Definition of Scientific Notation

10 min

Narrative

In this activity, students learn the definition of "scientific notation" as a way to write very large or very small numbers. Numbers can be written in scientific notation by multiplying a number between 1 and 10 by a power of 10.

Launch

Arrange students in groups of 2. Display the table and image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task

What do you notice? What do you wonder?

material speed (meters per second)
space 300,000,000
water 2.25×1082.25 \times 10^8
copper (electricity) 280,000,000
diamond 124×106124 \times 10^6
ice 2.3×1082.3 \times 10^8
olive oil 0.2×1090.2 \times 10^9

A zoomed number line.
A number line, 11 tick marks, 0, 1 times 10 to the power 8, 2 times 10 to the power 8, 3 times 10 to the power 8, 4 times 10 to the power 8, 5 times 10 to the power 8, 6 times 10 to the power 8, 7 times 10 to the power 8, 8 times 10 to the power 8, 9 times 10 to the power 8, 10 to the power 9. Two times 10 to the power 8 to 3 times 10 to the power 8 is zoomed out to a new number line with 9 tick marks between them. There are 3 points on the new line, 1 between the third and fourth tick mark, 1 at the fourth, and one at the ninth.

Sample Response

Students may notice:

  • This looks like the same set of number lines and table from a previous activity. 
  • The same materials that are labeled in the number lines are listed in the table.
  • The tick marks of the bottom number line are not labeled.
  • The speeds from the table are plotted on the number lines.
  • The number lines use a dot for multiplication while the table uses an “x” for multiplication.
  • Both the number lines and the table have powers of 10.

Students may wonder:

  • Is this the same set of number lines and table from a previous activity?
  • What is the speed of in the table?
  • Why are there some unlabeled dots on the number line?
  • Why do the number lines use a dot for multiplication while the table uses an “x” for multiplication?

Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image.

Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the fact that the number lines use a dot for multiplication and the table uses an “×\times” for multiplication does not come up during the conversation, ask students to discuss this idea.

Tell students that this is the same number line and table from a previous activity that examined the speed of light through different materials. Direct students' attention to an unlabeled point on the number line, such as 9⋅1089\boldcdot10^8. Explain that this number is written in scientific notation — when a number is written by multiplying a number between 1 and 10 by a power of 10. For example, “9” is between 1 and 10, and 10810^8 is a power of 10.
 
Explain 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 9⋅1089\boldcdot10^8 for all to see, and then rewrite it as 9×1089\times10^8. Emphasize that using ⋅\boldcdot is not incorrect, but that ×\times is the most common usage.

Standards
Building On
  • 5.NBT.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • 5.NBT.A.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Building Toward
  • 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

15 min