Lesson 12

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).

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. 

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. 

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.

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.

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.

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.