SWBAT to use scientific notation and the laws of exponents to compare the sizes of planets and stars

The only way to work fluently with large numbers is by using your number sense and the laws of exponents.

15 minutes

How big are the planets in our solar system? How can we comprehend these volumes?

Students start this class by reviewing a chart of our solar system and making notes in the margins. I ask them to write down any patterns they notice and make note of anything unexpected.

This is the chart I give: Planet Mean Radii

This is the wikipedia site I pulled it from:

http://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size

After we discuss their observations, I ask them a general question, "how do you find the number of times that one object will *fit* into another?" As we discuss this topic, I elaborate if students are stuck or unclear on what I mean, I ask, "what if you wanted to know how many times a ping pong ball would fit inside a basketball?" The goal is to get them to name the operation of division and why that is the perfect operation for the task.

The second goal in this conversation is to review how we find the volume of a sphere. I like to bring up the connection to a cylinder (a sphere fills 2/3 of a cylinder with an equal base and height), but any interpretation of the formula will do. I like to review this process with a sample problem. For example, the volume of a ping pong ball with a radius of 40mm would be....

Before we calculate, I mention that I want all volumes today in km. I would ask them to consider if they want to convert before or after they calculate volume. Here I would advise students to convert before, because converting with cubic units can be tricky.

25 minutes

On each table is a mystery bag. Inside students will find two images. They need to compare those images to an object in the solar system. The idea is to find out how many times bigger the volume of each object is than the one before it. For example, they might have a ping pong ball and the Star Sirius. If they choose to throw Earth into the mix, I want to know how many ping pong balls would fit into the volume of the Earth and how many would fit into the Star Sirius. I would also want to know how many times the volume of the Earth would fit into Sirius.

Reversing the process, I also want to know what fraction of Sirius would fit into the Earth and what fraction of Sirius would fit into the ping pong ball. I would also want know what fraction of the Earth would fit into the ping pong ball.

Essentially, I am asking students to make 6 multiplicative comparisons.

If you have spherical objects with radii a, b and c where a < b < c, find:

1) the number of times the volume of a fits into b and c

2) the number of times the volume of b fits into c

3) the number of times the volume of c fits into a and b

4) the number of times the volume of b fits into a

In the resources, I include photos of the objects that I use. I place one star and one non-star in each bag.

**Two important notes about the image resources**:

- The star files are named by star and the size of the radius of the star. For example, 883 R Antares.png is the file name for the Star Antares and 883 R is the radius of the star. The "R" stands for solar radius and is based off the size of the radius of our sun. Thus, students will also need: Solar Radius.png
- The non-star objects are named appropriately, but the measurement that is given is the diameter. For example, 3.4 meters twine ball.jpg is the file name for the largest ball of twine with a diameter of 3.4 meters. This was initially done by accident, but ended up working really nicely as an opportunity to engage in mathematical practices. Students now have to discuss the terms radius and diameter in a lesson about scientific notation.

20 minutes

The sizes of these stars are truly unbelievable. Each table has a different star but is perhaps unaware of how truly massive their star is. Unless they chose to compare it to the sun in our solar system, they might not have any reference for the incredible scale they are dealing with. I address with the summary by showing them this great image from wikipedia: Star-sizes.jpg

As students share their findings, we extend the math by seeing if this image is correct. As a class we compare how many times larger each of the stars are to our sun. By constantly referencing our sun in the discussion, students get a sense of the massive size of these stars.

Because this lesson is complex and deals with many aspects of scientific notation, I usually end with an exit ticket question, like how many times larger is this start than another. This is a good reference for me, because I like to follow up with a project. I need to know how well they handled this lesson in order to gauge how ready they are to start asking their own questions.