This is Common Core State Standards Support Video in mathematics. The standard is 8.NS.A.1. This standard states: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
First, let’s look at what standards would be connected to standard 8.NS.A.1. In seventh grade, we have standard 7.NS.A.2d that deals with converting a rational number to a decimal using long division. So, students already have some experience with doing division where we convert a fraction to a decimal. In eighth grade, we also have standard 8.NS.A.2 that deals with rational approximations of irrational numbers. Basically, what students will do here is compare the size of irrational numbers.
Let’s look at the idea of rational. By definition, a rational number is a number that can be expressed as a ratio of two integers. For example, if we have a and b both representing integers with b not equal to zero, this is how we would represent that. Since b can be equal to one, then any integer is also a rational number. Symbolically we can show it by a/1 would be equal to a. So again, any integer is also a rational number. An irrational number by definition is a number that’s not a rational number. What this is saying is that an irrational number cannot be expressed as a ratio of two integers.
Let’s look at this statement in the standard: understand informally that every number has a decimal expansion. What that is saying is that any number, be it rational or irrational, can be written in decimal form. Now, decimal expansions of rational numbers are either going to terminate or repeat in some type of pattern. As an example, 1/4 would be .25. One eleventh, on the other hand, has a repeating pattern where we have two digits that repeat, 0 and 9. Rather than write all those repeating numerals, one way of expressing a repeating decimal is to put a bar over the repeating digits, but we put that only over the repeating digits. That bar is called a vinculum.
Now, irrational numbers, the decimal expansions for them are going to neither terminate nor repeat. Common examples are a lot of our square roots. For example, the square root of 2 is 1.4142 and then it just keeps going. It does not terminate, and there is no repeating pattern. Pi is another common example. Pi would be equal to 3.1459 etcetera, etcetera. Again, it keeps going; no repeating pattern and it doesn’t terminate. The notion that irrational numbers have no repeating pattern doesn’t mean that digits cannot occur more than once. For example, here, the square root of 2, we have the digit 4 that appears several times, but again it’s not in any type of repeating pattern. What this idea of repeating digits means is that there is no set or group of digits that repeats infinitely. At this level, you cannot show that a given number is irrational via a non-repeating decimal expansion because it’s not possible to expand to an infinite number of decimal places. Instead, we need to use the fact that an irrational number is not rational. In other words, an irrational number cannot be expressed as a fraction with integers for the numerator and the denominator.
Now let’s look at this statement in the standard: for rational numbers show that the decimal expansion repeats eventually. For rational numbers, the decimal expansions will terminate or repeat. Now, there’s a little bit of confusion here. The standard says that the rational numbers will repeat eventually, but that seems to contradict the idea of terminate. The way to interpret that would be, for example, 1/4 is 25 hundredths. But what we can do is add zeros infinitely. So, in a sense, we do have 0 that is our repeating digit. So that’s how to interpret this piece of the standard, that when they say repeats eventually, they do mean that it is possible for it to terminate if that repeating digit is a 0.
Let’s look at some examples of rational numbers, 1/3. The decimal expansion for it has a repeating pattern of the numeral 3. So, we could express it in this form with the vinculum. Two over nine turns out to be a repeating pattern where we have twos that repeat, and again, this is how we can express it in a shorter manner. Three over 11; that’s a little bit different. We have a pattern where two digits repeat, 2 and 7, and again, this is how we can write that in a shorter way. One over seven—interesting pattern here. We have to go out six digits to see the repeating pattern. Here it’s the 142857 that repeat.
Now let’s look at the last part of the standard that deals with converting a decimal expansion which repeats eventually into a rational number. So, in other words, we have to take a decimal and put it in the form of a fraction. Converting a terminating decimal to a rational number is simply a matter of converting the number to a whole number and placing it over the appropriate power of 10. In other words, let’s say we have something like this. What we need to do then is pretend that there are no decimals or anything, and we take our number, and make it a whole number, and that’s our numerator. Then we put it over the appropriate number of tens. So, our denominator in this case needs to be 10 because there’s only one place to the right of the decimal point.
Let’s look at this example. Again, we make it into a whole number, and that’s going to be our numerator. Then we place it over the appropriate power of 10, which in this case would be 100. So, this is 75/100. Sometimes it might be necessary to simplify the fraction if it’s called for in the problem or in that context. So, in this case, we need to simplify 75/100 to 3/4. So, let’s express this in a general form. In order to convert this decimal to a rational number, we just make it into a whole number, and that’s our numerator. Then we put it over the appropriate power of 10, which in this case will be 10 thousandths. So our denominator would be 10,000.
Now we come to the tricky part, converting a repeating decimal to a rational number. That’s going to be a challenge. The dilemma lies in how to eliminate the repeating portion of that decimal expansion. The key is doing an appropriate subtraction. That’s how to do this. Let’s take this example. We have a decimal where we have a single digit that repeats. It’s a 6. So, let’s make this into an equation. We have a single digit that repeats, so we need to multiply this by 10 because we only have one digit that repeats. So, that’s one decimal place, which again would be a power of 10, in this case again, just 10. We multiply that by 10, and this is our new equation. Now we need to do our subtraction, so we do that. Notice that we have eliminated the repeating portion. In our difference, we have nothing but zeros. So, we can get rid of all those zeros, and now we have a nice simple equation. We need to divide by nine to solve, and so we have y is equal to 6/9. So, that simplifies to 2/3. So, either of those two fractions, 6/9 or 2/3 would be equivalent to the repeating decimal .66666...etcetera.
Let’s take another example. Let’s set up our equation. We notice that we have three digits that repeat; 1, 3, 4. Since it’s three digits, we need to multiply by 1,000, so let’s do that. Now we need to do our subtraction. Notice that it is set up to where when we subtract, we will eliminate the repeating decimals. Do our subtraction. We can eliminate all of the zeros. Now we simply have to divide by 999. Solve. We get 134/999. So, our fraction 134/999 is the rational equivalent of the repeating decimal .134134 and so forth.
So far it seems like we have a pattern. For 6/9 we had one repeating digit, and then we put that over a single 9. In the second example, we had 134 that repeated, and that’s three digits. So we put that over 3 nines. Hmm. Would it be that simple, just take our repeating digit and make that into our numerator, and then just put it over so many nines depending on how many digits there are that repeat? Let’s test our hypothesis. Let’s try another one. Make that into our equation. This time we have two repeating digits, 3 and 4. So, we have to multiply by 100 to get rid of our repeating digits. Now let’s do our subtraction. Notice that here are our repeating digits, and that should get eliminated when we subtract. Let’s get rid of all of those zeros. We now divide by 99, and we get that y is equal to 13.3/99.
That’s a little messy. We typically don’t express a rational number with a decimal in it. So, we need to continue on and simplify this to where we have no decimals. We can always multiply anything by one. In this case, we need to multiply by 10 so that we can get rid of the tenths in the numerator. So let’s multiply by 10. Let’s make the one 10 over 10. Our result now is 133/990. So, that is the rationale equivalent of our repeating decimal .134343434...etcetera. So, that blows our hypothesis that we take the number of repeating digits and put it over that many nines. It didn’t work this time because the denominator was 990. So, it wasn’t that simple.
Let’s take another example. Notice that this time, we’re using a number that’s greater than one. This is 5.322222 and so forth. Upon examination, we see that we have one repeating digit, so we have to multiply by 10. So, let’s take our equation, multiply by 10 to get this equivalent equation. Now we do our subtraction. We get rid of all the repeating twos. Simplify and get rid of all the zeros. Now we need to divide by nine, and here’s the result. But we have a mess again because we have a decimal within our fraction, so we need to get rid of that. We need to multiply by 10/10 so we can move our decimal one spot to the right. We do that and we get 479/90, which as a mixed number would be 5 29/90. So, here’s our result: 5 29/90 is the rational expression of 5.322222...
Let’s take that same example and work it in a slightly different manner. We can convert that to 5 + .3222 and so forth, and what we can do is, let’s not worry about that five for the time being. Let’s just set it aside. Our main concern is the decimal portion here. So, let’s just work with what we have left. Let’s set up our equation. We have one repeating digit, so we need to multiply by 10. We do that, and here’s the result. Now we do our subtraction. Here’s the result. Let’s get rid of our zeros. Now we need to divide by nine. We have the same problem that we did earlier. We need to get rid of our decimal. We multiply by 10/10 to accomplish that, and we get 29/90. Let’s set up our equivalent equation here, but remember, now we have to reinsert the 5. We can’t just throw it out. But, it’s just simple. Now we just add 5 to each side of the equation, and we get what we did earlier. We just did it in a slightly different manner where we didn’t worry about the whole number. We just worried about the decimal portion, and then we just reinstate the whole number part when we’re done.
Let’s look at the standards for mathematical practice. If we look at the first four, by doing some of the activities with this standard, students would reason abstractly and quantitatively, and they would construct viable arguments and critique the reasoning of others. Now we look at the last four of the standards for mathematical practice. Students would attend to precision, and by doing some of these activities, they also would look for and express regularity in repeated reasoning.