Lesson 4

Students continue to use repeated reasoning to discover the exponent rule \(\frac{10^n}{10^m} = 10^{n-m}\) (MP8). For now, students work with expressions where \(n\) and \(m\) are positive integers and \(n > m\). In the last activity, students extend to the case where \(n = m\) to make sense of why \(10^0\) is defined to be equal to 1 and critique a faulty argument that it should be defined to be equal to 0 (MP3). Students make sense of this rule when they recognize that separating the same number of factors from the numerator and denominator, then dividing has the effect of multiplying by 1. This essentially means that \(m\) factors are subtracted from the \(n\) factors in the numerator. For example \(\frac{10^3}{10^2} = \frac{10 \boldcdot 10}{10 \boldcdot 10} \boldcdot 10\) which is the same as \(1 \boldcdot 10 = 10\) or the same as \(10^{3-2} = 10^1\). In a subsequent lesson, students will extend this rule to include situations where \(n < m\).

Create a visual display for the rule \(\frac{10^n}{10^m} = 10^{n-m}\). For a guiding example, consider \(\frac{10^5}{10^2} = \frac{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}{10 \boldcdot 10} = \frac{10 \boldcdot 10}{10 \boldcdot 10} \boldcdot 10 \boldcdot 10 \boldcdot 10 = 1 \boldcdot 10^3 = 10^3\).

Create a visual display for the rule \(10^0 = 1\). For an example, you can show \(\frac{10^6}{1}=\frac{10^6}{10^0}=10^{6−0}=10^6\). Another possibility is to write \(10^5\boldcdot 1= 10^5\boldcdot 10^0=10^{5+0}=10^5\) and use visual aids to highlight that each of these examples implies \(10^0=1\).