# Lesson 11

Representing Small Numbers on the Number Line

## 11.1: Small Numbers on a Number Line (5 minutes)

### Warm-up

The purpose of this warm-up is for students to reason about expressions with negative exponents on a number line. Students explore a common misunderstanding about negative exponents that is helpful to address before scientific notation is used to describe very small numbers.

For students’ reference, consider displaying a number line from a previous lesson that shows powers of 10 on a number line.

### Launch

Give students 2 minutes of quiet work time, followed by a whole-class discussion.

### Student Facing

Kiran drew this number line. Andre said, “That doesn’t look right to me.”

Explain why Kiran is correct or explain how he can fix the number line.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The important idea to highlight during the discussion is that the larger the size (or absolute value) of a negative exponent, the closer the value of the expression is to zero. This is because the negative exponent indicates the number of factors that are $$\frac{1}{10}$$. For example, $$10^{\text-5}$$ represents 5 factors that are $$\frac{1}{10}$$ and $$10^{\text-6}$$ represents 6 factors that are $$\frac{1}{10}$$, so $$10^{\text-6}$$ is 10 times smaller than $$10^{\text-5}$$.

Ask one or more students to explain whether they think the number line is correct and ask for their reasoning. Record and display their reasoning for all to see, preferably on the number line. If possible, show at least two correct ways the number line can be fixed.

## 11.2: Comparing Small Numbers on a Number Line (10 minutes)

### Activity

This task is analogous to a previous activity with positive exponents. The number line strongly encourages students to think about how to change expressions so they all take the form $$b \boldcdot 10^k$$, where $$b$$ is between 1 and 10, as in the case of scientific notation. The number line also is a useful representation to show that, for example, $$29 \boldcdot 10^\text{-7}$$ is about half as much as $$6 \boldcdot 10^\text{-6}$$. The last two questions take such a comparison a step further, asking students to estimate relative sizes using numbers expressed with powers of 10.

As students work, look for different strategies they use to compare expressions that are not written as a product of a number and $$10^{\text-6}$$. Also look for students who can explain how they estimated in the last two problems. Select them to share their strategies later.

### Launch

Arrange students in groups of 2. Give students 5 minutes of quiet work time, followed by partner discussion and whole-class discussion. During partner discussion, ask students to share their responses for the first two questions and reach an agreement about where the numbers should be placed on the number line.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch take time to review the visual display of rules for exponents.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports. Display sentence frames to support students as they describe how they compared and estimated the difference in magnitude between the pairs of values $$29⋅10^{\text-7}$$ and $$6⋅10^{\text-6}$$ or $$7⋅10^{\text-8}$$ and $$3⋅10^{\text-9}$$. For example, “First I ____ , then I ____ .” or “_____ is (bigger/smaller) because ______ .”
Design Principle(s): Optimize output (for explanation)

### Student Facing 1. Label the tick marks on the number line.

2. Plot the following numbers on the number line:

A. $$6 \boldcdot 10^{\text -6}$$

B. $$6 \boldcdot 10^{\text -7}$$

C. $$29 \boldcdot 10^{\text -7}$$

D. $$(0.7) \boldcdot 10^{\text -5}$$

3. Which is larger, $$29 \boldcdot 10^{\text -7}$$ or $$6 \boldcdot 10^{\text -6}$$? Estimate how many times larger.
4. Which is larger, $$7 \boldcdot 10^{\text -8}$$ or $$3 \boldcdot 10^{\text -9}$$? Estimate how many times larger.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may have trouble comparing negative powers of 10. Remind these students that, for example, $$10^\text{-5}$$ is 5 factors that are $$\frac{1}{10}$$ and $$10^\text{-6}$$ is 6 factors that are $$\frac{1}{10}$$, so $$10^\text{-5}$$ is 10 times larger than $$10^\text{-6}$$.

Students may also have trouble estimating how many times one larger expression is than another. Offer these students an example to illustrate how representing numbers as a single digit times a power of 10 is useful for making rough estimations. We have that $$9 \boldcdot 10^\text{-12}$$ is roughly 50 times as much as $$2 \boldcdot 10^\text{-13}$$ because $$10^\text{-12}$$ is 10 times as much as $$10^\text{-13}$$ and 9 is roughly 5 times as much as 2. In other words,$$\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50$$

### Activity Synthesis

Select previously identified students to explain how they used the number line and powers of 10 to compare the numbers in the last two problems.

One important concept is that it’s always possible to change an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, we can think of $$29 \boldcdot 10^\text{-7}$$ as $$(2.9) \boldcdot 10 \boldcdot 10^\text{-7}$$ or $$(2.9) \boldcdot 10^\text{-6}$$. Another important concept is that powers of 10 can be used to make rough estimates. Make sure these ideas are uncovered during discussion.

## 11.3: Atomic Scale (20 minutes)

### Activity

Students convert a decimal to a multiple of a power of 10 and plot it on a number line. The first problem leads to a product of an integer and a power of 10, and the second leads to a product of a decimal and a power of 10. It is difficult to fit the numbers on the number line without using scientific notation. Again, students build experience with scientific notation before the term is formally introduced.

As students work, notice those who connect the number of decimal places to negative powers of 10. For example, they might notice that counting decimal places to the right of the decimal point corresponds to multiplying by $$\frac{1}{10}$$ a certain number of times.

### Launch

This is the first time students convert small decimals into a multiple of a power of 10 with a negative exponent. Before students begin the activity, review the idea that a decimal can be thought of as a product of a number and $$\frac{1}{10}$$. Explain, for example, that 0.3 is $$3 \boldcdot \frac{1}{10}$$ (3 tenths), 0.03 is $$3 \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}$$ (three hundredths). Similarly, 0.0003 is 3 multiplied by $$\frac{1}{10}$$ 4 times
(3 ten-thousandths). So $$(0.0003) = 3 \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = 3 \boldcdot 10^\text{-4}$$.

Arrange students in groups of 2. Give students 10 minutes to work, followed by whole-class discussion. Encourage students to share their reasoning with a partner and work to reach an agreement during the task.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support use of structure. For example, check in with students within the first 2-3 minutes of work time. Ask students to share how they decide what power of 10 to put on the right side of this number line.
Supports accessibility for: Visual-spatial processing; Organization
Writing: MLR3 Clarify, Critique, Correct. Display the incomplete statement: “I just count how many places and write the number in the exponent.” Prompt discussion by asking, “What is unclear?” or “What do you think the author is trying to say?” Then, ask students to write a more precise version to explain the strategy of converting. Improved statements should include reference to the relationship between the number of decimal places to the right of the decimal point and repeated multiplication of $$\frac{1}{10}$$. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Support sense-making

### Student Facing

1. The radius of an electron is about 0.0000000000003 cm.

1. Write this number as a multiple of a power of 10.
2. Decide what power of 10 to put on the right side of this number line and label it.

3. Label each tick mark as a multiple of a power of 10. 4. Plot the radius of the electron in centimeters on the number line.

2. The mass of a proton is about 0.0000000000000000000000017 grams.

1. Write this number as a multiple of a power of 10.
2. Decide what power of 10 to put on the right side of this number line and label it.

3. Label each tick mark as a multiple of a power of 10. 4. Plot the mass of the proton in grams on the number line.

3. Point $$A$$ on the zoomed-in number line describes the wavelength of a certain X-ray in meters. 1. Write the wavelength of the X-ray as a multiple of a power of 10.
2. Write the wavelength of the X-ray as a decimal.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

For the mass of the proton, students might find that it is equal to $$17 \boldcdot 10^\text{-25}$$, but 17 does not fit on the number line because there are not 17 tick marks. Ask students whether 1.7 would fit on the number line
(it does). Follow up by asking how to replace 17 with 1.7 times something ($$17 = (1.7) \boldcdot 10$$). So $$17 \boldcdot 10^\text{-25} = (1.7) \boldcdot 10 \boldcdot 10^\text{-25} = (1.7) \boldcdot 10^\text{-24}$$.

### Activity Synthesis

One key idea is for students to convert small decimals to scientific notation in the process of placing them on a number line. Select a student to summarize how they wrote the mass of the proton as a multiple of a power of 10. Poll the class on whether they agree or disagree and why. The discussion should lead to one or more methods to rewrite a decimal as a multiple of a power of 10. For example, students might count the decimal places to the right of the decimal point and recognize that number as the number of factors that are $$\frac{1}{10}$$. With this method, 0.0000003, for example, would equal $$3 \boldcdot 10^\text{-7}$$ because 3 has been multiplied by $$\frac{1}{10}$$ seven times.

## Lesson Synthesis

### Lesson Synthesis

The purpose of the discussion is to check that students know how to convert between decimal numbers and numbers expressed as multiples of powers of 10, and that they understand the order of numbers with negative exponents on the number line.

Some questions for discussion:

• “As we move to the right on the number line, what happens to the value of the numbers we encounter?” (They get larger.)
• “Would $$10^\text{-5}$$ appear to the left or to the right of $$10^\text{-4}$$ on a number line? Explain.” ($$10^\text{-5}$$ is smaller than $$10^\text{-4}$$, so it would be to the left.)
• “How does zooming in on the number line help express numbers between the tick marks?” (Zooming in allows us to subdivide the distance between two tick marks into 10 equal intervals, which allows us to to describe a number to an additional decimal place.)
• “Describe how to convert a number such as 0.000278  into a multiple of a power of 10.” (The number is equivalent to $$278 \boldcdot (0.000001)$$ or $$278 \boldcdot \frac {1}{100,000}$$. The fraction $$\frac {1}{100,000}$$ is $$\frac {1}{10^6}$$ or $$10^\text{-6}$$, so 0.000278 can be written as $$278 \boldcdot 10^\text{-6}$$.)

If time allows, give students other small numbers that are written as decimals and ask them to write them as multiples of powers of 10, and vice versa.

## 11.4: Cool-down - Describing Very Small Numbers (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

The width of a bacterium cell is about $$\displaystyle 2 \boldcdot 10^{\text -6}$$ meters. If we want to plot this on a number line, we need to find which two powers of 10 it lies between. We can see that $$2 \boldcdot 10^{\text -6}$$ is a multiple of $$10^{\text -6}$$. So our number line will be labeled with multiples of $$\displaystyle 10^{\text -6}$$ Note that the right side is labeled $$\displaystyle 10 \boldcdot 10^{\text -6} =10^{\text -5}$$

The power of ten on the right side of the number line is always greater than the power on the left. This is true for powers with positive or negative exponents.