# Lesson 12

Applications of Arithmetic with Powers of 10

## 12.1: What Information Do You Need? (5 minutes)

### Warm-up

The purpose of this warm-up is for students to reason about a real-world situation and consider the essential information required to solve problems (MP4).

### Launch

Arrange students in groups of 2. Give students 1 minute of quiet think time, followed by 1 minute to share their responses with a partner. Follow with a whole-class discussion.

### Student Facing

What information would you need to answer these questions?

1. How many meter sticks does it take to equal the mass of the Moon?
2. If all of these meter sticks were lined up end to end, would they reach the Moon?

### Activity Synthesis

Ask students to share their responses for each question. Record and display the responses for all to see.

Consider asking questions like these to encourage students to reason further about each question:

• “Why do you need that piece of information?”
• “How would you use that piece of information in finding the solution?”
• “Where would you look to find that piece of information?”

If there is time, ask students for predictions for each of the questions. Record and display their responses for all to see.

## 12.2: Meter Sticks to the Moon (20 minutes)

### Activity

The large quantities involved in these questions lend themselves to arithmetic with powers of 10, giving students the opportunity to make use of scientific notation before it is formally introduced. This activity was designed so students could practice modeling skills such as identifying essential features of the problem and gathering the required information (MP4). Students use powers of 10 and the number line as tools to make it easier to calculate and interpret results.

Notice the ways in which students use relevant information to answer the questions. Identify students who can explain why they are calculating with one operation rather than another. Speed is not as important as carefully thinking through each problem.

Teacher Notes for IM 6–8 Math Accelerated
Adjust time to 15 minutes. Give students 10 minutes of work time.

### Launch

From the warm-up, students have decided what information they need to solve the problem. Invite students to ask for the information they need. Provide students with only the information they request. Display the information for students to see throughout the activity. If students find they need more information later, provide it to the whole class then.

Here is information students might ask for in order to solve the problems:

• The mass of an average classroom meter stick is roughly 0.2 kg.
• The length of an average classroom meter stick is 1 meter.
• The mass of the Moon is approximately $$7 \boldcdot 10^{22}$$ kg.
• The Moon is roughly $$(3.8) \boldcdot 10^8$$ meters away from Earth.
• The distances to various astronomical bodies the students might recognize, in light years, as points of reference for their last answer. (Consider researching other distances in advance or, if desired, encouraging interested students to do so.)

Arrange students in groups of 2–4 so they can discuss how to use the information to solve the problem. Give students 15 minutes of work time.

### Student Facing

1. How many meter sticks does it take to equal the mass of the Moon? Explain or show your reasoning.
2. Label the number line and plot your answer for the number of meter sticks.

3. If you took all the meter sticks from the last question and lined them up end to end, will they reach the Moon? Will they reach beyond the Moon? If yes, how many times farther will they reach? Explain your reasoning.
4. One light year is approximately $$10^{16}$$ meters. How many light years away would the meter sticks reach? Label the number line and plot your answer.

### Student Facing

#### Are you ready for more?

Here is a problem that will take multiple steps to solve. You may not know all the facts you need to solve the problem. That is okay. Take a guess at reasonable answers to anything you don’t know. Your final answer will be an estimate.

If everyone alive on Earth right now stood very close together, how much area would they take up?

### Activity Synthesis

Select previously identified students to share how they organized their relevant information and how they planned to use the information to answer the questions. The important idea students should walk away with is that powers of 10 are a great tool to tackle challenging, real-world problems that involve very large numbers.

It might be illuminating to put 35 million light years into some context. It is over a thousand trillion times as far as the distance to the Moon, or about the size of a supercluster of galaxies. The Sun is less than $$1.6 \times 10^\text{-5}$$ light year away from Earth.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . ..”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion Supports. Give students additional time to make sure everyone in their group can explain their response to the first question. Then, vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

## 12.3: The “Science” of Scientific Notation (15 minutes)

### Activity

Students learn the definition of scientific notation and practice using it. Students attend to precision when determining whether or not a number is in scientific notation and converting numbers into scientific notation (MP6).

Throughout the activity, students use the usual $$\boldcdot$$ symbol to indicate multiplication, but the discussion establishes the standard way to show multiplication in scientific notation with the $$\times$$ symbol. Although these materials tend to avoid the $$\times$$ symbol because it is easy to confuse with $$x$$, the ubiquitous use of $$\times$$ for scientific notation outside of these materials necessitates its use here.

### Launch

Tell students, “Earlier, we examined the speed of light through different materials. We zoomed into the number line to focus on the interval between $$2.0 \times 10^8$$ meters per second and $$3.0 \times 10^8$$ meters per second as shown in the figure.” Display the following image notation for all to see.

Tell students, “We saw that the speed of light through ice was $$2.3 \times 10^8$$ meters per second. This way of writing the number is called scientific notation. Scientific notation is useful for understanding very large and very small numbers.”

Display and explain the following definition of scientific notation for all to see.

A number is said to be in scientific notation when it is written as a product of two factors:

• The first factor is a number greater than or equal to 1, but less than 10, for example 1.2, 8, 6.35, or 2.008.

• The second factor is an integer power of 10, for example $$10^8$$, $$10^\text{-4}$$, or $$10^{22}$$.

Carefully consider the first question and go through the list of numbers as a class, frequently referring to the definition to decide whether the number is written in scientific notation. When all numbers written in scientific notation have been circled, consider demonstrating or discussing how a number that was not circled could be written in scientific notation. Then, ask students to complete the second question (representing the other numbers in scientific notation). Leave 3–4 minutes for a whole-class discussion.

### Student Facing

The table shows the speed of light or electricity through different materials.

material speed (meters per second)
space 300,000,000
water $$2.25 \times 10^8$$
copper (electricity) 280,000,000
diamond $$124 \times 10^6$$
ice $$2.3 \times 10^8$$
olive oil $$0.2 \times 10^9$$

Circle the speeds that are written in scientific notation. Write the others using scientific notation.

### Activity Synthesis

Tell students that almost all books and information about scientific notation use the $$\times$$ symbol to indicate multiplication between the two factors, so from now on, these materials will use the $$\times$$ symbol in this same way. Display $$(2.8) \boldcdot 10^8$$ for all to see, and then rewrite it as $$2.8 \times 10^8$$. Emphasize that using $$\boldcdot$$ is not incorrect, but that $$\times$$ is the most common usage.

Ask students to come up with at least two examples of numbers that are not in scientific notation. Select responses that highlight the fact that the first factor must be between 1 and 10 and other responses that highlight that one of the factors must be an integer power of 10. Make sure students recognize what does and does not count as scientific notation.

Also make sure students understand how to write an expression that may use a power of 10 but is not in scientific notation as one that is in scientific notation. Consider using the speed of light through diamond as an example. Ask a series of questions such as:

• “In $$124 \times 10^6$$, how must we write the first factor for the expression to be in scientific notation?” (A number between 1 and 10, so 1.24 in this case)
• “How can we rewrite 124 as an expression that has 1.24?” (Write it as $$1.24 \times 100$$ or $$1.24 \times 10^2$$)
• “What is the equivalent expression in scientific notation?” ($$1.24 \times 10^2) \times 10^6$$, which is $$1.24 \times 10^8$$)
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary.  Include the following term and maintain the display for reference throughout the unit: scientific notation. Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Writing: MLR3 Clarify, Critique, Correct. Display a hypothetical student statement that represents a misunderstanding about how to write values in scientific notation, such as: “To write the speed of light through a diamond is $$12.4 \times 10^7$$.” Ask pairs of students to critique the response by asking, “Do you agree with the author? Why or why not?”Invite students to write feedback to the author that identifies the reasoning error and how to improve the statement. Listen for students who include in their feedback a need for the first factor to be between 1 and 10. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness; Support sense-making

## 12.4: Scientific Notation Matching (15 minutes)

### Activity

In this activity, students match cards written in scientific notation with their decimal values. The game grants advantage to students who distinguish between numbers written in scientific notation from numbers that superficially resemble scientific notation (e.g. $$0.43 \times 10^5$$).

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if students need additional practice identifying numbers written in scientific notation.

### Launch

The blackline master has three sets of cards: set A, set B, and set C. Set A is meant for demonstration purposes, so only a single copy of set A is necessary.

Arrange students in groups of 2. Consider giving students a minute of quiet time to read the directions. Then, use set A to demonstrate a round of the game for the class. Explain to students that a match can be made by pairing any two cards that have the same value, but it is favorable to be able to tell the difference between numbers in scientific notation and numbers that simply look like they are in scientific notation.

When students indicate that they understand how to play, distribute a set of cards (either set B or set C) to each group. Save a few minutes for a whole-class discussion.

Action and Expression: Internalize Executive Functions. Begin with a whole-class think aloud to demonstrate the steps of the game. Consider providing some groups with cards that contain more accessible values to begin with.
Supports accessibility for: Memory; Conceptual processing
Representing, Conversing: MLR8 Discussion Supports. Demonstrate the steps of how to play the game. To do this, select a student to play the game with you while the rest of the class observes. This will help clarify the expectations of the task, invite more student participation, and facilitate meta-awareness of the language involving scientific notation.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Your teacher will give you and your partner a set of cards. Some of the cards show numbers in scientific notation, and other cards show numbers that are not in scientific notation.

1. Shuffle the cards and lay them facedown.

2. Players take turns trying to match cards with the same value.

3. On your turn, choose two cards to turn faceup for everyone to see. Then:

1. If the two cards have the same value and one of them is written in scientific notation, whoever says “Science!” first gets to keep the cards, and it becomes that player’s turn. If it’s already your turn when you call “Science!”, that means you get to go again. If you say “Science!” when the cards do not match or one is not in scientific notation, then your opponent gets a point.

2. If both partners agree the two cards have the same value, then remove them from the board and keep them. You get a point for each card you keep.

3. If the two cards do not have the same value, then set them facedown in the same position and end your turn.

4. If it is not your turn:

1. If the two cards have the same value and one of them is written in scientific notation, then whoever says “Science!” first gets to keep the cards, and it becomes that player’s turn. If you call “Science!” when the cards do not match or one is not in scientific notation, then your opponent gets a point.

2. Make sure both of you agree the cards have the same value.
If you disagree, work to reach an agreement.

5. Whoever has the most points at the end wins.

### Student Facing

#### Are you ready for more?

1. What is $$9 \times 10^{\text-1} + 9 \times 10^{\text-2}$$? Express your answer as:
1. A decimal

2. A fraction

2. What is $$9 \times 10^{\text-1} + 9 \times 10^{\text-2} + 9 \times 10^{\text-3} +9 \times 10^{\text-4}$$? Express your answer as:
1. A decimal

2. A fraction

3. The answers to the two previous questions should have been close to 1. What power of 10 would you have to go up to if you wanted your answer to be so close to 1 that it was only $$\frac{1}{1,000,000}$$ off?
4. What power of 10 would you have to go up to if you wanted your answer to be so close to 1 that it was only $$\frac{1}{1,000,000,000}$$ off? Can you keep adding numbers in this pattern to get as close to 1 as you want? Explain or show your reasoning.
5. Imagine a number line that goes from your current position (labeled 0) to the door of the room you are in (labeled 1). In order to get to the door, you will have to pass the points 0.9, 0.99, 0.999, etc. The Greek philosopher Zeno argued that you will never be able to go through the door, because you will first have to pass through an infinite number of points. What do you think? How would you reply to Zeno?

### Activity Synthesis

The main idea is for students to practice using the definition of scientific notation and flexibly convert numbers to scientific notation. Consider selecting students to explain how they could tell whether two cards had the same value and whether they were written in scientific notation.

## Lesson Synthesis

### Lesson Synthesis

The purpose of the discussion is to reflect on the modeling process and make sure that students understand the definition of scientific notation. Here are some possible questions for discussion. Consider displaying student responses for all to see.

• “Describe your thinking as you planned a solution path for the problems. For example, did you ask for information first and then decide what to do with it, or did you decide what needs to be done first before asking for certain information?”
• “Once you had the information you needed, what were some difficulties you encountered? How did you work through them?”
• “How did exponent rules and powers of 10 make the calculations easier?” (Powers of 10 make the numbers easier to express and interpret. The rules of exponents were handy for comparing how many times as large or as small one number is compared to another number.)
• “Why might scientific notation be useful?” (We don’t have to use as many zeros to write very large or very small numbers. It is quicker to compare numbers written in scientific notation.)
• “Can you think of information in the real world that might be easier to work with in scientific notation?” (Comparing the population of large groups of things, like the number of ants on different continents.)
• “How would you write a very small number like 0.000021 in scientific notation?” ($$2.1 \times 10^{\text-5}$$)
• “How would you write a very large number like 21,000,000 in scientific notation?” ($$2.1 \times 10^7$$)
If time allows, arrange students in groups of 2 and ask students to create a small decimal or large number for a partner to rewrite with scientific notation.

## Student Lesson Summary

### Student Facing

The total value of all the quarters made in 2014 is 400 million dollars. There are many ways to express this using powers of 10. We could write this as $$400 \boldcdot 10^6$$ dollars, $$40 \boldcdot 10^7$$ dollars, $$0.4 \boldcdot 10^9$$ dollars, or many other ways. One special way to write this quantity is called scientific notation. In scientific notation,

400 million

dollars would be written as $$\displaystyle 4 \times 10^8$$ dollars. For scientific notation, the $$\times$$ symbol is the standard way to show multiplication instead of the $$\boldcdot$$ symbol. Writing the number this way shows exactly where it lies between two consecutive powers of 10. The $$10^8$$ shows us the number is between $$10^8$$ and $$10^9$$. The 4 shows us that the number is 4 tenths of the way to $$10^9$$.

Some other examples of scientific notation are $$1.2 \times 10^{\text-8}$$, $$9.99 \times 10^{16}$$, and $$7 \times 10^{12}$$. The first factor is a number greater than or equal to 1, but less than 10. The second factor is an integer power of 10.

Thinking back to how we plotted these large (or small) numbers on a number line, scientific notation tells us which powers of 10 to place on the left and right of the number line. For example, if we want to plot $$3.4 \times 10^{11}$$ on a number line, we know that the number is larger than $$10^{11}$$, but smaller than $$10^{12}$$. We can find this number by zooming in on the number line: