SCIENTIFIC NOTATION AND POWERS OF TEN
The objective for this lesson on Scientific notation and Powers of Ten is the student will translate between scientific notation and standard form in order to compare numbers and solve real world problems.
The skills students should have in order to help them in this lesson include exponents, exponential notation, and rules of exponents.
We will have three essential questions that will be guiding our lesson. Number 1, what is the difference between scientific notation and standard form? Number 2, why is it helpful to use scientific notation? And number 3; explain how to compare numbers in scientific notation or standard form.
We will begin by completing the warm-up on laws of exponents to simplify expressions to prepare for scientific notation and powers of ten in this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, scientists at NASA are able to approximate distances of planets and planetary orbs in the universe. Most recently, the distance from the sun to the Earth has been approximated at nine point two nine times ten to the seventh power miles. The distance from Mars to the sun is one hundred forty one million six hundred thousand miles. Which planet is farther from the sun? Explain your thinking.
We will begin by Studying the Problem. First we need to identify where the question is located within the problem and underline the question. Which planet is farther from the sun? Explain your thinking.
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 planet that is farther from the sun.
During this lesson we will learn how to translate between scientific and standard notation, to compare values and solve real world problems. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
DISCOVERING SCIENTIFIC NOTATION WITH LARGE NUMBERS
We are going to use our knowledge of laws of exponents to complete the graphic organizer seen here.
Look at the first column. What do you notice about the values? They are multiples of ten. What is the first value given in the table? It is the number ten.
Explain how to write the factors using tens for the number ten. The factors of ten are one and ten. We can multiply one times ten to equal ten. Let’s record this information in the second column, Factors Using Tens for the numeral ten. If we had to write ten using an exponent, how would we write this number? The number ten has one group of ten in it. So we write this with the exponent one. It is ten to the first power. In the last column of the graphic organizer, we can represent the Product Using Exponents, for the numeral ten as one times ten to the first power, because we have one group of ten.
How many tens are in one hundred? There are two tens in one hundred. Ten times ten equals one hundred. So let’s write out the factors of one hundred using tens. We can multiply one times ten times ten to equal one hundred. Let’s record this information in the second column of the graphic organizer for the number one hundred. How can we simplify the factors for one hundred using an exponent? Since there are two groups of ten, we need to write ten to the second power. So the product using exponents is one times ten to the second power.
Now complete the rest of the table. Let’s review the answers together.
For the numeral one thousand, the factors using tens are one times ten times ten times ten. We can write this product using exponents as one times ten to the third power. Because we have three groups of ten in one thousand.
For the numeral ten thousand, the factors using tens are one times ten times ten times ten times ten. We can write this product using exponents by writing one times ten to the fourth power. Because we have four groups of ten in ten thousand.
What do you notice about the information from the chart? One observation is, that as the numeral’s place value increases, so does the exponent of the ten.
What else do you notice about the chart? Another observation is that, the number of zeros in the numeral matches the exponent to which then ten is raised. For example: For the numeral one thousand there are three zero’s. When we look at the product using exponents we see that the exponent that the ten is raised to for the number one thousand is three.
Using the same strategy we used to complete the first chart, let’s take a look at the second chart. This chart starts with the numeral thirty. What is thirty written as a product of a factor and tens? We can multiply three times ten to give us thirty. Let’s record this information in the second column of the graphic organizer, for the numeral thirty. If we had to use an exponent with the ten to represent the number of tens multiplied, what would the product be? Here we have one ten, so the product using exponents is three times ten to the first power, because we have one group of ten. Let’s record this information in the last column of the graphic organizer.
Now let’s look at the numeral three hundred. What is three hundred written as a product of a factor and tens? We multiply three times ten times ten to equal three hundred. Record this information in the second column of the graphic organizer for the numeral three hundred. How can we simplify this product using exponents? This time we have two groups of ten. So our product using exponents is three times ten to the second power, to represent the two groups of ten as factors. Let’s record this information in the last column of the graphic organizer. Complete the rest of this table.
Now let’s review together. For the numeral three thousand the factors using tens are three times ten times ten times ten. And the product using exponents is three times ten to the third power, because we have three tens as factors for the numeral three thousand.
For the numeral thirty thousand the factors are three times ten times ten times ten times ten. We can write this product using exponents as three times ten to the fourth power, because for the number thirty thousand we have four groups of ten.
What do you notice about the information from the chart? One Observation is that as the numeral’s place value increases, so does the exponent of the ten. Another Observation is that the number of zeros in the numeral matches the exponent to which the ten is raised. For example: The numeral thirty thousand has four zeros. We can see in the product using exponents that the ten is raised to the fourth power. And another observation is that for each numeral there is a three multiplied by a group of tens.
If given the number thirty billion, predict how this numeral can be written in the same form that is in the third column of the previous page. Let’s predict that we can write this number as three times ten to the tenth power. Explain your thinking and justify your answer. The pattern is that the exponent represents the number of zeros in the numeral. Each zero represents a place value in the number thirty billion. There are ten zeros so we raised the ten to the tenth power to represent all of the zeros in the number thirty billion.
Why do you think we consider writing numbers in the form we provided for the solution to Question three that we just completed? Writing a number as large as thirty billion can be difficult because of the number of zeros. It is much easier to write it in terms of a power of ten.
What are some real world situations where we can use very large numbers? We use very large numbers in astronomy, when we’re looking at the distance from the earth, to stars, the sun and the moon, in aeronautics, engineering and other sciences.
What do you think we call numbers written in the form the numbers are written in the “Numeral” column on the previous page? We call these numbers written in Standard Form. Numbers written in standard form have the entire number written with all of the digits.
With many scientists and engineers using these large numbers, we can refer to the notation in the “Product Using Exponents” column as scientific notation.
If we had the number two hundred thousand how could we write this using scientific notation? We write the value of two hundred thousand in scientific notation by writing the digit of two multiplied by the power of ten. Two times ten times ten times ten times ten times ten equals two hundred thousand. There are five groups of ten that we are multiplying together. So we can simplify this by writing two times ten to the fifth power. The five as are exponent represent the five groups of ten that we need to multiply by two in order to create the number two hundred thousand.
If we had the number two hundred four thousand three hundred fifty one and we wanted to write this using scientific notation, how would it be different than the number two hundred thousand? In the number two hundred thousand all of the digits except the first one are zeros. In the number two hundred four thousand three hundred fifty one there are other values in the number besides the initial two and zeros. Can we just count the number of zeros in the number and place the exponent on the ten? Explain your thinking. No we can’t, because eliminating the zeros changes the place value of the significant digits. If we write this number in scientific notation and eliminate all the other digits other than two, will it be a correct representation of the number in standard form? Why or why not? No, because we would lose the value of the four, three, five and one in the number.
What do you notice about all the numbers we wrote in scientific notation? They are a single digit value less than ten which is multiplied by a power of ten.
Let’s apply what we know about writing a number in scientific notation for the number two hundred four thousand three hundred fifty one. Explain the first step in writing the value in scientific notation. We need to write the first part of the value as a number between one and ten. This means that the decimal point would be between the two and the zero, which makes this a number between one and ten. The number can be written as two point zero four three five one.
What is the next step in the process? We need to multiply by the power of ten based on the number of decimal places behind the decimal point. What is our final answer? Our final answer is two point zero four three five one times ten to the fifth power, because we have five decimal places behind the decimal point, when we write the number as a number between one and ten.
How can we check our answer? We can multiply to check our answer. Two point zero four three five one times ten to the fifth power is the same as two point zero four three five one times one hundred thousand, because ten to the fifth power is equal to one hundred thousand. When we multiply two point zero four three five one times one hundred thousand it equals two hundred four thousand three hundred fifty one. So our answer checks correctly.
Now let’s take a look at Problem one seen here. The number is twenty three billion eight hundred nine million. Where will we place the new decimal point in Problem one? We will need to place the decimal between the two and the three. Explain your thinking. When we write a number in scientific notation the value must be between one and ten and then multiplied by a power of ten. So placing the decimal between the two and the three in the number makes this number between one and ten.
Can we eliminate any zeros from the decimal? Yes, the last six zeros can be eliminated. Explain your thinking. Those zeros are represented by the power of ten. Should we eliminate the zero between eight and nine? Why or why not? No, because it is holding the place value between the eight and the nine and would ruin the value when converted back to standard form if we just took it out.
What decimal will we use for scientific notation? We will use the decimal two point three eight zero nine. Remember that we need to keep this zero between the eight and the nine to hold that place value in the original number.
What power of ten will we multiply the decimal by? Since we are placing the decimal between the two and the three there will be ten place values after the decimal. So the power of ten that we will multiply the decimal is ten. We multiply by ten to the tenth power.
What is the number written in scientific notation? The number is two point three eight zero nine times ten to the tenth power.
Now let’s take a look at this problem one. The number is forty five million. Where will we place the new decimal point in Problem one? Between the four and the five. Explain your thinking. When we write a value in scientific notation the value must be between one and ten and then multiplied by a power of ten. When we place the decimal point between the four and the five in our original number it makes this a number between one and ten.
Can we eliminate any zeros from the decimal? Yes, we can eliminate the last six zero’s. Explain your thinking. Those zeros are represented by the power of ten.
So what decimal will we use for scientific notation? We will use the decimal four point five. What power of ten will we multiply the decimal by? Since we placed the decimal point between the four and the five, there will be seven places after the decimal point in our new number. So we will multiply the power of ten by seven. We multiply by ten to the seventh power.
What is the number written in scientific notation? It is four point five times ten to the seventh power.
DISCOVERING SCIENTIFIC NOTATION WITH SMALL NUMBERS
We are going to use our knowledge of laws of exponents to complete the graphic organizer seen here.
Take a look at the first numeral listed in the graphic organizer. It is the numeral one tenth. How can we write one tenth as a fraction? We put a one in the numerator and a ten in the denominator. This is the fraction one tenth. Record this information in the second column in the graphic organizer.
If we had to place an exponent on the ten in the denominator, how would we write that fraction? We would write the fraction one over ten to the first power to represent that we have one group of ten in the denominator. Let’s record this information in the last column of the graphic organizer.
Now explain your thinking about your answer one over ten to the first power. Any value with no exponent is understood to have the exponent one. In this example, one times ten is equal to ten or ten to the first power is equal to ten.
How do we write the base with an exponent for the value of one over ten to the first power? We can move both the base and the exponent to the numerator by changing the exponent to a negative value. What is that value? One over ten to the first power is equal to ten to the negative first power. Let’s include this information in the last column of the graphic organizer. One over ten to the first power is equal to ten to the negative first power.
Justify your answer. The exponent represents how many places we move the decimal point in the value of one. When the exponent is a positive one, such as ten to the first power, we move the decimal point one place to the right to model multiplying by ten. When the exponent is negative, such as ten to the negative first power, we move the decimal point one place to the left to model multiplying by one tenth.
Now let’s take a look at the second numeral in the graphic organizer. This is the numeral one hundredth. How can we write one hundredth as a fraction? We write this as one over one hundred.
How can we write the denominator as a product of tens? One hundred is equal to ten times ten. So we can write the fraction as one over ten times ten. Record this information in the graphic organizer.