• Scientific Notation worksheet requires students to translate between standard notation and scientific notation. Plan necessary re
• Teaching according to the types of errors made by students.
• Activity worksheet indicates how well students are able to use their knowledge of place value to represent parts per thousand in terms of scientific notation.
• The Optional Quiz (M-8-4-2_Optional Short Quiz and KEY.docx) may also be used to assess student mastery.

Write a number on the chalkboard that contains at least ten digits. Ask students to multiply that number by 2, 3, and 10. After a few minutes ask whether students found this task difficult or time consuming. “There is a way to represent very small and very large numbers to complete calculations more quickly. Scientific notation allows us to write these numbers in a shorter, more manageable form.”

This section will review place value and powers of ten. Use the number written on the board and have volunteers come to the board and write the place value of any number.

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. In other words, if there were only billions, millions, . . . how many zeros would there be?” Allow 5–10 minutes for students to work. If some finish early, ask them to help students who might need additional 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)

“These place values all relate to powers of ten. Take a few minutes to do the following multiplication problems.” Distribute short worksheets with multiplication problems that have 10 multiplied by 10 numerous times or write 15 problems on the board.

Example:

10 × 10 = 10² = 100

10 × 10 × 10 = 10³ = 1,000

10 × 10 × 10 × 10 = 10⁴ = 10,000

Students can calculate by hand or use calculators. If they use calculators, have them copy their result onto their paper.

“Did you find any patterns with the number of zeros each answer had and the number of times you multiplied 10?” Students should respond that the number of times they multiplied by ten was the number of zeros. “The following chart shows how place value relates to powers of 10.”

Project the chart:

“Notice that for each place value, the number of zeros it has in standard form becomes the power of 10 when the place value is written in exponential form. For values above 1, the exponent is positive. But for values below 1 (our decimal place values), the exponent is negative.

“As you can see, any place value can be expressed as a power of 10. This means that if we are dealing with a very large or very small number (or in other words, a number with many digits), we may abbreviate the number by expressing it as the product of an integer and a power of ten. This way of expressing a number is called scientific notation.

“A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.”

Write on the board:

Scientific Notation

a × 10ⁿ, where 1 ≤ a < 10 and n is an integer

“Let’s look at an example. We’ll start with a large number, like 5 million.

“What is the largest place value in 5 million?” (millions) “How many zeros are in a million?” (6) “How can we express millions as a power of 10?” (10⁶) “So if 10⁶ represents 1 million, what should we multiply 1 million by to get 5 million?” (5)

5 million = 5,000,000 = 5  1,000,000 =  

Activity 1

“Please write the following number in scientific notation. As you try to do this, consider the numbers of zeros that follow each number and which power of 10 that represents.”

7,000,000,000                         (7 × 10⁹)

50,000                                     (5 × 10⁴)

20                                            (2 × 10¹)

0.0006                                     (6 × 10⁻⁴)

0.8                                           (8 × 10⁻¹)

900,000,000,000,000              (9 × 10¹⁴)

“Despite our recent examples, there are other numbers between 1 and 10 that are not whole numbers. We can also use decimals as the factors in scientific notation. For example, let’s consider the number 5,930,000,000. What is the highest place value in this number?” (billions) “According to our place value chart and what we have learned about corresponding powers of 10, what power of 10 is associated with one billion?” (9) “This implies that our scientific notation will look like: __ × 10⁹. So the only issue now is to identify the factor. This time, we cannot simply say that the factor is 5, because 5 × 10⁹ = 5 billion (or 5,000,000,000). Yet we want 5,930,000,000. If we call the factor 6, then we have 6 × 10⁹ = 6 billion (or 6,000,000,000), and this is too big. So our factor must be between 5 and 6. Is it 5.1, 5.2, 5.3, . . . ?” Guide students toward realizing that the factor must be 5.93.

5,930,000,000 = 5.93 × 10⁹

“Please convert the following numbers from standard notation to scientific notation.”

617                                          (6.17 × 10²)

9,125,600,000,000,000           (9.1256 × 10¹⁵)

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

This system will help students understand how they are moving the decimal place. “Now we will look at a slightly different strategy when it comes to converting between standard and scientific notation. In this strategy, we will use the exponent to count how many times to move the decimal place.” Write the following example on the board:

5,032,000

“First, locate the position of the decimal point in the standard form of the number.” (5,032,000.) “Now, we will move the decimal point until the number is between 1 and 10. What direction will we need to move the decimal point in order for 5,032,000 to become a number in between 1 and 10?” (left) “Start moving the decimal point left. Keep going until the number you see is between 1 and 10.”

        Think: Is 5,032,000 between 1 and 10? If no, keep going.

        Think: Is 503,200 between 1 and 10? If no, keep going.

     Think: Is 50,320 between 1 and 10? If no, keep going.

        Think: Is 5,032 between 1 and 10? If no, keep going.

          Think: Is 503.2 between 1 and 10? If no, keep going.

          Think: Is 50.32 between 1 and 10? If no, keep going.

           Think: Is 5.032 between 1 and 10? If yes, stop!

“By moving the decimal point left 1 place at a time, we eventually get to the number 5.032. This value is indeed greater than or equal to 1 and less than 10. This means that 5.032 is the factor in our scientific notation: 5.032 × 10^?. Now all we have to do is determine the exponent. To do so, think about how many times we needed to move the decimal point to get from 5,032,000 to 5.032. How many times was this?” (6 to the left) “Since we moved the decimal point 6 times to the left, our exponent is +6.”

5,032,000 = 5.032 × 10⁶

“Let’s look at another example.”

0.002705

“First, locate the position of the decimal point when the number is written in standard form.” (0.002705) “Now, we will move the decimal point until the number we see is between 1 and 10. What direction will we have to move the decimal point to accomplish this?” (right) “Start moving the decimal point to the right one place value at a time. Don’t stop until you see a number between 1 and 10.”

           Think: Is 0.002705 between 1 and 10? If no, keep going.

           Think: Is 0.02705 between 1 and 10? If no, keep going.

           Think: Is 0.2705 between 1 and 10? If no, keep going.

           Think: Is 2.705 between 1 and 10? If yes, stop!

“By moving the decimal point right one place at a time, we eventually get to the number 2.705. This value is indeed greater than or equal to 1 and less than 10. This means that 2.705 is the factor in our scientific notation: 2.705 × 10^?. Now all we have to do is determine the exponent. To do so, think about how many times we needed to move the decimal point to get from 0.002705 to 2.705. How many times was this?” (3 to the right) “Since we moved the decimal point 3 times to the right, our exponent is −3.”

0.002705 = 2.705 × 10⁻³

“The looping method can also be helpful when starting with a number in scientific notation and converting it to standard form. All we have to do is go backwards.”

3.8 × 10⁷

“Once again, we start by locating the position of the decimal point.” (3.8 × 10⁷) “This time, however, instead of taking a very large number and moving the decimal point so it becomes a number in between 1 and 10, we are starting with a factor in between 1 and 10 and moving the decimal point back so it is a very large number. What direction will we need to move the decimal point to create a very large number?” (right) “How many times will we have to move the decimal point to the right?” (7, because this is the value of the exponent)

3.8 × 10⁷ = 38,000,000

“Let’s convert another number from scientific notation into standard form.”

1.52 × 10⁻⁵

“As always, start by locating the position of the decimal point.” (1.52 × 10⁻⁵) “The negative exponent indicates that this is a very small number. Which direction will we have to move the decimal point in 1.52 to convert this back to a very small number?” (left) “How many times will we have to move the decimal point to the left?” (5, because this is the value of the exponent)

1.52 × 10⁻⁵ = 0.0000152

Activity 2

Divide the class into two groups. Each student in one group will get a piece of paper with a zero printed on it. Students in the second group will have a piece of paper with one number, 1 through 9. Show the class a number written in scientific notation. Have the group with whole numbers decide who to designate as the beginning digit of the standard form of the number. Have the second group determine how many of their members it will take to complete the standard form of the number. Have these students stand in a row to display the result.

Alternative ways to play:

• Use negative exponents and have the factor group be in charge of the decimal point.
• Give  students just the factor and the exponent without the full expression.
• As the class progresses, have the factor include tenths and hundredths, reinforcing that the factor must be between 0 and 10.

Calculator Notation

Note: Calculator display notation varies widely among different brands of calculators and different purposes and capacities of the calculator. Bring some examples of as many different ones as you can find to the classroom to show students directly. Scientific notation has a very specific representation on calculators that is different than handwritten or printed text. Have students choose a whole number 2 through 9 and have them multiply that number repeatedly on their calculators. Have students raise their hands when more than just numbers appear on the calculator.

“Calculators do not show scientific notation in the form that we have been learning. Look at your calculators. Does anyone have a number like this?”

5.477E8

“Thinking about what we have already learned, how can we write this number in scientific notation?”

5.477 is the factor

8 is the exponent

5.477 × 10⁸ = 547,700,000

“Other calculators have the factor in the main screen with the exponent of ten in the upper right.”

1.38412872¹⁰

1.38412872 is the factor
10 is the power of ten

1.38412872 × 10¹⁰ = 13,841,287,200

Activity 3

This activity will reinforce converting numbers into scientific notation in a real-world setting. Information is from the Pennsylvania Department of Environmental Protection (http://www.dep.state.pa.us/dep/DEPUTATE/AIRWASTE/AQ/standards/standards.htm#1).

“The Commonwealth of Pennsylvania has adopted air quality standards that regulate the amount of harmful pollutants in the air we breathe. I will pass the following table to each student so that we may study the standards together.” Hand out the Activity Sheet (M-8-4-2_Activity Sheet and KEY.doc). “Reviewing the standards, each pollutant is allowed in ppm, or parts per million. This means that for every million particles of air, the number listed is the amount of each pollutant allowed in those one million 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.” Write the equations on the board as the answers are calculated.

“The standard for carbon monoxide is 9 ppm.

“Nine particles of carbon monoxide per million particles of air + carbon monoxide is
9 parts per million”

  = 0.000001 or one millionth

9 × 0.000001 = 0.000009

0.000009 = 9 × 10⁻⁶

“So for each particle of air + carbon monoxide, 9 × 10⁻⁶ particles may be carbon monoxide.

The standard for nitrogen dioxide is 0.053 ppm or 0.053 particles per million.”

0.053 = 5.3 × 10⁻²

 = 0.000001 or one millionth

0.000001 = 1 × 10⁻⁶

“So we are looking for 0.053  0.000001.”

0.053  (1 × 10⁻⁶) = ?

“We have to make sure our factor is between 1 and 10. So if we move the decimal two places to the right, we have to make the exponent two smaller.

“So for each particle of air, 5.3 × 10⁻⁸ of it can be nitrogen dioxide.

“Complete the chart. If you have difficulties, consult your neighbor for help or raise your hand.

“Remember the rule for multiplying the same bases with exponents: Add the exponents. For example, 10³ × 10⁻⁴ = 10⁻¹ because 10⁽³⁺⁻⁴⁾ = 10⁽⁻¹⁾.”

Use the Scientific Notation worksheet (M-8-4-2_Scientific Notation and KEY.doc). To reinforce and evaluate the concepts introduced, list equations that look like scientific notation, but are not written correctly. Have students find the correct equations and explain why the original numbers were written incorrectly.

Examples:

0.002 × 10⁻⁶

5326 × 10⁻²

984 × 10⁹

64.1 × 10⁰

0.3 × 10⁶

53 × 10¹

Solutions:

2 × 10⁻⁹

5.326 × 10¹

9.84 × 10¹¹

6.41 × 10¹

3 × 10⁵

5.3 × 10²

Allow students to work alone or in pairs to find the correct expressions. When most are finished, have students write their work on the board. Go over the problems as a class. Students should respond that each of these is incorrect because the factor is not between 1 and 10, and therefore must be adjusted to be properly written in scientific notation.

Extension:

• Routine: Have students find their own real−world examples of the use of very small and very large numbers. Very large numbers can include the population of countries, the number of fish in the ocean, or even the amount of money in the stock exchange. Very small numbers might include the size of atoms or water quality requirements. Encourage students to find their own examples and write these numbers in scientific notation. Look for the opportunity to include scientific notation in journal writing in other subject areas.

Students may also review conversion using one of the games at the following web addresses:

http://janus.astro.umd.edu/astro/scinote/

http://www.aaastudy.com/dec71ix2.htm

• Small Group: Students who may benefit from additional practice with scientific notation and decimal conversion may be allowed to play the online King Kong game at the
  websites listed below:

http://www.quia.com/quiz/382466.html?AP_rand=1980576296

If students require additional instruction, the following websites may be used:

https://www.purplemath.com/modules/exponent3.htm

 http://www.nyu.edu/pages/mathmol/textbook/scinot.html

• Expansion: Students who are prepared for a challenge beyond the requirements of the standard may be given the activities below.

Have students write the following expressions in scientific notation.

5.6 × 10⁶ × 10⁻⁹

2.39 × 10² × 10⁵

9×  10⁻³ × 10⁻¹

87.75 × 10⁻⁶ × 10⁻³

Solutions for Expansion:

5.6 × 10⁻³

2.39 × 10⁷

9 × 10⁻⁴

8.775 × 10⁻⁸