Lesson 10

This warm-up prompts students to reason about values on a number line that end in a power of 10. It enables them to visualize and make sense of numbers expressed as a product of a single digit and a power of 10, which prepares them to begin working with scientific notation.

Expect student responses to include a variety of incorrect or partially-correct ideas. It is not important that all students understand the correct notation at this point, so it is not necessary to extend the time for this reason.

During the partner discussions, identify and select students who have partially-correct responses to share during the whole-class discussion.

Arrange students in groups of 2. Give students 1 minute of quiet work time and then 2 minutes to compare their number line with their partner. Tell the partners to try and come to an agreement on the values on the number line. Follow with a whole-class discussion. 

• Some students may count the tick marks instead of the segments and think \(10^7\) is being divided into 9 or 11 parts. Consider asking these students to mark each segment with a highlighter to count them.
• Some students may try to label the tick marks \(10^1\), \(10^2\), \(10^3\), etc., and as a result, they may say that the \(10^7\) is shown in the wrong place. Ask these students how many equal parts \(10^7\) is divided into, and how they would write that as a division problem or with a fraction.
• Some students may label the tick marks as \(10^6\), \(20^6\), \(30^6\), etc. Ask these students to expand \(20^6\) and \(10^6\) into their repeated factors and compare them so they see that \(20^6\) is not twice as much as \(10^6\).
• Some students may work with the idea of \(\frac{1}{2}\) and label the middle tick mark as \(10^{3.5}\). Explain that this tick mark would have a value of \(\frac12 \boldcdot 10^7\) and ask these students to use what they know about powers of 10 to find the value of the first tick mark after 0.

Ask selected students to explain how they labeled the number line. Record and display their responses on the number line for all to see. As students share, use their responses, correct or incorrect, to guide students to the idea that the first tick mark is \(1 \boldcdot 10^6\), the second is \(2 \boldcdot 10^6\), etc. This gives students the opportunity to connect the number line representation with the computational rules they developed in previous lessons. For example, if a student claims that the second tick mark is \(20^6\), they can check whether \(20^6\) is equal to \(2 \boldcdot 10^6\) by expanding both expressions.

If not uncovered in students' explanations, ask the following questions to make sure students see how to label the number line correctly: 

• “How many equal parts is \(10^7\) being divided into?” (10)
• “If the number at the end of this number line were 20, how would we find the value of each tick mark?” (Divide 20 by 10)
• “Can we use the same reasoning with \(10^7\) at the end?” (Yes)
• “What is \(10^7 \div 10\)?” (\(10^6\)) “What does this number represent?” (The distance between two tick marks)
• “Can we write \(10^6\) as \(1 \boldcdot 10^6\)?” (Yes). 
• “If the first tick mark is \(1 \boldcdot 10^6\), then what is the second tick mark?” (\(2 \boldcdot 10^6\)) “

This activity encourages students to use the number line to make sense of powers of 10 and think about how to rewrite expressions in the form \(b \boldcdot 10^n\), where \(b\) is between 1 and 10 (as in the case of scientific notation). It prompts students to use the structure of the number line to compare numbers, and to extend their use to estimate relative sizes of other numbers when no number lines are given. 

As students work, notice the ways in which they compare expressions that are not written as multiples of \(10^6\). Highlight some of these methods in the discussion. 

Some students may misplace expressions like \((0.6) \boldcdot 10^7\) or \(75 \boldcdot 10^5\) on the number line. For \(75 \boldcdot 10^5\), point out that 75 is the same as \((7.5) \boldcdot 10\), so \(75 \boldcdot 10^5\) is equivalent to \((7.5) \boldcdot 10 \boldcdot 10^5\). For \((0.6) \boldcdot 10^7\), point out that 0.6 is the same as \(6 \boldcdot 10^{\text-1}\), so \((0.6) \boldcdot 10^7\) is equivalent to \(6 \boldcdot 10^{\text-1} \boldcdot 10^7\). Alternatively, tell the student to think of \((0.6) \boldcdot 10^7\) as between 0 and \(10^7\) in the same way that 0.6 is between 0 and 1 to guide them to the correct placement on the number line.

Ask students, “How could you change 4,000,000, \(75 \boldcdot 10^5\), and \((0.6) \boldcdot 10^7\) so that all the expressions have the same power of 10?” Highlight the main idea that it’s always possible to rewrite an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, 75 can be written as \((7.5) \boldcdot 10\), and 0.6 can be written as \(6 \boldcdot 10^{\text-1}\).

If time allows, consider presenting a problem that allows students to use powers of 10 to estimate the relative sizes of large numbers and use them to answer a question in context:

• The population of the United States is roughly \(3 \boldcdot 10^8\) people. The global population is roughly \(7 \boldcdot 10^9\) people. Estimate how many times larger the global population is than the U.S. population. (\(7 \boldcdot 10^9\) is roughly 20 times as large as \(3 \boldcdot 10^8\), because 7 is roughly twice as large as 3, and \(10^9\) is 10 times as large as \(10^8\).)

This activity guides students to thinking in terms of scientific notation while investigating the properties of light. A number line that shows a power of 10 partitioned into 10 equal intervals is again used to illustrate the base-ten structure. Plotting numbers along it gives a clearer meaning to expressions that are a product of a single digit and a power of 10.

To distinguish more easily between the different speeds of light through various materials, the interval between \(2 \boldcdot 10^8\) and \(3 \boldcdot 10^8\) is magnified on the number line. This illustrates numbers with more decimal places and allows students to see how they are expressed in scientific notation.

Once a number line is labeled with powers of 10 and its structure is understood, numbers given in scientific notation can be placed on the number line fairly straightforwardly. This encourages students to look for ways to write the other numbers in scientific notation. 

Students may struggle to rewrite a number written using one power of 10 as a number with a different power, for example writing \(125 \boldcdot 10^6\) using \(10^8\). Some may know the relationship between the powers of 10 (say, between \(10^6\) and \(10^8\)), but may not know how expressing one in terms of the other affects the other factor. Help students make sense of the rewriting process with a series of questions such as these:

• “What do we need to multiply \(10^6\) by to get \(10^8\)?” (100 or \(10^2\)) 
• “If we multiply \(10^6\) by \(10^2\), what must we also do to maintain the value of the expression? (Divide the other factor in the expression by \(10^2\).)
• “What is the resulting expression?” (\((125 \div 10^2) \boldcdot (10^6 \boldcdot 10^2) = (1.25) \boldcdot 10^8\))
• “How do we know that the two expressions are equivalent?” (We multiplied the expression by \(10^2\) and then divided it by \(10^2\), which is equal to multiplying it by 1.)

Ask students to explain how they were able to compare the speeds of light. Students do not yet know the definition of scientific notation, but this activity should help them see that expressing values in this format allows us to more easily compare them. Consider using the applet to further illustrate how each number could be plotted on the number line.

Tell students, “We saw that the speed of light through ice can be written as \((2.3) \boldcdot 10^8\) meters per second, and the speed of electricity can be written as \((2.8) \boldcdot 10^8\) meters per second. When you write them both the same way like this, it makes it much easier to compare them.”