Match each expression to one or more diagrams that could represent it. For each match, explain what the value of a single small square would have to be.
E
2 large squares each composed of 100 blocks, 10 by 10, 4 small squares each 1 block.
F
2 large squares each composed of 100 blocks, 10 by 10, 4 rectangles composed of 10 blocks, 10 by 1.
G
2 rectangles composed of 10 blocks, 10 by 1, 4 small squares, each 1 block.
8.2
Activity
Comparing Large Numbers with a Number Line
Place the numbers on the number line. Be prepared to explain your reasoning.
4,000,000
Which is larger, 4,000,000 or ? Estimate how many times larger.
Compare number lines with a partner, and discuss how you each decided where each point should go. If you disagree about a placement, work to reach an agreement.
8.3
Activity
The Speeds of Light
The table shows how fast light waves can travel through different materials.
material
speed (meters per second)
A
space
300,000,000
B
water
C
copper wire (electricity)
280,000,000
D
diamond
E
ice
F
olive oil
200,000,000
Let’s zoom in to highlight the values between and .
A number line, 11 tick marks, 0, 1 times 10 the power 8, 2 times 10 the power 8, 3 times 10 the power 8, 4 times 10 the power 8, 5 times 10 the power 8, 6 times 10 the power 8, 7 times 10 the power 8, 8 times 10 the power 8, 9 times 10 the power 8, 10 to the power 9. Two times 10 the power 8, 3 times 10 the power 8 is zoomed out to a new number line with 9 blank tick marks between them.
Label the tick marks between and . Then plot a point for the speed of light through each material A–F on one of the number lines.
There is one material whose speed you cannot plot on the bottom number line. Which is it? If you haven’t already, plot the point for this material on the top number line.
Which travels faster—light through a diamond or light through ice? How can you tell from the given expressions for the speed of light? How can you tell from the number line?
Student Lesson Summary
Suppose we want to compare the number of pennies the U.S. Mint made in 2020, about 7,600,000,000, to the number of one dollar bills printed by the U.S. Bureau of Engraving and Printing in the same year (about 1.6 billion). There are many ways to do this.
We could write 1.6 billion as a decimal value, 1,600,000,000, and then we can tell that in 2020 there were more pennies made than one dollar bills printed.
Or we could use powers of 10 to write these numbers:
for the number of pennies and
for the number of one dollar bills.
Since both numbers are written using the same power of 10, we can compare 7.6 to 1.6 and confirm that there were more pennies made than one dollar bills printed in 2020.
We could also plot these two numbers on a number line. We would need to carefully choose our end points to make sure that both numbers can be plotted. Here is a number line with the two values plotted:
A number line, 11 tick marks, 0, 1 times 10 to the power 9, 2 times 10 to the power 9, 3 times 10 to the power 9, 4 times 10 to the power 9, 5 times 10 to the power 9, 6 times 10 to the power 9, 7 times 10 to the power 9, 8 times 10 to the power 9, 9 times 10 to the power 9, 10 to the power 10. A point labeled People is between 7 times 10 to the power 9, and 8 times 10 to the power 9. A point labeled Pennies is just to the left of 9 times 10 to the power 9.
Glossary
None
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