Prerequisite Knowledge and Objectives
Before beginning this lesson, students should already know how to evaluate expressions and find subsequent values in a recursive pattern.
By the end of this lesson, students will be able to organize real world data into tables and then graph the resulting ordered pairs. Students will initially demonstrate their understanding by completing two activities in this lesson and in later lessons by creating scatter plots from data that have either been given to them or from data they've collected.
Opening
For the opening, have the students complete a K-W-L chart for domain and range. The activity should be completed as a class.
Pre-Knowledge: What you already KNOW
Have you ever heard the words domain and range? What do these words mean to you? Where have you heard them?
Students may have heard these words in different contexts. It is often helpful when learning new vocabulary if students associate them with words or concepts they already know. Students may have heard the word domain used to talk about territory in history class or range used to discuss distance or a span of space. Students might also associate range as a measure of central tendency, as in when they studied mean, median, mode, and range in middle school.
Mini-Reading and Discussion: What you WANT to know
Have the students read this short blurb on domain and range. Then, proceed into a discussion on the terms.
The domain of an expression is all of the possible values for which x can be. For example, in the expression 2x, x can be any value positive, zero, or negative because no mathematical restrictions are placed on x. The domain of an expression is often all real numbers, unless a given mathematical operation becomes undoable using certain numbers. Look at the expression2/x. The operation in this expression is division. We know that we can divide by any number other than zero, so the domain can be any real number besides 0.
The range of an expression is the values for which the evaluated expression can be. For instance, the expression 2x can be any positive number (as long as x is positive), any negative number (as long as x is negative), or zero (as long as x is 0). Like domain, the range of an expression is sometimes restricted. Look at √x. No value of x can ever make this expression negative. Thus, the range of √x is zero and all positive numbers.
In mathematics, the definitions we came up with at the beginning of class are not wholly different than the ones we just read. The domain of an expression is indeed the territory or the space in which the x-values reside, while the range is the span of values for which the expression can be.
Practice: What you LEARNED
Put the following expressions on the board: x
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and x/4. Feel free to use other expressions, but remember that algebra 1 students have a limited knowledge of any functions that go beyond the five basic mathematical operations: exponents, multiplication, division, addition and subtraction. By the end of the year, students should also be to apply this to absolute value functions. Underneath each expression, write "domain:" and "range:". Give the students an opportunity to come to the board and determine what the domains and ranges are for each expression. After students have had a chance, review and discuss the solutions.
Activity 1: Building Tables
Provide the students with the following information. Then, ask them to complete the table one row at a time. Students should be able to hypothesize when coal reserves will run out.
The Energy Information Agency (EIA) reports that, in 2007, the total U.S. recoverable (mineable) coal reserves were 18,584 million short tons.
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Recoverable coal reserves refer to the amount of coal that can be mined from existing coal reserves. In 2008, the recoverable reserves in the U.S. fell to 17,875 million short tons. At what rate are the reserves being depleted each year? If we assume that the U.S. will continue to use coal at the same rate for the next 25 years, how many million short tons will remain by the end of this period? Students will use the data to determine whether or not the U.S. can continue to depend on domestic coal reserves at our current consumption level.
To complete this activity, have the students complete the activity handout (Lesson 1 Activity 1) found in the appendix. Initially, the students will be asked to simply subtract 709 million short tons (the difference of 18,584 and 17,875) for each year. After the first 5 years, however, the students will be asked to develop a rule that they can follow so they don't have to subtract one year at a time. This will be a short introduction to writing equations of a line, though it will not be introduced in those terms until Unit 5, following the district curriculum. By the end of this activity, students should have a completed table and a rule that states the reserve level x years after 2007 is 18,584 709x. The students already know how to evaluate an expression, so they should feel comfortable applying this rule.
Discussion and Short Lecture
After completing the above activity, transition into a discussion on coal and its connection with the carbon cycle. In the next activity, students will gain a better understanding of the carbon cycle along with learning to graph points. If you have time, you may also have your students read the following short articles on coal consumption; the third is actually a short clip from YouTube.com:
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"US Coal Consumption Trends": http://www.examiner.com/x-325-Global-Warming-Examiner~y2010m5d8-US-coal-consumption-trends (July 30, 2010) discusses expected American consumption rates and provides a brief discussion on the need to decrease our coal dependence for environmental reasons.
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"Oil Is Not the Climate Change Culprit It's All About Coal": http://www.wired.com/wiredscience/2008/12/oil-not-the-cli/ (July 30, 2010) discusses the negative impact of continued coal consumption.
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"A Convenient Truth": http://www.youtube.com/watch?v=71kckb8hhOQ (May 31, 2010) a comical satire of a public service announcement in favor of coal consumption.
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Before beginning the next activity, review with the students how to plot a point. Be sure to go over which axis is the x-axis and which is the y-axes.
Activity 2: Graphing Points and the Carbon Cycle
This activity is based on the two-person game Battleship, but has been modified to fit into a unit on energy. Students will each be given a board with the carbon cycle on it. A copy of the game play board can be found in the appendix. All freshmen students study the carbon cycle at Scholars in their Physical Chemistry class, so the transference of carbon in the cycle is understood at least on a basic level. In this game, students will be given 6 carbon molecules to place anywhere in the atmosphere. Their opponent will call out ordered pairs in an attempt to capture the free carbon in the atmosphere. If a player correctly guesses the coordinates of a carbon molecule, the carbon will move to another section of the carbon cycle. The first player to move 3 carbon molecules from the atmosphere to water, water to biomass (plants and animals), from biomass to fossil fuels, and then from fossil fuels back to the atmosphere wins. If a group finishes early, encourage them to play again.
The objective of this activity is not only to have the students practice plotting and reading points in the coordinate plane, but to also familiarize the students with the carbon cycle and reiterate some of the points about how coal consumption can impact the environment.
Discussion
After students have completed the game, have a quick discussion with the students about how carbon flows through the carbon cycle. Have a large playing board at the front of the class and ask students to identify where the carbon molecules are by naming the ordered pairs. Then, have the students explain where the carbon can potentially move within the cycle.
Exit Ticket
As the closing, have the students write down three things they learned about carbon in today's lesson.
