This is the first part of a new sub-chapter in the chapter on Seventh Grade math, in what will be a book called Traditional Math, to be published by John Catt Educational.
Chapter 7.2: Rational Numbers
I recall when teaching an accelerated seventh grade math class, introducing the topic of rational numbers. When I had finished, a boy with unusual insight into math made the following observation: “Since they can contain whole numbers as well as fractions, rational numbers are deeper than just fractions”.
I have kept his words in mind. Rational numbers can be confusing for students, particularly since they appear to be a fancy way of saying “fractions”. This notion is reinforced since most chapters on rational numbers focus on operations with fractions. Students are also confused by the notion of “irrational” numbers, since the topic is introduced before they know what square roots are, or what the number pi is all about.
This lesson provides the definition of rational numbers which is revisited later in the year. It is also revisited in later courses as students understand more of the structure of the real number system.
Review of Decimals and Fractions. I like to begin with a quick review of decimals and fractions. The warm-ups for the day typically will have questions such as “Which is greater? 0.099 or 0.2?” and “What is the decimal equivalent of 1/3?”, “What is 2 1/5 as a fraction?” “What is 13/5 as a mixed number?”
In previous grades they have worked with decimals and how to convert from fraction form to decimal form. Having said this, the students often act as if this is the first time they have worked with converting fractions to decimals. I therefore spend a few minutes on the first day going over how to express decimals as fractions, and vice versa. They should also know the decimal representations of certain fractions, such as 1/4, 1/5, 1/8, 3/8, 7/8, as well as the repeating decimal representations of 1/3, 2/3, and the ninths. (Some students will not know these; I therefore make one of the goals of this unit that students know these representations.
Rational Numbers. Because students are not yet fluent using symbols, starting with the formal definition of “rational number” will be a distraction. The definition is:
A rational number is a number that can be written as a/b where aand bare integers and b does not equal zero.
Instead, I begin with discussion and examples which will lead to the above definition of rational numbers. I first ask for examples of fractions. After the usual ones (e.g., 2/3, 5/6, etc.) I’ll ask whether -2/3, -5/6 are also fractions. Students will agree but there might be some hesitation. “What about 3/2, 5/3, 13/2. Are they fractions?”
Someone may point out that they are “improper fractions”. I like to make clear that what are referred to as “improper fractions” are still considered fractions. They can be expressed as mixed numbers as well, so I will show what those fractions are in that form, picking on students to do the conversions. In fact later in the year I make it known that unless directed to do so, students do not have to convert to mixed numbers; a fraction like 39/8 can be left in that form.
I then ask about numbers such as 6/3, 9/3, -25/5, -8/2, 4/1. Are these fractions? There will be mixed responses. I point out that they’ve learned that 5/5, 6/6 and so on are “one whole” or the number one, and then ask if people have changed their minds. I’ll state “I think we can safely say that some fractions can be mixed numbers, and others can be whole numbers.” (Were I to leave it as an open question, there is bound to be one student who disagrees, which while interesting, takes away from the momentum and direction of the lesson.) I then plot the numbers we have on a number line.
I will then give some examples of decimals: 0.333…, 0.75, -1.25, and plot those on the number line. I’ll point out that as we learned from the warm-up exercises, we can express decimals as fractions as well.
Now I present the definition of rational numbers, pointing out that as we’ve seen we can express whole numbers, mixed numbers and decimals in fractional form. First, an informal definition: Numbers that can be expressed in a fractional form are called “rational numbers”. I’ll follow it up with the mathematical definition (given earlier in this section). I make it clear that rational numbers include fractions and whole numbers. They also include negative numbers. And like all numbers, they all can be located on a number line.
In the spirit of my unusual student who saw rational numbers as “deeper” than fractions, I now point out that the term “rational number” expands the concept of fraction, so that it includes numbers that are between two whole numbers as well as whole numbers themselves.
Checking for Understanding. I’ll ask how they know that, say, 9/3 is a rational number, looking at the definition. This is a worked example and I’ll ask whether nine is an integer, and three. Hearing agreement, I summarize that there are two integers in the form a/b.
I’ll give other examples, asking the same question but without the hints. Mixed numbers are included as examples, so students will need to make the leap and put them into fractional form; e.g., 1 2/3 is 5/3 to show conformity with the definition. Also included in the examples are decimals. Again, they must make a leap and put them into fractional form.
I will ask whether decimals such as 0.333…., or 0.666… are rational numbers. Since I discussed repeating decimals at the beginning of the lesson, they should recognize at least one of these in fraction form. Then again, one learns to live with blank stares and temporary amnesia, and take it as part of the job of teaching. In any event, the question serves as a lead-in to the next topic.
Terminating and Repeating Decimals. Students are familiar with the two types of decimals, terminating (such as 0.12, 1.2, and so on), and non-terminating repeating (such as 0.333…, 5.222…, 0.0101… and so on).
Repeating decimals are those that will go on forever, but have a repeating pattern. Both the terminating and repeating decimals can be represented in fractional form, and are therefore, by definition, rational numbers. The decimal 0.333…. is 1/3, 0.0101… is 1/99.
Irrational Numbers. This brings us to the rather perplexing topic of irrational numbers. I once had a student who, thinking he was making a play on words, asked “If there are rational numbers, are there irrational numbers?”
He looked surprised when I told him that was an excellent question and in fact there are. An irrational number is one that cannot be expressed as a/b where a and b are integers. In such cases, the number will be a non-terminating and non-repeating decimal. Non-repeating decimals do not have a pattern that repeats. For example, the decimal 0.01020304…if continued in this fashion does not repeat, and it is non-terminating. It is therefore an irrational number. There is no fraction that will produce such a decimal representation.
Students will have to take such claims on faith since the proof that irrational numbers are non-terminating and non-repeating is typically covered in college level math courses. Similarly, the number pi, which students may have heard about is also irrational and students must take it on faith that the famous sequence of numbers does not repeat.
Irrational numbers will be revisited with respect to square roots in the unit where square roots are discussed.
Division by Zero. The final discussion of this lesson is to talk about division by zero and why it is impossible. I start by asking students what 0/5 equals and then brace myself for the shock of that some students will not know this. We go over that zero divided by anything is zero because zero multiplied by any number is zero. I then ask if 5/0 has an answer. Many will say “zero” to which I respond “So you’re saying that zero times zero equals five?” I continue by asking whether other numbers will do and they quickly see that no number will work. This demonstrates why division by zero is impossible and I explain that it is considered (and called) “undefined”. For any fraction, zero cannot be in the denominator, because it represents division by zero, and that’s where I leave it. This concept should be revisited at appropriate times during the course.