I was at a math workshop one summer and the facilitator showed us a number line and asked us where one million would go.
At first, I was stumped. My initial instinct was to place one million right in the middle between zero and one billion. Then I second guessed my reasoning and had to stop and think about what I knew about large numbers and their relationship with one another. How big is a million and how far away is it from a billion? How many millions are in one billion? These questions tested my number sense in a big way.
One billion is a really big number! Did you know that a billion seconds ago it was 1990? A billion minutes ago the Roman Empire was in full swing. A billion hours ago, our ancestors were living in the stone age. And a billion days ago, no human walked on Earth!
Grasping the magnitude of large numbers isn’t easy, yet we expect our students to understand really big numbers early on. For example, fourth graders must be able to read and write numbers to one million with an understanding of place value. That’s a big task for a nine or ten- year-old.
Using Children’s Literature to Introduce Large Numbers
Two of my favorite books to use when introducing large numbers is How Much, How Many, How Far, How Heavy, How Long, How Tall is 1000? by Helen Nolan and Tracy Walker, and How Much is a Million by David M. Schwartz. You can use the links to share the books with your students online if you are teaching virtually or face-to-face, but I encourage you to buy the books for your classroom library if your budget allows.
Both books help kids think about large numbers as the quantities they represent. In Nolan and Walker’s book, the authors wonder what one thousand looks like. They ask, “When is 1,000 a lot? When is it not?” Throughout their book they provide real world contexts that help children visualize one thousand. For example, if you collect a thousand acorns and put them in a pile, the pile won’t be very big. But if the thousand acorns grow into oak trees, they will make a whole forest!
David Schwartz helps children visualize one million by asking questions and providing real world referents. For example, if one million kids stood on one another’s shoulders, they would be taller than the tallest buildings and the highest mountains.
These books can set the stage for some interesting investigations into large numbers and help children develop their sense of number magnitude and scale.
Investigating One Thousand
After sharing Nolan and Walker’s book about 1,000 there are many things you can have students do; following are a few ideas.
•Pose the following question: “When is 1,000 a lot and when is 1,000 not a lot? Have students brainstorm ideas and then have them choose one and illustrate their work.
•Ask students what they are wondering about 1,000. Elicit their questions and then have students investigate and try to answer them. Here are a few examples from a group of fourth and fifth graders I worked with:
Notice how this student brainstormed a list of questions about 1,000 (top) that focus on a topic they’re interested in, before settling on one question to investigate (bottom).
The next step for this student might be to do some computing and then see how their initial estimate compares.
•Provide questions for students to investigate:
-What are some things that are about 1,000 feet tall?
-What are some things that weigh about 1,000 pounds?
-If you travelled 1,000 miles, where might you end up?
-How long does it take to count up to 1,000?
•Use Lima Beans (or another non-standard unit) to estimate and measure
Give small groups of students a bag of lima beans and some measuring tools (rulers, inch-square grid paper, calculators, balance scales, cm/gram cubes) and pose the following questions from which to choose. Using lima beans or other non-standard units is a way to make use of at home resources during distance learning.
How far would 1,000 lima beans stretch?
How tall would 1,000 lima beans be?
How much area would 1,000 lima beans cover?
How heavy would 1,000 lima beans weigh?
How much volume would 1,000 lima beans take up?
Because students don’t have 1,000 lima beans to measure with, they have to use their problem-solving skills to figure out the answers. For example, if students want to know how far 1,000 lima beans would stretch, they might figure out how far 50 would stretch as a benchmark. If they want to find out how heavy 1,000 lima beans are, they might first weigh 100 lima beans as a benchmark. This investigation tests and develops students’ number sense, ingenuity, and resourcefulness.
Investigating One Million
After reading the book How Much is a Million? I go back to the page that reads, “If you wanted to count to one million, it would take you 23 days.”
I then ask students, “How do you think the author went about making an estimate of how long it would take to count to one million?” This question is pretty intriguing because it’s clear David Schwartz didn’t stay up for 23 days counting!
The discussion that follows helps students think about using benchmarks to make estimates. For example, David Schwartz might have measured how long it takes to count to 1,000 and then use that information to estimate for one million. Using a benchmark in this case can be tricky, since it takes longer to say aloud larger numbers than smaller ones. This sort of ‘messiness’ makes these types of problems challenging and interesting.
This investigation is nearly the same as representing one thousand lima beans (see above), but with a twist. Students begin by choosing one of the following questions to explore:
How far would 1 million lima beans stretch?
How tall would 1 million lima beans be?
How much area would 1 million lima beans cover?
How much volume would 1 million lima beans take up?
How heavy would 1 million lima beans weigh?
Then they use measurement tools and a bag of lima beans to work on coming up with an estimate to respond to the following prompt:
It would take about 1 million lima beans to _____________.
As students work to come up with an estimate, the investigation engages them in lots of math reasoning, problem solving, computing, and working with benchmarks. Students end up using a variety of measurement units, tools, procedures, and conversions. For example, if students are trying to figure out how tall one million lima beans would be, they might do the following:
First figure out how many lima beans are in one foot (there are about 12 beans per foot).
If about 12 beans make a foot, then figure how many beans make a mile (5,280 x 12= about 63,000 beans).
1 million beans divided by 63,000 beans equals about 15 miles.
The twist is that once they know about how tall one million beans would be, they have to do some internet research to come up with a real-world referent. For example, students might find out that It would take about 1 million lima beans to equal the height of two Mount Everests!
Conceptualizing Large Numbers
One misconception we often have as educators is that if a student can accurately carry out a procedure, then they understand the numbers they are working with. If students are working with numbers that they can’t imagine or visualize, then maybe some of the activities and ideas in this post can help.
There’s a great resource called Population Education that includes video directions for a lesson plan called Millions to Billions that helps students understand how big a million and a billion is using real world scenarios, riddles, pattern blocks, and measurement investigations. Check it out!