Lesson 4

In this activity, students investigate fractions that are equal to 1. This is an important concept that helps students make sense of the exponent division rule explored later in this lesson. It is expected that students will try to compute the numerator and denominator of the fraction directly in the first problem. Notice any students who instead make use of structure to show multiplication by 1.

Give students 5 minutes of quiet work time. Expect students to attempt to work out all of the multiplication without using exponent rules. Follow with a brief whole-class discussion.

The key takeaway is that a fraction is often easier to analyze when dividing matching factors from the numerator and denominator to show multiplication by 1. Select any students who made use of structure in this way. If no students did this, provide the following example:\(\displaystyle \frac{2 \boldcdot 3 \boldcdot 7 \boldcdot 11}{5 \boldcdot 3\boldcdot 7 \boldcdot 11} = \frac{3\boldcdot 7 \boldcdot 11}{3\boldcdot 7 \boldcdot 11} \boldcdot \frac{2}{5} = 1 \boldcdot \frac{2}{5} = \frac25.\)

If time allows, consider the following questions for discussion:

• “What has to be true about a fraction for it to equal 1?” (The numerator and denominator must be the same value and something other than 0.)
• “Create your own fraction that is equivalent to 1 that has several bases and several exponents.”

Explore division of powers of 10 to derive the rule \(\frac{10^n}{10^m} = 10^{n-m}\). At this point, students will only be working in cases where \(n > m\) and will later extend the rule to include \(n = m\). The case of \(n < m\) is left for the next lesson. The rule arises from the fact that \(\frac{10^n}{10^m} = \frac{10^m}{10^m} \boldcdot 10^{n-m}\) which is equivalent to \(10^{n-m}\). These problems are also meant to underscore the connection between fractions and division, and students should be expected to use both fractions and division interchangeably as they work.

Before students begin working, ask students to notice and wonder about the “expanded” column for the expression \(10^4 \div 10^2\). 

It is important for students to understand that the “expanded” column shows each power of 10 expanded into factors, the division written as a fraction, and a certain number of factors in the numerator and denominator being grouped because their quotient is 1.

Give students 5–7 minutes quiet work time followed by a whole-class discussion.

The key idea for writing \(\frac{10^n}{10^m}\) with one exponent is that \(m\) factors in the numerator and denominator can be divided to make 1, leaving \(n-m\) factors in the numerator. Here are some sample questions to guide students to this idea:

• “Say we want to write \(10^6 \div 10^3\) with a single power of 10. What happens to the 3 factors that are 10 in the denominator? How many factors that are 10 are left in the numerator?” (The 3 factors that are 10 in the denominator are matched with 3 of the factors that are 10 in the numerator, then divided to make 1. Since 3 factors that are 10 from the numerator are used to make 1, there are still 3 left.)
• “If you wanted to write \(10^{80} \div 10^{20}\), what would happen with the 20 factors that are 10 in the denominator? How many factors that are 10 would still be left in the numerator?” (To make this connection explicit, you might show the calculation \(\frac{10^{80}}{10^{20}} = \frac{10^{20} \boldcdot 10^{60}}{10^{20}} = \frac{10^{20}}{10^{20}}\boldcdot 10^{60}\). Compare this to \(10^{80} \div 10^{20} = 10^{80-20} = 10^{60}\).)

Make another “Exponent Rule” poster for the rule \(\frac{10^n}{10^m} = 10^{n-m}\). An example to illustrate the rule could be \(\frac{10^5}{10^3} = \frac{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10} = \frac{10 \boldcdot 10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10} \boldcdot 10 \boldcdot 10 = 1 \boldcdot 10 \boldcdot 10 = 10^2\). Use colors and other visual aids to highlight the fact that the exponents are subtracted because \(m\) factors that are 10 in the numerator and denominator are used to make a factor of 1, leaving \(n-m\) factors that are 10.

Students extend exponent rules to discover why it makes sense to define \(10^0\) as 1. Students create viable arguments and critique the reasoning of others when discussing Noah’s argument that \(10^0\) should equal 0 (MP3).

Arrange students in groups of 2. Give students 5 minutes to answer the first 3 problems and a few minutes to discuss the last problem with a partner before a whole-class discussion.

The important concept is that \(10^0 = 1\) is a convenient definition that extends the usefulness of the exponent rules to a wider range of numbers. This idea is developed in the next lesson to further extend exponent rules to include negative exponents.

Ask students to share their thinking about what \(10^0\) means. Noah’s argument raises a valid concern that \(10^0\) doesn’t fit the definition that we use when the exponent is positive. One way to address this concern is to allow alternate definitions and see what happens to the exponent rules. For example, if we want to define \(10^0\) as 0, then we can choose to do that. However, when we look at an example like \(10^3 \boldcdot 10^0\), the result would be zero. This just means \(10^0\) does not match the pattern of the exponent rule, which says the result should be \(10^{3+0}\) which is equal to \(10^3\).

Introduce the visual display for the rule \(10^0 = 1\). Display for all to see throughout the unit. To illustrate the rule, consider displaying the example \(\frac{10^6}{10^0} = 10^{6-0} = 10^6\). For this to be true, the denominator \(10^0\) must be equal to 1. Another possibility is to write \(10^5 \boldcdot 10^0 = 10^{5+0} = 10^5\) which can only be true if \(10^0 = 1\).

This activity expands on a previous cool-down as students generate different representations of the same number to solidify what they have learned about exponent sum, product, and subtraction rules. Notice students who use the exponent rules they have learned in different ways to achieve an exponent of 6, especially those who combine different exponent rules together. It is not expected that students will make an exponent of 6 using negative exponents, but do not discourage it if they do. Explain to these students that, while the rules still work when using negative exponents, it is not yet clear what the value of, say, \(10^{\text- 2}\) is.

Arrange students in groups of 2. Give 5–7 minutes of quiet work time before asking students to share their response with their partner. Follow with a whole-class discussion.

Tell students to compare their response with their partner’s to discuss what differences there might be. In a whole-class discussion, select students to share a variety of responses. Look especially for examples that show creativity or that combine multiple rules together. The main idea is for students to show flexibility with the exponent rules they have learned so far. To include more students in the discussion, consider asking: “How can you build on __’s method to come up with another expression that makes a million?” If time allows, ask whether it’s possible to make an exponent of 6 using negative numbers and indicate that a future lesson will explore negative exponents in detail.