putting math to work AWS
MATH GRADE 6 UNIT 7
PUTTING MATH TO WORK
EXERCISES
Grade 6 Unit 7: Putting Math to Work
CONTENTS
EXERCISES EXERCISES
LESSON 1: LIGHTNING! ������������������������������������������������������������������������������� 4 LESSON 2: LIGHTNING! (KILOMETERS) ������������������������������������������������ 5 LESSON 3: CHOOSING A PROJECT �������������������������������������������������������� 6 LESSON 4: GUMMY BEARS ������������������������������������������������������������������������� 7 LESSON 5: COST OF GUMMY BEARS ����������������������������������������������������� 9 LESSON 6: PROJECT WORK DAY ����������������������������������������������������������� 10 ANSWERS
LESSON 1: LIGHTNING! ����������������������������������������������������������������������������� 12 LESSON 2: LIGHTNING! (KILOMETERS) ���������������������������������������������� 13 LESSON 4: GUMMY BEARS ����������������������������������������������������������������������� 14 LESSON 5: COST OF GUMMY BEARS ��������������������������������������������������� 16
Grade 6 Unit 7: Putting Math to Work
LESSON 1: LIGHTNING!
EXERCISES
EXERCISES 1.
How fast does sound travel according to the rule below? Write your answer in miles per hour. Rule 1: The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5.
2. a. The actual speed of sound at ground level is around 761 mi per hr. Compare this speed with your response to problem 1. b. How accurate is rule 1? Explain. 3.
For a more accurate estimate of your distance from a lightning strike, what value should you divide by?
Challenge Problem 4. The speed of sound varies with respect to altitude, among other things. Although most commercial jets are capable of flying over even the worst storms, high-altitude lightning strikes are not unheard of. Suppose a certain passenger plane cruises at an altitude of 30,000 ft. a. The pilot of this aircraft says that at this altitude he divides the time between sight and sound of a lightning strike by 5.3 for an accurate estimate of his distance from the strike. Without making any calculations, say whether the speed of sound increases or decreases at this altitude. Explain. b. Calculate the speed of sound at this altitude. c. How does this value compare to the speed of sound at ground level? Explain your thinking.
Grade 6 Unit 7: Putting Math to Work
LESSON 2: LIGHTNING! (KILOMETERS)
EXERCISES
EXERCISES 1.
How fast does sound travel according to the rule below? Write your answer in meters per second. Rule 2: The distance of lightning in kilometers is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 3.
2. a. The actual speed of sound at ground level is 340.34 m per sec. Compare this speed with your response to problem 1. b. How accurate is rule 2? Explain. 3.
What are the units of the value (3) you divide the number of seconds by? Explain how you know.
Challenge Problem 4. a. Which rule gives a more accurate estimation? Rule 1: The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5. Rule 2: The distance of lightning in kilometers is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 3. b. Explain your answer, using the fact that there are 1.61 km per mi. Remember, the true speed of sound at ground level is around 761 mi per hr.
Grade 6 Unit 7: Putting Math to Work
LESSON 3: CHOOSING A PROJECT
EXERCISES
•
Based on what your classmates shared about their projects, revise your project proposal.
•
Make a plan for how you will complete your project in 2 weeks.
•
If you have a partner or are working with a group, make a list of assignments for each person.
•
Complete any exercises that you have not finished from this unit.
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS
EXERCISES
EXERCISES 1.
Look at this ratio table. Quantity A
24
6
54
Quantity B
400
100
900
a. Explain how the values 6 : 100 and 54 : 900 were calculated. b. What is the unit ratio, where quantity A is 1? c. What are two other pairs of values you could add to the table? 2. Here are the ingredients to make 30 cookies. The quantities of ingredients needed to make any number of cookies are in proportion to these quantities. 1 1 c milk c butter 2 c flour 4 3 1 3 tsp vanilla c sugar 3 eggs 2 4 a. Denzel wants to make some cookies, and he has 5 eggs. What is the maximum number of cookies he can make? What quantities of each ingredient would he need to make that number of cookies? b. Which measurements might Denzel have to round up or down? Explain your answer. c. Denzel also considered making batches of 20, 60, and 40 cookies. To help him, copy and complete this ratio table.
Number of Cookies Cups of Flour Cups of Milk Cups of Butter Eggs Cups of Sugar Teaspoons of Vanilla
30
20
60
40
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS 3.
EXERCISES
Denzel has a 2 lb (32 oz) bag of flour. He looked in a cookbook and found that 1 c of flour weighs 4 oz. a. How many cups of flour are in Denzel’s bag of flour? b. Denzel wants to use the whole bag of flour to make cookies. How many cookies can he make if he buys more eggs? c. How much of each ingredient will Denzel need to be able to make the number of cookies you calculated in part b?
Challenge Problem 4. A machine has two gear wheels that interconnect. The larger gear wheel has 10 teeth and the smaller gear wheel has 6 teeth.
teeth
a. What is the unit ratio of the number of teeth on the large wheel to the number of teeth on the small wheel? b. The rate at which the larger wheel turns is 20 revolutions (complete turns) per second. At what rate does the smaller wheel turn?
Grade 6 Unit 7: Putting Math to Work
LESSON 5: COST OF GUMMY BEARS
EXERCISES
EXERCISES Mia is shopping for a party she will be giving. The grocery store is selling three different types of club soda. Type of Soda Mr. K’s Thirsty Theo Mexico Dry 1.
Cost of Package $5.95 $7.99 $2.69
Number in Package six 10 fl oz bottles twelve 12 fl oz cans one 1 L bottle
What is the unit price of each type of club soda? (There are 33.8 fl oz in 1 L.)
2. Mia figures she’ll need the equivalent of seventeen 12 fl oz cans for her party. What should she buy if she wants to minimize both her total cost and the amount of club soda she will have left over? 3.
There is a bulk sale on Mexico Dry club soda: $10.99 for six 1 L bottles. (The sale price only applies to each group of six bottles.) What should Mia buy if she wants to minimize both her total cost and the amount of club soda she will have left over? Explain how you know.
Challenge Problem 4. Draw a graph of the bulk sale price of Mexico Dry club soda versus the number of bottles.
Grade 6 Unit 7: Putting Math to Work
LESSON 6: PROJECT WORK DAY
EXERCISES
•
Continue to work on your project.
•
Check that you have addressed all the rubric criteria for your project.
•
Complete any exercises that you have not finished from this unit.
MATH GRADE 6 UNIT 7
PUTTING MATH TO WORK
ANSWERS FOR EXERCISES
Grade 6 Unit 7: Putting Math to Work
LESSON 1: LIGHTNING!
ANSWERS
ANSWERS 1.
Sound travels at 720 mi per hr. The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5. 1 This is mi per sec; 12 mi per min; and 720 mi per hr. 5
2. a. The rule assumes a lower speed of sound. b. The rule is fairly accurate. 3.
You would divide by 4.73 for a more accurate estimate. 1 1 1 = = 761 mi per hr 12.7 mi per min 0.2114 mi per sec
Challenge Problem 4. a. A greater time between sight and sound means a greater difference between the speeds at which they travel. Since the speed of light is constant (at least for the purposes of this exercise), the speed of sound must be slower at this altitude. b. 679.25 ≈ 680 mi per hr The speed of sound at an altitude of 30,000 ft is approximately 680 mi per hr. c. The speed of sound at ground level is approximately 761 mi per hr. The speed of sound decreases as the altitude increases.
Grade 6 Unit 7: Putting Math to Work
LESSON 2: LIGHTNING! (KILOMETERS)
ANSWERS
ANSWERS 1.
Sound travels at 333.3 m per sec. 1 km = 333.3 m per sec 3
2. a. The rule assumes a lower speed of sound. b. The rule is fairly accurate; the rule speed is 2% lower. 3.
The units are seconds per kilometer because seconds divided secondsby÷
seconds = kilometers . kilometers
Challenge Problem 4. a. Rule 2 (divide by 3 for the distance in kilometers) is more accurate. b. Answers will vary. Possible answer: One method is to write both assumed speeds in the same units as the actual speed. The assumed speed for this lesson (kilometers) is 745.34 mi per hr. This is closer to the actual speed of sound than the assumed speed from rule 1 (720 mi per hr).
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS
ANSWERS
ANSWERS 1.
a. The ratio 6 : 100 was calculated by dividing each quantity in the first ratio in the table by 4. The ratio 54 : 900 was calculated by first dividing each quantity in the first ratio by 4, and then multiplying each resulting quotient by 9. b. The unit ratio is 16.6 : 1, where quantity A equals 1. c. Answers will vary. Two possible pairs of values are 12 : 200 and 48 : 800.
2. a. The maximum number of cookies Dwayne can bake is 50, because he has only 5 eggs. The quantities of the ingredients are as follows: 1 5 5 c milk c butter 3 c flour 3 12 9 5 1 tsp vanilla 1 c sugar 5 eggs 6 4 In actuality, cup measurements are usually given in quarter-, third-, or half‑cup increments. If a quantity other than these is required, the measurement is usually 1 given as tablespoons or teaspoons: 2 tbsp are equivalent to c and 3 tsp are 8 equivalent to 1 tbsp. 1 b. To make 50 cookies Dwayne would round the amount of milk up to c or 2 1 5 1 down to c. He would round c of butter down to c or up to 3 9 2 2 c. He would also round the amount of vanilla up to 1 tsp. It is highly 3 unlikely that any cook would use exactly the amount calculated! c.
Number of Cookies
30
20
60
40
Cups of Flour
2
1
1 3
4
2
Cups of Milk
1 4
1 6
1 2
1 3
Cups of Butter
1 3
2 9
2 3
4 9
Eggs
3
2
6
4
Cups of Sugar
3 4
1 2
1 2
1
Teaspoons of Vanilla
1 2
1 3
1
2 3
1
2 3
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS 3.
ANSWERS
a. There are 8 c in Dwayne’s bag of flour. b. Each of the original values is multiplied by 4. He can make 120 cookies.
c.
Number of Cookies
30
120
Cups of Flour
2
8
Cups of Milk
1 4
1
Cups of Butter
1 3
1
Eggs
3
12
Cups of Sugar
3 4
3
Teaspoons of Vanilla
1 2
2
1 3
Challenge Problem 4. a. The ratio of the number of teeth on the large gear to the number of teeth on the 5 small gear is : 1. 3 5 2 b. The small gear turns (or 1 ) times when the large gear turns once. 3 3 5 1 If the large gear turns 20 times, then the small gear turns 20 • = 33.3 = 33 3 3 revolutions per second.
Grade 6 Unit 7: Putting Math to Work
LESSON 5: COST OF GUMMY BEARS
ANSWERS
ANSWERS 1.
Mr. K’s: $0.10 per fluid ounce ($5.95 ÷ 60 = $0.099 ≈ $0.10) Thirsty Theo: $0.06 per fluid ounce ($7.99 ÷ 144 = $0.055 ≈ $0.06) Mexico Dry: $0.08 per fluid ounce ($2.69 ÷ 33.8 = $0.079 ≈ $0.08)
2. Mia needs seventeen 12 fl oz cans, which is 204 fl oz (17 • 12 = 204). One package of Thirsty Theo is 144 fl oz (12 • 12 = 144). She still needs 60 fl oz (204 – 144 = 60). She should buy 2 bottles of Mexico Dry, which is approximately 68 fl oz and minimizes the amount left over (2 • 33.8 =67.6). 3.
Mia needs 204 fl oz, but the 6 bottles of Mexico Dry is not enough (6 • 33.8 = 202.8). Since the sale price only applies to 6 bottles, she should still buy 1 package of Thirsty Theo plus 2 bottles of Mexico Dry.
Challenge Problem y
Price ($)
4.
30 20 10 0
0
2
4
6
8
10
12
Bottles of Soda
14
16
x
PUTTING MATH TO WORK
EXERCISES
Grade 6 Unit 7: Putting Math to Work
CONTENTS
EXERCISES EXERCISES
LESSON 1: LIGHTNING! ������������������������������������������������������������������������������� 4 LESSON 2: LIGHTNING! (KILOMETERS) ������������������������������������������������ 5 LESSON 3: CHOOSING A PROJECT �������������������������������������������������������� 6 LESSON 4: GUMMY BEARS ������������������������������������������������������������������������� 7 LESSON 5: COST OF GUMMY BEARS ����������������������������������������������������� 9 LESSON 6: PROJECT WORK DAY ����������������������������������������������������������� 10 ANSWERS
LESSON 1: LIGHTNING! ����������������������������������������������������������������������������� 12 LESSON 2: LIGHTNING! (KILOMETERS) ���������������������������������������������� 13 LESSON 4: GUMMY BEARS ����������������������������������������������������������������������� 14 LESSON 5: COST OF GUMMY BEARS ��������������������������������������������������� 16
Grade 6 Unit 7: Putting Math to Work
LESSON 1: LIGHTNING!
EXERCISES
EXERCISES 1.
How fast does sound travel according to the rule below? Write your answer in miles per hour. Rule 1: The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5.
2. a. The actual speed of sound at ground level is around 761 mi per hr. Compare this speed with your response to problem 1. b. How accurate is rule 1? Explain. 3.
For a more accurate estimate of your distance from a lightning strike, what value should you divide by?
Challenge Problem 4. The speed of sound varies with respect to altitude, among other things. Although most commercial jets are capable of flying over even the worst storms, high-altitude lightning strikes are not unheard of. Suppose a certain passenger plane cruises at an altitude of 30,000 ft. a. The pilot of this aircraft says that at this altitude he divides the time between sight and sound of a lightning strike by 5.3 for an accurate estimate of his distance from the strike. Without making any calculations, say whether the speed of sound increases or decreases at this altitude. Explain. b. Calculate the speed of sound at this altitude. c. How does this value compare to the speed of sound at ground level? Explain your thinking.
Grade 6 Unit 7: Putting Math to Work
LESSON 2: LIGHTNING! (KILOMETERS)
EXERCISES
EXERCISES 1.
How fast does sound travel according to the rule below? Write your answer in meters per second. Rule 2: The distance of lightning in kilometers is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 3.
2. a. The actual speed of sound at ground level is 340.34 m per sec. Compare this speed with your response to problem 1. b. How accurate is rule 2? Explain. 3.
What are the units of the value (3) you divide the number of seconds by? Explain how you know.
Challenge Problem 4. a. Which rule gives a more accurate estimation? Rule 1: The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5. Rule 2: The distance of lightning in kilometers is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 3. b. Explain your answer, using the fact that there are 1.61 km per mi. Remember, the true speed of sound at ground level is around 761 mi per hr.
Grade 6 Unit 7: Putting Math to Work
LESSON 3: CHOOSING A PROJECT
EXERCISES
•
Based on what your classmates shared about their projects, revise your project proposal.
•
Make a plan for how you will complete your project in 2 weeks.
•
If you have a partner or are working with a group, make a list of assignments for each person.
•
Complete any exercises that you have not finished from this unit.
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS
EXERCISES
EXERCISES 1.
Look at this ratio table. Quantity A
24
6
54
Quantity B
400
100
900
a. Explain how the values 6 : 100 and 54 : 900 were calculated. b. What is the unit ratio, where quantity A is 1? c. What are two other pairs of values you could add to the table? 2. Here are the ingredients to make 30 cookies. The quantities of ingredients needed to make any number of cookies are in proportion to these quantities. 1 1 c milk c butter 2 c flour 4 3 1 3 tsp vanilla c sugar 3 eggs 2 4 a. Denzel wants to make some cookies, and he has 5 eggs. What is the maximum number of cookies he can make? What quantities of each ingredient would he need to make that number of cookies? b. Which measurements might Denzel have to round up or down? Explain your answer. c. Denzel also considered making batches of 20, 60, and 40 cookies. To help him, copy and complete this ratio table.
Number of Cookies Cups of Flour Cups of Milk Cups of Butter Eggs Cups of Sugar Teaspoons of Vanilla
30
20
60
40
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS 3.
EXERCISES
Denzel has a 2 lb (32 oz) bag of flour. He looked in a cookbook and found that 1 c of flour weighs 4 oz. a. How many cups of flour are in Denzel’s bag of flour? b. Denzel wants to use the whole bag of flour to make cookies. How many cookies can he make if he buys more eggs? c. How much of each ingredient will Denzel need to be able to make the number of cookies you calculated in part b?
Challenge Problem 4. A machine has two gear wheels that interconnect. The larger gear wheel has 10 teeth and the smaller gear wheel has 6 teeth.
teeth
a. What is the unit ratio of the number of teeth on the large wheel to the number of teeth on the small wheel? b. The rate at which the larger wheel turns is 20 revolutions (complete turns) per second. At what rate does the smaller wheel turn?
Grade 6 Unit 7: Putting Math to Work
LESSON 5: COST OF GUMMY BEARS
EXERCISES
EXERCISES Mia is shopping for a party she will be giving. The grocery store is selling three different types of club soda. Type of Soda Mr. K’s Thirsty Theo Mexico Dry 1.
Cost of Package $5.95 $7.99 $2.69
Number in Package six 10 fl oz bottles twelve 12 fl oz cans one 1 L bottle
What is the unit price of each type of club soda? (There are 33.8 fl oz in 1 L.)
2. Mia figures she’ll need the equivalent of seventeen 12 fl oz cans for her party. What should she buy if she wants to minimize both her total cost and the amount of club soda she will have left over? 3.
There is a bulk sale on Mexico Dry club soda: $10.99 for six 1 L bottles. (The sale price only applies to each group of six bottles.) What should Mia buy if she wants to minimize both her total cost and the amount of club soda she will have left over? Explain how you know.
Challenge Problem 4. Draw a graph of the bulk sale price of Mexico Dry club soda versus the number of bottles.
Grade 6 Unit 7: Putting Math to Work
LESSON 6: PROJECT WORK DAY
EXERCISES
•
Continue to work on your project.
•
Check that you have addressed all the rubric criteria for your project.
•
Complete any exercises that you have not finished from this unit.
MATH GRADE 6 UNIT 7
PUTTING MATH TO WORK
ANSWERS FOR EXERCISES
Grade 6 Unit 7: Putting Math to Work
LESSON 1: LIGHTNING!
ANSWERS
ANSWERS 1.
Sound travels at 720 mi per hr. The distance of lightning in miles is the time in seconds between the appearance of the lightning and the sound of the thunder divided by 5. 1 This is mi per sec; 12 mi per min; and 720 mi per hr. 5
2. a. The rule assumes a lower speed of sound. b. The rule is fairly accurate. 3.
You would divide by 4.73 for a more accurate estimate. 1 1 1 = = 761 mi per hr 12.7 mi per min 0.2114 mi per sec
Challenge Problem 4. a. A greater time between sight and sound means a greater difference between the speeds at which they travel. Since the speed of light is constant (at least for the purposes of this exercise), the speed of sound must be slower at this altitude. b. 679.25 ≈ 680 mi per hr The speed of sound at an altitude of 30,000 ft is approximately 680 mi per hr. c. The speed of sound at ground level is approximately 761 mi per hr. The speed of sound decreases as the altitude increases.
Grade 6 Unit 7: Putting Math to Work
LESSON 2: LIGHTNING! (KILOMETERS)
ANSWERS
ANSWERS 1.
Sound travels at 333.3 m per sec. 1 km = 333.3 m per sec 3
2. a. The rule assumes a lower speed of sound. b. The rule is fairly accurate; the rule speed is 2% lower. 3.
The units are seconds per kilometer because seconds divided secondsby÷
seconds = kilometers . kilometers
Challenge Problem 4. a. Rule 2 (divide by 3 for the distance in kilometers) is more accurate. b. Answers will vary. Possible answer: One method is to write both assumed speeds in the same units as the actual speed. The assumed speed for this lesson (kilometers) is 745.34 mi per hr. This is closer to the actual speed of sound than the assumed speed from rule 1 (720 mi per hr).
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS
ANSWERS
ANSWERS 1.
a. The ratio 6 : 100 was calculated by dividing each quantity in the first ratio in the table by 4. The ratio 54 : 900 was calculated by first dividing each quantity in the first ratio by 4, and then multiplying each resulting quotient by 9. b. The unit ratio is 16.6 : 1, where quantity A equals 1. c. Answers will vary. Two possible pairs of values are 12 : 200 and 48 : 800.
2. a. The maximum number of cookies Dwayne can bake is 50, because he has only 5 eggs. The quantities of the ingredients are as follows: 1 5 5 c milk c butter 3 c flour 3 12 9 5 1 tsp vanilla 1 c sugar 5 eggs 6 4 In actuality, cup measurements are usually given in quarter-, third-, or half‑cup increments. If a quantity other than these is required, the measurement is usually 1 given as tablespoons or teaspoons: 2 tbsp are equivalent to c and 3 tsp are 8 equivalent to 1 tbsp. 1 b. To make 50 cookies Dwayne would round the amount of milk up to c or 2 1 5 1 down to c. He would round c of butter down to c or up to 3 9 2 2 c. He would also round the amount of vanilla up to 1 tsp. It is highly 3 unlikely that any cook would use exactly the amount calculated! c.
Number of Cookies
30
20
60
40
Cups of Flour
2
1
1 3
4
2
Cups of Milk
1 4
1 6
1 2
1 3
Cups of Butter
1 3
2 9
2 3
4 9
Eggs
3
2
6
4
Cups of Sugar
3 4
1 2
1 2
1
Teaspoons of Vanilla
1 2
1 3
1
2 3
1
2 3
Grade 6 Unit 7: Putting Math to Work
LESSON 4: GUMMY BEARS 3.
ANSWERS
a. There are 8 c in Dwayne’s bag of flour. b. Each of the original values is multiplied by 4. He can make 120 cookies.
c.
Number of Cookies
30
120
Cups of Flour
2
8
Cups of Milk
1 4
1
Cups of Butter
1 3
1
Eggs
3
12
Cups of Sugar
3 4
3
Teaspoons of Vanilla
1 2
2
1 3
Challenge Problem 4. a. The ratio of the number of teeth on the large gear to the number of teeth on the 5 small gear is : 1. 3 5 2 b. The small gear turns (or 1 ) times when the large gear turns once. 3 3 5 1 If the large gear turns 20 times, then the small gear turns 20 • = 33.3 = 33 3 3 revolutions per second.
Grade 6 Unit 7: Putting Math to Work
LESSON 5: COST OF GUMMY BEARS
ANSWERS
ANSWERS 1.
Mr. K’s: $0.10 per fluid ounce ($5.95 ÷ 60 = $0.099 ≈ $0.10) Thirsty Theo: $0.06 per fluid ounce ($7.99 ÷ 144 = $0.055 ≈ $0.06) Mexico Dry: $0.08 per fluid ounce ($2.69 ÷ 33.8 = $0.079 ≈ $0.08)
2. Mia needs seventeen 12 fl oz cans, which is 204 fl oz (17 • 12 = 204). One package of Thirsty Theo is 144 fl oz (12 • 12 = 144). She still needs 60 fl oz (204 – 144 = 60). She should buy 2 bottles of Mexico Dry, which is approximately 68 fl oz and minimizes the amount left over (2 • 33.8 =67.6). 3.
Mia needs 204 fl oz, but the 6 bottles of Mexico Dry is not enough (6 • 33.8 = 202.8). Since the sale price only applies to 6 bottles, she should still buy 1 package of Thirsty Theo plus 2 bottles of Mexico Dry.
Challenge Problem y
Price ($)
4.
30 20 10 0
0
2
4
6
8
10
12
Bottles of Soda
14
16
x
