Lesson 13

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying by a power of 10. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to work with numbers in scientific notation. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. After the last problem, ask students, “How could we rewrite each expression as a product of a number and a power of 10?” Record and display their responses next to each of the original expressions for all to see. 

To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

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.

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.

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 \(1.24 \times 10^8\))

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. \(0.43 \times 10^5\)).

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.

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.