# Lesson 9

Describing Large and Small Numbers Using Powers of 10

## 9.1: Thousand Million Billion Trillion (5 minutes)

### Warm-up

In this warm-up, students connect thousand, million, billion, and trillion to their respective powers of ten—$$10^3, 10^6, 10^9$$, and $$10^{12}$$. Understanding powers of 10 associated with these denominations will help students reason about quantities in real-world contexts such as the number of cells in a human body (trillions), world population (billions), etc.

### Launch

Arrange students in groups of 2. Give students 2 minutes of quiet work time and then 1 minute to compare their responses with their partner. Given the limited time, it may not be possible for students to create examples for each of the values in the second question. Tell students to try to at least find 1 or 2 examples and then to find others as time allows. Follow with a whole-class discussion.

### Student Facing

1. Match each expression with its corresponding value and word.
expression
$$10^{\text-3}$$
$$10^6$$
$$10^9$$
$$10^{\text-2}$$
$$10^{12}$$
$$10^3$$

value
1,000,000,000,000
$$\frac{1}{100}$$
1,000
1,000,000,000
1,000,000
$$\frac{1}{1,000}$$

word
billion
milli-
million
thousand
centi-
trillion

2. For each of the numbers, think of something in the world that is described by that number.

### Anticipated Misconceptions

Some students may think that, for example, $$1,\!000,\!000 = 10^7$$ because the number 1,000,000 has 7 digits. Ask these students if it is true that $$10 = 10^2$$ .

Some students may confuse the prefix “milli-” with the word “million.” The word “million” literally means “a big thousand,” and so both “million” and “mille” are related to the Latin “mille,” meaning “thousand.” While “milli-” is talking about a thousand parts (thousandths), “million” is talking about a thousand thousands.

### Activity Synthesis

Ask students to share the corresponding expressions, words, and values. Record and display their responses for all to see. If time is limited, consider displaying the completed table for all to see and discussing any questions or disagreements. Then, invite students to share their examples for the final question. After each student shares, ask the class whether they agree that the given example could be described by that value.

If students struggle to find something that could be described by each value, consider sharing some of the following examples:

• Thousand ($$10^3$$)
• Number of students in a school
• Population of an endangered species
• Cost of a car that barely runs
• Number of brain cells of a jellyfish
• Million ($$10^6$$)
• Population of a state
• Cost of the most expensive car in the world
• Number of brain cells of a cockroach
• Billion ($$10^9$$)
• Population of India
• Population of China
• Number of students in the world
• Number of brain cells of a monkey
• Number of trees in the U.S.
• Trillion ($$10^{12}$$)
• Amount of wealth produced by a developed country in a year in U.S. dollars
• Number of stars in a large galaxy
• Number of brain cells of 10 students

## 9.2: Base-ten Representations Matching (20 minutes)

### Activity

In this activity, students use their understanding of decimal place value and base-ten diagrams to practice working with the structure of scientific notation before it is formally introduced. They express numbers as sums of terms, each term being multiples of powers of 10. For example, 254 can be written as $$2 \boldcdot 10^2 + 5 \boldcdot 10^1 + 4 \boldcdot 10^0$$

Notice students who choose different, yet correct, diagrams for the first and last problems. Ask them to share their reasoning in the discussion later.

### Launch

Display the following diagram for all to see. Pause for quiet think time after asking each question about the diagram. Ask students to explain their thinking. Here are some questions to consider:

• “If each small square represents 1 tree, what does the whole diagram represent?” (121 trees)
• “If each small square represents 10 books, what does the whole diagram represent?” (1,210 books)
• “If each small square represents 1,000,000 stars, what does the whole diagram represent?” (121,000,000 stars)
• “If each small square represents 0.1 seconds, what does the whole diagram represent?” (12.1 seconds)
• “If each small square represents $$10^3$$ people, what does the whole diagram represent?” (121,000 people)

Give students 10 minutes to work on the task, followed by a brief whole-class discussion.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, display only the first expression and ask students what they notice before inviting them to select one or more diagrams that could represent it. Ask 1-2 students to explain their match before revealing the remaining expressions.
Supports accessibility for: Conceptual processing; Organization

### Student Facing

1. Match each expression to one or more diagrams that could represent it. For each match, explain what the value of a single small square would have to be.
1. $$2 \boldcdot 10^{\text -1} + 4 \boldcdot 10^{\text -2}$$

2. $$2 \boldcdot 10^{\text -1} + 4 \boldcdot 10^{\text -3}$$

3. $$2 \boldcdot 10^3 + 4 \boldcdot 10^1$$

4. $$2 \boldcdot 10^3 + 4 \boldcdot 10^2$$

1. Write an expression to describe the base-ten diagram if each small square represents $$10^{\text -4}$$. What is the value of this expression? 2. How does changing the value of the small square change the value of the expression? Explain or show your thinking.
3. Select at least two different powers of 10 for the small square, and write the corresponding expressions to describe the base-ten diagram. What is the value of each of your expressions?

### Anticipated Misconceptions

Some students may think that the small square must always represent one unit. Explain to these students that, as in the launch, the small square might represent 10 units, 0.1 units, or any other power of 10.

Some students may have trouble writing the value of expressions that involve powers of 10, especially if they involve negative exponents. As needed, suggest that they expand the expression into factors that are 10, and remind them that $$10^{\text-1}$$ corresponds to the tenths place, $$10^{\text-2}$$ corresponds to the hundredths place, etc.

### Activity Synthesis

Select students to share their responses to the first and last problems. Bring attention to the fact that diagrams B or C could be used depending on the choice of the value of the small square.

The main goal of the discussion is to make sure students see the connection between decimal place value to sums of terms that are multiples of powers of 10. To highlight this connection explicitly, consider discussing the following questions:

• “How are the diagrams related to our base-ten numbers and place value system?” (In base-ten numbers, each place value is ten times larger than the one to its right; so every 1 unit of a place value can be composed of 10 units of the next place value to its right. The diagrams work the same way: each shape representing a base-ten unit can be composed of 10 that represent another unit that is one tenth of its value.)
• “How are the diagrams related to numbers written using powers of 10?” (We can think of each place value as a power of ten. So a ten would be $$10^1$$, a hundred would be $$10^2$$, a tenth would be $$10^\text{-1}$$, and so on.)
• “If each large square represents $$10^2$$, what do 2 large squares and 4 long rectangles represent?” (A long rectangle is a tenth of the large square, so we know the long rectangle represents $$10^1$$. This means 2 large squares and 4 long rectangles represent $$2 \boldcdot 10^2 + 4 \boldcdot 10^1$$.)
• “Why is it possible for one base-ten diagram to represent many different numbers?” (Because of the structure of our place value system—where every group of 10 of a base-ten unit composes 1 of the next higher unit—is consistent across all place values.)
Speaking, Listening: MLR7 Compare and Connect. As students prepare to share their responses to the first and last problems, look for those using different strategies for matching base-ten representations. During the discussion, ask students to share what worked well in a particular approach. Listen for and amplify any comments that describe each representation (i.e., place value, base-ten unit, powers of 10). Then encourage students to make the connection between decimal place value to sums of terms that are multiples of powers of 10. This will foster students’ meta-awareness and support constructive conversations as they compare strategies for describing large and small quantities.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

## 9.3: Using Powers of 10 to Describe Large and Small Numbers (15 minutes)

### Activity

This activity motivates students to find easier ways to communicate about very large and very small numbers, using powers of 10 and working toward using scientific notation. Students take turns reading aloud and writing down quantities that involve long strings of digits, noticing the challenges of expressing such numbers.

As students work, monitor the different ways students communicate the number of zeros precisely to their partners. Some might use standard vocabulary (billion, ten-thousandth, etc.), or some may communicate the number of zeros after the decimal point or after the significant digits. Select students using different strategies to share later.

### Launch

Arrange students in groups of 2. Distribute a pair of cards (one for Partner A and one for Partner B) from the blackline master to each group. Ask partners not to show their card to each other.

Tell students that one partner should read an incomplete statement in the materials and the other partner should read aloud the missing information on the card. The goal is for each partner to write down the missing quantity correctly. Partners should take turns reading and writing until all four statements for each person are completed.

Consider explaining (either up front or as needed during work time) that students who have the numbers can describe or name them in any way that they think convey the quantities fully. Likewise, those writing the numbers can write in any way that captures the quantities accurately.

Give students 10 minutes to work. Leave a few minutes for a whole-class discussion.

Action and Expression: Internalize Executive Functions. Begin with a whole-class think aloud to demonstrate how students should work together.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Your teacher will give you a card that tells you whether you are Partner A or B and gives you the information that is missing from your partner’s statements. Do not show your card to your partner.

Partner A’s statements:

2. The mass of a proton is ______________________ kilograms.
3. The population of Russia is about ______________________ people.
4. The diameter of a bacteria cell is about ______________________ meter.

Partner B’s statements:

1. Light waves travel through space at a speed of ______________________ meters per second.
2. The population of India is about ______________________ people.
3. The wavelength of a gamma ray is _______________________ meters.
4. The tardigrade (water bear) is ______________ meters long.

### Student Facing

#### Are you ready for more?

A “googol” is a name for a really big number: a 1 followed by 100 zeros.

1. If you square a googol, how many zeros will the answer have? Show your reasoning.
2. If you raise a googol to the googol power, how many zeros will the answer have? Show your reasoning.

### Activity Synthesis

Ask previously identified students to share how they described their partner's numbers or recorded those given to them. The purpose of the discussion is for students to hear different strategies for communicating very small and very large numbers. For example, 150,000,000,000 can be described as “one hundred fifty billion,” as “15 followed by 10 zeros,” as $$150 \boldcdot 10^9$$, or as $$(1.5) \boldcdot 10^{11}$$.

For negative powers, discuss the idea that multiplying by $$10^\text{-1}$$ means multiplying by $$\frac{1}{10}$$, which in turn increases the number of decimal places. For example, consider $$48 \boldcdot 10^{\text-13}$$. We know that $$1 \boldcdot 10^{\text-13}$$ is 1 multiplied by $$\frac{1}{10}$$, 13 times, which is $$0.0000000000001$$. So $$48 \boldcdot 10^{\text-13}$$ is $$48 \boldcdot (0.0000000000001)$$, which is $$0.0000000000048$$. Display the following table for all to see and briefly explain to students how to write each number using powers of 10.

quantities using powers of 10
150,000,000,000 meters $$150 \boldcdot 10^9$$ meters
300,000,000 meters per second $$300 \boldcdot 10^6$$ meters per second
0.0000000000048 meters $$48 \boldcdot 10^{\text-13}$$ meters
0.00000000000000000000000000167 kilogram $$167 \boldcdot 10^{\text-29}$$ kilogram
Conversing, Representing: MLR2 Collect and Display. While pairs are working, circulate and listen to students talk about very large and very small numbers. Write down common or important phrases you hear the ways students communicate the number of zeros to their partners. Listen for students who use standard vocabulary (billion, ten-thousandth, etc.), provide the number of zeros after the decimal point or after the significant digits, or refer to powers of 10. Display their representations together with the students’ language onto a visual display. This will help students use mathematical language when communicating about very small and very large numbers during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

## Lesson Synthesis

### Lesson Synthesis

The focus of the discussion is the structure of our place value system and the rationale and usefulness of describing large and small numbers in different ways. Consider asking some of the following questions:

• “How do base-ten diagrams help us make sense of (or explain) the exponents in powers of 10?” (When using diagrams, grouping 10 of the next smaller unit means multiplying by 10. When dealing with powers of 10, multiplying by 10 increases the exponent by 1. Likewise, decomposing a base-ten unit into 10 of the next smaller unit means multiplying by $$\frac{1}{10}$$, so the exponent in the power of 10 goes down by 1.)
• “How does using powers of 10 make it easier to communicate about very large or very small numbers?” (We can write in a smaller space. It’s also faster to read and easier to understand the size of a number and to compare numbers. Using powers of 10 helps us avoid errors of missing zeros or extra zeros.)
• “What are some different ways to describe a large number like 123 billion?” ($$123 \boldcdot (1,\!000,\!000,\!000$$ or $$123 \boldcdot 10^9$$.)
• “What are some different ways to describe a small number like 0.0000000789?” (789 ten-billionths, $$789 \boldcdot \frac{1}{10,000,000,000}$$, or $$789 \boldcdot 10^\text{-10}$$.)

## Student Lesson Summary

### Student Facing

Sometimes powers of 10 are helpful for expressing quantities, especially very large or very small quantities. For example, the United States Mint has made over

500,000,000,000

pennies. In order to understand this number, we have to count all the zeros. Since there are 11 of them, this means there are 500 billion pennies. Using powers of 10, we can write this as: $$\displaystyle 500 \boldcdot 10^9$$ (five hundred times a billion), or even as: $$\displaystyle 5 \boldcdot 10^{11}$$ The advantage to using powers of 10 to write a large number is that they help us see right away how large the number is by looking at the exponent.

The same is true for small quantities. For example, a single atom of carbon weighs about

0.0000000000000000000000199

grams. We can write this using powers of 10 as $$\displaystyle 199 \boldcdot 10^{\text-25}$$ or, equivalently, $$\displaystyle (1.99) \boldcdot 10^{\text-23}$$ Not only do powers of 10 make it easier to write this number, but they also help avoid errors since it would be very easy to write an extra zero or leave one out when writing out the decimal because there are so many to keep track of!