2.5 cups or 1.3 pints?

Step 1 Conversion factor Working in cups: or Working in pints:

Convert to cups (c)1 pt = 2 c

Convert to pints (pt)2 c = 1 pt


.

pt ptcx

=2 5 2

1.

c cptx

=

Step 3 Unit analysis ptc

ptc

= ,

pt cpt

c:

=cpt

cpt

= ,

c ptc

pt:=



1 3 21

2 6

.

.

.

pt c pt

pt cpt

c

: :

:

=

=

=

x

x

x

1.3 pints = 2.6 cups

2 5 1 2

2 5 12

1 25

.

.

.

c pt c

c ptc

pt

: :

:

=

=

=

x

x

x

2.5 cups = 1.25 pints


2 6 12

1 3

.

.

.

c cpt

c ptc

pt 1.3pt

x

x

x

=

=

= =

:

1 25 12

1 25 21

2 5

.

.

.

pt ptc

pt cpt

c 2.5c

x

x

x

=

=

= =

:

Answer: 1.3 pints = 2.6 cups, more than 2.5 cups, so 1.3 pints is more

than 2.5 cups.

2.5 cups = 1.25 pints, less than 1.3 pints, so 1.3 pints is more

than 2.5 cups.

Model 2A: Metric Measures (length)

Which is longer: 58 centimeters or 0.59 meters?

Step 1 Conversion factor Working in meters: or Working in centimeters:Convert to meters (m)

100 cm = 1 mConvert to centimeters (cm)

1 m = 100 cm


cm cmmx

=0 59 1

100.

m mcmx

=

Step 3 Unit analysis cmm

cmm

= , cm m

cmm:

=mcm

mcm

= ,

m cmm

cm:=

Step 4 Calculate 58 1 100

58 1100

0 58

cm m cm

cm mcm

m

: :

:

=

=

=

x

x

x.

58 centimeters = 0.58 meters

0 59 100 1

0 59 1001

59

.

.

m cm m

m cmm

cm

: :

:

=

=

=

x

x

x

0.59 meters = 59 centimeters

Shortcut: cm to m: move the decimal 2 places to the left (replace cm with m): 58.0 cm = 0.58 m

m to cm: move the decimal 2 places to the right (replace m with cm): 0.59 m = 59.0 cm

Step 5 Validate 0.58 m × 100 = 58 cm58 cm = 58 cm

59 cm ÷ 100 = 0.59 m0.59 m = 0.59 m

Answer: 58 centimeters = 0.58 meters, shorter than 0.59 meters, so

0.59 meters is longer than 58 centimeters.

0.59 meters = 59 centimeters, longer than 58 centimeters, so 0.59 meters is longer than 58

centimeters.


Model 2B: Metric Measures (weight)

Which is heavier: 975 grams or 1.01 kilograms?

Step 1 Conversion factor Working in grams: or Working in kilograms:

Convert to grams (g)1 kg = 1000 g

Convert to kilograms (kg)1000 g = 1 kg


.

kg kggx

=975 1000

1

g gkgx

=

Step 3 Unit analysis kgg

kgg

= ,

kg gkg

g:

=gkg

gkg

= ,

g kgg

kg:

=


1 01 10001

1010

.

.

kg g kg

kg gkg

g

: :

:

=

=

=

x

x

x

1.01 kilograms = 1010 grams

975 1 1000

975 11000

0 975

g kg g

g kgg

kg

: :

:

=

=

=

x

x

x.

975 grams = 0.975 kilograms

Shortcut: kg to g: move the decimal 3 places to the right (replace kg with g): 1.01 kg = 1010 g

g to kg: move the decimal 3 places to the left (replace g with kg): 975 g = 0.975 kg

Step 5 Validate 1010 g ÷ 1000 = 1.01 kg1.01 kg = 1.01 kg

0.975 kg × 1000 = 975 g975 g = 975 g

Answer: 1.01 kilograms = 1010 grams, heavier than 975 grams, so 1.01 kilograms is heavier

than 975 grams.

975 grams = 0.975 kilograms, lighter than 1.01 kilograms,

so 1.01 kilograms is heavier than 975 grams.


Model 2C: Metric Measures (volume)

Which is more: 253 milliliters or 0.3 liters?

Step 1 Conversion factor Working in milliliters: or Working in liters:

Convert to milliliters (mL)1 L = 1000 mL

Convert to liters (L)1000 mL = 1 L


.

L LmLx

=253 1000

1

mL mLLx

=

Step 3 Unit analysis LmL

LmL

= ,

L mLL

mL:= mL

LmLL

= ,

mL LmL

L:=


0 3 10001

300

.

.

L mL L

L mLL

mL

: :

:

=

=

=

x

x

x

0.3 liters = 300 milliliters

253 1 1000

253 11000

0 253

mL L mL

mL LmL

L

: :

:

=

=

=

x

x

x.

253 milliliters = 0.253 liters

Shortcut: L to mL: move the decimal 3 places to the right (replace L with mL): 0.3 L = 300 mL

mL to L: move the decimal 3 places to the left (replace mL with L): 253 mL = 0.253 L

Step 5 Validate 300 mL ÷ 1000 = 0.3 L0.3 L = 0.3 L

0.253 L × 1000 = 253 mL253 mL = 253 mL

Answer: 0.3 liters = 300 milliliters, more than 253 milliliters, so 0.3 liters is more than 253

milliliters.

253 milliliters = 0.253 liters, less than 0.3 liters, so 0.3 liters is more than 253 milliliters.

Did you notice?To change units within the metric system use place value. Multiply or divide by the power of ten that separates the two measurements. When converting from a smaller unit to a larger unit, multiply. When converting from a larger unit to a smaller unit, divide.


Model 3a: Converting between Metric and English Measures

Feet to Meters: Convert 4.5 feet to meters

Step 1 Conversion factor 1 foot ≈ 0.3048 meters

Step 2 Proportion

4 5 10 3048

..

ft ftmx

=

Step 3 Unit analysis ftm

ftm

= ,

ft mft

m:=

Step 4 Calculate 4 5 0 3048 1

4 5 0 30481

1 372

. .

. .

.

ft m ft

ft mft

m

: :

:

=

=

x

x

x .

Answer: 4.5 feet ≈ 1.372 meters

Step 5 Validate 1 372 0 30481

1 372 10 3048

4 501

. .

..

.

m mft

m ftm

f

x

x

x

=

=:

. tt 4.5 ft .

Model 3b: Converting between Metric and English Measures

Kilometers to Miles: Convert 120 kilometers to miles

Step 1 Conversion factor 1 kilometer ≈ 0.62 miles


km kmmix

=.

Step 3 Unit analysis kmmi

kmmi

= ,

km mikm

mi:=


120 0 621

74 4

km mi km

km mikm

mi

: :

:

.

.

.

=

=

x

x

x .

Answer: 120 kilometers ≈ 74.4 miles


74 4 10 62

120

. .

..

mi mikm

mi kmmi

km 12

x

x

x

=

=

=

:

. 00 km


Model 4: Double Conversion

Convert 3 gallons to liters

While we can convert gallons to quarts and quarts to liters, we have no direct conversion for gallons to liters. We could break this up into two separate conversions, but because we are able to perform a unit analysis, we can set up a proportional equation that will allow us to perform both of these conversions in Step 4. We will simply create a longer proportional relationship in Step 2.

Step 1 Conversion factors 1 gallon = 4 quarts1 quart ≈ 0.946 liters


0 9461

galL

= galqt

qtLx

:.

Step 3 Unit analysis galL

= galqt

qtL

:

Step 4 Calculate 3 1 0 9464 1

0 9464

3 40 9

gal =gal qt

qt L= gal

L

=gal L

x

x

:

:

:

. .

. 446120 946

12 68

gal= L L

.. .

Answer: 3 gallons ≈ 12.68 liters

Shortcut: Because we use unit analysis to verify and validate our starting and ending units, we can

string together multiple conversion proportions:

x

= gal

qtgal

Lqt

= L L341

10 946

120 946

12 68: :. .

. .

We always arrange our proportions so that all units cancel except the ones we need for our answer.

Step 5 Validate Convert 12.68 liters to gallons:

x( ..

. .

in gal) = Lqt

Lgalqt

=

12 680 946

114

12 68 0 9464

: :

: . 22 99. gal


Model 5: Problem Solving

You have a planter that is 5 feet long and 18 inches wide. You want to put an attractive border around the planter. The border is only sold in 30-centimeter sections. How many do you need to buy? First, draw a diagram.

The first computation is converting 5.25 feet to the equivalent number of inches in order to calculate the perimeter.

Now we need to convert 162 inches to centimeters:

We know that the perimeter ≈ 411.48 cm. Because the border is only sold in sections that are 30 centimeters long, we must divide the

perimeter by 30 to find out how many sections to buy.

411 4830

13 716. .cmcm

=

Therefore we need to buy 14 pieces of border.

5.25 feet

18 inches

Step 1 Conversion factor 1 foot = 12 inches Step 4 Calculate 5 25 12 1

5 25 121

63

.

.

ft in ft

ft inft

in

: :

:

=

=

=

x

x

x

5.25 feet = 63 inches

Step 5 Validate 63 121

in inft

5.25 ft x

x

=

=


.

ft ftinx

=

Step 3 Unit analysis ftin

ftin

= ,

ft inft

in:=

P l wPPP

= +

= +

= +

=

2 22 63 2 18126 36162

( ) ( )in inin inin

We now know that 5.25 feet = 63 inches. Next, we need to find the perimeter of the planter, using the converted measurement:

Step 1 Conversion factor

1 in ≈ 2.54 cm Step 4 Calculate 162 2 54 1

162 2 541

411 48

in cm in

in cmincm

: :

:

.

.

.

=

=

x

x

x .

162 in ≈ 411.48 cm


. .

cm cmin

162 in x

x

=

=


in incmx

=.

Step 3 Unit analysis incm

incm

= ,

in cmin

cm :

=


Addressing common errors

Issue Incorrect Process Resolution Correct

Process

Not converting to a common unit for comparison

Which is more: two 2-liter bottles of cola or a 12 pack of 12 oz cans? Two 2-liter colas because they’re large bottles, rather than small cans.

You must use common units when making comparisons. You cannot simply trust appearance or your intuition.

Convert 12 times 12 ounces (144 ounces) to quarts: 144 32

1

oz ozqtx

=

x = 4.5 qt

and then quarts to

liters:

4 5 10 946

..

qt qtLx

=

x ≈ 4.3 L

OR

x( .

. .

in qt) = ozqtoz

Lqt

=

1441

320 946

1

144 0 94632

4 3

: :

: . L

Compare 4.3 L to two 2-liters or 4 liters. The 12 pack of 12 oz sodas is more.

Validation

We worked the problem in quarts; to validate our answer, we should convert all our measures into ounces and perform the comparison that way: Two 2 L bottles = 4 L

4 0 9461

4 23

L Lqt

qtx

x

=.

..

Convert 4 L to quarts:

Then quarts to ounces:4 23 1

32135 36

.

.

qt qtoz

ozx

x

=

.

OR

x(.

..

in oz) = Lqt

Lozqt

=

41

0 946321

4 320 946

135 31

: :

: . ooz

Compare 135 oz (the cola in the bottles) with 144 oz (the cola in the cans). The cans contain more cola.

Did you notice?Validating the solution in two separate conversions meant that we had to approximate twice, thus compounding rounding-off errors.

This second method yields a more precise answer because we only round off once.


Validation

The problem did not specify which units were needed for the answer. We worked in inches for the correct process, so we can rework the problem in feet and compare our answers in order to validate the calculation of area. We begin by converting 10 inches to feet and the calculate the area in square feet.

10 121

0 83

in inft

ftx

x

=

. .

A = lwA = 1.5 ft • 0.83 ftA ≈ 1.25 ft2

In order to compare this answer with the answer in square inches, we need convert our answer in square feet to the equivalent answer in square inches:

12 in × 12 in = 1 ft2 = 144 in2 1 25 1144

179 28

2 2

2

2

.

.

ft ft in

inx

x

=

. 179.28 in2 ≈ 180 in2


Process Validation

Using incorrect conversion information

Convert 30 cm to inches.

30 31

30 1 3

30 13

10

cm cmin

cm in cm

cm incm

in

xx

x

x

=

=

=

=

: :

:

Converting from metric units to English units (or visa versa) requires the use of conversion factors that can be difficult to remember. Check your conversion factors.

From the chart, 1 in ≈ 2.54 cm, not 3 cm

30 2 541

30 1 2 54

30 12 54

cm cmin

cm in cm

cm incm

xx

x

x

=

=

=

.

.

.

: :

:

.. in 11 81.

To validate, we will convert our answer in inches back to centimeters and compare:11 81 1

2 5429 99

..

.

in incm

cmx

x

=

.

Compare 29.99 cm with the original measurement of 30 cm:29.99 cm ≈ 30 cm


Process

Not converting to a common unit before calculating area

A picture frame measures 1.5 ft by 10 in. What is the area of the picture?

1.5 ft • 10 in =15 sq in

Units must be the same before multiplying for area.

Convert 1.5 feet to inches.

1 5 1121 5 12

18

.

.

ft ftin

ft ft inin

xxx

=

=

=

: :

A = lwA = 18 in • 10 inA = 180 in2


PrePArAtion inventory

Before proceeding, you should be able to use conversion tables to:

Look up appropriate equivalent measures

Build proportions to convert between units

Use unit analysis to validate conversions


Process Validation

Moving the decimal point the wrong direction when converting metric measures

Convert 120 L to mL.

120 L = 0.12 mL

If you cannot remember which direction to move the decimal point, use the proportion methodology for conversions.

1 1000120 1

1000120 1000

120 1000

L mLL L

mLL L mL

L

=

=

=

=

xx

x

: :

: mmLLmL x =120 000,

Convert mL back to L and compare:

1000 1120 000 1000

1120 000 1

10001

mL LmL mL

LmL LmL

=

=

=

=

,

,x

x

x

:

220 L

Not validating by unit analysis

Convert 1.6 yards to inches.

1 6 112

19 2

.

.

yd ftin

inx

x

=

=

Always perform a unit analysis in Step 2 to make sure the units in the proportion match up correctly.

yd ftin in

≠

so the proportion was not set up properly.

Step 2

11

1 1 65

yd 36in1.6yd yd

36inydin

ydin

yd yd 36in

=

=

=

=

=

x

xx: :.

77 6. in

To validate calculations, convert inches back to yards and compare answers:

3636

57 6 1

in 1yd57.6in in

1ydinyd

inyd

in yd 36in

=

=

=

=

x

xx

. : :

==1 6. yd

1.6 yd = 1.6 yd

�00

Activity

Section 4.2

PerformAnce criteriA

• Produce translated values within a system of measurement.– accuracy– unit Analysis– validation– proper presentation of answer

• Produce translated values from a given system to a new system of measurement– accuracy– unit Analysis– validation– proper presentation of answer

Using Tables for Conversions

criticAL thinking Questions

1. What are the two components of a measurement such as 2.6 m?

2. Some rulers have English and metric units on them. When you use such a ruler to record a length, how do you ensure that someone else knows what your number means?

�0�Sect�on �.� — Us�ng Tables for Convers�ons

3. When converting from a smaller unit to a larger unit, what happens to the numerical part of the measurement?

4. When converting from a larger unit to a smaller unit, what happens to the numerical part of the measurement?

5. How do you locate a conversion table for your needs?

6. How does checking units help you to make sure your conversion was performed correctly?

�0� Chapter � — Tables and S�mple Stat�st�cs

7. What are three “short cuts” you can use to convert from one measure to another?

8. Why do multiple measurement systems exist for a given instrument like a ruler or kitchen measuring cup?

tiPs for success

• Converting units by setting up proportions offers two ways to validate your work. You can examine the units to confirm that the proportions have been set up correctly and the final unit is the one you want (this is called unit analysis). You can also validate the solution by setting it back into the proportion and test for equivalency. (See Foundations of Math, Section 4.2.)

• In the examples given to compare two measures, notice that one way to work the proportion uses division to reach the solution and the other one uses multiplication. Before calculators, division was much harder to do, so scientists frequently chose the conversion that required multiplication. Conversion tables are often set up so that all unit changes can be made using multiplication. These tables list conversion factors (a factor being part of a product). To convert from one unit to another simply multiply the measurement by the appropriate conversion factor.


Problem Worked Solution Validation

a) 12 pints to gallons

b) 6 quarts to pints

c) 6 miles to yards

d) 60 inches to feet

1. Convert the following to the units indicated:

demonstrAte your understAnding



e) 12,400 pounds to tons

f) 100 yards to feet

2. Convert the following to the units indicated:


a) 1.23 meters to centimeters

b) 176 millimeters to meters

c) 500 milliliters to liters



d) 456 grams to kilograms

e) 34.6 kilograms to grams

f) 23.5 centimeters to millimeters

3. Perform the following conversions:


a) 15 liters to quarts

b) 28 feet to meters



c) 185 pounds to kilograms

d) 26.5 kilometers to miles

4. Perform the following double conversions:


a) 2 miles to yards



b) 550 grams to pounds

5. If a baby weighs 6 lb 7 oz and her car seat weighs 3.2 kg, how much do they weigh together?

6. A recipe calls for 750 mL of orange juice, 50 mL lemon juice, 1 liter of pineapple juice and 500 mL of rum. How large should the container be to hold the punch?


7. Help Maria convert the following recipe ingredients from English to Metric.

English Metric

2 cups to _____________ all-purpose flour

2 cups to _____________ sugar

1 teaspoon to _____________ baking soda

1/2 teaspoon to _____________ salt

1/2 pound to _____________ butter

1/3 cup to _____________ cocoa

1 cup to _____________ water

1/2 cup to _____________ buttermilk

2 eggs, lightly beaten

1 teaspoon to _____________ vanilla

8 ounces to _____________ mini marshmallows


In the second column, identify the error(s) in the worked solution or validate its answer. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer.

Worked Solution Identify Errors or Validate Correct Process

1) Allison wanted to know how many cups of tomato sauce to add to a 1.2 L jar of sauce to make 2 liters.2 Ll – 1.2 L = 0.8 LShe needs to convert 0.8 L to cups.1 liter ≈ 4.23 cups

0 8 14 23

3 4

..

.

L Lc

Lc

Lc

c c

x=

=

.

Allison will use 3 ½ cups of tomato sauce.

2) I have a friend who describes himself as a “2 meter” man. How tall is he in inches?

2 3 2812

7 3

m ftin

ftx

x

=

=

.

.

3) Micha needed 11.2 g of iron for his experiment. He measured out 0.0112 mg.

identify And correct the errors

��0 Chapter � — Tables and S�mple Stat�st�cs

Worked Solution Identify Errors or Validate Correct Process

4) Convert 12.5 gallons to liters.

12 5 4 0 946

3 784 12 50 33

. .

. ..

gal qtL

L

xx

x

=

=

:

:

.

5) What is the perimeter of a lot that measures 57.3 m by 0.34 km?

19.5 m2

Documents

Pre ctivity Using Tables for Conversions PrePArAtion · 2013-08-07 · Pre-Activity PrePArAtion Using Tables for Conversions Section 4.2 • Understand that a measurement may be expressed