IDENTIFY COMPARE AND ORDER IRRATIONAL NUMBERS
INTRODUCTION
The objective for this lesson on identify, compare and order irrational numbers is, the student will create rational approximations on irrational numbers in order to compare and order them on a number line.
The skills students should have in order to help them in this lesson include rational numbers and plotting rational numbers on a number line.
We will have three essential questions that will be guiding our lesson. Number 1, explain how to find the rational approximation for the irrational numbers in the form of square roots. Number 2, why is it helpful to know how to find rational approximations for irrational numbers? Justify your thinking. And number 3, how can you compare irrational values that are written in different forms? Explain your thinking.
We will begin by completing the warm-up on determining the square roots of perfect squares, identifying decimal equivalence of fractions and mixed numbers and categorizing them as terminating or repeating to prepare for identifying, comparing and ordering irrational numbers in this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Matthew cuts square pieces of glass that are used as materials for furniture and home décor. He currently offers three different sizes where the sides measure three inches, five inches, and eight inches. To create more options, he will offer two more squares. One new square has a total area of forty five inches squared, and another square with an area of seventeen inches squared. To organize all of the different sizes, he needs a place all five options in order and correctly on a number line. What is the order of the squares from least to greatest? Use a number line to show the difference in sizes.
In Step S, we will Study the Problem. First we need to identify where the question is located within the problem and underline the question. What is the order of squares from least to greatest? Use a number line to show the difference in sizes.
Now that we have identified the question, we want to put this question in our own words in the form of a statement. This problem is asking me to find the order, from least to greatest, of the five different glass squares and to create a number line to show the different sizes.
During this lesson we will learn how to use rational approximations for irrational numbers to compare and order to values and plot them on a number line. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
SQUARE ROOTS OF IRRATIONAL NUMBRS – CONCRETE AND PICTORIAL
Use twenty five of the small yellow algebra tiles to make a square. What is the square root of twenty five? We can see that the square that we created using the twenty five small yellow algebra tiles has a length of five on each side. So the square root of twenty five is five.
Now write in your own words what a square root is. The square root of a number is a number that when multiplied by itself will yield the number under the radical. The square roots that we explored in Lesson four are called perfect squares because each number’s square root is a whole number.
Try to make a square using twelve yellow algebra tiles. Were you able to make a perfect square? No. What is the closest shape to a square that we can create? We can create a three by four rectangle. Let’s fill the area below the rectangle with red tiles to make a perfect square. We need to use four red algebra tiles to make our perfect square. What are the dimensions of the new square? The new square is four by four.
Now let’s draw a square around the largest square that is completely yellow. How many tiles are in the square? There are nine tiles in the square. What is the square root of your perfect square? The square root of the perfect square is three. Explain how you determined the square root. The measure of one side of the square is three. Three times three equals nine. So the square root of nine is three.
How many tiles are there that are not part of your perfect square? Let’s count them: one, two, three, four, five, six, seven. There are seven tiles that are not part of your perfect square. How many of these tiles are yellow? Let’s count them: one, two, three. There are three yellow tiles that are not part of are perfect square. We can approximate the square root of twelve using a mixed number. What is the whole number of the square root of the largest perfect square? It is three. The fraction is the number of yellow tiles over the total number of tiles outside of the perfect square. There are three yellow tiles outside of the perfect square and a total of seven tiles outside of the perfect square. So this gives us a fraction of three- sevenths. The approximate value of the square root of twelve is equal to three and three-sevenths. What is the mixed number is decimal form? Use a calculator to divide three by seven to find the decimal portion of the number. Three and three-sevenths is equal to three point four, two, eight, five, seven, one, four, two, nine, when we use a calculator.
Now using a calculator, find the square root of twelve. Whenwe put the square root of twelve into a calculator we get three point four, six, four, one, zero, one, six, one, five. Let’s record these. Our decimal using the tiles is equivalent to, three point, four, two, eight, five, seven, one, four, two, nine. And our decimal using the calculator for the square root of twelve is, three point four, six, four, one, zero, one, six, one, five. How does your decimal from the tiles compare to the calculator results? They are very close and differ in the hundredths column. Each is a bit less than three and five tenths. Explain why the two values are different. When we divide three by seven, it is an approximate value and the calculator gives a more exact value when we enter the square root of twelve.
Now take a look at Question one on the next page. It is the square root of six. How many yellow tiles do you have to use to try to create a square? We need six tiles since we’re looking for the square root of six. Try to make a square using six yellow algebra tiles. Were you able to make a perfect square using six tiles? No. Now draw the six tiles in the box for Question one. Now, box in the largest perfect square that you can make. How many tiles are in the largest perfect square? There are four tiles. What is the square root of this perfect square? Two. Explain how you know this. The square root is the measure of one side of the square. Explain how we use the value of the square root of the perfect square. The square root of the perfect square will be the whole number in our mixed number approximation.
Add red tiles to complete the next largest perfect square. We need three red tiles to create the next largest perfect square. How many total tiles are outside the perfect square? Let’s count them: one, two, three, four, five. There are five tiles outside the perfect square. How many of the tiles are yellow? Let’s count: one, two. Two of the tiles are yellow. What is the fraction of yellow tiles outside of the perfect square over the total tiles outside of the perfect square? It is two over five. So what is the mixed number that approximates the square root of six? Two and two fifths.
Now take a look at Question three. Question three asks us to find the square root of thirteen. How many yellow tiles do you have to use to try to create a square? We need thirteen yellow tiles. Try to make a square using thirteen yellow algebra tiles. Were you able to make a perfect square using thirteen tiles? No. Draw the thirteen tiles in the box for Question three. What is the largest perfect square that you can make? A three by three. Now box in the largest perfect square that you can make. How many tiles are in the largest perfect square? Nine. So what is the square root of this perfect square? The square root is three. Explain how we use the value of the square root of the perfect square. The square root of the perfect square will be the whole number in our mixed number approximation.
Add red tiles to complete the next largest perfect square. We need three red tiles. How many total tiles are outside of the perfect square? Let’s count them: one, two, three, four, five, six, seven. There are seven total tiles outside of the perfect square. How many of the tiles are yellow? Let’s count: one, two, three, four. There are four yellow tiles outside of the perfect square. So what is the fraction of yellow tiles outside of the perfect square over the total tiles outside of the perfect square? Four over seven. What is the mixed number that approximates the square root of thirteen? Three and four sevenths.
RATIONAL APPROXIMATIONS WITH NUMBER LINES
You can estimate the square root of a number that is not a perfect square without the algebra tiles. For example, if you were asked to find the square root of ninety four you would not be able to find the whole number that when multiplied by itself equals ninety four. You could, however find the two closest perfect squares that are less than ninety four and greater than ninety four. What is a perfect square that is close to, but less than ninety four? The square root of eighty one equals nine. What is a perfect square that is close to, but greater than ninety four? The square root of one hundred equals ten. What can we conclude about the square root of ninety four? It is between nine and ten. Is ninety four closer to eighty one or one hundred? It is closer to one hundred. Explain what this means. This means that the square root of ninety four is closer to ten. Sometimes we need a more exact answer than simply the range. In this case we can apply our understanding of the tiles.
By completing the chart, we will be able to find an approximation without using the algebra tiles. Let’s take a look at the chart together. The second column of this chart asks for the perfect square and its square root close to, but less than the number. What is this number when we’re considering the square root of ninety four? It is the square root of eighty one which equals nine. The third column asks for the perfect square and its square root close to, but greater than the number. What is this number? We said this number was the square root of one hundred which equals ten.
When we were using tiles to find an approximation, explain how we created the denominator of the fraction. We found the total number of tiles that were necessary to build the next largest square. Identify what operation we can use to find the number of tiles necessary to build the next largest perfect square. We can use subtraction. We subtract the perfect square above the number mine the perfect square below.
In the fourth column, find the difference between the perfect squares above and below the number. We need to subtract one hundred take away eighty one which equals nineteen.
When we were using the tiles to find an approximation, explain how we created the numerator of the fraction. We found the total number of yellow tiles that were outside of the perfect square that contributed to building the next square. Identify the operation we can use to find how far away the number is from the lower perfect square. We can use subtraction. We subtract the lower perfect square from the number. Let’s do that now. We will subtract eighty one from ninety four which equals thirteen. The difference between the number and the lower perfect square is thirteen.
Now explain how we wrote the fraction part of the mixed number which represented the approximation of our square root. We wrote the numerator as the difference between the number and lower perfect square, which in this case is thirteen and the denominator as the difference between the perfect squares, which in this case is nineteen. What is the fraction we create? For this example, the fraction is thirteen over nineteen. Let’s add this information in our graphic organizer in the last column. Now this needs to be a mixed number. So we need to figure out what the whole number part of this number is. What is the whole number that should accompany the fraction? We need to look back at the perfect square that is close to but less than ninety four which was nine. So the whole number in our mixed number is nine. The rational approximation in the form of a mixed number is nine and thirteen-nineteenths.
By creating a mixed number, we are creating a rational number that is approximately the same value as the number that we started with. We created a rational approximation. We can also plot the rational approximation on a number line.
Explain how we can create a decimal from our rational approximation. We need to divide the numerator by the denominator and add the whole number to the decimal. What is the decimal form of this rational approximation? Nine and thirteen-nineteenths is equal to nine point six, eight, four, two, one, zero, five, two, two, six, three.
Use the number line to plot a point to show the location of the rational approximation of the number. Label it with the number, the rational approximation and the decimal. We will begin our number line with nine and end with ten. The number line is broken up into ten sections, each one representing a tenth between nine and ten. For the square root of ninety four we found that the rational approximation is about nine and thirteen-nineteenths, which was about nine and sixty eight hundredths. Let’s place a dot on the number line to represent the point. This point on the number line represents the square root of ninety four which is about nine and thirteen-nineteenths or about nine and sixty eight hundredths. Where is the square root of ninety four located on the number line? It is between nine and six-tenths and nine and seven tenths.
Let’s use the same process we previously used to complete the first example. What is the square root that we are using to find the rational approximation? It is the square root of twenty two. What is the square root and perfect square that is closest to but less than the square root of twenty two? The square root of sixteen which equals four. What is the square root and perfect square that is closest to but greater than the square root of twenty two? The square root of twenty five which equals five. Next we need to find the difference between the perfect squares.
Explain how to find the difference between the perfect squares. We need to subtract twenty five minus sixteen which equals nine. We take the two perfect squares above and below our square root of twenty two and subtract them from each other.
Now explain how to find the difference between the number and the lower perfect square. The lower perfect square was sixteen and our number was the square root of twenty two. So we will take sixteen from twenty two which equals six.
Now explain how to find the rational approximation of the square root of twenty two using the information from the chart. We can create a fraction that represents the difference between the number and the lower perfect square, which in this case was six, which will be our numerator, over the difference between the perfect squares, which in this case is nine, which will be the denominator. This makes our fraction six over nine. Let’s record this in the last column of the graphic organizer. We now need to figure out what the whole number is that is a part of our mixed number. What do we need to write with the fraction when we write the rational approximation in the last column? We will need to use the whole number four. Because it is the perfect square that is close to but less than the number. Our rational approximation in the form of a mixed number is four and six-ninths. What is the rational approximation for the form of the mixed number? Four and six-ninths.