Lesson 11

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. 

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

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.

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.

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.

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,\(\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50\)

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.

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.

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.

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 \(17 \boldcdot 10^\text{-25} = (1.7) \boldcdot 10 \boldcdot 10^\text{-25} = (1.7) \boldcdot 10^\text{-24}\).

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.