• Observe student participation during class discussions and number line activities to see whether students (individuals and groups) demonstrate that number lines are infinitely long in both positive and negative directions and that there are infinitely many spaces between each interval, no matter how small the interval.
• In the evaluation of student responses on the Lesson 1 Exit Ticket, look for similar classification as real and rational for and . Recognizing that both terminating and repeating decimals are rational is an indication of how well students understand the difference between numbers that can be expressed as numerator and denominator and those that cannot.

Part 1: Real Numbers: Overview

Ask students to define the number system we operate in for most purposes. Students should provide the answer: the real number system.

Next ask the class the following questions to get them discussing among themselves (with appropriate amounts of time between questions for discussion): “What is a real number?  [IS.4 - Struggling Learners] What types of numbers are included in the real number system? Are these numbers subsets of the real numbers? How can we recognize certain categories of real numbers? What methods of comparison do we have at our disposal? If you were asked to provide an example of a real number, what would you say? Where in the real-world do real numbers appear?”

A real number is either rational or irrational. Ask the class: “What is a rational number?  [IS.5 - Struggling Learners] What is an irrational number?” Give students time to suggest definitions and examples before proceeding. Once students have had time to define rational and irrational numbers, provide the following definitions. “A rational number can be written as the ratio of a/b, where b ≠ 0. We can also state that a rational number is terminating or repeating. An irrational number is a number that is not rational, or cannot be written as the ratio of a/b. An irrational number is nonterminating and nonrepeating.”

Now ask the class: “When saying that a rational number can be written as a/b, with b not equal to 0, what does that really mean? Why don’t all numbers fit that description?” Give students time to discuss this, while prompting them to think about certain numbers. Good ones to ask students to consider are 2 and . 2 is a rational number, but does not look like one. Ask students why it is a rational number before proceeding. (2 is rational because it can be written as ,  but simplifies to 2.)

“When asking if a number is rational or not, what we are really asking is this: If we have a decimal number, can we write that number as a fraction or ratio? If so, what constraints are placed upon the decimal? To be a rational number, the decimal number must either repeat or terminate. Let’s look at a couple of cases to demonstrate this.” [IS.6 - Struggling Learners]

“Suppose we have the repeating decimal . Can we write this number as a fraction? The answer is yes, but what is the fraction? To answer this, we first assign a variable to the repeating part of the decimal (only the repeating part). What we call this variable does not matter, so let’s call it a. Let a = the repeating part of that decimal, or a = .9292929...” (Explain that it only matters that you have a full repetition here, so a = .92... or a = .9292..., two repetitions, would work as well.)

“Now we need to do a trick with a. By itself, a is not going to tell us what the fraction is, so we need to cancel out the repeating part somehow. The way to do this is: Compute 10ⁿ · a, where n = the number of digits under the bar. Here n is 2, so we compute 100a. Since a = .929292... that gives us 100a = 92.9292... Then you can compute 99a, which gives us:

which no longer has any repeating part, as all the repeating digits are canceled with the subtraction.”

“We can make this into a fraction by solving for a to get:

We can write  as . Thus we have written the number as a ratio, with b not equal to 0. Therefore, the number is a rational number.” [IS.7 - Struggling Learners]

Now ask the class: “Suppose we have the terminating decimal 12.68. Can we write this number as a fraction?” (Yes, it is ).

“This is another example of a rational number. These examples provide an illustration of how we can take a decimal and determine whether or not it is rational.”

“Let’s look at an example of a number that is not rational. Suppose we have the decimal 2.9345612… Nothing repeats and it doesn’t terminate.” Ask students, “How do we decide whether this number is rational, whether we can write the decimal as a fraction?”

Continue by explaining that: “We can use a guess and check method. The repeating decimal conversion method used earlier only works when a portion of the decimal repeats, which does not happen with this example. We also know that we can’t simply write the decimal portion as a numerator over a specific denominator or power of 10 because the number doesn’t end! So, let’s try to find a fraction representing this number.” [IS.8 - Struggling Learners]

“We know that 0.9 is . We also know that 0.94 would be written as . Therefore, our fraction would be somewhere between  and .”

“Trial and error gives:

Keep investigating and try to find one fraction that equals 2.9345612…” (Students should realize that they always need another number in between two close fractions and can never exactly replicate the nonrepeating, nonterminating decimal.)

“This means that this number is irrational, since we cannot find a fraction that represents it. It is nonterminating, nonrepeating, and cannot be written as the ratio .”

Activity 1: Rational/Irrational Number Examples

Place students into groups and have each group provide a list of five rational numbers and five irrational numbers; at least one of the rational numbers must be a repeating decimal. Include decimal and fractional forms. Assist students with finding fractional forms of the repeating decimals as necessary. When students are ready, have some of them share their examples and explain why the numbers they picked are rational or irrational.

Once the activity is done, remind the class, “We have so far approached the real number system by giving an overview of the two main subcategories of numbers: rational and irrational. Within the rational subset, we have other categories of numbers.” Ask the class, “What are these other categories of numbers? What are some examples?” (integers, whole numbers, and natural numbers)

Once students have provided the sets, ask if anyone can provide an example of an integer. After a few examples, give the class the official definition of integer. An integer includes both the positive counting numbers and additive inverse of each (negative counting numbers), as well as zero. The set of integers thus looks like this:

The numbers continue infinitely in both directions.

Now ask the class, “What is a whole number? Can anyone give me an example of a whole number?”  [IS.9 - Struggling Learners] Once answers have been provided, give the definition of a whole number. A whole number includes 0 and all counting numbers, or natural numbers. The set of whole numbers looks like this:

“We have already used the term ‘counting number’ in the two previous definitions. A counting number is simply a number that shows a one-to-one correspondence between the number and the amount shown. Therefore, we cannot have a negative counting number. We also cannot have 0, since we start counting with the number 1. Zero (0) is representative of an amount, ‘nothing’ or ‘zero pieces.’ However, 0 is not used to count. The set of counting, or natural, numbers looks like this:

Activity 2: Relationships Between Numbers

Have students create a diagram using the definitions of subsets within the real number system to illustrate the relation of each subset to the universal set of the real number system. “Universal set” simply means “encompassing set” or “umbrella set.” Within each set, provide at least three numerical representations. (Students should draw a Venn diagram representation.) Make sure the diagram has whole numbers, natural numbers, integers, rationals, and irrationals. You may need to assist some students with this diagram; as needed, assist those who are having trouble.

Once the activity is done, tell the class that rational and irrational numbers do not always come in fraction and/or decimal form: “In some cases, you might be presented with a square root. You must determine whether or not the square root is a rational or irrational number and further identify what kind of rational number it is, when appropriate. For example, suppose we randomly write the following square roots:

Are these numbers rational or irrational? How can we tell?” Give the class time to discuss before continuing. Tell the class that each one of these is irrational because they are nonterminating and nonrepeating.

Now ask the class: “When would we get a rational square root, and can you provide an example of one?”After giving the class some time to answer, say, “We will have a rational square root when we have a perfect square under the square root sign. Therefore, we would either need a perfect square natural number, or a perfect square real number in decimal form. For example, [IS.10 - Struggling Learners]

and

are both examples of rational square roots.”

Activity 3: Exit Ticket

Ask students to complete the Lesson1 Exit Ticket (M-A1-3-1_Lesson 1 Exit Ticket and KEY.docx). Students complete a chart by identifying all categories that apply to a given number, including real number, rational number, irrational number, integer, whole number, and natural number. Square roots are included.

Part 2: Graphing and Comparing Real Numbers

This portion of the lesson’s focus is two-fold:

1. Graph real numbers, both on a number line and a coordinate grid.
2. Compare and order real numbers.

“We will also revisit one of our initial questions, regarding the occurrence of real numbers in the world around us. Specifically, where can we find integers? Where can we find irrational numbers? Where can we find rational numbers?” and so on.

Start off this part of the lesson by asking the class to place the following numbers on the number line, using estimation where needed. (Placement of irrational numbers will need to be estimated.)

Draw this number line on the board without the locations of each number. Ask students in turn to mark each location. [IS.11 - Struggling Learners]

Activity 4: Human Number Line

To start out this activity you will need to make an area into a number line. Using masking tape to make straight lines, set up chairs or desks in a straight line with numbers attached to each chair or desk. Once the grid is set up, have the class participate in a human number line activity as follows: Give students each a number and tell them to place themselves in the right spot on the number line. (Note: If you use desks, you may have a problem of spacing with fractions, though you could keep them in the activity and see how creative your students get.)

Begin with an easy one. Assign the number  to one student and  to a second student. Ask both students to arrange themselves in proper order between the two chairs that mark their locations. While the two students are deliberating about where to stand, discuss with the class the similarities and differences between the two numbers. Similarity: Their values are close to each other. Difference: five halves is rational and the square root of 6 is irrational.

By this time, one or more students will have found an approximation for  as 2.449, and that is enough information to show that the first student should be standing closer to 3 and the second student should be standing closer to 2.

Once the activity is complete, ask the class: “When might we need to graph ordered pairs of real numbers?”

After giving students a chance to answer, say, “We need to graph ordered pairs of real numbers in any situation in which we relate the value of one variable to the value of another variable. For example, if we look at the cost of a particular vehicle over x number of years, we relate the price of the vehicle to each year. With this example, real numbers, rational numbers, and natural numbers are all involved.”

Now ask the class the following questions:

• “We might also plot the distance traveled as a result of time. In this case, which numbers are involved?” (rational numbers)
• “What if we wanted to examine weight of babies at different months? Which numbers are involved?” (positive rational numbers)
• “What if we wanted to examine the amount of money in a checking account in a particular week? Which numbers are likely involved? Which numbers could be involved?” (rational numbers)

After the class has answered these questions, have them graph the following ordered pairs on a coordinate grid. [IS.12 - All Students]

Assist students as needed with graphing these points.

Activity 5: Human Point Plotting

Present another activity similar to the human-number-line activity. Before the activity starts, set up a small grid and designate one point as the origin. You can use a lot of duct tape for this grid, or arrange desks and chairs into a grid. Students will be given an x-value or y-value and must work together to plot points on a large grid, with the condition that points be as close together as possible. This is open-ended and requires deeper thinking. Students will experience the distance formula indirectly, while comparing and ordering real numbers during this exercise.

Once the exercise is done, explain this problem to the class: “Suppose we wished to graph the weights of babies at various months of age. Depending on the scale, we could either have rational or irrational weights. For example, it is possible to weigh 18.473521… units. However, weight can be rounded to the nearest tenth of a decimal or written in fractional form (either as a precise weight or rounded). Weights cannot be negative or zero. They can be rational or irrational. They can also be natural numbers, as when a baby weighs exactly 18 pounds, or rounding is used.”

Activity 6: Ordered Pairs

Have students create a data set with 20 ordered pairs, [IS.13 - Struggling Learners] indicating the birth weight of one particular baby over the course of 12 months. For example, month 1 and a weight of 7 pounds 3 ounces would be graphed as approximately (1, 7.2). Suppose the medical staff used alternate methods of recording the baby’s weight. Therefore, include rational and irrational numbers as natural numbers (when rounding), rational numbers as decimals and fractions, and irrational numbers. Write a summary statement comparing the growth of the baby from month to month. Between which two months did the baby gain the most? Gain the least? Sample data is available at http://www.infantchart.com.

Activity 7: Real-World Data Points

Decide on a real-world example where integers, both positive and negative, can be gathered. Gather a sample of 20 data points with two variables involved. Example: record 20 average weekly temperatures over the course of 20 weeks, with week number as the independent variable and temperature as the dependent variable.

For review of the lesson, hold a class discussion related to any questions, difficulties, or epiphanies experienced during the lesson. The ending activities are higher-level and require some analysis and critical thinking. As a result, students may have several questions on the topics presented here.

Extension:

• Have students create a chart outlining the different subsets of real numbers found in the real world. Students should include at least five real-world examples for each number type. Example: A bank encounters real numbers, rational numbers, integers, whole numbers, and natural numbers.