Objectives

Students will learn how to apply exponents properly. Students will: - use scientific notation to express very large and very small numbers, as well as the relationship between place value and positive and negative exponents. - use scientific notation as a tool to compute, represent, and resolve expressions using real-world data.

Core Questions

- How can mathematics help to quantify, compare, depict, and model numbers?

Vocabulary

- Exponent: A numeral that tells how many times a number or variable is used as a factor. For example, in \(2^7\), 2 is the base and 7 is the exponent; this means 2 is multiplied by itself 7 times.  - Scientific Notation: A way of writing a number of terms of an integer power of  10 multiplied by a number greater than or equal to 1 and less than 10.

Materials

- note cards with numbers on them (1–9 and many with zeros)  - Scientific Notation worksheet (M-8-4-2_Scientific Notation and KEY) for each student  - Activity Sheet (M-8-4-2_Activity Sheet and KEY) for each student  - Optional Quiz (M-8-4-2_Optional Short Quiz and KEY) for any student who may benefit

Assignment

- The Scientific Notation worksheet asks students to translate between standard notation and scientific notation. Plan re-teaching strategies based on student errors.  - The activity worksheet evaluates students' ability to use their knowledge of place value to represent parts per thousand in scientific notation.  - The Optional Quiz (M-8-4-2_Optional Short Quiz and KEY) can also be used to check student understanding.

Supports

Active Engagement, Modeling, Explicit Instruction, Formative Assessment W: This lesson teaches students about scientific notation. Students will use calculators to convert numbers from scientific notation to conventional notation, do computations, and portray scientific notation.  H: Encourage students to use scientific notation as a beneficial tool for faster, more efficient, and error-free math. Having students complete extensive computing tasks using both conventional and scientific notation provides them with practical experience in discovering the benefits of using it effectively.  E: Students examine the relationship between place value names and their corresponding exponential power of ten. Students use something familiar to increase their comprehension of scientific notation. Students can count each iteration of multiplication by ten and compare it to the expression's exponent.  R: In Activity 3, students review course concepts by applying scientific notation to real-world data and governmental standards. By linking air quality data to specific objects like particles, students can identify it with something that affects their quality of life.  E: Completing a scientific notation worksheet demonstrates students' comprehension of the relationship between scientific and standard notation, converting between the two, and evaluating their responses.  T: The Extension can be used to customize the lesson to match the needs of students. The Routine section includes strategies for reviewing course concepts throughout the year. The Small Group portion is meant for students who could benefit from further practice. The Expansion section contains suggestions for students who are ready for a challenge beyond the requirements of the standard.  O: The lesson starts with a discussion of very small and very large numbers. It teaches scientific notation as a means of representing these numbers, starting with a review of place value. Place value is then compared to powers of 10. Scientific notation is introduced, and students are taught a few strategies for representing large and small numbers using the looping and calculator notations. A real-world example of air quality supports this knowledge.

Procedures

On the blackboard, write a number with at least ten digits. Have them multiply that number by 2, 3, and 10. After a few minutes, ask students if they found the task difficult or time-consuming. "To speed up calculations, it is possible to express very small and very large values. Scientific notation helps us to write these numbers in a shorter and more manageable format." This part will go over powers of ten and place value. Utilize the number that is put on the board, and ask volunteers to write any number's place value on it. Example:   1,324,890,625 1 – billions 3 – hundred millions 2 – ten millions 4 – millions 8 – hundred thousands 9 – ten thousands 0 – thousands 6 – hundreds 2 – tens 5 – ones "At your desk, please write how many zeros each place value has following it. Or, in other terms, how many zeros would there be if there were only billions or millions,...?" Allow 5-10 minutes for students to work. If any finish early, ask them to assist those who need more practice. 1 – billions (9 zeros) , 3 – hundred millions (8 zeros) 2 – ten millions (7 zeros) 4 – millions (6 zeros) , 8 – hundred thousands (5 zeros) 9 – ten thousands (4 zeros) 0 – thousands (3 zeros) , 6 – hundreds (2 zeros) 2 – tens (1 zero) 5 – ones (0 zeros) "All of these place values are related to powers of ten. Take a few minutes to complete the following multiplication problems." Distribute short worksheets with multiplication problems containing 10 multiplied by 10 many times, or write 15 problems on the board. Example: 10 × 10 = \(10^2\) = 100. 10 × 10 × 10 = \(10^3\) = 1,000 10 × 10 × 10 × 10 = \(10^4\) = 10,000 Students can calculate by hand or with calculators. If students use calculators, ask them to copy their results onto paper. "Did you observe any patterns with the number of zeros in each answer and the number of times you multiplied 10?" Students are expected to respond that the number of zeros was the result of multiplying by ten. "The chart below illustrates the relationship between powers of 10 and place value." Project the chart:<figure class="image"><img style="aspect-ratio:620/573;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_39.png" width="620" height="573"></figure><figure class="image"><img style="aspect-ratio:616/381;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_40.png" width="616" height="381"></figure><figure class="image"><img style="aspect-ratio:619/388;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_41.png" width="619" height="388"></figure>"Observe that the number of zeros in standard form for each place value is transformed into a power of 10 when the place value is expressed in exponential form. Positive exponents are used for values greater than 1 . However, for numbers less than 1 decimal point, the exponent is negative. "As you can see, any place value can be represented as a power of ten. This indicates that if we are dealing with a very large or extremely small number (i.e., a number with many digits), we can shorten it by expressing it as the product of an integer and a power of ten. This way of expressing a number is known as scientific notation. "A number is stated in scientific notation by writing it as the product of a factor and a power of ten. The component must be more than or equal to 1, but less than 10." Write on the board: Scientific Notation a × \(10^n\), where 1 ≤ a < 10 and n is an integer<figure class="image"><img style="aspect-ratio:514/129;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_42.png" width="514" height="129"></figure>"Let's examine an illustration. We'll start with a large number, such as 5 million." "What is the largest place value in 5 million?" (millions) "How many zeros are there in a million?" (6) "How do we represent millions as a power of ten?" (10^6) "What should we multiply one million by to get five million if 10^6 is one million?" (5)  5 million = 5,000,000 = 5  1,000,000 =<figure class="image"><img style="aspect-ratio:305/46;" src="https://storage.googleapis.com/worksheetzone/images/image.png" width="305" height="46"></figure> Activity 1 "Please provide the following number in scientific notation: Consider the number of zeros that follow each number, as well as the power of 10 that indicates." 7,000,000,000                         (7 × \(10^9\)) 50,000                                     (5 × \(10^4\)) 20                                            (2 × \(10^1\)) 0.0006                                     (6 × \(10^{-4}\)) 0.8                                           (8 × \(10^{-1}\)) 900,000,000,000,000              (9 × \(10^{14}\)) "Regardless of our recent examples, there are other numbers between 1 and 10 that are not whole numbers. Decimals can also be used as factors in scientific notation. Let's use 5,930,000,000 as an example. Which place value in this number is the highest?" (billions) "Which power of 10 is associated with one billion, based on our place value chart and our knowledge of corresponding powers of 10?" (9) " This suggests that __ × \(10^9\) will be the format of our scientific notation. Finding the factor is now the only problem. This time, it would be incorrect to state that the factor is 5, as 5 x \(10^9\) = 5 billion, or 5,000,000,000. However, we want 5,930,000,000. The number 6 × \(10^9\) = 6 billion (or 6,000,000,000) is too big if we name the factor 6. Therefore, we need to use a factor of five to six. Which number is it—5.1, 5.2, 5.3,..." Help students understand that 5.93 is the required factor. 5,930,000,000 = 5.93 × \(10^9\) "Please convert the following numerical values to scientific notation from standard notation." 617                                          (6.17 × \(10^2\)) 9,125,600,000,000,000           (9.1256 × \(10^{15}\)) 0.000345                                 (3.45 × \(10^{-4}\))   [Note that the LARGEST place value in 0.000345 is the ten thousandths, NOT the thousand thousandths, as 0.0001 > 0.0000001.] The Looping Method Students will gain a better understanding of moving the decimal place with the help of this system. "We'll now examine a somewhat different approach to switching between scientific and standard notation. Using the exponent, we will determine how many times to move the decimal position in this method." On the board, write the following example: 5,032,000 "First, find the decimal point's location in the number's standard form." (5,032,000). "The decimal point will now be moved until the number is between 1 and 10. In what direction should the decimal point be moved to make 5,032,000 a number between 1 and 10? (left) “Move the decimal point to the left. Continue until you see a number between 1 and 10.”<figure class="image"><img style="aspect-ratio:140/45;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_44.png" width="140" height="45"></figure>        Think: Is 5,032,000 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:135/51;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_45.png" width="135" height="51"></figure>        Think: Is 503,200 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:147/51;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="147" height="51"></figure>        Think: Is 50,320 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:140/49;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_47.png" width="140" height="49"></figure>        Think: Is 5,032 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:143/56;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="143" height="56"></figure>        Think: Is 503.2 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:143/50;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="143" height="50"></figure>        Think: Is 50.32 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:138/51;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="138" height="51"></figure>        Think: Is 5.032 between 1 and 10? If yes, stop!   "We eventually arrive at 5.032 by moving the decimal point left one place at a time. This number is, in fact, less than 10 and greater than or equal to 1. Thus, the factor in our scientific notation, 5.032 × \(10^?\), is 5.032. Finding the exponent is all that has to be done. Consider the number of times we had to shift the decimal point from 5,032,000 to 5.032 to accomplish this. How many times was this? " (6 to the left) " Our exponent is +6 because we moved the decimal point six times to the left." 5,032,000 = 5.032 × \(10^6\) “Let’s come to another example.” 0.002705 "To start, find where the decimal point is when writing the number in standard form." (0.002705) "At this point, we'll move the decimal point till the displayed value falls between 1 and 10. In order to do this, which way should the decimal point be moved?" (Right) “Move the decimal point one place value at a time to the right. Till you see a number between 1 and 10, keep going.”<figure class="image"><img style="aspect-ratio:124/41;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="124" height="41"></figure>           Think: Is 0.002705 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:126/50;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="126" height="50"></figure>           Think: Is 0.02705 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:127/50;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_53.png" width="127" height="50"></figure>           Think: Is 0.2705 between 1 and 10? If not, keep going.<figure class="image"><img style="aspect-ratio:124/52;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_54.png" width="124" height="52"></figure>           Think: Is 2.705 between 1 and 10? If yes, stop! "We eventually arrive at 5.032 by moving the decimal point left one place at a time. This number is, in fact, less than ten and higher than or equal to one. Thus, the factor in our scientific notation, 2.705 × \(10^?\), is 2.705. Finding the exponent is all that has to be done. Consider the number of times we had to move the decimal point from 0.002705 to 2.705 to accomplish this. How many times was this?" (3 to the left) "Our exponent is -3 because we moved the decimal point three times to the left." 0.002705 = 2.705 × \(10^{-3}\) "When translating a number in scientific notation to standard form, the looping method can also be useful. All we need to do is reverse our direction." 3.8 × \(10^7\) "Once more, we begin by determining where the decimal point is." (3.8 × \(10^7\)) "This time, however, we start with a factor in between 1 and 10 and move the decimal point back so it is a very large number. Rather than starting with a very large number and moving the decimal point so it becomes a number between 1 and 10, as we did previously, How far should the decimal point be moved to produce a very big number?" (right) "How many times does the decimal point need to be moved to the right?" (7, since this is the exponent's value)<figure class="image"><img style="aspect-ratio:147/59;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_55.png" width="147" height="59"></figure>3.8 × \(10^7\) = 38,000,000  "Let's convert one more number from scientific notation to standard form." 1.52 × \(10^{-5}\) "As usual, begin by figuring out where the decimal point is." (1.52 × \(10^{-5}\)) "This is a very small number, as indicated by the negative exponent. How far should we shift the decimal point to 1.52 to get this back to a very small number?" (left) "How many times must the decimal point be moved to the left?" (5, since this is the exponent's value)<figure class="image"><img style="aspect-ratio:152/61;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_56.png" width="152" height="61"></figure> 1.52 × \(10^{-5}\) = 0.0000152 Activity 2 Divide the students into two groups. Each student in one group will receive a piece of paper with a zero printed on it. The second group's students will each have a piece of paper with one number on it, from 1 to 9. Present a number in scientific notation to the class. Decide who should be the first digit in the number's standard form by asking the group of whole numbers to vote. Ask the second group to calculate the number of members needed to complete the number in standard form. Ask these students to line up in a row to present the outcome. Other games to play: Assign the decimal point to the factor group and use negative exponents. Give the students only the exponent and the factor—not the entire expression. As the lesson goes on, reinforce that the factor must be between 0 and 10 by adding tenths and hundredths to the factor. Calculator Notation Note: There are significant differences in the display notation of calculators between different brands, as well as across the many functions and capacities of the device. Bring in as many different kinds of examples as you can to demonstrate them directly to students. Unlike handwritten or printed language, scientific notation is represented on calculators in a very specific way. Ask students to select a whole number between 2 and 9 and use their calculators to multiply it over and over again. When the calculator displays something other than numbers, ask the students to raise their hands. "Calculators do not display scientific notation in the format we have been taught. Take a look at your calculators. Does anybody have a number like this?" 5.477E8 "Thinking about what we have already learned, how can we represent this number in scientific notation?" 5.477 is the factor. 8 is the exponent. 5.477 × \(10^8\) = 547,700,000 "Many other calculators display the factor on the main screen, with the exponent of ten in the upper right." \(1.38412872^{10}\) 1.38412872 is the factor. 10 is the power of ten. 1.38412872 × \(10^{10}\) = 13,841,287,200 Activity 3 The conversion of numerical values into scientific notation will be reinforced in a practical context through this activity. The information is from the Pennsylvania Department of Environmental Protection (<a href="http://www.dep.state.pa.us/dep/DEPUTATE/AIRWASTE/AQ/standards/standards.htm#1">http://www.dep.state.pa.us/dep/DEPUTATE/AIRWASTE/AQ/standards/standards.htm#1</a>). "The Commonwealth of Pennsylvania has enacted air quality regulations that limit the number of harmful pollutants in our air. To help us all study the standards, I will pass each student the following table.” Hand out the Activity Sheet (M-8-4-2_Activity Sheet and KEY). "After going over the guidelines, it is allowed for each pollutant to be measured in parts per million, or ppm. This indicates that the number listed is the maximum amount of each pollutant permitted in one million air particles. We are going to convert the numbers to scientific notation, imagining what part of one particle of air can be made up of each pollutant." As the solutions are determined, write the equations on the board. "The standard for carbon monoxide is 9 parts per million. "Nine particles of carbon monoxide per million particles of air + carbon monoxide = 9 parts per million." \(9 \over 1,000,000\) \(1 \over 1,000,000\) = 0.000001 or one millionth 9 × 0.000001 = 0.000009 0.000009 = 9 × \(10^{-6}\) "Therefore, 9 × \(10^{-6}\) particles of carbon monoxide may be present for every particle of air plus carbon monoxide. 0.053 parts per million, or 0.053 ppm, is the standard for nitrogen dioxide." \(0.053 \over 1,000,000\) 0.053 = 5.3 × \(10^{-2}\) \(1 \over 1,000,000\) = 0.000001, or one millionth 0.000001 = 1 × \(10^{-6}\) “So we are finding 0.053  0.000001.” 0.053  (1 × \(10^{-6}\)) = ? "We must ensure that our factor falls between 1 and 10. Therefore, we must reduce the exponent by two if we move the decimal two places to the right. "Thus, nitrogen dioxide can make up 5.3 × \(10^{-8}\) of each air particle. "Finish the chart. Raise your hand or ask your neighbor for assistance if you're having trouble. Keep in mind that you must add the exponents when multiplying the same bases by exponents. For example, \(10^3\) × \(10^{-4}\) = \(10^{-1}\) because \(10^{3+-4}\) = \(10^{-1}\).” Use the Scientific Notation worksheet (M-8-4-2_Scientific Notation and KEY) for scientific notation. List equations that appear to be in scientific notation but are not written correctly in order to test and reinforce the concepts that were given. Ask students to locate the right equations and provide an explanation for the inaccurate writing of the original figures. Examples: 0.002 × \(10^{-6}\) 5326 × \(10^{-2}\) 984 × \(10^9\) 64.1 × \(10^0\) 0.3 × \(10^6\) 53 × \(10^1\) Solutions: 2 × \(10^{-9}\) 5.326 × \(10^1\) 9.84 × \(10^{11}\) 6.41 × \(10^1\) 3 × \(10^5\) 5.3 × \(10^2\) Permit students to identify the right expressions by working in pairs or on their own. Once the majority have completed their assignment, ask them to write it on the board. Discuss the issues as a class. Students should reply that each of these needs to be changed in order to be represented correctly in scientific notation because the factor is not between 1 and 10. Extension: Routine: Ask students to find their own instances of using very small and very large numbers in the actual world. The population of a nation, the quantity of fish in the ocean, or even the amount of money traded on the stock exchange are examples of very huge numbers. The size of atoms or the standards for water purity are examples of extremely small numbers. Encourage students to find examples on their own and put these numbers in scientific notation. Seek out opportunities to write in journals on other subjects where scientific notation can be included. Students have the option to review conversion using one of the games available at the following web addresses: <a href="http://janus.astro.umd.edu/astro/scinote/">http://janus.astro.umd.edu/astro/scinote/</a>  <a href="http://www.aaastudy.com/dec71ix2.htm">http://www.aaastudy.com/dec71ix2.htm</a>  Small Group: Students who could use more experience with decimal conversion and scientific notation might be permitted to access the online King Kong game at the webpages mentioned below: <a href="http://www.quia.com/quiz/382466.html?AP_rand=1980576296">http://www.quia.com/quiz/382466.html?AP_rand=1980576296</a>  Additional instruction may be offered to students through the utilization of the subsequent websites: <a href="https://www.purplemath.com/modules/exponent3.htm">https://www.purplemath.com/modules/exponent3.htm</a>  <a href="http://www.nyu.edu/pages/mathmol/textbook/scinot.html">http://www.nyu.edu/pages/mathmol/textbook/scinot.html</a>  Expansion: Students may be assigned the following assignments if they are ready for a challenge that goes above and beyond the standards. Students should compose the following sentences using scientific notation: 5.6 × \(10^6\) × \(10^{-9}\) 2.39 × \(10^2\) × \(10^5\) 9 × \(10^{-3}\) × \(10^{-1}\) 87.75 × \(10^{-6}\) × \(10^{-3}\) Solutions for Expansion: 5.6 × \(10^{-3}\) 2.39 × \(10^7\) 9 × \(10^{-4}\) 8.775 × \(10^{-8}\)

Layla Nguyen

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Scientific Notation (M-8-4-2)

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Layla Nguyen

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Description

Students will learn how to apply exponents properly. Students will:
- use scientific notation to express very large and very small numbers, as well as the relationship between place value and positive and negative exponents.
- use scientific notation as a tool to compute, represent, and resolve expressions using real-world data.

Grade

Grade 8