Beginning Algebra: How Big is a Billion

Lesson: Introduction to Scientific Notation

Essential Questions for Students (Objectives):
Can you compare extremely large (small) numbers?
Can you form a pictorial reference for extremely large (small) numbers?
Why is it useful to use scientific notation?

Common Core Standards: 8.EE.1-4

Supplies:
Book: "How Much is a Million?" by David M. Schwartz
One 10lb bag of rice
assorted containers
plastic ridged cups
Two 2.5 gallon buckets
paper plates
large number line

Instructional Format:
Teacher-led instruction, student partner work

Vocabulary for a Word Wall:
scientific notation, exponent, base

Prior Knowledge/ Possible Warm-up Activities:
Students need to know basic place value and multiplication by 10

Time needed:
1 hour and 20 minutes

Description:

1)  Warm-up: Scavenger Hunt (see attached worksheet). I usually give students 4 minutes to complete (as timed by my playing The Pet Shop Boys - One in a Million). The rules are that they can only have each person in the class on their scavenger hunt once. If they sign a person's name to a box, then that person must convince them they know the answer to topic listed. The winner has to have a name in each box along a row. I usually give the winner a $100,000 candy bar.

2) Debrief Scavenger Hunt: Ask students to work in partners to discuss which of the boxes piques their curiosity. Share out as a class.

3)  Rice Activity (see attached worksheet): This demonstration allows students to predict and help measure a million grains of rice.

    a. Give students Rice Activity Worksheet and put out multiple containers. I always use a clear 6 oz cup, a large big-gulp cup, a large juice tupperware, a giant milk carton (or orange carton) with the top cut off for easy pouring, different size buckets (especially a gallon bucket), and a large trash can. Your question to the students is how many containers (or container) would it take to hold one million grains of rice. I usually hold up an individual grain and show it to the students. The need to write their guess on the "first estimate" line of their worksheet.

    b. Next, I hand out a total of 10 paper plates (to groups of 3-4 students) and pour some rice on each plate. I ask the students to work together to count out 100 grains of rice and put it in the clear plastic cup. (Make sure that you only have 10 plastic cups spread around the groups). When the students have the 100 grains counted out, ask them to re-think their estimate. Now that they know how much space 100 grains of rice takes up, how much would 1 million take up (from the original containers that you have shown)? They put their new estimate in the second estimate line on their worksheet.

    c. Holding one of the clear plastic cups of 100 rice grains, ask the students to chorally count by 100 as you empty all the groups 100 grains into your cup. You now have 1,000 grains of rice in a container (it goes exactly to the top of the indented edge that rings the bottom of the cup). The students have a chance now to change their estimate for 1 million again and put it on the third estimate line.

    d. Pass out more rice to the students on their paper plates and ask them to fill their plastic cup to 1,000 by measuring using the indented edge on the bottom of their plastic cup. (Remember, there will only be 9 groups to fill the cup, since you already have one) Now, ask the students to count chorally by 1,000 as you empty the cups into your plastic cup. It should fill the plastic cup to about one finger-joint from the top. Students have one last chance to change their estimate, putting it on the fourth estimate line on the worksheet.

    e. Pass out more rice to the students on their paper plates and ask them to fill their plastic cup to 10,000 by measuring against your plastic cup. (Remember, there will only be 9 groups to fill the cup, since you already have one ) Ask the students to count chorally by 10,000 until they get to 100,000 as you have them pour their plastic full cups into a larger container (for this I use my orange juice container with the top cut off). Once you have counted to 100,000, mark the outside of the container.

    f. At this point, I ask two volunteers to come up and start pouring 100,000 grains into the 2½ gallon bucket using the mark on the larger container as a measurement tool. Ask the class to chorally count by 100,000 as the students fill the bucket. It takes 5 times (or 500,000) to fill one bucket. Usually I stop there and ask how far are we to a million, in which the students can see that we are half way there, so the final answer will be 2 -2½ gallon buckets.

    g. At this point, we explore the middle of the worksheet as a class and I introduce powers of 10 to describe our larger numbers instead of writing out all the zeros. I also introduce ppm (parts per million) and ppb (parts per billion) so that a student can get a visual for when they hear news reports, "there is 2 ppm of arsenic in the well water." There is a line on the worksheet for students to determine how many buckets would we need for one billion grains of rice (which is 2,000 buckets).

    h. From the estimates reached above, ask the students to imagine the grains of rice represent years. Have the students predict the answers at the bottom of the activity sheet.

4)  Read How Much is a Million? By David Schwartz. Ask students to write 2 of the most surprising facts from the book on the back of their worksheet. Share with a partner.

5)  Show Powerpoint slides 1-5 to see some other visuals of large numbers. Extension exercises for students to explore from these slides:

    a. Estimate the mass of any of the pictures.

    b. Predict the volume of 1 billion in $10 bills, $100 bills, or coins.

    c. How could you cover the surface area of a dollar with various coins?

    d. How would changing the denominations to coins change the mass?

    e. How can you maintain the same volume but change one of the dimensions?

6)  Find Your Number Activity.

    1) Copy and cut the large numbers so that each number has it's corresponding partner on the back. For example 106 would have 10-6 on the back. Quintillion would have one quintillionth on the back. (or you can shuffle so that 106 on the front may have 10-4 on the back) Hand out either a number card or a name card to each student. I always start with the positive exponents and large numbers (to help the students with this, I copy the large numbers and names on one color paper and the small numbers and names on another color - then glue them together. This way I can say - look at the green side now).

    2) Using a roll of smaller butcher paper, I created a giant number which starts with a 1 followed by 63 zeros. Show the students where the decimal point would be on the large number and ask them to stand where they think they belong. The students with the numbers have better luck than the names. After a little while, I hand them the Large (small) number name sheet and have them try to locate their place. When they stop moving, I start at the one's place value and call out the numbers and names for all students to hear. Then I use the Large Number location sheet to show them some common numbers that they might attach to in their mind. For example I stop at 109 and point out it is Oprah's net worth in 2007.

    3) At this point in the lesson, I like to switch to small numbers, so I cover up the '1' on my sheet of butcher paper and add a decimal point and a 1 at the opposite end of the number line. Students turn over their number/name cards and try to find their spot. This is a good activity for students to review decimals and place value in a decimal. The numbers and names have to shift one place, because there is no 1's place value in a decimal. After I walk the line and say where every one should be, we stop and discuss some small number locations such as the diameter of a neutron or proton at 10-15.

7)  Extension to science - There is a great activity that involves unrolling toilet paper to show when events happened in time (for example - how long ago were dinosaurs on the earth). You can look up this activity at http://serc.carleton.edu/quantskills/activities/TPGeoTime.html.

8)  Finish the Powerpoint slides on small numbers. In the slides, I have frozen on pictures to highlight the size of small numbers. If you have access to the internet, the sites provided would be a great resource to explore.

9)  Closure - looking back at the scavenger hunt, are there any questions students didn't get answered? Maybe it would be a time to have students choose a topic to explore on his/her own.

10)  Taking it back to algebra - After this activity, I spend time converting numbers into scientific notation and then doing algebraic manipulations (adding, multiplying, dividing) once numbers are in scientific notation.

Assessment (Acceptable Evidence):
Student Rice Activity Sheet

Attached worksheets or documents:
Scavenger Hunt (pdf)
Rice Activity Worksheet (pdf)
Large (small) Number Name Sheet (pdf)
Large (small) Number Location Sheet (pdf)
Name Cards (pdf)
Number Cards (pdf)
Presentation (PowerPoint .ppt *note: you may need to right-click this link and use "save as" to save the file to your computer)

Cautionary notes/ misconceptions:
It is important to get the right kind of plastic cups for the rice activity to work. They need to be clear cups with a small indented bottom (diameter of 2 inches) across the bottom. The height of the cup should be at least 3 inches tall. I usually have luck finding them at party supply stores (see picture at right). In addition, the buckets that are the right size are 2½ gallon.

When doing the rice activity, make sure you hide your container of rice so that students can't guess the size of the container by looking at how much rice you brought. I only fill up one bucket, so I can get away with one 10lb bag.

In order to make the large number for the "number line" activity, I just got a small roll of butcher paper with a height of 1½ feet. I rolled it out down the longest hallway in the school, painted a 1 at the beginning and started painting zeros with a comma after every three (about 63 zeroes). I just roll it back up and re-use it every year.

Acknowledgements:
The rice activity is one that I learned in a Music Inservice Class. I wish I could remember the names of the ladies who showed it to me, but I didn't write it down anywhere (sorry ladies). So, if this activity was yours - here is your credit! Also, the powerpoint was researched and designed for a science/math presentation in coordination with another regional trainer, Yvette Deighton. All the science applications are her doing (I don't even know what most of the things are that she referenced)! 

Music:
I actually made a music playlist that I played on my ipod while we were doing the activities and going over slides from the powerpoint. Students really enjoy my choices and laugh at the large number references!
1) If I Had a Million Dollars by BareNaked Ladies
2) How To Be A Millionaire by ABC
3) A Million Miles Away by The Plimsouls
4) One in a Million by The Platters
5) One In A Million by The Pet Shop Boys
6) Space Mountain by Disneyland musicians
7) Money Changes Everything by Cyndi Lauper
8) Space Age Love Song by A Flock of Seagulls
9) D.N.A. by A Flock of Seagulls
10) Money Can't Buy It by Annie Lennox
11) Time by Culture Club

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