Synopsis

In “We the People,” students are challenged to go far beyond the computational aspect of coordinate geometry and into the critical stage of analysis and synthesis. This lesson provides a structure for exploration of real data. Students utilize actual statistical data from the U.S. Census Bureau to explore the rate at which populations of various countries rise and fall. Students are asked to critically analyze data and to develop their own ideas about cause and effect, extending their ideas to conditional statements. Students will find political, economic, global, and/or health related issues that have played significant roles in the population dynamics of the world.

Narrative

Consider inertia. Alfred North Whitehead suggests that ideas are inert if they are simply received, but not utilized. In education, so many ideas pass from teacher to student without any external force. This knowledge is perhaps transferred on some basic level, but, having never been utilized, it may very well only be a facade of the actual knowledge. Alfred North Whitehead argues that “education with inert ideas is not only useless: it is, above all things, harmful…” Newmann, Bryk, and Nagaoka (2001) make the distinction between didactic and interactive instruction. In didactic instruction, students simply regurgitate what they have learned (such as facts and algorithms) and are really only asked to apply what they have learned in assessments. However, in interactive instruction,

Students are often asked to formulate problems, to organize their knowledge and experiences in new ways to solve them, to test their ideas with other students, and to express themselves using elaborated statements, both orally and in writing.

To successfully calculate the midpoint between any two points is significant in the sense that a student has successfully made appropriate substitutions and has correctly followed the order of operations. According to Oklahoma Priority Academic Student Skills (PASS), these two objectives are being taught as early as 6th grade.[1] Yet, just as words, sentences, and paragraphs only begin to have purpose as the story unravels, simple calculations based on the foundational aspects of 6th grade mathematics are only the minor players unless there is context, or utilization.

To be sure, foundational mathematics (i.e., arithmetic) cannot be ignored. In Geometry, foundational mathematics should be utilized continuously as students are challenged to consider context in all cases. The National Council for Teachers of Mathematics (NCTM) adopted the Principles and Standards for School Mathematics in 2000, replacing the original Curriculum and Evaluation Standards for School Mathematics.[2] In the Process Standards, NCTM chose Connections as a standard and stated, “Instructional programs from prekindergarten through grade 12 should enable all students to… recognize and apply mathematics in contexts outside of mathematics.”

In “We the People,” students are challenged to go far beyond the computational aspect of coordinate geometry and into the critical stage of analysis and synthesis.[3] This lesson provides a structure for exploration of real data. Students utilize actual statistical data from the U.S. Census Bureau[4] to explore the rate at which populations of various countries rise and fall. Students are asked to critically analyze data and to develop their own ideas about cause and effect, extending their ideas to conditional statements. Students will find political, economic, global, and/or health related issues that have played significant roles in the population dynamics of the world.

As with most lessons that I write, my focus is primarily on the application of the material and the autonomy of the student as he or she explores the context for meaning. “We the People” demonstrates this bent for recognizing Value Beyond School as students are guided to explore and understand real data.[5] Students are first asked to make connections from the data to the context as they explore whether or not the growth rate over 20 years (according to the Census data) is, on average, representative of the actual data at the 10 year mark.

An example of this is given where the global population from 1950 and 1970 would suggest that the global population growth rate would be 57.5 million each year, placing the population in 1960 at 3.135 billion. However, according to the Census data, the actual global population in 1960 was 3.04 billion. This .095 billion (95 million) difference represents a deficit in the population. The reason for this deficit is the context that is so critical. Why do you think the global population growth in 1960 was 95 million less than it should have been had growth been steady? The potential for this question to guide profound conversations in the classroom is why students are asked to process this very same question in an actual research paper.

As mathematics teachers search for the right context by which they engage their students, consider the development of the knowledge, which in large part can be refined and standardized to be most effective. Students are not initially presented with the formula or the answer, but with a fairly difficult question that requires much exploration. Only when students begin considering how to determine that growth rate of the population, do they realize that they need to create a ratio comparing population growth to time. This development of slope as a rate is simple and effective. Next, students look at the data and ask if it is consistent with the rate according to the average of the data points. Students are amazed that they can simply average the population and the year to determine the midpoint. Now, explaining that process as the actual Midpoint Formula seems like elementary math for students (and for teachers!).

As the lesson concludes, students gain a sense of purpose. They have not just done equations and simplified numbers so that they can box it up and turn it in. They are investigating. They are inquiring. Whitehead further explains the utilization of an idea as “relating it to that stream, compounded of sense perceptions, feelings, hopes, desires, and of mental activities adjusting thought to thought, which forms our life.” This lesson, although mathematically simple, does just that; it connects students to a situation. If you’ve ever thought that this type of lesson just doesn’t fit in your classroom; perhaps you don’t believe it will be beneficial for your lower students, consider this, Newmann, Bryk, and Nagaoka found that high-quality assignments in mathematics actually have greater impact on low achieving students as compared to their high achieving peers. [6] This type of interactive lesson can, and will, impact your students. 


Snapshot

1. Engage

a. Population Discussion [10 minutes] – Have students explore what they know about their immediate surroundings regarding population. Expand this idea to the world and begin to discuss factor that have influence on population.

2. Explore

a. Look at the Numbers [25 minutes] – Students explore the growth rates of 4 countries.
b. The Story [25 minutes] – Students further analyze the data, making an argument as to whether or not the growth rate of the countries is actually a good indication of the actual population of those countries. Students also engage in research to help understand what factors may have increased or decreased the country’s population.
c. Compare, Contrast, and Estimation [20 minutes] – Students attempt to find countries with similar and opposite growth rates.

3. Explain

a. Understanding Rates and Averages [20 minutes] – Students look at their previous work and connect it to slope, midpoint, and parallel and perpendicular lines.

4. Extension

a. Logic [20 minutes]– Student prepare conditional statements related to the country they focused on in the exploration phase.
b. Research [1 hour]– Students now conduct research to determine the validity of their conditional statements, highlighting social, economic, health and other issues that support or negate their claim.

Materials

Pencil
Population Data Handout – 1 per student
Ruler, protractor, or other straightedge – 1 per student
Graph paper
Colored pencils
Internet (if completing the Research portion of the lesson)

Vocabulary

Coordinate Pair – Refers to two numbers that define the location of a point on a 2-dimensional plane. That is, (x,y) such that x and y signify the distance from 0 on the x- and y-axis, respectively.
Horizontal –
An aspect of a line that is parallel to the x-axis, or the horizon.
Midpoint –
Defined as a point C on a segment AB, such that AC=CB. Found by taking the average of the x- and y-coordinates of the coordinates that define the segment.
Parallel –
Coplanar lines that extend forever in both directions without intersecting are considered parallel. That is, .
Perpendicular –
Lines that intersect to create 4 right angles are considered perpendicular. That is, .
Skew –
Lines that are not coplanar and do not intersect.
Vertical –
An aspect of a line that is parallel to the y-axis.

Student Objectives

Students will…

  • Critically analyze the population dynamics of various countries around the world;
  • Explore the growth rates of the countries by applying slope concepts using coordinate pairs;
  • Determine similar and contrasting growth rates by categorizing qualities of the population statistics including rate, population size, and more;
  • Make inferences about political, economic, global, and/or health related issues that affect population;
  • Develop understanding of how to find the midpoint of a segment, first by estimation and then by explicit calculation;
  • Create estimates about the cause and effect related to world issues and population while exploring deeper understanding of those issues that influence the population of the countries in question.

Engage

Population Discussion [10 minutes] – Have students explore what they know about their immediate surroundings regarding population. Expand this idea to the world and begin to discuss factor that have influence on population.


Lead student through a discussion about local, regional, national, and global population. Focus on estimates for the following areas:

School
Town
State     |  Actual in 2006 – 3,579,212
USA      |  Actual in 2006 – 299,398,484
World   |  Actual in 2006 – 6,554,167,609

If students have access to computers have them find actual data. Otherwise, utilize http://quickfacts.census.gov/qfd/states/40000.html to find the statistics.

Have students estimate the population of the world over the last 50 or 60 years. Utilizing a chart similar to the one below, have students estimate the population of the world. After each estimate, allow students to carefully consider their next estimation. Depending on the classroom, you may consider simplifying the values written as 6,857,124,831 to 6.9 billion.

Estimated World Population Actual World Population
2010 6,857,124,831 2010 6,857,124,831*
2000 2000 6,092,409,072
1990 1990 5,284,486,614
1980 1980 4,452,557,135
1970 1970 3,711,996,957
1960 1960 3,041,685,851
1950 1950 2,555,955,393

* According to http://www.census.gov/main/www/popclock.html. Retrieved 16:42 UTC (EST+5) Jul 20, 2010.

 

Ask students to find the growth rate of the global population. As students determine the rate, ask them if what they find is consistent for each ten-year period.

Students will think of many ways to represent the growth rate. The teacher should guide students to consider plotting the values on a coordinate plane. Students can find the difference between any two values and divide by the number of years between them.

For example, students may look at the growth rate from 1960 and 1980. The population change is 1,410,871,284. This occurs over 20 years. Therefore, the growth rate is 1,410,871,284 ÷ 20 = 70,543,564 per year.

After students determine the growth rate, reiterate the question of whether it is consistent. Have them explain how they know that it is or is not, and what my account for their finding. For a detailed look at the global growth rate, check out http://www.census.gov/ipc/www/idb/worldpopchggraph.php

 

 

Explore

Look at the Numbers [25 minutes] – Students explore the growth rates of 4 countries.


On the Population Data Handout, statistics are provided for eleven countries. The twelfth spot has been left empty to allow students the opportunity to explore a country they want to analyze.

Have students choose a total of four countries. For each country, have students:

  1. Plot the data on a graph
  2. Find the growth rate for 5 segments using the data or graph. (Since students will be utilizing the midpoint later in the lesson, have them choose non-consecutive data points. For instance, choose 1950 and 1970 instead of 1960 and 1970. Or, perhaps choose 1970 and 2010.
The Story [25 minutes] – Students further analyze the data, making an argument as to whether or not the growth rate of the countries is actually a good indication of the actual population of those countries. Students also engage in research to help understand what factors may have increased or decreased the country’s population.


Ask students to explore whether or not the growth rate is, on average, representative of the actual census data.

One possible approach to answering the question is the midpoint formula. Students can look at two non-consecutive years, perhaps 1960 and 1980. The point exactly between the two population values and between the low and high years is the midpoint.

Instead of teaching the midpoint through a formula, have students think about the following problem:

A convenience store is considering adding a third option to their drink sizes. Currently, they offer a small and a large option. The small drink holds exactly 12 ounces. The large drink holds exactly 40 ounces. The convenience store wants the size of the medium drink to be exactly in between the small and the large drink. How many ounces should they medium drink hold.

Now, the convenience store wants to know how much to charge for the drink. They want the price to be exactly in between the small and the large drink. The small drink is $1.00 and the large drink is $2.00.

Which drink is the best deal for the customer?

—————————-—————————-—————————-—————————-

The medium drink should be 26 ounces.

The cost of the medium should be $1.75.

The small drink is $1.00 for 12 ounces or 

The medium drink is $1.50 for 26 ounces or 

The large drink is $2.50 for 40 ounces or

 

As an example, we can look at the segment between 1960 and 1980. The two points can be written as (1960, 3,041,685,851) and (1980, 4,452,557,135), respectively. To find the midpoint, we can take the average of both the years and the populations:
and 

Compare, Contrast, and Estimation [20 minutes] – Students attempt to find countries with similar and opposite growth rates.


Give students about 20 minutes to begin looking at relationships between their four countries. Students can organize the similarities and differences on a chart or Venn diagram. Make sure that students clearly label each similarity or difference. For example, under Similarities students may write, “Between 1983 and 1989, Romania grew by about 400,000 people. Also, between 1996 and 1998, Chad grew by about 400,000 people.”

Let students discuss what similarities or differences they recognized. This will give other students ideas if their observations were limited.

As students complete their comparison, have students consider how they made their conclusions. Allow students enough time to consider the validity of their guesses.

Explain

Understanding Rates and Averages [20 minutes] – Students look at their previous work and connect it to slope, midpoint, and parallel and perpendicular lines.

After students have discussed the visual cues that led them to realize similarities and/or differences, give students another 20 minutes to utilize slope to determine actual growth rates.

With new growth rates, have students re-create their charts or Venn diagrams. Considering the example of Romania and Chad, students would recognize that the growth rate of Romania was actually , while the growth rate of Chad was actually . With the math clearly defining the growth rate, it is clear that these two seemingly similar time periods are, in fact, very different.

If time allows, challenge students to find a time period where the growth rates are very similar. This is a great way to discuss parallel lines. If they find a time period that is similar, have them plot the two segments and discuss what they see. The segments, if extended will not intersect. Lead the conversation from growth rate to slope by discussing how m1=m2.

Student may also attempt to find periods of time in which growth rates are essentially opposite. Discuss with students whether this is even possible.

Suppose a country has a negative growth rate. For example, in Romania, from 1995 and 2001, the population decreased by about 44,000 per year. Mathematically speaking, a line parallel to 44,000/1 would -1/44,000 (the negative reciprocal). Have students interpret the meaning behind this statement. -1/44,000 implies that a country loses only 1 person in 44,000 years!

Since it is unlikely that actual parallel segments will be found, have students find segments that represent the extremes of growth and decline.

Extend

Logic [20 minutes]– Student prepare conditional statements related to the country they focused on in the exploration phase.


The extension to this lesson starts with logical statements, but could actually be carried out in a social studies class. Have students explore their historical data to create cause and effect statements. An example of a cause and effect idea about a country might be:

Cause: Economic crisis in Romania
Effect:
Growth rate of the country decreases

Have students develop If-Then statements using the cause and effect ideas.

If Romania is having an economic crisis, then the growth rate of Romania will decrease.

This is a great chance to talk about converse, inverse, and contrapositive.

Statement: If Romania is having an economic crisis, then the growth rate of Romania will decrease.
Converse:
If the growth rate of Romania is decreasing, then Romania is having an economic crisis.
Inverse:
If Romania is not having an economic crisis, then the growth rate of Romania will not decrease.
Contrapositive:
If the growth rate of Romania is not decreasing, then Romania is not having an economic crisis.


Research [Individually]– Students now conduct research to determine the validity of their conditional statements, highlighting social, economic, health and other issues that support or negate their claim.


The structure of this lesson is such that it allows students to pursue their curiosity about certain countries and the history of that country. If a student finds a particular growth rate interesting and is unable to determine a very obvious cause for the circumstance (increase, decreased, or simply sustained population), guide the student to explore the country’s history in more detail.

PASS

Content Standards

Note: Asterisks (*) have been used to identify standards and objectives that must be assessed by the local school district. All other skills may be assessed by the Oklahoma School Testing Program (OSTP).

1.2 – Logical Reasoning: State, use, and examine the validity of the converse, inverse, and contrapositive of “if-then” statements.
2.1* – Properties of 2-D Figures: Use geometric tools (for example, protractor, compass, straightedge) to construct a variety of figures.
5.1 – Coordinate Geometry: Use coordinate geometry to find the distance between two points; the midpoint of a segment; and to calculate the slopes of parallel, perpendicular, horizontal, and vertical lines.

Process Standards

2.1 – Communication: Use mathematical language and symbols to read and write mathematics and to converse with others.
2.2 – Communication: Demonstrate mathematical ideas orally and in writing.
2.3 – Communication: Analyze mathematical definitions and discover generalizations through investigations.
3.1 – Reasoning: Use various types of logical reasoning in mathematical contexts and real-world situations.
4.1 – Connections: Link mathematical ideas to the real world.
5.1 – Representation: Use algebraic, graphic, and numeric representations to model and interpret mathematical and real-world situations.
5.2 – Representation: Use a variety of mathematical representations as tools for organizing, recording, and communicating mathematical ideas.


[1] The specific objectives are Mathematics 6th Grade PASS 1.3 and 2.2.e. You can find more about PASS at http://sde.state.ok.us/curriculum/PASS/default.html.

[2] NCTM published this book in 1989. It transformed the role of the national organizations for all of the content areas.

[3] Bloom’s Taxonomy guides most of my lessons. It is paramount that teachers recognize how the style of questions promote deeper learning.

[4] The official Census Bureau website can be found at http://census.gov.

[5] NCTM has put together a really simple list of ways to use real data in the classroom. Visit the site at http://bit.ly/k20alt-realdata.

[6] Newmann, F. M., Bryk, A. S., & Nagaoka, J. K. (2001). Authentic intellectual work and standardized tests: conflict or coexistence. Chicago, IL: Consortium on Chicago School Research.