Representing Small Numbers on the Number Line
5 min
Teacher Prep
Setup
Narrative
In this Warm-up students reason about expressions with negative exponents on a number line. They attend to the meaning of the numbers used to label the tick marks of the number line and how their values are related to the tick marks directly to the right or left (MP2).
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Task
Kiran drew this number line.
Andre said, “That doesn’t look right to me.”
Explain why Kiran is correct or explain how he can fix the number line.
Sample Response
Sample response: Kiran can change each instance of 10-4 into 10-6 or change 10-5 to 10-3.
Activity Synthesis (Teacher Notes)
The goal of this discussion is to highlight the idea that the larger the size (or absolute value) of a negative exponent, the closer the value of the expression is to zero. This is because the negative exponent indicates the number of factors that are 101. For example, 10-5 represents 5 factors that are 101, and 10-6 represents 6 factors that are 101, so 10-6 is 10 times smaller than 10-5.
Invite students to share their reasoning about whether the number line is correct or not. If not brought up in students’ explanations, show at least two correct ways in which the number line can be fixed—by changing the exponents for each tick mark to be one power of 10 smaller than 10-5 (10-6), by changing 10-5 to be one power of 10 greater than the tick marks (10-3), or by changing all of the exponents so that the tick marks are one power of 10 less than the power of 10 at the end of the number line.
Standards
- 8.EE.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <em>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger.</em>
- 8.EE.A.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <span>For example, estimate the population of the United States as <span class="math">\(3 \times 10^8\)</span> and the population of the world as <span class="math">\(7 \times 10^9\)</span>, and determine that the world population is more than <span class="math">\(20\)</span> times larger.</span>
10 min
20 min
Knowledge Components
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Representing Small Numbers on the Number Line
5 min
Teacher Prep
Setup
Narrative
In this Warm-up students reason about expressions with negative exponents on a number line. They attend to the meaning of the numbers used to label the tick marks of the number line and how their values are related to the tick marks directly to the right or left (MP2).
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Task
Kiran drew this number line.
Andre said, “That doesn’t look right to me.”
Explain why Kiran is correct or explain how he can fix the number line.
Sample Response
Sample response: Kiran can change each instance of 10-4 into 10-6 or change 10-5 to 10-3.
Activity Synthesis (Teacher Notes)
The goal of this discussion is to highlight the idea that the larger the size (or absolute value) of a negative exponent, the closer the value of the expression is to zero. This is because the negative exponent indicates the number of factors that are 101. For example, 10-5 represents 5 factors that are 101, and 10-6 represents 6 factors that are 101, so 10-6 is 10 times smaller than 10-5.
Invite students to share their reasoning about whether the number line is correct or not. If not brought up in students’ explanations, show at least two correct ways in which the number line can be fixed—by changing the exponents for each tick mark to be one power of 10 smaller than 10-5 (10-6), by changing 10-5 to be one power of 10 greater than the tick marks (10-3), or by changing all of the exponents so that the tick marks are one power of 10 less than the power of 10 at the end of the number line.
Standards
- 8.EE.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <em>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger.</em>
- 8.EE.A.3·Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <span>For example, estimate the population of the United States as <span class="math">\(3 \times 10^8\)</span> and the population of the world as <span class="math">\(7 \times 10^9\)</span>, and determine that the world population is more than <span class="math">\(20\)</span> times larger.</span>