IM 8.7.10 Lesson: Representing Large Numbers on the Number Line

Label the tick marks on the number line.
[size=150]Explain your reasoning.[/size]
Drag the points to their proper places on the number line.
[size=150]Explain your reasoning.[/size]
Discuss with a partner how you decided where each point should go.[br][br]
Which is larger, 4,000,000 or [math]75\cdot10^5[/math]? Estimate how many times larger.
The table shows how fast light waves or electricity can travel through different materials:
[img]data:image/png;base64,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[/img][br][br]Which is faster, light through diamond or light through ice? 
How can you tell from the expressions for speed?
[size=150]Let’s zoom in to highlight the values [math](2.0)\cdot10^8[/math] between  and [math](3.0)\cdot10^8[/math].[/size]
Plot a point for each speed on both number lines, and label it with the corresponding material.
There is one speed that you cannot plot on the bottom number line. Which is it? Plot it on the top number line instead.
Which is faster, light through ice or light through diamond?
How can you tell from the number line?
Find a four-digit number using only the digits 0, 1, 2, or 3 where:
[list][*]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.[/*][/list][size=150]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.[br][/size][br]Can you find more than one number like this?
How many solutions are there to this problem? Explain or show your reasoning.
Close

Information: IM 8.7.10 Lesson: Representing Large Numbers on the Number Line