Lesson 2 M3 ALGEBRA II

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

, Lesson 2: Base 10 and Scientific Notation

Student Outcomes . Students review place value and scientific notation. . Students use scientific notation to compute with large numbers.

Lesson Notes This lesson reviews how to express numbers using scientific notation. Students first learned about scientific notation in Grade 8 where they expressed numbers using scientific notation (8.EE.A.3) and computed with and compared numbers expressed using scientific notation (8.EE.A.4). Refer to Grade 8, Module 1, Topic B to review the approach to introducing scientific notation and its use in calculations with very large and very small numbers. This lesson sets the stage for the introduction of base-10 logarithms later in this module by focusing on the fact that every real number can be expressed as the product of a number between 1 and 10 and a power of 10. In the paper-folding activity in the last lesson, students worked with some very small and very large numbers. This lesson opens with these numbers to connect these lessons. Students also compute with numbers using scientific notation and discuss how the properties of exponents can simplify these computations (N-RN.A.2). In both the lesson and the Problem Set, students define appropriate quantities for the purpose of descriptive modeling (N-Q.A.2). The lesson includes a demonstration that reinforces the point that using scientific notation is convenient when working with very large or very small numbers and helps students gain some sense of the change in magnitude when different powers of 10 are compared. This is an excellent time to watch the 9-minute classic film “Powers of 10” by Charles and Ray Eames, available at https://www.youtube.com/watch?v=0fKBhvDjuy0, which clearly illustrates the effect of adding another zero. The definition of scientific notation from Grade 8, Module 1, Lesson 9 is included after Example 1. Consider allowing students to use a calculator for this lesson.

Classwork Opening (2 minutes) In the last lesson, Example 1 gave the thickness of a sheet of gold foil as 0.28 millionth of a meter. In the Exit Ticket, students calculated the size of a square sheet of paper that could be folded in half thirteen times, and this area was very large. These numbers are used in the Opening Exercise of this lesson. Before students begin the Opening Exercise, briefly remind them of these numbers that they saw in the previous lesson, and tell them that this lesson provides them with a way to conveniently represent very small or very large numbers. If there was not an opportunity to share one of the news stories in Lesson 1, that could be done at this time as well.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

Opening Exercise (5 minutes) Students should work these exercises independently and then share their responses with a partner. Ask one or two students to explain their calculations to the rest of the class. Be sure to draw out the meaning of 0.28 millionth of a meter and how that can be expressed as a fraction of a meter. Check to make sure students know the place value for large numbers such as billions. Students should be able to work the first exercise without a calculator, but on the second exercise, students should definitely use a calculator. If they do not have access to a calculator, give them the number squared, and simply have them write the rounded value.

Opening Exercise

In the last lesson, you worked with the thickness of a sheet of gold foil (a very small number) and some very large numbers that gave the size of a piece of paper that actually could be folded in half more than ퟏퟑ times.

a. Convert ퟎ. ퟐퟖ millionth of a meter to centimeters, and express your answer as a decimal number. ퟎ. ퟐퟖ ퟏퟎퟎ 퐜퐦 ퟐퟖ 퐦 ∙ = 퐜퐦 = ퟎ. ퟎퟎퟎퟎퟐퟖ 퐜퐦 ퟏ , ퟎퟎퟎ, ퟎퟎퟎ ퟏ 퐦 ퟏ, ퟎퟎퟎ, ퟎퟎퟎ

b. The length of a piece of square notebook paper that can be folded in half ퟏퟑ times is ퟑ, ퟐퟗퟒ. ퟐ 퐢퐧. Use this number to calculate the area of a square piece of paper that can be folded in half ퟏퟒ times. Round your answer to the nearest million.

(ퟐ ∙ ퟑퟐퟗퟒ. ퟐ)ퟐ = ퟒퟑ, ퟒퟎퟕ, ퟎퟏퟒ. ퟓퟔ Rounded to the nearest million, the area is ퟒퟑ, ퟎퟎퟎ, ퟎퟎퟎ square inches.

c. Match the equivalent expressions without using a calculator.

ퟑ. ퟓ × ퟏퟎퟓ −ퟔ −ퟔ × ퟏퟎퟎ ퟎ. ퟔ ퟑ. ퟓ × ퟏퟎ−ퟔ ퟑ, ퟓퟎퟎ , ퟎퟎퟎ ퟑퟓퟎ, ퟎퟎퟎ ퟔ × ퟏퟎ−ퟏ ퟎ. ퟎퟎퟎퟎퟎퟑퟓ ퟑ. ퟓ × ퟏퟎퟔ

ퟑ. ퟓ × ퟏퟎퟓ is equal to ퟑퟓퟎ, ퟎퟎퟎ. −ퟔ × ퟏퟎퟎ is equal to −ퟔ. ퟔ × ퟏퟎ−ퟏ is equal to ퟎ. ퟔ. ퟑ. ퟓ × ퟏퟎퟔ is equal to ퟑ, ퟓퟎퟎ, ퟎퟎퟎ. ퟑ. ퟓ × ퟏퟎ−ퟔ is equal to ퟎ. ퟎퟎퟎퟎퟎퟑퟓ.

While reviewing these solutions, point out that very large and very small numbers require students to include many digits to indicate the place value as shown in parts (a) and (b). Also, based on part (c), it appears that integer powers of 10 can be used to express a number as a product.

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Example 1 (7 minutes) Scaffolding: . Have students working Write the following statement on the board, and ask students to consider whether or not above grade level write they believe it is true. Have them discuss their thoughts with a partner, and then ask for the number 245 as the volunteers to explain their thinking. product of a number and a Every positive decimal number can be expressed as the product of a number power of 10 three between 1 and 10 and a power of 10. different ways. Students should explain that every positive decimal number can be expressed as the . To challenge students product of a number between 1 and 10 and a power of 10. Consider using the following working above grade level, prompts to direct student thinking: have them make a convincing argument . Think of an example of a decimal number between 1 and 10. Write it down. regarding the truth of a MP.3  2.5 statement such as the . Think of a power of 10. Write it down. following:  100 or 102 Every decimal number can . What does the word product mean? be expressed as the product of another  It means the result of multiplying two numbers together. decimal number and a . Compute the product of the two numbers you wrote down. power of 10.  2.5 ∙ 102 = 2500

First, have students share their answers with a partner. Then, put a few of the examples on the board. Finally, demonstrate how to reverse the process to express the following numbers as the product of a number between 1 and 10 and a power of 10. At this point, explain to students that when numbers are written in this fashion, it is said that they are written using scientific notation. This is an especially convenient way to represent extremely large or extremely small numbers that otherwise would have many zero digits as placeholders. Make sure to emphasize that using this notation is simply rewriting a numerical expression in a different form, which helps students to quickly determine the size of the number. Students may need to be reminded of place value for numbers less than 1 and writing equivalent fractions whose denominators are powers of 10. The solutions demonstrate how numbers can be expressed using scientific notation.

Example 1

Write each number as a product of a decimal number between ퟏ and ퟏퟎ and a power of ퟏퟎ.

a. ퟐퟑퟒ, ퟎퟎퟎ

ퟐ. ퟑퟒ ∙ ퟏퟎퟎ, ퟎퟎퟎ = ퟐ. ퟑퟒ × ퟏퟎퟓ

b. ퟎ. ퟎퟎퟑퟓ ퟑퟓ ퟑ. ퟓ ퟏ = = ퟑ. ퟓ ∙ = ퟑ. ퟓ × ퟏퟎ−ퟑ ퟏퟎ ퟎퟎퟎ ퟏퟎퟎퟎ ퟏퟎퟎퟎ

c. ퟓퟑퟐ, ퟏퟎퟎ, ퟎퟎퟎ ퟓ. ퟑퟐퟏ ∙ ퟏퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ = ퟓ. ퟑퟐퟏ × ퟏퟎퟖ

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d. ퟎ. ퟎퟎퟎퟎퟎퟎퟎퟎퟏퟐ

ퟏퟐ ퟏ. ퟐ ퟏ = = ퟏ. ퟐ ∙ = ퟏ. ퟐ × ퟏퟎ−ퟗ ퟏퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ ퟏ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ ퟏ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ

e. ퟑ. ퟑퟑퟏ ퟑ. ퟑퟑퟏ ∙ ퟏ = ퟑ. ퟑퟑퟏ × ퟏퟎퟎ

. Our knowledge of the integer powers of 10 enables us to understand the concept of scientific notation. . Consider the estimated number of stars in the universe: 6 × 1022. This is a 23-digit whole number with the leading digit (the leftmost digit) 6 followed by 22 zeros. When it is written in the form 6 × 1022, it is said to be expressed in scientific notation. Students may recall scientific notation from previous grades. Take time to review the definition of scientific notation provided below.

A positive, finite decimal 푠 is said to be written in scientific notation if it is expressed as a product 푑 × 10푛, where 푑 is a finite decimal number so that 1 ≤ 푑 < 10, and 푛 is an integer.

The integer 푛 is called the order of magnitude of the decimal 푑 × 10푛.

Exercises 1–6 (4 minutes) Students should work these exercises independently as their progress is monitored. Encourage students to work quickly and begin generalizing a process for quickly writing numbers using scientific notation (such as counting the number of digits between the leading digit and the ones digit). After a few minutes, share the solutions with students so they can check their work.

Exercises 1–6

For Exercises 1–6, write each number in scientific notation.

1. ퟓퟑퟐ, ퟎퟎퟎ, ퟎퟎퟎ ퟓ. ퟑퟐ × ퟏퟎퟖ

2. ퟎ. ퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟏퟐퟑ (ퟏퟔ zeros after the decimal place)

ퟏ. ퟐퟑ × ퟏퟎ−ퟏퟕ

3. ퟖ, ퟗퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ (ퟏퟒ zeros after the ퟗ)

ퟖ. ퟗ × ퟏퟎퟏퟓ

4. ퟎ. ퟎퟎퟎퟎퟑퟑퟖퟐ ퟑ. ퟑퟖퟐ × ퟏퟎ−ퟓ

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5. ퟑퟒ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ (ퟐퟒ zeros after the ퟒ) ퟑ. ퟒ × ퟏퟎퟐퟓ

6. ퟎ. ퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟒ (ퟐퟏ zeros after the decimal place) ퟒ × ퟏퟎ−ퟐퟐ

To help students quickly write these problems using scientific notation, the number of zeros is written above for each problem. Be very careful that students are not using this number as the exponent on the base 10. Lead a discussion to clarify that difference for all students who make this careless mistake.

Exercises 7–8 (5 minutes) After students practice writing numbers in scientific notation, emphasize the usefulness of scientific notation for working with very large or very small numbers by showing this demonstration: http://joshworth.com/dev/pixelspace/pixelspace_solarsystem.html, which illustrates just how far the planets in our solar system are from each other. After the demonstration, write down the distances between Earth and the sun, between Jupiter and the sun, and between Pluto and the sun on the board, and have students work with a partner to answer Exercise 7. Be sure to mention that these distances are averages; the distances between the planets and the sun are constantly changing as the planets complete their orbits. The average distance from the sun to Earth is 151,268,468 km. The average distance from the sun to Jupiter is 780,179,470 km. The average distance between the sun and Pluto is 5,908,039,124 km. In these exercises, students round the distances to the nearest tenth to minimize all the writing and help them focus more readily on the magnitude of the numbers relative to one another.

Exercises 7–8

7. Use the fact that the average distance between the sun and Earth is ퟏퟓퟏ, ퟐퟔퟖ, ퟒퟔퟖ 퐤퐦, the average distance between the sun and Jupiter is ퟕퟖퟎ, ퟏퟕퟗ, ퟒퟕퟎ 퐤퐦, and the average distance between the sun and Pluto is ퟓ, ퟗퟎퟖ, ퟎퟑퟗ, ퟏퟐퟒ 퐤퐦 to approximate the following distances. Express your answers in scientific notation (풅 × ퟏퟎ풏), where 풅 is rounded to the nearest tenth. a. Distance from the sun to Earth:

ퟏ. ퟓ × ퟏퟎퟖ 퐤퐦

b. Distance from the sun to Jupiter:

ퟕ. ퟖ × ퟏퟎퟖ 퐤퐦

c. Distance from the sun to Pluto:

ퟓ. ퟗ × ퟏퟎퟗ 퐤퐦

d. How much farther Jupiter is from the sun than Earth is from the sun:

ퟕퟖퟎ, ퟏퟕퟗ, ퟒퟕퟎ 퐤퐦 − ퟏퟓퟏ, ퟐퟔퟖ, ퟒퟔퟖ 퐤퐦 = ퟔퟐퟖ, ퟗퟏퟏ, ퟎퟎퟐ 퐤퐦 ퟔ. ퟑ × ퟏퟎퟖ 퐤퐦

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e. How much farther Pluto is from the sun than Jupiter is from the sun:

ퟓ, ퟗퟎퟖ, ퟎퟑퟗ, ퟏퟐퟒ 퐤퐦 − ퟕퟖퟎ, ퟏퟕퟗ, ퟒퟕퟎ 퐤퐦 = ퟓ, ퟏퟐퟕ, ퟖퟓퟗ, ퟔퟓퟒ 퐤퐦

ퟓ. ퟏ × ퟏퟎퟗ 퐤퐦

8. Order the numbers in Exercise 7 from smallest to largest. Explain how writing the numbers in scientific notation helps you to quickly compare and order them.

MP.7 The numbers from smallest to largest are ퟏ. ퟓ × ퟏퟎퟖ, ퟔ. ퟑ × ퟏퟎퟖ, ퟕ. ퟖ × ퟏퟎퟖ, ퟓ. ퟏ × ퟏퟎퟗ, and ퟓ. ퟗ × ퟏퟎퟗ. The power of ퟏퟎ helps to quickly sort the numbers by their order of magnitude, and then it is easy to quickly compare the numbers with the same order of magnitude because they are only written as a number between one and ten.

Example 2 (10 minutes): Arithmetic Operations with Numbers Written Using Scientific Notation

Model the solutions to the following example problems. Be sure to emphasize that final answers should be expressed using the scientific notation convention of a number between 1 and 10 and a power of 10. On part (a), it may be necessary to provide some additional scaffolding if students are struggling to rewrite the numbers using the same order of magnitude. Have students practice writing a number as a product of a power of 10 in three different ways. For example, 15 000 = 1.5 × 104 = 15 × 103 = 150 × 102. The lessons of Algebra I, Module 1, Topic B provide some suggestions and fluency exercises if students need additional practice on arithmetic operations with numbers in scientific notation. Be sure that students understand that the properties of exponents allow them to quickly perform the indicated operations.

Scaffolding: Example 2: Arithmetic Operations with Numbers Written Using Scientific Notation For Example 2, part (a), a. (ퟐ. ퟒ × ퟏퟎퟐퟎ) + (ퟒ. ퟓ × ퟏퟎퟐퟏ) students may want to add the exponents as they do when (ퟐ. ퟒ × ퟏퟎퟐퟎ) + (ퟒퟓ × ퟏퟎퟐퟎ) = ퟒퟕ. ퟒ × ퟏퟎퟐퟎ = ퟒ. ퟕퟒ × ퟏퟎퟐퟏ multiplying numbers written using scientific notation. Take b. (ퟕ × ퟏퟎ−ퟗ)(ퟓ × ퟏퟎퟓ) time to discuss the differences (ퟕ ∙ ퟓ) × (ퟏퟎ−ퟗ ∙ ퟏퟎퟓ) = ퟑퟓ × ퟏퟎ−ퟒ = ퟑ. ퟓ × ퟏퟎ−ퟑ in the three expressions if students are making this type

of mistake. ퟏ.ퟐ × ퟏퟎퟏퟓ c. ퟑ × ퟏퟎퟕ ퟏ. ퟐ × ퟏퟎퟏퟓ−ퟕ = ퟎ. ퟒ × ퟏퟎퟖ = ퟒ × ퟏퟎퟕ ퟑ

Debrief with the questions designed to help students see that the order of magnitude and the properties of exponents greatly simplify calculations. . How do the properties of exponents help to simplify these calculations? . How can you quickly estimate the size of your answer?

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

Exercises 9–10 (5 minutes)

Exercises 9–10

9. Perform the following calculations without rewriting the numbers in decimal form. a. (ퟏ. ퟒퟐ × ퟏퟎퟏퟓ) − (ퟐ × ퟏퟎퟏퟑ)

ퟏퟒퟐ × ퟏퟎퟏퟑ − ퟐ × ퟏퟎퟏퟑ = (ퟏퟒퟐ − ퟐ) × ퟏퟎퟏퟑ = ퟏퟒퟎ × ퟏퟎퟏퟑ = ퟏ. ퟒ × ퟏퟎퟏퟓ

b. (ퟏ. ퟒퟐ × ퟏퟎퟏퟓ)(ퟐ. ퟒ × ퟏퟎퟏퟑ)

(ퟏ. ퟒퟐ ∙ ퟐ. ퟒ) × (ퟏퟎퟏퟓ ∙ ퟏퟎퟏퟑ) = ퟑ. ퟒퟎퟖ × ퟏퟎퟐퟖ

ퟏ.ퟒퟐ × ퟏퟎ−ퟓ c. ퟐ × ퟏퟎퟏퟑ ퟏ. ퟒퟐ × ퟏퟎ−ퟓ = ퟎ. ퟕퟏ × ퟏퟎ−ퟓ−ퟏퟑ = ퟎ. ퟕퟏ × ퟏퟎ−ퟏퟖ = ퟕ. ퟏ × ퟏퟎ−ퟏퟗ ퟐ × ퟏퟎퟏퟑ

10. Estimate how many times farther Jupiter is from the sun than Earth is from the sun. Estimate how many times farther Pluto is from the sun than Earth is from the sun.

Earth is approximately ퟏ. ퟓ × ퟏퟎퟖ 퐤퐦 from the sun, and Jupiter is approximately ퟕ. ퟖ × ퟏퟎퟖ 퐤퐦 from the sun. Therefore, Jupiter is about ퟓ times as far from the sun as Earth is from the sun. Pluto is approximately ퟓ. ퟗ × ퟏퟎퟗ 퐤퐦 from the sun. Therefore, since ퟓퟗ × ퟏퟎퟖ ퟓퟗ = ≈ ퟑퟗ. ퟑퟑ, ퟏ. ퟓ × ퟏퟎퟖ ퟏ. ퟓ Pluto is approximately ퟑퟗ times as far from the sun as Earth is from the sun.

Closing (3 minutes) Have students discuss the following question with a partner and record the definition of scientific notation in their mathematics notebooks. Debrief by asking a few students to share their responses with the entire class. . List two advantages of writing numbers using scientific notation.  You do not have to write as many zeros when working with very large or very small numbers, and you can quickly multiply and divide numbers using the properties of exponents.

Exit Ticket (4 minutes)

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Name Date

Lesson 2: Base 10 and Scientific Notation

Exit Ticket

1. A sheet of gold foil is 0.28 millionth of a meter thick. Write the thickness of a gold foil sheet measured in centimeters using scientific notation.

2. Without performing the calculation, estimate which expression is larger. Explain how you know. 4 × 1012 (4 × 1010)(2 × 105) and 2 × 10−4

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Exit Ticket Sample Solutions

1. A sheet of gold foil is ퟎ. ퟐퟖ millionth of a meter thick. Write the thickness of a gold foil sheet measured in centimeters using scientific notation.

The thickness is ퟎ. ퟐퟖ × ퟏퟎ−ퟔ 퐦. In scientific notation, the thickness of a gold foil sheet is ퟐ. ퟖ × ퟏퟎ−ퟕ 퐦, which is ퟐ. ퟖ × ퟏퟎ−ퟓ 퐜퐦.

2. Without performing the calculation, estimate which expression is larger. Explain how you know.

ퟒ × ퟏퟎퟏퟐ (ퟒ × ퟏퟎퟏퟎ)(ퟐ × ퟏퟎퟓ) and ퟐ × ퟏퟎ−ퟒ The order of magnitude on the first expression is ퟏퟓ, and the order of magnitude on the second expression is ퟏퟔ. The product and quotient of the number between ퟏ and ퟏퟎ in each expression is a number between ퟏ and ퟏퟎ. Therefore, the second expression is larger than the first one.

Problem Set Sample Solutions

1. Write the following numbers used in these statements in scientific notation. (Note: Some of these numbers have been rounded.) a. The density of helium is ퟎ. ퟎퟎퟎퟏퟕퟖퟓ gram per cubic centimeter.

ퟏ. ퟕퟖퟓ × ퟏퟎ−ퟒ

b. The boiling point of gold is ퟓ, ퟐퟎퟎ°퐅.

ퟓ. ퟐ × ퟏퟎퟑ

c. The speed of light is ퟏퟖퟔ, ퟎퟎퟎ miles per second.

ퟏ. ퟖퟔ × ퟏퟎퟓ

d. One second is ퟎ. ퟎퟎퟎퟐퟕퟖ hour.

ퟐ. ퟕퟖ × ퟏퟎ−ퟒ

e. The acceleration due to gravity on the sun is ퟗퟎퟎ 퐟퐭/퐬ퟐ.

ퟗ × ퟏퟎퟐ

f. One cubic inch is ퟎ. ퟎퟎퟎퟎퟐퟏퟒ cubic yard.

ퟐ. ퟏퟒ × ퟏퟎ−ퟓ

g. Earth’s population in 2012 was ퟕ, ퟎퟒퟔ, ퟎퟎퟎ, ퟎퟎퟎ people.

ퟕ. ퟎퟒퟔ × ퟏퟎퟗ

h. Earth’s distance from the sun is ퟗퟑ, ퟎퟎퟎ, ퟎퟎퟎ miles.

ퟗ. ퟑ × ퟏퟎퟕ

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

i. Earth’s radius is ퟒ, ퟎퟎퟎ miles.

ퟒ × ퟏퟎퟑ

j. The diameter of a water molecule is ퟎ. ퟎퟎퟎퟎퟎퟎퟎퟐퟖ 퐜퐦.

ퟐ. ퟖ × ퟏퟎ−ퟖ

2. Write the following numbers in decimal form. (Note: Some of these numbers have been rounded.) a. A light year is ퟗ. ퟒퟔ × ퟏퟎퟏퟓ 퐦.

ퟗ, ퟒퟔퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ

b. Avogadro’s number is ퟔ. ퟎퟐ × ퟏퟎퟐퟑ 퐦퐨퐥−ퟏ.

ퟔퟎퟐ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ

퐦 ퟐ ퟔ. ퟔퟕퟒ × ퟏퟎ−ퟏퟏ 퐍 ( ) c. The universal gravitational constant is 퐤퐠 . ퟎ. ퟎퟎퟎퟎퟎퟎퟎퟎퟎퟎퟔퟔퟕퟒ

d. Earth’s age is ퟒ. ퟓퟒ × ퟏퟎퟗ years.

ퟒ, ퟓퟒퟎ, ퟎퟎퟎ, ퟎퟎퟎ

e. Earth’s mass is ퟓ. ퟗퟕ × ퟏퟎퟐퟒ 퐤퐠.

ퟓ, ퟗퟕퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ, ퟎퟎퟎ

f. A foot is ퟏ. ퟗ × ퟏퟎ−ퟒ mile.

ퟎ. ퟎퟎퟎퟏퟗ

g. The population of China in 2014 was ퟏ. ퟑퟓퟒ × ퟏퟎퟗ people.

ퟏ, ퟑퟓퟒ, ퟎퟎퟎ, ퟎퟎퟎ

h. The density of oxygen is ퟏ. ퟒퟐퟗ × ퟏퟎ−ퟒ grams per liter.

ퟎ. ퟎퟎퟎퟏퟒퟐퟗ

i. The width of a pixel on a smartphone is ퟕ. ퟖ × ퟏퟎ−ퟐ 퐦퐦.

ퟎ. ퟎퟕퟖ

j. The wavelength of light used in optic fibers is ퟏ. ퟓퟓ × ퟏퟎ−ퟔ 퐦.

ퟎ. ퟎퟎퟎퟎퟎퟏퟓퟓ

Lesson 2: Base 10 and Scientific Notation 39

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

3. State the necessary value of 풏 that will make each statement true. a. ퟎ. ퟎퟎퟎ ퟎퟐퟕ = ퟐ. ퟕ × ퟏퟎ풏 −ퟓ

b. −ퟑ. ퟏퟐퟓ = −ퟑ. ퟏퟐퟓ × ퟏퟎ풏 ퟎ

c. ퟕ, ퟓퟒퟎ, ퟎퟎퟎ, ퟎퟎퟎ = ퟕ. ퟓퟒ × ퟏퟎ풏 ퟗ

d. ퟎ. ퟎퟑퟑ = ퟑ. ퟑ × ퟏퟎ풏 −ퟐ

e. ퟏퟓ = ퟏ. ퟓ × ퟏퟎ풏 ퟏ

f. ퟐퟔ, ퟎퟎퟎ × ퟐퟎퟎ = ퟓ. ퟐ × ퟏퟎ풏 ퟔ

g. ퟑퟎퟎퟎ × ퟎ. ퟎퟎퟎퟑ = ퟗ × ퟏퟎ풏 −ퟏ

h. ퟎ. ퟎퟎퟎퟒ × ퟎ. ퟎퟎퟐ = ퟖ × ퟏퟎ풏 −ퟕ

ퟏퟔퟎퟎퟎ i. = ퟐ × ퟏퟎ풏 ퟖퟎ ퟐ

ퟓퟎퟎ j. = ퟐ. ퟓ × ퟏퟎ풏 ퟎ.ퟎퟎퟐ ퟓ

4. Perform the following calculations without rewriting the numbers in decimal form. a. (ퟐ. ퟓ × ퟏퟎퟒ) + (ퟑ. ퟕ × ퟏퟎퟑ)

ퟐ. ퟖퟕ × ퟏퟎퟒ

b. (ퟔ. ퟗ × ퟏퟎ−ퟑ) − (ퟖ. ퟏ × ퟏퟎ−ퟑ)

−ퟏ. ퟐ × ퟏퟎ−ퟑ

Lesson 2: Base 10 and Scientific Notation 40

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

c. (ퟔ × ퟏퟎퟏퟏ)(ퟐ. ퟓ × ퟏퟎ−ퟓ)

ퟏ. ퟓ × ퟏퟎퟕ

ퟒ.ퟓ × ퟏퟎퟖ d. ퟐ × ퟏퟎퟏퟎ ퟐ. ퟐퟓ × ퟏퟎ−ퟐ

5. The wavelength of visible light ranges from ퟔퟓퟎ nanometers to ퟖퟓퟎ nanometers, where ퟏ 퐧퐦 = ퟏ × ퟏퟎ−ퟕ 퐜퐦. Express the range of wavelengths of visible light in centimeters.

Convert ퟔퟓퟎ nanometers to centimeters: (ퟔ. ퟓ × ퟏퟎퟐ)(ퟏ × ퟏퟎ−ퟕ) = ퟔ. ퟓ × ퟏퟎ−ퟓ Convert ퟖퟓퟎ nanometers to centimeters: (ퟖ. ퟓ × ퟏퟎퟐ)(ퟏ × ퟏퟎ−ퟕ) = ퟖ. ퟓ × ퟏퟎ−ퟓ

The wavelength of visible light in centimeters is ퟔ. ퟓ × ퟏퟎ−ퟓ 퐜퐦 to ퟖ. ퟓ × ퟏퟎ−ퟓ 퐜퐦.

6. In 1694, the Dutch scientist Antonie van Leeuwenhoek was one of the first scientists to see a red blood cell in a microscope. He approximated that a red blood cell was “ퟐퟓ, ퟎퟎퟎ times as small as a grain of sand.” Assume a grain ퟏ of sand is 퐦퐦 wide, and a red blood cell is approximately ퟕ micrometers wide. One micrometer is ퟏ × ퟏퟎ−ퟔ 퐦. ퟐ Support or refute Leeuwenhoek’s claim. Use scientific notation in your calculations.

Convert millimeters to meters: (ퟓ × ퟏퟎ−ퟏ)(ퟏ × ퟏퟎ−ퟑ) = ퟓ × ퟏퟎ−ퟒ. A medium-size grain of sand measures ퟓ × ퟏퟎ−ퟒ 퐦 across. Similarly, a red blood cell is approximately ퟕ × ퟏퟎ−ퟔ 퐦 across. Dividing these numbers produces

ퟓ × ퟏퟎ−ퟒ = ퟎ. ퟕퟏퟒ × ퟏퟎퟐ = ퟕ. ퟏퟒ × ퟏퟎퟏ. ퟕ × ퟏퟎ−ퟔ So, a red blood cell is ퟕퟏ. ퟒ times as small as a grain of sand. Leeuwenhoek’s claim was off by approximately a factor of ퟑퟓퟎ.

7. When the Mars Curiosity Rover entered the atmosphere of Mars on its descent in ퟐퟎퟏퟐ, it was traveling roughly ퟏퟑ, ퟐퟎퟎ 퐦퐩퐡. On the surface of Mars, its speed averaged ퟎ. ퟎퟎퟎퟕퟑ 퐦퐩퐡. How many times faster was the speed when it entered the atmosphere than its typical speed on the planet’s surface? Use scientific notation in your calculations. ퟏ. ퟑퟐ × ퟏퟎퟒ = ퟎ. ퟏퟖ × ퟏퟎퟖ = ퟏ. ퟖ × ퟏퟎퟕ ퟕ. ퟑ × ퟏퟎ−ퟒ The speed when it entered the atmosphere is greater than its surface speed by an order of magnitude of ퟕ.

8. Earth’s surface is approximately ퟕퟎ% water. There is no water on the surface of Mars, and its diameter is roughly half of Earth’s diameter. Assume both planets are spherical. The radius of Earth is approximately ퟒ, ퟎퟎퟎ miles. The surface area of a sphere is given by the formula 푺푨 = ퟒ흅풓ퟐ, where 풓 is the radius of the sphere. Which has more land mass, Earth or Mars? Use scientific notation in your calculations.

The surface area of Earth: ퟒ흅(ퟒퟎퟎퟎ 퐦퐢)ퟐ ≈ ퟐ × ퟏퟎퟖ 퐦퐢ퟐ

The surface area of Mars: ퟒ흅(ퟐퟎퟎퟎ 퐦퐢)ퟐ ≈ ퟓ × ퟏퟎퟕ퐦퐢ퟐ

Thirty percent of Earth’s surface area is approximately ퟔ × ퟏퟎퟕ square miles. Earth has more land mass by approximately ퟐퟎ%.

Lesson 2: Base 10 and Scientific Notation 41

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M3 ALGEBRA II

9. There are approximately ퟐퟓ trillion (ퟐ. ퟓ × ퟏퟎퟏퟑ) red blood cells in the human body at any one time. A red blood cell is approximately ퟕ × ퟏퟎ−ퟔ 퐦 wide. Imagine if you could line up all your red blood cells end to end. How long would the line of cells be? Use scientific notation in your calculations.

Because (ퟐ. ퟓ × ퟏퟎퟏퟑ)(ퟕ × ퟏퟎ−ퟔ) = ퟏ. ퟕퟓ × ퟏퟎퟖ, the line of cells would be ퟏ. ퟕퟓ × ퟏퟎퟖ 퐦 long, which is ퟏ. ퟕퟓ × ퟏ.ퟕퟓ × ퟏퟎퟓ ퟏퟎퟓ 퐤퐦. One mile is equivalent to ퟏ. ퟔ 퐤퐦, so the line of blood cells measures 퐤퐦, which is ퟏ.ퟔ approximately ퟏퟎퟗ, ퟑퟕퟓ 퐦퐢, which is almost halfway to the moon!

10. Assume each person needs approximately ퟏퟎퟎ square feet of living space. Now imagine that we are going to build a giant apartment building that will be ퟏ mile wide and ퟏ mile long to house all the people in the United States, estimated to be ퟑퟏퟑ. ퟗ million people in ퟐퟎퟏퟐ. If each floor of the apartment building is ퟏퟎ feet high, how tall will the apartment building be?

Since (ퟑ. ퟏퟑퟗ × ퟏퟎퟖ)(ퟏퟎퟎ ) = ퟑ. ퟏퟑퟗ × ퟏퟎퟏퟎ, we need ퟑ. ퟏퟑퟗ × ퟏퟎퟏퟎ 퐟퐭ퟐ of living space.

Next, divide the total number of square feet by the number of square feet per floor to get the number of needed floors. Remember that ퟏ 퐦퐢ퟐ = ퟓퟐퟖퟎퟐ 퐟퐭ퟐ = ퟐퟕ ퟖퟕퟖ ퟒퟎퟎ 퐟퐭ퟐ. ퟑ. ퟏퟑퟗ × ퟏퟎퟏퟎ 퐟퐭ퟐ ≈ ퟏ. ퟏퟐퟔ × ퟏퟎퟑ ퟐ. ퟕퟖퟕ ퟖퟒ × ퟏퟎퟕ 퐟퐭ퟐ Multiplying the number of floors by ퟏퟎ feet per floor gives a height of ퟏퟏ, ퟐퟔퟎ feet, which is approximately ퟐ. ퟏퟑ miles.

Lesson 2: Base 10 and Scientific Notation 42