Fluency with Small Numbers

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SWBAT to convert small numbers in and out of scientific notation

Big Idea

We can use basic number sense to work with small numbers in scientific notation

Start Up

15 minutes

The microscopic world often looks alien and the quantum world seems to hold the mysteries of our universe. So how could there be a more exciting way to use mathematics than to make sense of this mysterious and tiny world?


I start off the lesson by introducing some type of string that highlights place value and scientific notation (with small numbers). I also introduce some basic units (although we return to units again). 

I ask students to write each term in scientific notation:













After students have had a chance to work out these problems, we share and I add some context to the conversation by showing a creature that fits these measurements. We do that by adding units. The way in which you add your units is key, so only add units that you think enhance the conversation. For example, you might only mention the micrometers here to show the image associated with it.

5.68 meters

.568 meters ( 5.68 decimeters)

.0568 meters (5.68 centimeters)

.00568 meters (5.68 millimeters)

.000568 meters (568 micrometers) Hydrothermal Worm 568 micro meters

.0000568 meters (56.8 micrometers)

.00000568 meters (5.68 micrometers) 

.000000568 meters (568 nanometers)

.0000000568 meters (56.8 nanometers)

.00000000568 meters (5.68 nanometers)


After we show the hydrothermal worm, I show them another image and ask where we would place it on this list (this starts them thinking about the ordering in scientific notation). Here is the image, its size in 25 nanometers: Parvovirus 25nm




30 minutes

Here students work through a variety of scientific notation problems and submit to me for a grade.

This is the collection I like to use:Fluency with Small Numbers The grade is a formative assessment and meant to give some informal feedback. I often ask students to swap and grade (I give them the answers). 

I make sure to circulate and identify common misconceptions. This is not only to help the individual students but to share with the class at the end (almost always anonymously). This will help the class address the issue and talk about a misperception that they may also have. 


15 minutes

After students have a chance to work through the various problems, we share common misconceptions and look at some other images that capture the amazing microscopic world around us. 


I start with the two virus images, since we still have our measurements from the start up problem. 

Influenza and 200 nanometers:

Influenza 200nm


Rotavirus at 75 nanometers:

rotavirus 75nm


We do elaborate on comparisons between numbers in future lessons, but I like to ask that question here as well. "Which one is larger?" The simple logic here is that "the more nanometers you have, the large the object." This is a nice feature of our number system. No matter how insane the number, if it is positive, a higher amount means "bigger." 


Then we move to the Pompeii Worm, another sea dwelling creature:

pompeii worm 13 cm


And finally a demodex mite, which is .1 mm. I save this one for last because it is truly gross and is found on most humans (around our eyebrows). The references and stats are in the resource section. Don't show this one if you don't think your class will respond well. However, use the shock factor to discuss this number and how it can represented in scientific notation. As for all the measurements here, we are converting to scientific notation when we represent these measurements in meters. 

demodex mite point 1 mm