In order to be thoughtful participants in society, people need to be able to comprehend and make decisions based on data and numbers. Numeracy plays a role in everyday tasks, financial literacy, technology use, societal issues, and many other facets of life. At times, the numbers we use in our everyday lives can be immense in size and difficult to understand. This blog discusses different strategies for young children, adolescents, and adults to make sense of large numbers.
How to Help Students Understand Large Numbers: Visual and Tactile Activities
Hands-on learning and kinesthetic teaching strategies are particularly useful for developing number sense in young students. In a previous blog, we shared examples of books, videos and art that help us visualize large numbers to aid in understanding. What other specific strategies make big numbers more accessible?
Estimation Demonstration
As part of the Yale National Initiative, Lynette Shouse, a Gifted and Talented Coordinator at Grissom Elementary in Tulsa, Oklahoma, utilized hands-on demonstrations and resourceful textbooks to teach her elementary students how to estimate and understand significantly large numbers. Based on the texts Great Estimations and Greater Estimations, Shouse created an estimation demonstration that uses the common household items of measuring cups, jelly beans, rice, and sugar to put students’ estimation skills to work.
First, students are given ¼, ½, and 1 cup measuring cups and jelly beans. They are instructed to fill the ¼ cup with jelly beans and estimate how many it took to fill it. After making estimations, they will count the actual number of jelly beans in the ¼ cup and find that it took 10 jelly beans to fill it. This would lead into an instructive discussion where students realize that 20 would fit into the ½ cup and 40 would fit into the 1 measuring cup.
The next step of the demonstration involves using grains of rice. Students will compare the size of a single grain of rice and a single jelly bean, then estimate how many grains of rice it would take to fill a ¼ cup before actually counting it. Students will realize that it took many more grains of rice than jelly beans to fill the ¼ cup based on their size. This process would continue for the remaining measuring cups.
The final step of the demonstration involves estimating the number of granules of sugar it would take to fill the measuring cups. Students will compare the size of a single granule of sugar to the size of a single grain of rice and jelly bean, then make estimations about how many granules of sugar it would take to fill the measuring cups. After doing the same procedure of estimating and counting, the granules of sugar will reach into the millions of billions. As stated in David Adler’s book, Million, Billions, and Trillions, Understanding Big Numbers, ¼ cup of sugar is equal to approximately 1 million granules of sugar, thus being a strong visual and kinesthetic representation of 1 million.
Guessing Jars
This concept can be easily adapted into classrooms by using a guessing jar. Guessing Jars are a great tool for young learners to build their estimation skills and visualize large numbers. The jar can be filled with common household or classroom items, such as candy, beads, or stickers, and students will attempt to guess how many are in the jar. Students can practice estimation strategies such as eye training, clump counting, and the box and count method as illustrated by Bruce Goldstein’s books Great Estimations and Greater Estimations.
Based on similar estimation strategies, Population Education offers a lesson called “How Many Flowers in the Field?” Using visual and tactile activities such as estimation demonstrations, guessing jars, and PopEd lessons are great for teaching young students about large numbers.
How to Better Understand Large Numbers Using Number Lines
Number lines can serve as both a physical and visual model of the magnitude and scale of large numbers. While using a number line, students can further grasp the concept of enormous numbers such as a million, a billion, and a trillion.
Another part of Shouse’s curriculum unit utilizes a number line for students to determine the size of large multi-digit numbers. The number line, marked with tick marks in units of 10, begins at the classroom door and spreads around the school. Students mark on the number line where they believe one thousand, ten thousand, and one hundred thousand are located. Later in the unit, students extend the number line and estimate where one million and one billion are located. Students can also estimate where even larger numbers are located, such as 10 million, 500 billion, or 1 trillion. To make real-world connections, students can also place their estimate of how many seconds old they are, how old the earth is, or lifetime earnings on the number line.
In the article, Millions, Billions, or Trillions: How to Partition Large Numbers into Friendly Figures, the authors share how partitioning number lines is useful when differentiating between a million, a billion, and a trillion. They discuss a study, published in Cognitive Science, where participants were instructed to locate 1 million on a number line from 0 to 1 billion.
Many participants incorrectly placed one million halfway in the number line. Since one million is one-thousandth of a billion, 1 billion should be placed close to 0. Maher and colleagues state that a partitioned number line where 500 million, the accurate midpoint between 0 and billion, is displayed on the number line can combat this misconception.
To further debunk misconceptions, researchers suggest partitioning the number line even further and display where 10 million, 50 million, and 100 million are located. This strategy gives a strong visual presentation of the accurate relativity of these large numbers and further corrects common misconceptions about the relationship between a million, a billion, and a trillion.
Making Big Numbers Meaningful
Adding meaning to large numbers and putting them in a real-life context is another strategy to aid in understanding. In one study, researchers Dan Goldstein, Jake Hofman, and Pablo Barrio looked at how using meaningful contextual clues, such as dollar amounts, time, sizes, and more, can make large numbers more comprehensible. Hofman stated, “We found that perspectives were pretty helpful in helping people recall unfamiliar numbers that they had read, in estimating numbers that they hadn’t seen before, and also in detecting errors in potentially manipulated numbers.” Putting large numbers in perspectives with everyday things, such as time, money, or weight, makes them much more understandable. For example:
- If someone made one million dollars a year, they would make about $480.77 per hour and $3,846.15 per day. If someone made a billion dollars per year they would make about $480,769 per hour and $3,846,153.85 per day.
- One million ants would weigh a little over 6 pounds. One billion ants would weigh over 3 tons—a little less than the weight of an elephant or about the weight of a baby blue whale. One trillion ants would weigh over 3,000 tons—the weight of nearly 430 elephants.
- Using $100 dollar bills, $10,000 could fit into your pocket. 1 million dollars would fit in a standard shopping bag. 1 billion dollars would occupy a small room in a house. 1 trillion dollars would take up an entire football field.
Population Education’s lesson plan “Population Riddles” builds on this concept of putting meaning behind large numbers and helps students both understand and appreciate the difference between millions and billions.
Strategies for Understanding Large Numbers
Understanding large numbers is a vital skill in modern day society. Kinesthetic and visual tools like guessing jars are a great resource, particularly for younger learners, when it comes to building their estimation skills. The number line is a tool that is prominent in K-12 Common Core standards and is invaluable for comprehending large numbers. Connecting the numbers of a million, billion, and trillion to things in everyday life gives them much more meaning and understandability. Additionally, Population Education provides a wealth of lessons and resources that helps people of all ages understand and simplify the concept of large numbers.
Image credits: Measuring cups (Vimkay, CC BY-SA 4.0, via Wikimedia Commons); Jar of candy (Photo by Robert Anasch on Unsplash)