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Part of the Lesson Stem Teachers Script Hello Mathematicians. After Part of the Stem Teachers Script Lesson In this lesson you Introduction Hello mathematicians. After this lesson, you will be able to compare an irrational are going to number to a rational number or another irrational number. learn…by 10-20 sec doing/using… Remember to compare two rational numbers in decimal form, you must line up your decimals, then begin with the digits in the place furthest to the left and compare from Connection left to right until you find the number that is greater. (Define Terms/ Building on Prior You know that… Knowledge) Also remember, that when you are comparing two negative numbers, the greater 30-60 sec negative would be further to the left of 0 on a number line, and is therefore the smaller number. Comparing irrational numbers to rational numbers or to other irrational numbers requires the same process as long as the numbers you want to compare are in decimal form. If you need to compare an irrational number to a fraction, you must change the fraction to a decimal first. Demonstration I’m going to explain this idea One type of irrational number, the one with the ... to indicate the number continues 1-3 minc by showing you¦ forever without repeating is already in decimal form. So let's compare 1.124...with 1.2. First we line up the decimals, then begin checking the digits in the place values furthest to the left. In the units place for both numbers is a 1, so that's a tie. In the tenths place, the irrational number has a 1, while the rational number has a 2. So 1.2 is greater than 1.124. Another type of irrational number, pi, can quickly be changed into a rational estimation in decimal form, 3.14159, which can now be compared to other decimals. So lets compare pi and 3.14. Wait, aren't they equal? No, 3.14 is a rational approximation of pi. Let's change pi to 3.14159, line up the decimals, and then compare. It takes until the thousandths place before we see a winner. Pi has a 1 where 3.14 has nothing, so pi is greater. The type of irrational number that will be the most challenging is the square root of a number that is not a perfect square. In a previous lesson, you learned how to locate the square root of a number that is not a perfect square between two integers. I will now show you how to be a little more precise. Radical 7. Radical 7 is in between radical 4, which is 2 and radical 9 which is 3. So radical 7 is somewhere between 2 and 3. But since 7 is closer to 9 than it is to 4, radical 7 will be closer to 3 than it is to 2. So radical 7 is a decimal greater than 2.5. That should be as precise as we need to be without a calculator. So let's compare radical 18 and 4.5. 18 is not a perfect square, but it is in between the perfect squares 16 and 25. So radical 18 is in between 4 and 5. Since 18 is closer to 16, radical 18 will be closer to 4 than it is to 5, which means radical 18 is less than 4.5 Application Let’s see how this Here are some practice problems. works in a 1-2 min problem¦ For each problem, decide which of the two numbers is greater and place a > or < sign to indicate the greater number: (1) 1.341 _____ 1.341... (2) 1/3 ______ 0.3132... (3) - pi ____ -3.2 (4) Radical 85 ____ 9.7 (5) pi _____ radical 14 (1) For number 1, once you line up the decimals you have a tie all the way to the ten thousandths place. The number on the left is out of digits, so it is like have a 0 in the ten thousandths place, while the number on the right has the ... indicating it will continue forever, so even though we do not know any more digits, we know there is at least 1 digit somewhere to the right that will be greater than the zeros that continue for the first number. (2) The numbers tie until we get to the hundredths digit. The rational 2.3 repeating will have a 3 in that place, while the irrational 2.3132...has a 1 in that place. So 2.3 repeating is larger. (3) Once we line up the decimals, 3.2 wins in the tenths place because 2 tenths are greater than 1 tenth. But, both numbers were negative, and a number that is more negative is actually further to the left of zero on a number line and therefore smaller. So -pi is greater than -3.2. (4) 85 is in between 81 and 100, so radical 85 is in between 9 and 10...closer to 9, so it is less than 9.5. If it is less than 9.5, it must be less than 9.7 (5) 14 is in between 9 and 16, so radical 14 is in between 3 and 4. It is closer to 4, so it's greater than 3.5, which means it is greater than 3.14. Conclusion Now you can compare an irrational number to a rational number or another irrational So, now you know how to…by… number. 10-20 sec .
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