Lesson 10Representing Large Numbers on the Number Line

Learning Goal

Let’s visualize large numbers on the number line using powers of 10.

Learning Targets

  • I can plot a multiple of a power of 10 on such a number line.

  • I can subdivide and label a number line between 0 and a power of 10 with a positive exponent into 10 equal intervals.

  • I can write a large number as a multiple of a power of 10.

Warm Up: Labeling Tick Marks on a Number Line

Problem 1

Label the tick marks on the number line. Be prepared to explain your reasoning.

A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the seventh power, and the remaining tick marks are blank.

Activity 1: Comparing Large Numbers with a Number Line

Problem 1

  1. Drag the points to their proper places on the number line. Be prepared to explain your reasoning.

    open applet in presentation mode

  2. Discuss with a partner how you decided where each point should go.

  3. Which is larger, 4,000,000 or ? Estimate how many times larger.

Print Version

  1. Place the numbers on the number line. Be prepared to explain your reasoning.

    A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the seventh power, and the remaining tick marks are blank.

  2. Trade number lines with a partner, and check each other’s work. How did your partner decide how to place the numbers? If you disagree about a placement, work to reach an agreement.

  3. Which is larger,  or ? Estimate how many times larger.

Activity 2: The Speeds of Light

Problem 1

The table shows how fast light waves or electricity can travel through different materials.

material

speed (meters per second)

space

water

copper wire (electricity)

diamond

ice

olive oil

  1. Which is faster, light through diamond or light through ice? How can you tell from the expressions for speed?

  2. Let’s zoom in to highlight the values between and .

    open applet in presentation mode

    Plot a point for each speed on both number lines, and label it with the corresponding material.

  3. There is one speed that you cannot plot on the bottom number line. Which is it? Plot it on the top number line instead.

  4. Which is faster, light through ice or light through diamond? How can you tell from the number line?

Print Version

The table shows how fast light waves or electricity can travel through different materials.

material

speed (meters per second)

space

water

copper wire (electricity)

diamond

ice

olive oil

  1. Which is faster, light through diamond or light through ice? How can you tell from the expressions for speed?

  2. Let’s zoom in to highlight the values between and .

    A zoomed in number line where the top and bottom number lines have eleven evenly spaced tick marks. The first tick is labeled 0 and the last tick mark is labeled 10 to the ninth power. Starting with the second tick mark, the ticks are labeled 1 times 10 to the eighth power, 2 times 10 to the eighth power, and so on, until 9 times 10 to the eighth power. Two lines extend downward from the third and fourth tick marks, pointing to the first and last tick marks on the bottom number line, representing a zoomed in portion of the top number line. The first tick mark is labeled 2 point 0 times 10 to the eighth power, the last tick mark is labeled 3 point 0 times 10 to the eighth power, and the remaining tick marks are blank.

    • Label the tick marks between and .

    • Plot a point for each speed on both number lines, and label it with the corresponding material.

  3. There is one speed that you cannot plot on the bottom number line. Which is it? Plot it on the top number line instead.

  4. Which is faster, light through ice or light through diamond? How can you tell from the number line?

Are you ready for more?

Problem 1

Find a four-digit number using only the digits 0, 1, 2, or 3 where:

  • the first digit tells you how many zeros are in the number,

  • the second digit tells you how many ones are in the number,

  • the third digit tells you how many twos are in the number, and

  • the fourth digit tells you how many threes are in the number.

The number 2,100 is close, but doesn’t quite work. The first digit is 2, and there are 2 zeros. The second digit is 1, and there is 1 one. The fourth digit is 0, and there are no threes. But the third digit, which is supposed to count the number of 2’s, is zero.

  1. Can you find more than one number like this?

  2. How many solutions are there to this problem? Explain or show your reasoning.

Lesson Summary

There are many ways to compare two quantities. Suppose we want to compare the world population, about 7.4 billion to the number of pennies the U.S. made in 2015, about 8,900,000,000.

There are many ways to do this. We could write 7.4 billion as a decimal, 7,400,000,000, and then we can tell that there were more pennies made in 2015 than there are people in the world! Or we could use powers of 10 to write these numbers: for people in the world and for the number of pennies.

For a visual representation, we could plot these two numbers on a number line. We need to carefully choose our end points to make sure that the numbers can both be plotted. Since they both lie between and , if we make a number line with tick marks that increase by one billion, or , we start the number line with 0 and end it with , or . Here is a number line with the number of pennies and world population plotted:

A number line from 0 to 10 to the tenth. Point people between 7 x 10 to the ninth and 8 x 10 to the ninth. Point pennies at almost 9 x 10 to the ninth