## Where is Everybody?

• Lesson
6-8
1

Using two online activities, students use ratios and percents to compare population density and explore various statistical measures.

Prior to this lesson, students should have familiarity with the following concepts and skills:

• The relationship between fractions, decimals and percents
• Calculation and interpretation of measures of central tendency (mean, median, mode)
• Creating data representations using box-and-whisker plots
• Geographical awareness about the United States and Canada
• Basic understanding of both metric and customary systems of measurement

Prior to conducting this lesson with students, open the State Data Map and Canada Data Map, to familiarize yourself with the activities. You may also wish to bookmark the activities for easy access by students.

### Know Thy Neighbor: An Introduction for American (or non-Canadian) Classes

Start with a discussion of what your students know about Canada. You might prepare some interesting facts to share, including a list of famous Canadians, such as Avril Lavigne, Wayne Gretzky, Steve Nash, Justin Bieber, Michael J. Fox, Nelly Furtado, Alexander Graham Bell, Kiefer Sutherland, Pamela Anderson, Shania Twain, Keanu Reeves, Alanis Morissette, Neil Young, Jim Carrey, William Shatner (Captain Kirk). For fun, present the list without introduction, and then ask students what all of these people have in common. Students might be also interested to know that the following originated from Canadian inventors:

• IMAX technology
• Blackberry® PDAs
• the telephone
• basketball
• Trivial Pursuit®

Students should also know that Canada consists of 10 provinces and 3 territories.

Use this introduction as an opportunity to introduce the "Rough Rule of 10" — that is, population‑related quantities in the United States tend to be roughly 10 times larger than those in Canada. However, do not refer to it as the Rough Rule of 10 yet! Students will discover this rule on their own. Ask students what the U.S. population is (approximately 309 million). Have students guess what the Canadian population is (approximately 34 million).

Distribute the Rough Rule Activity Sheet. (As this is meant to be a quick introduction to the lesson, you can save paper by displaying the same information on a chalkboard, overhead projector, or interactive whiteboard instead of making copies for all students.) Working in pairs, students should calculate values for the last two columns. To save time, you could have different parts of the room complete a different row, have a representative report their answer, and then aggregate the data. The completed table is shown below.

 Year Canadian Population (millions) U.S. Population (millions) Canadian Population as Share of U.S. Population (decimal form) Canadian Population as Share of U.S. Population (Percentage) 2010 34.095 309.163 0.110282 11.0% 2000 30.689 281.421 0.109050 10.9% 1990 27.512 248.710 0.110619 11.1% 1980 24.517 226.546 0.108221 10.8% 1970 21.297 203.212 0.104802 10.4%

Ask students if they can develop a rule for comparing the U.S. and Canadian populations using their conclusions from the table. Students should recognize that for the last 40 years, the Canadian population has been roughly 1/10 the U.S. population. For the remainder of the lesson, this concept can be referred to as the "Rough Rule of 10."

Conduct a discussion about the introductory activity. Ask students if the Rough Rule of 10 over or underestimates the Canadian population, if the U.S. population is known? What if they had the actual Canadian population and used the rule to estimate the U.S. population? Is the rule likely to under or overestimate the population, according to what was observed in the data?

Tell students that the rule can be extended in a few other cases, including the size of the economy. Introduce students to the concept of the gross domestic product, known by its acronym GDP. Simply put, GDP measures the total sale of all goods and services within an economy, including everything that consumers, businesses and governments purchase as well as net exports from trade and investment spending. (You might point out that this is the final sale of goods and services so that the measurement avoids double or triple counting.)

Inform students that 2009 U.S. GDP was $14.3 trillion. Using the Rough Rule of 10, ask students to predict the Canadian GDP for 2009. Canadian GDP for 2009 was$1.3 trillion. (Note that both the U.S. and Canadian GDP are given in U.S. dollars.) Student estimates should be close to this number.

Explain to students that per capita means "per person." Using the population data and GDP data, ask questions that allow students to articulate how to calculate the per capita GDP. This will help them later when working with the data maps. Ask students if it would make sense to use the Rough Rule of 10 for per capita numbers, such as when calculating per capita GDP, and to explain their reasoning. Students should realize this estimation rule only works to estimate a rough magnitude of quantifiable characteristics which are likely to be similar in U.S. and Canada. To reinforce this idea you could provide a list of quantifiable categories, such as fast food consumption or home heating expenditures, and ask them if they think the rule would apply. Ask them who might be interested in estimating this type of information.

### Where is Everybody? Population Density in Canada and the United States

Distribute a copy of the Where is Everybody Activity Sheet to each student. Students can work in pairs, and one method to randomly assign partners is to write a simple equation or expression in one corner of each activity sheet. When students solve the equation or simplify the expression, students with the same solution become partners. Have the students find their match. Alternatively, if the computers are numbered, the solution could be the computer number, which would tell students immediately where to go if the handouts are distributed as they enter the computer lab. In general, students can be paired either randomly or deliberately, but when choosing a pairing strategy, the following questions should be considered: Are my students always working with the same partners? Is there a benefit or downside to grouping by ability in this activity? How much time do I have to organize the groups?

Once students are in pairs and at a computer, they should open the State Data Map and Canada Data Map so that they can easily navigate between the two maps.

Have students observe the Canada Data Map. Using a projector, demonstrate that students can find data sets in the upper left corner. Have them select the Land Area map. Ask them what province has the largest land area in Canada. [Nunavut.] If you are in a non-Canadian classroom have a list of provinces with their abbreviations in a place where all students can see it so the students can readily identify the Canadian provinces and increase their geographical awareness. Have the students select the Population map and ask them which province is the most populated. [Ontario.]

Before switching to the Population Density map, ask students to predict which province has the greatest population density. Students may need to have the term population density defined for them. Explain that population density is a measure of the number of people per unit area, and it is commonly measured as people per square mile or people per square kilometer. Once students understand the concept, ask them again to predict which province will have the greatest population density. Without judging their responses, ask them to select the Population Density map from the Canada Data Map. Then ask students which province has the greatest population density. [Prince Edward.] Ask them to explain why there was a different province to the answer for the questions about population, area, and density. This introduction will provide students with an understanding of population density and familiarize them with the activity so that they can continue to work on the activity sheet.

Circulate as students work through the activity sheet, inquiring how students arrived at their answers and asking probing questions if they are stuck, such as, "How can the mean be used to help you find the total population?"

Check that students are using precise mathematical language. For example, encourage them to use the word denominator instead of "bottom" in their responses. If a students' language or calculations are inaccurate, have them rewrite a correct response. Use questioning to help them identify where they might have gone wrong. For example, when working with the U.S. data maps, a student may perform calculations with 50 rather than 51, overlooking the fact that Washington, DC, is included in the set. Asking, "How many data points are there?" might help the student realize the mistake without explicitly explaining where they went wrong.

### In the Classroom

Read about how this lesson came to fruition in this author's Success Story.

Assessment Options

1. Have students write a short letter to a Canadian or American pen pal explaining what they learned about the differences and similarities between Canada in the US. Have them include statistics that include ratios, decimals and percentages. Teachers can connect with classrooms in other countries through online education focused global communities.
2. Have students present their results to the class.

Extensions

1. Ask students if think that the Rough Rule of 10 was true 100 years ago. Do they think it will be true 100 years from now? …200 years? Have students share their thinking. If time permits, this discussion could lead into an exploration of the exponential nature of population growth.
2. Provide students with a new mean, median, range, and upper and lower quartile for population or land area and have students develop a data set that would produce such results. Students can enter a fictitious set of data into one of the Data Maps on Illuminations. Can they find more than one possibility?

Questions for Students

1. Notice that the population data is from different years. Is it reasonable to use these data sets to make comparisons between U.S. and Canada?

[Yes. The population data from the U.S. and Canada only differs by a couple years, and population doesn't change quickly enough to have a great influence on the results.]

2. When using the computer activity, the sum of all data points was not displayed. How can you quickly calculate the sum of all data points without adding them together?

[Take the number of data points, and multiply by the average.]

Teacher Reflection

• What was the level of student engagement in this activity? Could the activity have been structured differently to increase engagement?
• Were the students sufficiently prepared to be successful in this activity?
• What areas of strength or weaknesses did you observe among your students during this lesson?

### Other Related Resources

• Success Story
Read about how this lesson came to fruition in this author's Success Story.

### Learning Objectives

Students will:

• Make estimates about real-life data.
• Work flexibly with fractions, decimals and percents to solve problems and make comparisons.
• Develop number sense.
• Find, use and interpret measures of center and spread.
• Discuss and understand the correspondence between data sets and their graphical representations.

### NCTM Standards and Expectations

• Work flexibly with fractions, decimals, and percents to solve problems.
• Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
• Find, use, and interpret measures of center and spread, including mean and interquartile range.
• Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.
• Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken.

### Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

• CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 6, Stats & Probability

• CCSS.Math.Content.6.SP.A.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Grade 6, Stats & Probability

• CCSS.Math.Content.6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Grade 7, The Number System

• CCSS.Math.Content.7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.

Grade 7, Expression/Equation

• CCSS.Math.Content.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Grade 6, Stats & Probability

• CCSS.Math.Content.6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.