SWBAT to use their understanding of exponents and place value to translate very small numbers between standard decimal form and scientific notation.

Students can accept that 1 micron = 10^-6, but they often struggle with a measurement that is 1000 microns. Here we help students make sense of these types of numbers.

15 minutes

I start students with a string that points out some common misconceptions about small numbers.

Write each measurement in meters for both standard form and scientific notation:

**1 mm**

**12 mm**

**123 mm**

**1234 mm**

In this string, we start with something the students all recognize. They know that 1 mm = .001 m = 1 x 10^-3 meters. But what does it mean to have 12 mm? Its tough to think about. Do we write .012 or .0012? To support students, I ask students to count up.

I write this list out on the board:

**1mm = .001 m**

**2mm = .002 m**

**3mm = .003 m**

**4mm = .004 m**

**5mm = .005 m**

**.**

**10mm = .010 m = .01 m**

With each step I pause and ask the group what I should write next. Each new number is a key to seeing the pattern. Students need to think carefully about each step in order to recognize that 10mm = .010 m = .01 m.

Students are often unsure about why it isn't .0010 m (a common misconception). To help them I show the string and I ask about the *place value* of each number. This will help them see that .01 m > .001 m and .01 m > .05 m. I reason with students here by asking, "Do we want the larger value or the smaller value? If we are counting up, should we should pick the larger value?"

Finally, I show them that .001 m is what we started with. I ask, "how can 10 mm and 1 mm both equal .001m?"

I also check-in to confirm they all know that 1 mm = 1/1000 m = .001 m

25 minutes

First I give students a reference point for a discussion: Microns References.

This conversation is brief, but it gives context for the investigation and summary. We show what a micron is numerically and visually. We compare it to something they recognize (an inch). I tell students that their job today is to investigate the size of some particles in microns.

For the investigation, my students become "particle collectors." I print out or write out each particle and display them on the board.

These are the particle cards: Particle Cards

**Note**: Some cards have size ranges and some don't. Since not having a ranged measurement makes this process a bit easier, I use this to differentiate the lesson by giving the ranged measurements to higher performing students.

Each partnership or group sends up a member to quickly pick 3 particles from the list.

I ask students to:

1) Write the size range for their particles on the particle reference sheet: Particle Size Reference Sheet

2) Pick at least one particle and assume the measurements given represent the diameter of a spherical particle. Then, answer the following list of questions for your particle(s).

- What is the most particles you could fit in a meter?
- What is the least particles you could fit in a meter?
- What is the most particles you could fit in a square meter?
- What is the least particles you could fit in a square meter?
- What is the most particles you could fit in cubic meter?
- What is least particles you could fit in a cubic meter?
- What is most particles you could fit in a cubic kilometer?

3) If a nanometer = 10^-9 m, how many nanometers is each particle you chose?

20 minutes

In bringing closure to this lesson, I ask some students to share their findings on the standard form of each particle. Students share with the class on a projector. We give the class about 30 seconds to see if they disagree with each answer. We ask for these objections in a kind and respectful way. If there is a disagreement, we ask students to explain their reasoning. "Convince us that you are right!"

The conversation around the most particles and the least particles is important. I want students to recognize that a larger measurement means that less particles will fit. This will help them develop a natural sense as to the size of the number. When they look at a range of 5 - 50 microns, they write that it is between .000005 and .000050 meters. They need to recognize that .00050 meters is larger and will fit less in a measurement. Then they need to divide 1 m by this amount to answer the first question of how many will fit across the meter.

As the task involves square and cubic meters, it is especially important to have students draw a model to explain their thinking.

We finish the discussion around the nanometer. The goal is for students to reason that if 1 micon has 1000 nanometers, then a measurement of 50 microns would have 50,000 nanometers and be written as 50,000 x 10^-9 = 5 x 10^4 x 10^-9 = 5 x 10^-5 = .00005 meters. In other words, the number of nanometers would be different than the number of microns, but they both are the *exact same number of meters. *