Opening the book…
MAKING CONNECTIONS TO THE COLLABORATION
(1) Which of the following was one of the main mathematical ideas of the collaboration? The questions refer to the following quantities:
|
50 people/mi2 |
20 ft/5 sec |
34% |
(i) All three of the figures are ratios.
(ii) The first two figures are ratios.
(iii) The second figure could be written as 4 ft/sec.
(iv) The first figure is a population density.
Writing About Connections Among Mathematical Ideas
In this module you will be asked to write statements about connections among the mathematical ideas in lessons. For this first assignment, you are given statements to choose from. In future assignments, you may want to refer back to this as an example of how to write about connections.
(2) Refer back to Collaboration 1.6, Question 6. Choose the statement that explains how the mathematical ideas in Question 6 connect to the work in Collaboration 2.1.
(i) In Collaboration 1.6, you learned about the water footprints of the United States and China. In Collaboration 2.1, you used the population density of the United States and China. Both lessons compared the United States and China.
(ii) In Collaboration 1.6, you had to calculate the water footprint per capita in the United States. This is like splitting up all the water equally among the people. In Collaboration 2.1, you calculated the population density of the United States. This is like splitting up all the people equally over an area of land. These are both ratios.
(iii) In Collaboration 1.6, you used the population of a country to find the water footprint per capita. In Collaboration 2.1, you used the population of a country to find a population density. Both lessons used population in their calculations.
DEVELOPING SKILLS AND UNDERSTANDING
(3) Use the picture below to answer the following questions.
(a) What are the units for the density of the stars in Rectangle B? Select the proper units.
(i) stars
(ii) stars/ft
(iii) stars2/ft
(iv) stars/ft2
(b) What is the density of the stars in Rectangle B? Write your answer as a fraction or as a number rounded to the nearest hundredth. Do not give your answer as a mixed fraction. Include the proper units.
(c) What are the units for the density of the stars in Rectangle A? Note that the stars in Rectangle B are also in Rectangle A. Select the proper units.
(i) stars
(ii) stars/ft
(iii) stars2/ft
(iv) stars/ft2
(d) What is the density of the stars in Rectangle A? Note that the stars in Rectangle B are also in Rectangle A. Write your answer as a fraction or as a number rounded to the nearest hundredth. Do not give your answer as a mixed fraction. Include the proper units.
(e) Suppose three new stars were added in the gray part of Rectangle A. Which of the following statements would be correct?
(i) The density of Rectangle A would increase. The density of Rectangle B would stay the same.
(ii) The density of Rectangle A would decrease. The density of Rectangle B would stay the same.
(iii) The density of Rectangle A would increase. The density of Rectangle B would increase.
(iv) The density of Rectangle A would decrease. The density of Rectangle B would decrease.
(v) The density of Rectangle A would stay the same. The density of Rectangle B would increase.
(4) Wikipedia states the following about Anchorage, Alaska: “The Anchorage metropolitan area … accounts for more than half the state’s population.”1
(a) Calculate the population density for Anchorage, based on a 2020 population of 398,328 people living on 1,968.6 square miles. Round to the nearest person per square mile.
(b) Wikipedia also says that the small Alaskan town of Ketchikan has the densest population in Alaska. Ketchikan had a population of 8,192 in 2020 and a land area of 4.9 square miles. Calculate the population density of Ketchikan. Round to the nearest person per square mile.
(5) The following information comes from the collaboration:
|
Population |
Land Area (sq. miles) |
|
|
United States |
333,000,000 |
3,794,000 |
|
China |
1,412,000,000 |
3,705,000 |
In India, about 1,406,500,000 people live on 1,269,000 square miles of land.2 Which of the following statements is false?
(i) The population density of India is approximately three times the population density of China.
(ii) The population density of India is approximately 12 times the population density of the United States.
(iii) The population densities of these three countries ranked from smallest to largest are the United States, India, and China.
(iv) The population density of the United States is approximately 88 people per square mile. That is smaller than the population densities for China and India.
(6) Which of the following population densities are equivalent to a density of 20 people/mi2? There may be more than one correct answer.
(i) 1 person/0.05 mi2
(ii) 1 person/0.2 mi2
(iii) 1 person/0.5 mi2
(iv) 100 people/5 mi2
(v) 200 people/2 mi2
(7) One way to measure a country’s economy is per capita gross domestic product, or per capita GDP. This is the value of all the products and services produced in a country over the course of a year divided by its population.
(a) According to WorldData.info, in 2021, the United States had a per capita GDP of $70,249.3 If the population was about 333 million, which of the following is a reasonable estimate for the GDP of the United States?
(i) $2.3 trillion
(ii) $21 billion
(iii) $21 trillion
(iv) $60 billion
(v) $60 trillion
(b) Also according to WorldData.info, in 2021, China had a per capita GDP of $12,556 and a population of around 1,412,000,000. Which of the following is a reasonable estimate for the GDP of China?
(i) $1.3 trillion
(ii) $17 billion
(iii) $17 trillion
(iv) $50 billion
(v) $50 trillion
MAKING CONNECTIONS ACROSS THE COURSE
(8) In April 2020, the population of Nebraska was 1,961,504, and its population density was 25.5 people per mi2. The population of New Hampshire was 1,377,518, and its population density was 153.9 people per mi2.4 Which of the following statements is a valid interpretation of this information?
(i) Nebraska is approximately six times more densely populated than New Hampshire.
(ii) The population of Nebraska is approximately 71% of the population of New Hampshire.
(iii) New Hampshire is approximately six times more densely populated than Nebraska.
(iv) The population of New Hampshire is approximately 141% of the population of Nebraska.
(9) Cholera, a serious intestinal disease, broke out in London in the mid-19th century. Research by a physician named John Snow indicated that the disease was caused by germs in polluted water. A vaccine for cholera was then developed in the late 1800s. A vaccine is a drug that helps a person become immune to a disease. Scientists studied 818 people to determine the effects of a cholera vaccine.5 The study lasted from 1894 to 1896.
|
Infected |
Not Infected |
|
|
Vaccinated |
3 |
276 |
|
Not Vaccinated |
66 |
473 |
Use the information in the table to complete the following four statements. Round to the nearest tenth of a percent.
(a) ______% of those vaccinated were infected
(b) ______% of those not vaccinated were infected
(c) ______% of those infected were vaccinated
(d) ______% of those infected were not vaccinated
(e) Which of the following statements are correct based on the information in the table? There may be more than one correct answer.
(i) A person who was not vaccinated was 12 times more likely to get cholera than someone who got the vaccine.
(ii) A person who was not vaccinated was four times more likely to get cholera than someone who got the vaccine.
(iii) If a person was infected, it was four times more likely that (s)he was not vaccinated rather than vaccinated.
(iv) If a person was infected, it was more than 20 times more likely that (s)he was not vaccinated rather than vaccinated.
________________________________________
1 http://en.Wikipedia.org/wiki/Anchorage,_Alaska
2 http://www.indiastat.com/default.aspx
3 https://www.worlddata.info/largest-economies.php
4 https://www.census.gov/quickfacts
5 Major Greenwood and G. Udny Yule, “The Statistics of Anti-Typhoid and Anti-Cholera Inoculations, and the Interpretation of such Statistics in general,” Proceedings of the Royal Society of Medicine 8 (1915): 113-194.
This page titled 2.1.2: Exercise 2.1 was last modified on Wed, 15 May 2024 07:37:11 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) .
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