CHAPTER 3 Measurement, Perimeter, Area, and Volume
3.1 Systems of Measurement
Learning Objectives
By the end of this section, you will be able to:
- Make unit conversions in the imperial system
- Use mixed units of measurement in the imperial system
- Make unit conversions in the metric system
- Use mixed units of measurement in the metric system
- Convert between the imperial and the metric systems of measurement
- Convert between Fahrenheit and Celsius temperatures
Make Unit Conversions in the Imperial System
There are two systems of measurement commonly used around the world. Most countries use the metric system. Canada uses the metric system, and the United States use the imperial system of measurement. However, people in Canada often use imperial measurements as well. We will look at the imperial system first.
The imperial system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the imperial system and the metric system measure time in seconds, minutes, and hours.
The equivalencies of measurements are shown in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.
Identity Property of Multiplication
For any real number :
1 is the multiplicative identity.
To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction . When we multiply by this fraction we do not change the value, but just change the units.
But also equals 1. How do we decide whether to multiply by or ? We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert inches to feet, which multiplication will eliminate the inches?
The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.
EXAMPLE 1
MaryAnne is 66 inches tall. Convert her height into feet.
TRY IT 1.1
Lexie is 30 inches tall. Convert her height to feet.
Show answer
2.5 feet
TRY IT 1.2
Rene bought a hose that is 18 yards long. Convert the length to feet.
Show answer
54 feet
HOW TO: Make unit conversions
- Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
- Multiply.
- Simplify the fraction.
- Simplify.
When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
EXAMPLE 2
A female orca in the Salish Sea weighs almost 3.2 tons. Convert her weight to pounds.
We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction .
| Multiply the measurement to be converted, by 1. | |
| Write 1 as a fraction relating tons and pounds. | |
| Simplify. | |
| Multiply. | |
| The female orca weighs almost 6,400 pounds. |
TRY IT 2.1
Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.
Show answer
8,600 pounds
TRY IT 2.2
The Carnival Destiny cruise ship weighs 51,000 tons. Convert the weight to pounds.
Show answer
102,000,000 pounds
Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.
EXAMPLE 3
Juliet is going with her family to their summer home. She will be away from her boyfriend for 9 weeks. Convert the time to minutes.
Solution
To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we will multiply by conversion factors of 1.
| 9 weeks | |
| Write 1 as , , and . | |
| Divide out the common units. | |
| Multiply. | |
| Multiply. |
Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).
TRY IT 3.1
The distance between the earth and the moon is about 250,000 miles. Convert this length to yards.
Show answer
440,000,000 yards
TRY IT 3.2
The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.
Show answer
151,200 minutes
EXAMPLE 4
How many ounces are in 1 gallon?
We will convert gallons to ounces by multiplying by several conversion factors. Refer to the table on Imperial Systems of Measurement.
| 1 gallon | |
| Multiply the measurement to be converted by 1. | |
| Use conversion factors to get to the right unit. Simplify. |
|
| Multiply. | |
| Simplify. | There are 128 ounces in a gallon. |
TRY IT 4.1
How many cups are in 1 gallon?
Show answer
16 cups
TRY IT 4.2
How many teaspoons are in 1 cup?
Show answer
48 teaspoons
Use Mixed Units of Measurement in the Imperial System
We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.
Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!
EXAMPLE 5
Seymour bought three steaks for a barbecue. Their weights were 14 ounces; 1 pound, 2 ounces; and 1 pound, 6 ounces. How many total pounds of steak did he buy?
We will add the weights of the steaks to find the total weight of the steaks.
| Add the ounces. Then add the pounds. | |
| Convert 22 ounces to pounds and ounces. | 1 pound, 6 ounces |
| Add the pounds and ounces. | 2 pounds + 1 pound + 6 ounces |
| Answer | Seymour bought 3 pounds 6 ounces of steak. |
TRY IT 5.1
Laura gave birth to triplets weighing 3 pounds 3 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?
Show answer
9 lbs. 8 oz
TRY IT 5.2
Stan cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?
Show answer
21 ft. 6 in.
EXAMPLE 6
Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?
We will multiply the length of one plank to find the total length.
| Multiply the inches and then the feet. | |
| Convert the 16 inches to feet. Add the feet. |
|
| Anthony bought 25 feet and 4 inches of wood. |
TRY IT 6.1
Henri wants to triple his vegan spaghetti sauce recipe that uses 1 pound 8 ounces of black beans. How many pounds of black beans will he need?
Show answer
4 lbs. 8 oz.
TRY IT 6.2
Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?
Show answer
11 gallons 2 qt.
Make Unit Conversions in the Metric System
In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a metre. One kilometre is 1,000 metres; the prefix kilo means thousand. One centimetre is of a metre, just like one cent is of one dollar.
The equivalencies of measurements in the metric system are shown in the table below. The common abbreviations for each measurement are given in parentheses.
| Length | Mass | Capacity |
|---|---|---|
| 1 kilometre (km) = 1,000 m
1 hectometre (hm) = 100 m 1 dekametre (dam) = 10 m 1 metre (m) = 1 m 1 decimetre (dm) = 0.1 m 1 centimetre (cm) = 0.01 m 1 millimetre (mm) = 0.001 m |
1 kilogram (kg) = 1,000 g
1 hectogram (hg) = 100 g 1 dekagram (dag) = 10 g 1 gram (g) = 1 g 1 decigram (dg) = 0.1 g 1 centigram (cg) = 0.01 g 1 milligram (mg) = 0.001 g |
1 kilolitre (kL) = 1,000 L
1 hectolitre (hL) = 100 L 1 dekalitre (daL) = 10 L 1 litre (L) = 1 L 1 decilitre (dL) = 0.1 L 1 centilitre (cL) = 0.01 L 1 millilitre (mL) = 0.001 L |
| 1 metre = 100 centimetres
1 metre = 1,000 millimetres |
1 gram = 100 centigrams
1 gram = 1,000 milligrams |
1 litre = 100 centilitre s
1 litre = 1,000 millilitre s |
To make conversions in the metric system, we will use the same technique we did in the Imperial system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a 5K or 10K race? The length of those races are measured in kilometres. The metric system is commonly used in Canada when talking about the length of a race.
EXAMPLE 7
Nick ran a 10K race. How many metres did he run?
We will convert kilometres to metres using the identity property of multiplication.
| 10 kilometres | |
| Multiply the measurement to be converted by 1. | 10 kilometres × 1 |
| Write 1 as a fraction relating kilometres and metres. | |
| Simplify. | |
| Multiply. | 10,000 metres |
| Nick ran 10,000 metres. |
TRY IT 7.1
Sandy completed her first 5K race! How many metres did she run?
Show answer
5,000 metres
TRY IT 7.2
Herman bought a rug 2.5 metres in length. How many centimetres is the length?
Show answer
250 centimetres
EXAMPLE 8
Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?
We will convert grams into kilograms.
| Multiply the measurement to be converted by 1. | |
| Write 1 as a function relating kilograms and grams. | |
| Simplify. | |
| Multiply. | |
| Divide. | 3.2 kilograms The baby weighed 3.2 kilograms. |
TRY IT 8.1
Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?
Show answer
2.8 kilograms
TRY IT 8.2
Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?
Show answer
4.5 kilograms
As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.
To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.
We can apply this pattern when we make measurement conversions in the metric system. In Example 8, we changed 3,200 grams to kilograms by multiplying by (or 0.001). This is the same as moving the decimal three places to the left.
EXAMPLE 9
Convert a) 350 L to kilolitres b) 4.1 L to millilitre s.
- We will convert litres to kilolitres. In the Metric System of Measurement table, we see that
Multiply by 1, writing 1 as a fraction relating litres to kilolitres. Simplify. Move the decimal 3 units to the left. - We will convert litres to millilitre s. From Metric System of Measurement table we see that
Multiply by 1, writing 1 as a fraction relating litres to millilitre s. Simplify. Move the decimal 3 units to the right.
TRY IT 9.1
Convert: a) 725 L to kilolitres b) 6.3 L to millilitre s
Show answer
a) 7,250 kilolitres b) 6,300 millilitre s
TRY IT 9.2
Convert: a) 350 hL to litres b) 4.1 L to centilitre s
Show answer
a) 35,000 litres b) 410 centilitre s
Use Mixed Units of Measurement in the Imperial System
Performing arithmetic operations on measurements with mixed units of measures in the imperial system requires the same care we used in the Canadian system. Make sure to add or subtract like units.
EXAMPLE 10
Ryland is 1.6 metres tall. His younger brother is 85 centimetres tall. How much taller is Ryland than his younger brother?
We can convert both measurements to either centimetres or metres. Since metres is the larger unit, we will subtract the lengths in metres. We convert 85 centimetres to metres by moving the decimal 2 places to the left.
| Write the 85 centimetres as metres. |
Ryland is 0.75 m taller than his brother.
TRY IT 10.1
Mariella is 1.58 metres tall. Her daughter is 75 centimetres tall. How much taller is Mariella than her daughter? Write the answer in centimetres.
Show answer
83 centimetres
TRY IT 10.2
The fence around Hank’s yard is 2 metres high. Hank is 96 centimetres tall. How much shorter than the fence is Hank? Write the answer in metres.
Show answer
1.04 metres
EXAMPLE 11
Dena’s recipe for lentil soup calls for 150 millilitre s of olive oil. Dena wants to triple the recipe. How many litres of olive oil will she need?
We will find the amount of olive oil in millileters then convert to litres.
| Triple 150 mL | |
| Translate to algebra. | |
| Multiply. | 450 mL |
| Convert to litres. | |
| Simplify. | 0.45 L |
| Dena needs 0.45 litres of olive oil. |
TRY IT 11.1
A recipe for Alfredo sauce calls for 250 millilitre s of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many litres of milk will she need?
Show answer
2 litres
TRY IT 11.2
To make one pan of baklava, Dorothea needs 400 grams of filo pastry. If Dorothea plans to make 6 pans of baklava, how many kilograms of filo pastry will she need?
Show answer
2.4 kilograms
Convert Between the Imperial and the Metric Systems of Measurement
Many measurements in Canada are made in metric units. Our soda may come in 2-litre bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.
The table below shows some of the most common conversions.
| Length | Mass | Capacity |
|---|---|---|
(Figure.2) shows how inches and centimetres are related on a ruler.
(Figure.3) shows the ounce and millilitre markings on a measuring cup.
(Figure.4) shows how pounds and kilograms marked on a bathroom scale.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.
EXAMPLE 12
Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.
| 500 mL | |
| Multiply by a unit conversion factor relating mL and ounces. | |
| Simplify. | |
| Divide. | 16.7 ounces. |
| The water bottle has 16.7 ounces. |
TRY IT 12.1
How many quarts of soda are in a 2-L bottle?
Show answer
2.12 quarts
TRY IT 12.2
How many litres are in 4 quarts of milk?
Show answer
3.8 litres
EXAMPLE 13
Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometres. How many miles until the next rest stop?
| 100 kilometres | |
| Multiply by a unit conversion factor relating km and mi. | |
| Simplify. | |
| Divide. | 62 ounces. |
| Soleil will travel 62 miles. |
TRY IT 13.1
The height of Mount Kilimanjaro is 5,895 metres. Convert the height to feet.
Show answer
19,335.6 feet
TRY IT 13.2
The flight distance from Toronto to Vancouver is 3,364 kilometres. Convert the distance to miles.
Show answer
2,090 miles
Convert between Fahrenheit and Celsius Temperatures
Have you ever been in a foreign country and heard the weather forecast? If the forecast is for °F what does that mean?
The Canadian and imperial systems use different scales to measure temperature. The Canadian system uses degrees Celsius, written °C. The imperial system uses degrees Fahrenheit, written°F. (Figure.5) shows the relationship between the two systems.
Temperature Conversion
To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula
.
To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula
.
EXAMPLE 14
Convert 50° Fahrenheit into degrees Celsius.
Solution
We will substitute 50°F into the formula to find C.
| Simplify in parentheses. | |
| Multiply. | |
| So we found that 50°F is equivalent to 10°C. |
TRY IT 14.1
Convert the Fahrenheit temperature to degrees Celsius: 59° Fahrenheit.
Show answer
15°C
TRY IT 14.2
Convert the Fahrenheit temperature to degrees Celsius: 41° Fahrenheit.
Show answer
5°C
EXAMPLE 15
While visiting Paris, Woody saw the temperature was 20° Celsius. Convert the temperature into degrees Fahrenheit.
Solution
We will substitute 20°C into the formula to find F.
| Multiply. | |
| Add. | |
| So we found that 20°C is equivalent to 68°F. |
TRY IT 15.1
Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was 15° Celsius.
Show answer
59°F
TRY IT 15.2
Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was 10° Celsius.
Show answer
50° F
Key Concepts
- Metric System of Measurement
- Length
- Mass
- Capacity
- Length
- Temperature Conversion
- To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula
- To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula
Practice Makes Perfect
Make Unit Conversions in the Imperial System
In the following exercises, convert the units.
| 1. A park bench is 6 feet long. Convert the length to inches. | 2. A floor tile is 2 feet wide. Convert the width to inches. |
| 3. A ribbon is 18 inches long. Convert the length to feet. | 4. Carson is 45 inches tall. Convert his height to feet. |
| 5. A football field is 160 feet wide. Convert the width to yards. | 6. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet. |
| 7. Ulises lives 1.5 miles from school. Convert the distance to feet. | 8. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles. |
| 9. A killer whale weighs 4.6 tons. Convert the weight to pounds. | 10. Blue whales can weigh as much as 150 tons. Convert the weight to pounds. |
| 11. An empty bus weighs 35,000 pounds. Convert the weight to tons. | 12. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons. |
| 13. Rocco waited hours for his appointment. Convert the time to seconds. | 14. Misty’s surgery lasted hours. Convert the time to seconds. |
| 15. How many teaspoons are in a pint? | 16. How many tablespoons are in a gallon? |
| 17. JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces. | 18. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces. |
| 19. Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons. | 20. Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons. |
| 21. Jon is 6 feet 4 inches tall. Convert his height to inches. | 22. Faye is 4 feet 10 inches tall. Convert her height to inches. |
| 23. The voyage of the Mayflower took 2 months and 5 days. Convert the time to days. | 24. Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours. |
| 25. Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces. | 26. Baby Audrey weighted 6 pounds 15 ounces at birth. Convert her weight to ounces. |
Use Mixed Units of Measurement in the Imperial System
In the following exercises, solve.
| 27. Eli caught three fish. The weights of the fish were 2 pounds 4 ounces, 1 pound 11 ounces, and 4 pounds 14 ounces. What was the total weight of the three fish? | 28. Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. How many pounds of nuts did Judy buy? |
| 29. One day Anya kept track of the number of minutes she spent driving. She recorded 45, 10, 8, 65, 20, and 35. How many hours did Anya spend driving? | 30. Last year Eric went on 6 business trips. The number of days of each was 5, 2, 8, 12, 6, and 3. How many weeks did Eric spend on business trips last year? |
| 31. Renee attached a 6 feet 6 inch extension cord to her computer’s 3 feet 8 inch power cord. What was the total length of the cords? | 32. Fawzi’s SUV is 6 feet 4 inches tall. If he puts a 2 feet 10 inch box on top of his SUV, what is the total height of the SUV and the box? |
| 33. Leilani wants to make 8 placemats. For each placemat she needs 18 inches of fabric. How many yards of fabric will she need for the 8 placemats? | 34. Mireille needs to cut 24 inches of ribbon for each of the 12 girls in her dance class. How many yards of ribbon will she need altogether? |
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
| 35. Ghalib ran 5 kilometres. Convert the length to metres. | 36. Kitaka hiked 8 kilometres. Convert the length to metres. |
| 37. Estrella is 1.55 metres tall. Convert her height to centimetres. | 38. The width of the wading pool is 2.45 metres. Convert the width to centimetres. |
| 39. Mount Whitney is 3,072 metres tall. Convert the height to kilometres. | 40. The depth of the Mariana Trench is 10,911 metres. Convert the depth to kilometres. |
| 41. June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams. | 42. A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams. |
| 43. One stick of butter contains 91.6 grams of fat. Convert this to milligrams. | 44. One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams. |
| 45. The maximum mass of an airmail letter is 2 kilograms. Convert this to grams. | 46. Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams. |
| 47. A bottle of wine contained 750 millilitre s. Convert this to litres. | 48. A bottle of medicine contained 300 millilitre s. Convert this to litres. |
Use Mixed Units of Measurement in the Metric System
In the following exercises, solve.
| 49. Matthias is 1.8 metres tall. His son is 89 centimetres tall. How much taller is Matthias than his son? | 50. Stavros is 1.6 metres tall. His sister is 95 centimetres tall. How much taller is Stavros than his sister? |
| 51. A typical dove weighs 345 grams. A typical duck weighs 1.2 kilograms. What is the difference, in grams, of the weights of a duck and a dove? | 52. Concetta had a 2-kilogram bag of flour. She used 180 grams of flour to make biscotti. How many kilograms of flour are left in the bag? |
| 53. Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms? | 54. One glass of orange juice provides 560 milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in 30 days? |
| 55. Jonas drinks 200 millilitre s of water 8 times a day. How many litres of water does Jonas drink in a day? | 56. One serving of whole grain sandwich bread provides 6 grams of protein. How many milligrams of protein are provided by 7 servings of whole grain sandwich bread? |
Convert Between the Imperial and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
| 57. Bill is 75 inches tall. Convert his height to centimetres. | 58. Frankie is 42 inches tall. Convert his height to centimetres. |
| 59. Marcus passed a football 24 yards. Convert the pass length to metres | 60. Connie bought 9 yards of fabric to make drapes. Convert the fabric length to metres. |
| 61. According to research conducted by the CRC, Canadians regrettably produce more garbage per capita than any other country on earth, at 2,172.6 pounds per person annually. Convert the waste to kilograms. | 62. An average Canadian will throw away 163,000 pounds of trash over his or her lifetime. Convert this weight to kilograms. |
| 63. A 5K run is 5 kilometres long. Convert this length to miles. | 64. Kathryn is 1.6 metres tall. Convert her height to feet. |
| 65. Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds. | 66. Jackson’s backpack weighed 15 kilograms. Convert the weight to pounds. |
| 67. Ozzie put 14 gallons of gas in his truck. Convert the volume to litres. | 68. Bernard bought 8 gallons of paint. Convert the volume to litres. |
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
| 69. 86° Fahrenheit | 70. 77° Fahrenheit |
| 71. 104° Fahrenheit | 72. 14° Fahrenheit |
| 73. 72° Fahrenheit | 74. 4° Fahrenheit |
| 75. 0° Fahrenheit | 76. 120° Fahrenheit |
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.
| 77. 5° Celsius | 78. 25° Celsius |
| 79. -10° Celsius | 80. -15° Celsius |
| 81. 22° Celsius | 82. 8° Celsius |
| 83. 43° Celsius | 84. 16° Celsius |
Everyday Math
| 85. Nutrition Julian drinks one can of soda every day. Each can of soda contains 40 grams of sugar. How many kilograms of sugar does Julian get from soda in 1 year? | 86. Reflectors The reflectors in each lane-marking stripe on a highway are spaced 16 yards apart. How many reflectors are needed for a one mile long lane-marking stripe? |
Writing Exercises
|
87. Some people think that 65° to 75° Fahrenheit is the ideal temperature range. a) What is your ideal temperature range? Why do you think so? b) Convert your ideal temperatures from Fahrenheit to Celsius. |
88.
a) Did you grow up using the Canadian. or the Imperial system of measurement? b) Describe two examples in your life when you had to convert between the two systems of measurement. |
Answers
| 1. 72 inches | 3. 1.5 feet | 5. yards |
| 7. 7,920 feet | 9. 9,200 pounds | 11. tons |
| 13. 5,400 s | 15. teaspoons | 17. ounces |
| 19. gallons | 21. 76 in. | 23. 65 days |
| 25. 115 ounces | 27. 8 lbs. 13 oz. | 29. 3.05 hours |
| 31. 10 ft. 2 in. | 33. 4 yards | 35. 5,000 metres |
| 37. 155 centimetres | 39. 3.072 kilometres | 41. 1.5 grams |
| 43. 91,600 milligrams | 45. 2,000 grams | 47. 0.75 litres |
| 49. 91 centimetres | 49. 91 centimetres | 49. 91 centimetres |
| 53. 2.1 kilograms | 55. 1.6 litres | 57. 190.5 centimetres |
| 59. 21.9 metres | 61. 985.5 kilograms | 63. 3.1 miles |
| 65. 44 pounds | 67. 53.2 litres | 69. 30°C |
| 71. 40°C | 73. 22.2°C | 75. -17.8°C |
| 77. 41°F | 79. 14°F | 81. 71.6°F |
| 83. 109.4°F | 85. 14.6 kilograms | 87. Answers may vary. |
Attributions
This chapter has been adapted from “Systems of Measurement” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.


2.7: Effective Collaboration
Creating OER is no small feat, and usually involves concerted efforts of many individuals who have a shared vision and goal. Collaboration is at the heart of open work and has allowed the Open Education community to flourish in numerous ways. In the OER projects that Rebus has supported, engagement and collaboration have been key factors in determining the project’s advancement or success. Most projects saw some initial interest, buzz, and activity during the first few weeks. However, we noticed that this can die down fairly quickly, and in some cases, where a project has stretched out for too long, we’ve lost a few contributors along the way. It falls on the project leads to make a conscious effort to keep the team engaged.
Our most successful projects have had an active community of participants – authors, editors, proofreaders, copy editors, reviewers, formatters, leads, other volunteers – who conversed with each other often with updates about their tasks or the project more generally. These projects found a way to ensure participants knew the work they were doing was valuable and part of a larger effort. They also enabled contributors to connect with a larger community of practice in their subject. We found that public-facing projects with a lot of activity acted as social proof for current and potential collaborators: encouraging new people to get involved with the project, and fuelling those already on board to complete the tasks that they had signed up to do.
While engagement helped drive projects forward, for some participants, it was the connections they made with others, and the conversations/discussions they had that they valued more than the final product. Contact between authors, editors, advisors, etc. create a sense of community, even mentorship, among participants. Ultimately, teams are made up of people, and if you focus on the humanity at the core of our teams, this will lead to deeper engagement across your group and more fulfillment in your work. The project managers for the Media Innovation and Entrepreneurship open textbook formed a community of practice and held fortnightly calls with those adopting or teaching with the book. One of their authors notes:
“Many thanks…for the gift to join you in such a fascinating intellectual and collaborative journey for making the Open Textbook on Media Innovation & Entrepreneurship a dynamic learning, writing, and mentoring experiment. Along with the open textbook and the upcoming instructors’ manual, I think you are also helping to create a very dynamic network and community of practice.”
—Betty Tsakarestou, author and assistant professor at Panteion University
Considerations for effective collaboration
To help you on your OER journey, we’ve put together suggestions for effective collaboration so that the team around your OER project remains engaged and committed to seeing the resource all the way through to publication. Apply these tips to ensure the experience is an enjoyable and memorable one for everyone! Remember that there is no clear cut formula or strategy that works for everyone — so take some time as you read to think, experiment, and rely on your team and collective to share some ideas to find what works best for you and your team!
Introduce your team to one another
This sounds very straightforward, but make sure to introduce members of the team to one another, regardless of whether their roles intersect or have too much in common. It’s not uncommon to be juggling several lines of communication between yourself as the team lead and many participants, but creating connections across the project is just as important for keeping people engaged. If you’re located in the same city, you could try to arrange an in-person meet. However, most projects tend to involve participants from all around the world – so a simple email or video call to introduce the team to one another will go a long way!
If you’re not the project lead, but an editor of a particular part or section of the book, you could send a similar email to the authors in your part. The email should welcome individuals to the team, and could even include links to shared resources such as the project’s timeline.
Doing so can remind team members that they are part of something bigger!
Communicate and make plans with your team
If you’re the project lead, it’s especially important to keep in mind that members of your team may not have the same working or communication style. These kinds of things vary across people and cultures, and if you have a global team, you can’t expect everyone to switch to your default. Instead, you should discuss preferences and expectations at the start of the project, and invite members to help plan the project timeline and tasks with you. This may help to avoid conflict or disengagement down the line, and help you better gauge when an issue might be arising.
You should also discuss communication preferences with team members early on – is email okay, or are calls better? What kind of response time is expected? Respect these preferences, and as project lead, ask your team to do the same. Sharing this information up front, to whatever extent possible, will help your team understand one another, leading towards smoother interactions and fewer frustrations.
Share models and examples
Make sure everyone has model chapters/outlines/abstracts or whatever else they’re being asked to submit. This can ease the cognitive load of getting started, and help ensure consistency in what’s submitted. Where possible, it’s also good to select this model content from what has already been submitted or created by a project lead and share it with the others – this also adds the little nudge that others are further ahead, which can help get people moving!
In addition to modelling expectations through documents, we also encourage you to do so in your behaviours and engagement with the team. Participate in team conversations fully, communicate clearly, and treat your fellow team members with respect (this is especially critical if you are a project manager or other lead on the team). You set the stage for your fellow teammates, so model what you’d like to see on your team!
Encourage sharing drafts/posting questions in a shared and central communication space
Wherever possible, we ask contributors to share their drafts or ask any questions that they have on a public-facing channel, such as the Rebus Community forum or in Google Docs shared with other authors. If you find the same questions or concerns are coming up from multiple people, consider putting together an FAQ to share with other contributors. You can give contributors editing privileges, and encourage them to add to the list, so the burden of updating the list doesn’t fall solely on you. Also consider whether any of these questions could be addressed in your author/editor/contributor guide or project summary and add to them as needed. This helps other members of the team keep track of various tasks, reduces the work of having to answer the same questions many times over, gives people examples to work from, and acts as a subtle reminder to those who may be behind on their deadlines.
Keep in regular communication with your team
Communication is the key to a successful project – remember that teams are made up of people!
You may want to hold a call with your contributors, or an office hours-style drop in session. Encouraging attendance at these can also help develop the community feeling, which helps keep people invested and responsible to each other to fulfill their commitments. These calls may be structured around each chapter, connecting authors and instructors giving feedback for example, or each phase of the project (“Two weeks until submission deadline! Help!”) or be an open space for feedback and questions as needed.
If general reminder emails aren’t getting much of a response, try individual emails or one-on-one calls with the team leads. If a particular contributor has dropped off the radar, or you’re not sure they’re seeing the emails you’re sending to the whole team, check in with them personally to see how they’re getting on. A short note can do the trick, and it may just be that they’re having a busy couple of weeks, but the response rate is generally higher with this approach!
Dissuade the drop-outs: instead, encourage participation in less time-consuming tasks…
If at any point it looks like someone might be backing out, suggest that they could instead take on a different role, perhaps acting as a reviewer instead of an author, or help to promote the book down the line. And always ask if they want to stay up to date as things progress! You can contact them again when you’re looking for other kinds of contributions or releasing the book. Sharing updates or notifications when the book is available is a good way to acknowledge their interest in the project, even if they haven’t been able to commit to it entirely. Continue reading to learn about navigating and addressing any challenges that may crop up in the production process.
…but remember that not everyone will stay involved, and that’s okay!
Contributors always have the option to say no and withdraw if they’re not able to participate in a project. If they’re interested enough to have signed up in the first place, it’s worth trying to keep them in the loop, but they might not always want that. It can be disappointing, but if someone does have to bow out completely, remember that we all have competing priorities, and you never know what may be happening on their end. Be respectful of that, and wish them well.
Navigating and addressing challenges
No project is perfect, and things are sometimes bound to go differently than anticipated. If this happens, don’t worry – it’s normal. Things aren’t always super smooth on every project. Small things are bound to crop up. As team members, you need to be responsive and able to adapt to situations. Remember, good teams require nurturing, attention, and problem-solving!
When these tough situations do arise, here are some strategies you might look to:
- As project manager, you are responsible for the team overall, and should stand-up against anyone disobeying the team’s ground rules. If a problem emerges with a volunteer, try to set up some time to meet them and talk through the issue face-to-face, instead of over text or email. Your first goal should be to attain clarity over the situation, and not to make a decision or pass judgment. If a volunteer is acting in a crass or rude manner to you or anyone else on the project, it’s within your rights to ask them to leave the project. You can do so politely, but stand your ground and protect your fellow collaborators. Again, look for backup from others on your team if you need it.
- Depending on the nature of the issue, it might help to go back to your preparatory documents, such as the project summary, outline, MOU, etc. Consider these documents as evidence of your project work, and sift through them for clarity and reassurance. They also provide confirmation if there is disagreement on something that was set out early on.
- If an issue related to content crops up, for instance if an author wants to back out of the project during the editing/review phase, or if an author doesn’t want to make important changes requested by a reviewer or editor, look back to your MOU and confirm the license on content submitted. More often than not, if the content was licensed CC BY, it can be salvaged in some way to be useful for the project if you decide that’s best. However, depending on the situation, it might not be good form to include contested content even if you have the legal right to do so. Use your judgement, and defer to the content creator.
- It’s good to acknowledge that in some situations, you might be the person who messed up. In this case, be up front about where things went wrong, and take responsibility for your actions. Then, do what you can to look for solutions and to fix the error. Don’t be afraid to ask for help, from your team and from the community at large!
- Lastly, look to the community not only for help in situations, but also for advice and counsel. Talk to other project managers, find out their experiences, learn from their successes and failures, and share your own stories. Remember that even if you’re the only manager on your project, you’re not alone in the community!
| Challenge | Suggestion |
| Not showing up to meetings | Check in with them – is something technical or otherwise preventing them from attending? |
| Boundary setting (hesitating to pick up tasks) |
|
| Trouble prioritizing work | Rely on your documents & spreadsheets |
| Deliverables and timing |
|
| Confusion with tools |
|
| Unclear expectations of the role or assigned work |
|
Modeling being a good team member
Being intentional and genuine in your role as project manager will set an example for your team and establish a relationship of trust with them. It’s also good for you to model the engagement you want to see – in meetings and other spaces. If you are helping create a resource that will be accessible, inclusive, and culturally relevant, it’s important that these values show up in your practice with your team – so that they can embody this into what they are creating! A few simple suggestions for ways to start this:
- Learn how to pronounce team members’ names
- Start with a territorial acknowledgment
- Turn on captioning services during meetings
- Take into account communication preferences
- Accommodate team accessibility needs
- Assume the best in people, as OER projects are in production for a long period and you never know the full context of a team member’s life or challenges outside the project
We also encourage you to reflect on other ways in which you can demonstrate your commitment to equity with your team. These are a start, and we hope you can see the value in structuring and leading an equitable publishing process rooted in community and care!
Self-care
No matter your role in the project, be sure to take care of yourself! OER projects should never come at the cost of you or anyone else on your team. We wanted to end this section with a short reminder to make sure you’re managing your own time and well-being carefully too. Saying this explicitly at the get-go of your project may help you practice more self-care throughout! Be mindful of Finally, your language and communication also counts as you support other team members to take care of themselves or grant them the grace they need if they have to pause or step away from the project for a period of time.
Managing a team is hard work, and you might get used to putting yourself in the backseat and keeping the project or volunteers at the forefront. While the project and other volunteers are important, you are too!
Be nice to yourself along the way, and remember that you are also volunteering your energy on the project. Whatever role you may occupy within your team, be it paid or volunteer, respect your own physical and mental boundaries. Take breaks when you need them and understand that communication is essential – not to share delicate details, but to let others know when challenges interfere with your project work.


M2 | Spinal Mechanisms for Sensorimotor Integration
Introduction to Neural Control of Movement
Neuroscientist Daniel Wolpert starts from a surprising premise: the brain evolved, not to think or feel, but to control movement. In this entertaining, data-rich talk he gives us a glimpse into how the brain creates the grace and agility of human motion.
Transcript: Daniel Wolpert at TEDGlobal 2011
References
Wolpert, D. M. and J. R. Flanagan (2001). “Motor prediction.” Current Biology 11(18): R729-R732. PubMed | PDF
What are some of the complexities involved in the control of movement?
Sensorimotor integration
Motor signals are generated by the central nervous system (CNS). The generation of motor commands requires up-to-date sensory information from peripheral receptors. Consider for example a limb movement to reach to grasp an object. Visual receptors in the retina provide information about where the object is located; muscle and joint receptors provide information about the limb’s initial position. Knowledge of both target location and limb position is necessary to accurately move the limb to the target. Thus the seemingly simple task of reaching to grasp an object is really a complex process that involves integration of sensory and motor signals. Understanding how reach-to-grasp movements are controlled by the central nervous system will ultimately require understanding of the multiple stages of information processing that occur between the retinal image of a visual target in a given spatial location and the motor output signals that move the hand to the target. This process is generally assumed to involve a sequence of sensorimotor transformations that produce signals for the appropriate patterns of muscle activation.
Degrees of freedom problem
The degrees of freedom problem refers to the problem of how a complex system constrains the system’s many degrees of freedom to produce a specific result. Lets consider movement of the thumb as an example of the difficulties in describing movements fully and precisely. The thumb can move in 3 planes: flexion/extension, adduction/abduction, and movements to oppose the remaining fingers. There are 3 additional planes of movement at the wrist, 1 at the elbow, 3 at the shoulder. So the tip of the thumb can move in 27 different planes of movement. Adding to this complexity is the fact that movements in each plane are affected by at least 2 muscles, one for each direction of movement. It is clear that the task of simultaneously controlling all of these variables is formidable.
Voluntary movements require postural adjustments
Voluntary limb movements (for example, extending an arm to catch a ball) are accompanied by activation of musculature of the legs and trunk to compensate for the disturbance in posture that results from the shift in weight produced by the arm movement.
How does the motor system deal with these complexities?
Different parts of the motor system are specialized for different stages of movement production. For example, the structures involved in the idea to move are different from those involved in generating the movement plan, and these in turn are different from the ones that actually execute the movement.
What neural structures are involved in the control of movement?
The motor system is organized as a hierarchy with three main levels of control.
Motor control hierarchy
Highest level: association areas of cerebral cortex, basal ganglia. Function: strategy, goal of the movement and strategy that best achieves the goal.
Intermediate level: cerebral cortical motor areas and cerebellum. Function: tactics, the sequences of muscle contractions, in space and time, required to smoothly and accurately achieve the strategic goal.
Lowest level: brain stem and spinal cord. Function: execution, activation of the motor neuron and interneuron pools that generate the goal-directed movement and make any necessary adjustments in posture.
Levels of control are connected in series as well as in parallel. For example, cerebral cortical motor areas project directly to the spinal cord via the corticospinal tracts, and indirectly via descending motor pathways that originate from brainstem nuclei.
Key Takeaways
Three hierarchical levels of control:
- Cerebral cortex: voluntary movements, least automatic movements.
- Brain stem: posture, rhythmical movements.
- Spinal cord: reflexes, most automatic movements.
Two subcortical systems provide connections between hierarchical levels of control:
- Basal ganglia receive input from all regions of cerebral cortex and output targets motor, premotor and prefrontal cortex. The basal ganglia are involved in movement initiation and in cognitive aspects of movements.
- Cerebellum receives input from all hierarchical levels and its output targets all structures involved in the production of movement. The cerebellum is involved in movement coordination and motor learning.


67 Worked Example: Supply and Demand
Supply and Demand
The example we just considered showed a shift to the left in the demand curve, as a change in consumer preferences reduced demand for newspapers. Often changes in an economy affect both the supply and the demand curves, making it more difficult to assess the impact on the equilibrium price. Let’s review one such example.
First, consider the following questions:
- Suppose postal workers are successful in obtaining a pay raise from the U.S. Postal Service. Will this affect the supply or the demand for first-class mail? Why? Which determinant of demand or supply is being affected? Show graphically with before and after curves on the same axes. How will this change the equilibrium price and quantity of first-class mail?
- How do you imagine the invention of email and text messaging affected the market for first-class mail? Why? Which determinant of demand or supply is being affected? Show graphically with before and after curves on the same axes. How will this change the equilibrium price and quantity of first-class mail?
- Suppose that postal workers get a pay raise and email and text messaging become common. What will the combined impact be on the equilibrium price and quantity of first-class mail?
In order to complete a complex analysis like this it’s helpful to tackle the parts separately and then combine them, while thinking about possible interactions between the two parts that might affect the overall outcome.
Part 1: A Pay Raise for Postal Workers
A pay raise for postal workers would represent an increase in the cost of production for the Postal Service. Production costs are a factor that influences supply; thus, the pay raise should decrease the supply of first-class mail, shifting the supply curve vertically by the amount of the pay raise. Intuitively, all else held constant, the Postal Service would like to charge a higher price that incorporates the higher cost of production. That is not to say the higher price will stick. From the graph (Figure 1), it should be clear that at that higher price, the quantity supplied is greater than the quantity demanded—thus there would be a surplus, indicating that the price the Postal Service desires is not an equilibrium price. Or to put it differently, at the original price (P1), the decrease in supply causes a shortage driving up the price to a new equilibrium level (P2). Note that the price doesn’t rise by the full amount of the pay increase. In short, a leftward shift in the supply curve causes a movement up the demand curve, resulting in a lower equilibrium quantity (Q2) and a higher equilibrium price (P2).
Part 2: The Effect of Email and Text Messaging
Since many people find email and texting more convenient than sending a letter, we can assume that tastes and preferences for first-class mail will decline. This decrease in demand is shown by a leftward shift in the demand curve and a movement along the supply curve, which creates a surplus in first-class mail at the original price (shown as P2). The shortage causes a decrease in the equilibrium price (to P3) and a decrease in the equilibrium quantity (to Q3). Intuitively, less demand for first-class mail leads to a lower equilibrium quantity and (ceteris paribus) a lower equilibrium price.
Part 3: Combining Factors
Parts 1 and 2 are straightforward, but when we put them together it becomes more complex. Think about it this way: In Part 1, the equilibrium quantity fell due to decreased supply. In Part 2, the equilibrium quantity also fell, this time due to the decreased demand. So putting the two parts together, we would expect to see the final equilibrium quantity (Q3) to be smaller than the original equilibrium quantity (Q1). So far, so good.
Now consider what happens to the price. In Part 1, the equilibrium price increased due to the reduction in supply. But in Part 2, the equilibrium price decreased due to the decrease in demand! What will happen to the equilibrium price? The net effect on price can’t be determined without knowing which curve shifts more, demand or supply. The equilibrium price could increase, decrease, or stay the same. You just can’t tell from graphical analysis alone.


Transcription with Notation Shorthand
The system of notation shorthand, developed by pedagogue Jenine Lawson Brown, can be useful in tracking the relationships between rhythm and meter in a way that is helpful in writing rhythmic notation. Even if an instructor doesn’t require you to use this system, you are encouraged to do so if you struggle with notating rhythm.
In notehead shorthand, every beat is represented by stems and beams as appropriate, showing how it divides. For example, each beat of a simple meter with 4 on the bottom (4/4, 3/4, 2/4, etc.) would be represented by a pair of beamed eighth note stems with no noteheads; each beat of a compound meter with 8 on the bottom (6/8, 9/8, 12/8, etc.) would be represented by a trio of beamed eighth note steps with no noteheads. Measures are shown with barlines, just as in traditional notation.
The first step of transcribing music using notation shorthand is to determine the meter, count the number of measures, and set up your barlines and placeholder stems/beams.  Give the stems noteheads where a new note occurs.
Example: Notehead shorthand in simple meter
Completed notehead shorthand:
Example: Protonotation in compound meter
*the melody in this example starts at 0:19
Completed notehead shorthand:
To transform notation shorthand into traditional rhythmic notation:
- Find the first notehead you’ve written.
- Count how many beat divisions that note lasts by counting stems till the next notehead.
- Determine an appropriate rhythmic value by adding up these beat divisions.
As recommended in the previous section, make sure to clarify the meter in your notation by using beaming where appropriate. The general rule of thumb is that beams should join together notes smaller than a quarter note when they lie within the same beat; examples can be found here.


X. Orchestration
Core Principles of Orchestration
Mark Gotham
Key Takeaways
- “simultaneously” (or “vertically” if you prefer) for voicing chords, doubling melody lines, and handling texture;
- “successively” (or “horizontally”) for effects like orchestral crescendos.
Simultaneous (Vertical) Combinations
To b(lend) or not to b(lend) … that is the question … for “simultaneous” orchestration. For most tonal orchestrators, the default answer is a very clear “yes, blend!”
Voicing chords
As a rule of thumb for tutti orchestration, blend by treating each section as if it were self-contained. This applies to the large sections (winds, brass, strings), and often also to the smaller sub-sections (e.g., flutes) in the case of larger orchestras where there are greater numbers of each. For each section, follow the principles you learned in four-part writing. That’s right, the principles you’ve already learned about chordal writing still apply. Specifically, you’ll need to consider:
- The inclusion of all (or at least the most important) harmonic pitches.
- The number and prominence of each pitch, including avoiding the overdoubling of the third in a triad.
- The gaps between voices:
- We now see the densest packing in the middle.
- Continue to use larger gaps in the bass register (e.g., the octaves given by cello and double bass notated on the same pitch but sounding an octave apart).
- Continue to use smaller gaps higher up in the treble clef range.
- We tend to see more activity in the very highest registers in orchestral scores than in most other contexts; it’s usually best to return to wider spacing in those highest registers.
We have terminology for the different types of chord spacing as summarized in
:
Context determines the best choice among these. For a blended sound, we often use integrated “interlocking” voicing, taking care to balance the different strengths of each instrument according to the register; conversely, to achieve a different color for each note, use wider spacing. [1]
Doubling lines
The approach to blending in doubling lines again involves finding a compatible balance of (equally strong) instruments and registers. The second part (m.1 4ff.) of Mozart 40/iii is effectively a two-part invention and an intriguing example of blended, balanced lines. Both of the two parts are played in each of the three octaves, and by both winds and strings.
As always, this default is not a law like that of gravity in physics; composers will diverge from this blended practice for creative and compositional effect in just the same way that they do with any other parameter. Perhaps the most famous example of an odd doubling is provided by Schubert 8/i. The first theme (m. 13) sees the extraordinary doubling of a single oboe and clarinet in unison. While some commentators have seen this as an error, it’s probably deliberate on Schubert’s part as a way of creating an unsettled feel that entirely befits the mood of the music. It also contrasts with a second theme (reh. A, m. 42): a cello theme accompanied by the optimally blended viola and clarinet in thirds.
Texture
The choice of instruments, register, and more can help support and define the wider texture. For instance:
- Polyphonic writing: Tends to see instruments and groups balanced and blended. Examples include fugues and the Mozart invention just discussed.
- Melody and accompaniment: Foregrounds the melodic instrument by placing it in its best range, often at the top of the texture (as in other contexts), and with other accompanimental parts contrasting in:
- Timbre, accompanying a solo woodwind line by full strings, for instance.
- Register (as discussed).
- Texture, with less or a different kind of rhythmic activity. This is similar to the kind of complementary rhythmic patterns you often find in polyphonic music: in both cases, we’re trying to have more than one element at once, but keep them noticeably distinct.
Successive (Horizontal) Combinations
We’ll look here at forms of succession on two timescales: at the level of individual notes (small scale) and of whole sections (large scale).
Firstly, once again, everything you’ve learned about voice-leading still applies. (Yes, this will be a recurring mantra.) Most of the time, it will be eminently possible to reduce apparently complex orchestral scores down to a two-, three-, or four-part texture with clear and consistent doublings and with “correct” voice-leading.
In handling successive sections, antiphony is a highly favored method for distributing material, creating timbral variety, and articulating the form. This can take the form of whole blocks of material being passed from strings to winds (common in early orchestral scores of the Baroque era), or of more subtle effects. For instance, look at the first page of “In the Hall of the Mountain King” from Grieg’s Peer Gynt suite:
At the repeat of the tune, the only variation is antiphony (strings-to-winds). The material (both tune and accompaniment) and register are identical. Later, this strings-to-winds principle continues to the upper voices as the register expands upward. Let’s take a closer look at that crescendo.
Crescendos
We have a range of tools at our disposal for achieving an effective orchestral crescendo. Apart from mere dynamic markings, we can:
- Add instruments, often moving from the “lightest” to “heaviest” instruments: broadly, strings-wind-brass-percussion (e.g., Grieg rehearsal mark A).
- Expand the overall range, often upward from the bass or outward from the center (e.g., Grieg rehearsal marks A and B).
- Diversify the note values used, adding sustained, longer notes and/or tremolo, shorter ones, perhaps through note repetitions (e.g., Grieg rehearsal mark B).
Take a look at rehearsal mark B in the Grieg, and notice the:
- Forces and register: We now have the full orchestral and full register in use except for one instrument, the piccolo, which is reserved for the top C♯, four measures later. The piccolo is removed at rehearsal mark C and reintroduced at rehearsal mark D for a similar effect. Likewise, the final page briefly removes the bass for the piano passage before the final fff chord.
- Wider range of note values: The accompanimental figure (16th-16th-8th on the weak beats), first introduced by the viola part at rehearsal mark A, has now become focal, adding rhythmic density to the score. This is further enhanced by the upper strings playing the theme with repeated (sixteenth) notes and the timpani roll. Conversely, the horns introduce longer (half) notes, lending a more sustained sound to the previously staccato accompaniment.
As an aside, it’s worth noting that moving from lightest to heaviest instruments (strings-winds-brass-percussion) is related to overall playing time: the strings generally play the most, the percussion the least. This is partly for pragmatic reasons: brass players would keel over from exhaustion if you had them play the string parts—there’s just too much playing time and not enough resting time in between. You’ll find that the best choices for pragmatic reasons and musical ones often go together in this way.
Holding a timbre back for effect
Percussion parts are the clearest indication of the practice of holding a timbre back for effect. One of the essential skills of the percussionist is being able to count hundreds of measures of rest and then come in with a bang at the crucial, climatic moment. Here’s an extract from the triangle part for Mahler 4/iii. At reh. 12, a crash cymbal marks the moment of arrival, and the triangle leads a glorious texture, full of all the rich effects described above (register, texture, sustain, and note density). The triangle is one of those instruments that can be pretty well guaranteed to cut through most any texture and make an impact. The glockenspiel and piccolo are also good in this role.
This holding back of a timbre is very often for loud and/or climactic moments, but not always. For instance, reserving an entirely new sound until near the end of a work can be extremely effective. Perhaps the most iconic example is to be found in Debussy’s Prélude à l’après-midi d’un faune, where the antique cymbal is not used until near the end. (This is a “new” sonority for the piece, and even for Western audiences of the time.) The crucial moment comes at reh. 10, (8’56” in the following recording), but clearly the effect depends on listening to the wider context.
Dovetailing parts
Finally, in an approach somewhat complementary to antiphony, orchestrators will often “dovetail” parts: share a continuous line out between different voices. This can be:
- related to antiphony (the melodic line moves from part to part)
- or the opposite: ensuring continuity/cohesion within a line on instruments for which that would be impractical
- tailored in subtle ways to demarcate a rhythm
Take a look at Smetana’s Vltava to this effect:
- At the opening, the flutes dovetail, overlapping by one note on the downbeat. This is easy to play, and it allows them to share the long line. It could be more seamless, though, by slurring onto the downbeat, rather than separating that note with a staccato. The resultant effect is of a fluid line, but with the beats clearly demarcated.
- When the clarinets enter, we have dovetailing both within and between those subsections: the flute continues as before, the clarinets follow suit, and this leads to further pairs of Fl.1 + Cl.2 and Cl.1, Fl.2.
- The strings take a different approach at their entrance at m. 36. The line continues to be shared out among the string parts, but the groupings break it up into clear 6+3+3 (2+1+1) rhythms. This follows a period of syncopated accents in the winds.
This gives some sense of how much variety can be achieved within the simple dovetail pattern. It can get even more detailed, as in the following passage from Rachmaninoff 3/ii (reh. 42). The example shows the dense dovetailing among the oboes, bassoons, and violin parts. (Click here for scores on IMSLP to see this accompaniment in context). This rhythm saturates the four options: every sixteenth sees the entry of one part. It is both:
- dovetailed within the wind and string sections without overlap: i.e., one or other part is always playing, but never both (oboes or bassoons; violins 1 or violins 2)
- and dovetailed between those sections with winds patterns starting on the 2nd and 4th sixteenths, and violins starting on the 1st and 3rd (after the first beat).
The texture makes great use of the resources at hand, sparkles with detail, and gives the players interesting parts to think about (an important and easily overlooked consideration).
Symphony No.3, Op.44 – Sergei Rachmaninoff by openmusictheory
- Dovetailing: transcribe the sixteenth-notes part (piano right hand) of Louise Reichardt’s Unruhiger Schlaf (12 Gesänge, no. 6) for two clarinets, dovetailing regularly every quarter or half note. A score is available online here and for download here.
- Look at the extract from Hensel’s Overture in C major below. How does the combination of bassoon and cello relate to topics discussed in this chapter?
Overture in C Major – Fanny Hensel by openmusictheory
- For a recent article on this topic (even more recent than this chapter!), see McAdams, Stephen, Meghan Goodchild, and Kit Soden. 2022. ‘A Taxonomy of Orchestral Grouping Effects Derived from Principles of Auditory Perception’. Music Theory Online 28 (3). https://mtosmt.org/issues/mto.22.28.3/mto.22.28.3.mcadams.html.
Media Attributions
- C3 Chord Spacing Types
- For more textbook examples, see Adler (2002, 253); Piston (1955, 396). ↵
A call-and-response texture in which musical material is passed from group to group.


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10 On My Path: Jalayiah Goins College and Career Pathways My path led me through hurt it has led me Wondering why the person I expected safety from didn’t protect and comfort me I studied and discovered a little bit of the reasons of absent fathers and what triggers the kind of reactions that fathers that don’t want to be in their child’s life have. But what I have learned and realize is that it’s hard for a father to be a father when they have their own issues it’s hard for a father to be a father when he is afraid to Be face to face with the wrongdoings that he chose to makeI worked as a employee at Ivers in Renton and I still do and did discover that working with and trying to help customers can be difficult and I realize that As long as you mostly do your job right and are very polite and get things done but things are wrong you will get credit for it Cashier Cleaner Cook A path of being able to take care of my daughter and to never to start working it’s hard to balance and times but I never gave up losing hope now my life is here might go to clear For the trade:Accomplish high school Take care of my daughter into the chart to love and also to make sure she is healthy throughout her life Taking care of myself Strengths of my spirit:Discipline Patience Respectfulness Empathy Compassion My path is mine alone it’s that far it is Mine to take – I will succeed at RTC


1.6 Review Questions
- What factors contribute to a growing interest in entrepreneurship?
-
- unemployment levels
- low job turnover rates
- increasing levels of creativity
- decreasing rates of discrimination
2. Angel investors contribute to the entrepreneurial world by ________.
-
- outpacing venture capitalists in access to startup funds
- directly investing in entrepreneurial startups
- holding seminars to teach people how to become an entrepreneur
- forming groups of other investors to supply greater quantities of investment funds to entrepreneurs.
3.An entrepreneur is someone who ________.
-
- creates a small business based on other existing businesses
- creates a home-based business that sells products available through other retailers
- creates a business around the idea of solving a problem
- is highly creative
4. Roxanne Quimby of Burt’s Bees is an example of an entrepreneur primarily for which of the following reasons?
-
- She saw available resources and recognized the resources could be used to create a business.
- She acted on her passions.
- She solved a significant societal problem.
- She had the time and energy to dedicate to starting business.
5. In the early stages of developing an entrepreneurial venture, the entrepreneur might seek funding from which group?
-
- venture capitalists
- the local bank
- friends and family
- shark tank
6. The acronym SMART for goal setting stands for ________.
-
- short-term, measurable, accurate, relevant, and timely
- specific, measurable, accurate, relevant, and timely
- specific, measurable, achievable, realistic, and time oriented
- specific, measurable, achievable, results, and time oriented
7. A vision statement does not include ________.
-
- a picture of the future described in words
- concepts and ideas that inspire and focus people’s actions toward a united outcome
- both creative and exploratory process in creating the vision
- the scope of the venture
8. Which one of the following is not a quantifiable result?
-
- 5 percent annual growth in sales
- increase market share by 2 percent
- decrease collection of accounts receivable by $5,000.00
- increase creativity throughout the venture
9. Which of the following is not a creativity building technique or tool?
-
- conducting a case study
- brainstorming activities
- mind map creation
- brain writing
10. Disruptive technologies include all but which of the following?
-
- new methods or approaches that make established methods obsolete
- self-booking of air travel
- restaurant food delivery services
- a change from ownership of resources to the ability to have access to resources
11. The entrepreneurial mindset includes ________.
-
- a willingness to develop ideas that are not always initially popular
- a willingness to wait for an idea to become popular before moving forward with that idea
- an ability to focus on past successes
- a belief in following tradition and established patterns


3 Critical thinking and information
Learning Objectives
In this chapter you will learn
- How to evaluate information and why this is important
Being able to evaluate information is an important part of critical thinking. Critical thinking is an essential skill to develop and apply during your third-level education and will come in useful as a skill for employment and in general for making your way in a world awash with fake news, deep fakes, and misinformation. Scholarly information is not immune – some of the world’s most long-standing hoaxes are rooted in bad science, for example in poorly conducted research that has had to be retracted from scholarly journals.
Evaluating Information
The following video explains the importance of ‘information literacy,’ which is critical thinking applied in particular to the world of information. It explains concepts such as ‘currency’ and ‘authority’ as part of a test that you can use to help think through the quality of information to evaluate it: the CRAAP test.
‘Scholarly’ information is information written by specialists of a particular field, usually someone holding a PhD in that field and who works at a third-level institution. In terms of the CRAAP test, scholarly information scores particularly high on ‘A’ for ‘Authority.’ This is the case not just because it is written by specialists, but because it usually goes through a process of what is called ‘peer-review.’ Peer-review refers to when where other specialists in the same field do a thorough check of a scholarly researchers work before it is then published as a journal article, book chapter, or occasionally other formats, as explained in the previous page.
The CRAAP Test
What is the CRAAP test? Developed by the Library at California State University, Chico, the CRAAP test is a handy checklist for critically evaluating web sources (or indeed ANY sources) for use in your assignments. The test provides a list of questions to ask yourself when deciding whether or not a source is reliable and credible enough to use in your assignment. CRAAP stands for Currency, Relevance, Authority, Accuracy, and Purpose. For more information, please see below.


Chapter 2 Electric Charge and Electric Field
2.11 Conductors and Electric Fields in Static Equilibrium
Summary
- List the three properties of a conductor in electrostatic equilibrium.
- Explain the effect of an electric field on free charges in a conductor.
- Explain why no electric field may exist inside a conductor.
- Describe the electric field surrounding Earth.
- Explain what happens to an electric field applied to an irregular conductor.
- Describe how a lightning rod works.
- Explain how a metal car may protect passengers inside from the dangerous electric fields caused by a downed line touching the car.
- Explain how a Faraday Cage can protect your digital privacy
Conductors contain free charges that move easily. When excess charge is placed on a conductor or the conductor is put into a static electric field, charges in the conductor quickly respond to reach a steady state called electrostatic equilibrium.
Figure 1 shows the effect of an electric field on free charges in a conductor. The free charges move until the field is perpendicular to the conductor’s surface. There can be no component of the field parallel to the surface in electrostatic equilibrium, since, if there were, it would produce further movement of charge. A positive free charge is shown, but free charges can be either positive or negative and are, in fact, negative in metals. The motion of a positive charge is equivalent to the motion of a negative charge in the opposite direction.
A conductor placed in an electric field will be polarized. Figure 2 shows the result of placing a neutral conductor in an originally uniform electric field. The field becomes stronger near the conductor but entirely disappears inside it.
Misconception Alert: Electric Field inside a Conductor
Excess charges placed on a spherical conductor repel and move until they are evenly distributed, as shown in Figure 3. Excess charge is forced to the surface until the field inside the conductor is zero. Outside the conductor, the field is exactly the same as if the conductor were replaced by a point charge at its centre equal to the excess charge.
Properties of a Conductor in Electrostatic Equilibrium
- The electric field is zero inside a conductor.
- Just outside a conductor, the electric field lines are perpendicular to its surface, ending or beginning on charges on the surface.
- Any excess charge resides entirely on the surface or surfaces of a conductor.
The properties of a conductor are consistent with the situations already discussed and can be used to analyze any conductor in electrostatic equilibrium. This can lead to some interesting new insights, such as described below.
How can a very uniform electric field be created? Consider a system of two metal plates with opposite charges on them, as shown in Figure 4. The properties of conductors in electrostatic equilibrium indicate that the electric field between the plates will be uniform in strength and direction. Except near the edges, the excess charges distribute themselves uniformly, producing field lines that are uniformly spaced (hence uniform in strength) and perpendicular to the surfaces (hence uniform in direction, since the plates are flat). The edge effects are less important when the plates are close together.
Earth’s Electric Field
A near uniform electric field of approximately 150 N/C, directed downward, surrounds Earth, with the magnitude increasing slightly as we get closer to the surface. What causes the electric field? At around 100 km above the surface of Earth we have a layer of charged particles, called the ionosphere. The ionosphere is responsible for a range of phenomena including the electric field surrounding Earth. In fair weather the ionosphere is positive and the Earth largely negative, maintaining the electric field (Figure 5(a)).
In storm conditions clouds form and localized electric fields can be larger and reversed in direction (Figure 5(b)). The exact charge distributions depend on the local conditions, and variations of Figure 5(b) are possible.
If the electric field is sufficiently large, the insulating properties of the surrounding material break down and it becomes conducting. For air this occurs at around 3 x 106 N/C Air ionizes ions and electrons recombine, and we get discharge in the form of lightning sparks and corona discharge.
Electric Fields on Uneven Surfaces
So far we have considered excess charges on a smooth, symmetrical conductor surface. What happens if a conductor has sharp corners or is pointed? Excess charges on a nonuniform conductor become concentrated at the sharpest points. Additionally, excess charge may move on or off the conductor at the sharpest points.
To see how and why this happens, consider the charged conductor in Figure 6. The electrostatic repulsion of like charges is most effective in moving them apart on the flattest surface, and so they become least concentrated there. This is because the forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surfaces are different. The component parallel to the surface is greatest on the flattest surface and, hence, more effective in moving the charge.
The same effect is produced on a conductor by an externally applied electric field, as seen in Figure 6 (c). Since the field lines must be perpendicular to the surface, more of them are concentrated on the most curved parts.
Applications of Conductors
On a very sharply curved surface, such as shown in Figure 7, the charges are so concentrated at the point that the resulting electric field can be great enough to remove them from the surface. This can be useful.
Lightning rods work best when they are most pointed. The large charges created in storm clouds induce an opposite charge on a building that can result in a lightning bolt hitting the building. The induced charge is bled away continually by a lightning rod, preventing the more dramatic lightning strike.
Of course, we sometimes wish to prevent the transfer of charge rather than to facilitate it. In that case, the conductor should be very smooth and have as large a radius of curvature as possible. (See Figure 8.) Smooth surfaces are used on high-voltage transmission lines, for example, to avoid leakage of charge into the air.
Another device that makes use of some of these principles is a Faraday cage. This is a metal shield that encloses a volume. All electrical charges will reside on the outside surface of this shield, and there will be no electrical field inside. A Faraday cage is used to prohibit stray electrical fields in the environment from interfering with sensitive measurements, such as the electrical signals inside a nerve cell.
A simple metal covering will protect your digital privacy.
https://en.wikipedia.org/wiki/Faraday_cage
During electrical storms if you are driving a car, it is best to stay inside the car as its metal body acts as a Faraday cage with zero electrical field inside. If in the vicinity of a lightning strike, its effect is felt on the outside of the car and the inside is unaffected, provided you remain totally inside. This is also true if an active (“hot”) electrical wire was broken (in a storm or an accident) and fell on your car.
Section Summary
- A conductor allows free charges to move about within it.
- The electrical forces around a conductor will cause free charges to move around inside the conductor until static equilibrium is reached.
- Any excess charge will collect along the surface of a conductor.
- Conductors with sharp corners or points will collect more charge at those points.
- A lightning rod is a conductor with sharply pointed ends that collect excess charge on the building caused by an electrical storm and allow it to dissipate back into the air.
- Electrical storms result when the electrical field of Earth’s surface in certain locations becomes more strongly charged, due to changes in the insulating effect of the air.

Conceptual Questions
1: Is the object in Figure 9 a conductor or an insulator? Justify your answer.
2: If the electric field lines in the figure above were perpendicular to the object, would it necessarily be a conductor? Explain.
3: The discussion of the electric field between two parallel conducting plates, in this module states that edge effects are less important if the plates are close together. What does close mean? That is, is the actual plate separation crucial, or is the ratio of plate separation to plate area crucial?
4: Would the self-created electric field at the end of a pointed conductor, such as a lightning rod, remove positive or negative charge from the conductor? Would the same sign charge be removed from a neutral pointed conductor by the application of a similar externally created electric field? (The answers to both questions have implications for charge transfer utilizing points.)
5: Why is a golfer with a metal club over her shoulder vulnerable to lightning in an open fairway? Would she be any safer under a tree?
6: Can the belt of a Van de Graaff accelerator be a conductor? Explain.
7: Are you relatively safe from lightning inside an automobile? Give two reasons.
8: Discuss pros and cons of a lightning rod being grounded versus simply being attached to a building.
9: Using the symmetry of the arrangement, show that the net Coulomb force on the charge q at the centre of the square below (Figure 10) is zero if the charges on the four corners are exactly equal.
10: (a) Using the symmetry of the arrangement, show that the electric field at the centre of the square in Figure 10 is zero if the charges on the four corners are exactly equal. (b) Show that this is also true for any combination of charges in which qa = qd and qb = qc
11: (a) What is the direction of the total Coulomb force on q in Figure 10 if q is negative, qa = qc and both are negative, and qb = qd and both are positive? (b) What is the direction of the electric field at the centre of the square in this situation?
12: Considering Figure 10, suppose that qa = qd and qb = qc. First show that q is in static equilibrium. (You may neglect the gravitational force.) Then discuss whether the equilibrium is stable or unstable, noting that this may depend on the signs of the charges and the direction of displacement of q from the centre of the square.
13: If qa = 0 in Figure 10, under what conditions will there be no net Coulomb force on q?
14: In regions of low humidity, one develops a special “grip” when opening car doors, or touching metal door knobs. This involves placing as much of the hand on the device as possible, not just the ends of one’s fingers. Discuss the induced charge and explain why this is done.
15: Tollbooth stations on roadways and bridges usually have a piece of wire stuck in the pavement before them that will touch a car as it approaches. Why is this done?
16: Suppose a woman carries an excess charge. To maintain her charged status can she be standing on ground wearing just any pair of shoes? How would you discharge her? What are the consequences if she simply walks away?
Problems & Exercises
1: Sketch the electric field lines in the vicinity of the conductor in Figure 11 given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
2: Sketch the electric field lines in the vicinity of the conductor in Figure 12 given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
3: Sketch the electric field between the two conducting plates shown in Figure 13, given the top plate is positive and an equal amount of negative charge is on the bottom plate. Be certain to indicate the distribution of charge on the plates.
4: Sketch the electric field lines in the vicinity of the charged insulator in Figure 14 noting its nonuniform charge distribution.
5: What is the force on the charge located at x=8.00 cm in Figure 15(a) given that q = 1.00 μC?
6: (a) Find the total electric field at x=1.00 cm in Figure 15(b) given that q = 5.00 nC. (b) Find the total electric field at x = 11.00 cm in Figure 15(b). (c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, will there be a single charge, double charge, etc., and what will its value(s) be?)
7: (a) Find the electric field at x = 5.00 cm in Figure 15(a), given that q = 1.00μC. (b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for -2q alone? (c) Can the electric field be zero anywhere between 0.00 and 8.00 cm? (d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidly approach zero and why? (e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity? (Hint: A graphing calculator can yield considerable insight in this problem.)
8: (a) Find the total Coulomb force on a charge of 2.00 nC located at x=4.00 cm in Figure 15 (b), given that q = 1.00 μC. (b) Find the x-position at which the electric field is zero in Figure 15 (b).
9: Using the symmetry of the arrangement, determine the direction of the force on q in the figure below, given that qa = qb = +7.50 μC and qc = qd = -7.50 μC. (b) Calculate the magnitude of the force on the charge q, given that the square is 10.0 cm on a side and q = 2.00 μC.
10: (a) Using the symmetry of the arrangement, determine the direction of the electric field at the centre of the square in Figure 16, given that qa = qb = – 1.00 μC and qc = qd = +1.00 μC . (b) Calculate the magnitude of the electric field at the location of q, given that the square is 5.00 cm on a side.
11: Find the electric field at the location of qa in Figure 16 given that qb = qc = qd = +2.00 nC, q= -1.00 nC, and the square is 20.0 cm on a side.
12: Find the total Coulomb force on the charge q in Figure 16, given that q = 1.00 μC, qa = 2.00 μC, qb = -3.00 μC , qc = -4.00 μC and q d = +1.00 μC. The square is 50.0 cm on a side.
13: (a) Find the electric field at the location of qa in Figure 17, given that q_b = +10.00 μC and q c = -5.00 μC . (b) What is the force on qa, given that qa = +1.50 nC?
14: (a) Find the electric field at the centre of the triangular configuration of charges in Figure 17, given that qa = +2.50 nC , qb = -8.00 nC , and q c= +1.50 nC. (b) Is there any combination of charges, other than qa = qb = qc, that will produce a zero strength electric field at the centre of the triangular configuration?
Glossary
- conductor
- an object with properties that allow charges to move about freely within it
- free charge
- an electrical charge (either positive or negative) which can move about separately from its base molecule
- electrostatic equilibrium
- an electrostatically balanced state in which all free electrical charges have stopped moving about
- polarized
- a state in which the positive and negative charges within an object have collected in separate locations
- ionosphere
- a layer of charged particles located around 100 km above the surface of Earth, which is responsible for a range of phenomena including the electric field surrounding Earth
- Faraday cage
- a metal shield which prevents electric charge from penetrating its surface
Solutions
Concept Questions
11) to the right
Problems & Exercises
6: (a)
(b) 2.03 x 105 N/C
(c) one charge of +q = +5 nC
8: (a) 0.252 N to the left (b) x = 6.07 cm
10: (a)The electric field at the centre of the square will be straight up, since qa and qb are positive and qc and qd are negative and all have the same magnitude.
(b) 2.04 x 107 N/C C (upward)
12: 0.102 N , 0.102N, in the -y direction
14: (a) , below the horizontal. (b) No