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Worksheet Grade 8 Math

Pennifold Membrane Test: Very Small Numbers in Powers of Ten

A grade 8 worksheet on a fictional water district's membrane test, where students write tiny and huge quantities as a single digit times a power of ten.

Objectives

  1. Students will write a very small or very large quantity as a single digit times an integer power of ten.
  2. Students will find how many times as large one quantity is than another using powers of ten.
  3. Students will estimate a count of treatment units by dividing two quantities written in powers of ten.

Vocabulary

Parts per trillion
A unit for measuring extremely small amounts of a chemical in water. PFAS limits are set in parts per trillion because these chemicals matter even in tiny doses.

Also: power of ten, negative exponent, micrometer, membrane

Teacher overview

The fictional Pennifold Regional Water District is testing membrane sheets for a new treatment building, and its lab sheet lists everything in micrometers, from the openings in the membrane to a grain of sand. Those sizes run from ten to the negative fourth power up to ten to the second power of a micrometer, which is the situation the grade 8 standard was written for. The worksheet also uses the federal limit for two PFAS compounds, 4.0 parts per trillion each, so students can practice with a real regulatory number written as a power of ten. A common error here is treating a more negative exponent as a larger number, so item 7 asks students to put all five sizes in order by exponent.

Pennifold Membrane Test: Very Small Numbers in Powers of Ten — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

In our scenario, the fictional Pennifold Regional Water District has rounded every size on its lab sheet to one digit. A micrometer is one millionth of a meter.

Item Size in micrometers
Membrane pore 0.0004
PFAS molecule 0.001
Bacterium 2
Human hair 70
Fine sand grain 200

Write each answer as a single digit times a power of ten when the item asks for that form.

  1. Write the membrane pore size, 0.0004 micrometers, as a single digit times a power of ten.
  2. Write the sand grain size, 200 micrometers, as a single digit times a power of ten.
  3. Write the hair width, 70 micrometers, as a single digit times a power of ten.
  4. How many times as wide as the membrane pore is the sand grain?
  5. How many times as wide as the bacterium is the human hair?
  6. How many times as wide as the membrane pore is the PFAS molecule?
  7. List the five items in order from smallest to largest.
  8. The district's raw water tested at 12 parts per trillion of one PFAS compound. The federal limit is 4.0 parts per trillion. How many times the limit is that test result?
  9. One part per trillion is 1 times 10 to the negative twelfth power. Write 4 parts per trillion as a single digit times a power of ten.
  10. One part per billion is 1 times 10 to the negative ninth power. Write 8 parts per billion as a single digit times a power of ten.
  11. How many times as large as your answer to item 9 is your answer to item 10?
  12. The district must treat 9 times 10 to the sixth power gallons in a 30-day month. One membrane element treats 3 times 10 to the third power gallons per day. About how many elements are needed?
  13. Explain why powers of ten make items 4 and 11 easier than working with the decimals.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 4 x 10^-4 micrometers. Move the decimal point four places to the right to reach 4, so the exponent is negative four.
  2. Answer: 2 x 10^2 micrometers. The value 200 is 2 times 100, and 100 is 10 to the second power.
  3. Answer: 7 x 10^1 micrometers. The value 70 is 7 times 10, and 10 is 10 to the first power.
  4. Answer: 5 x 10^5 times. Divide 2 x 10^2 by 4 x 10^-4. Dividing 2 by 4 gives 0.5, and subtracting the exponents gives 10 to the sixth power. Then 0.5 times 10 to the sixth power is 5 x 10^5, or 500,000 times.
  5. Answer: 35 times. Divide 70 by 2 to get 35, so the hair is 35 times as wide as the bacterium.
  6. Answer: 2.5 times. Divide 0.001 by 0.0004. Dividing 1 by 4 gives 0.25, and the exponents give 10 to the first power, so the result is 2.5 times.
  7. Answer: Membrane pore, PFAS molecule, bacterium, human hair, fine sand grain. Ordering by exponent works because each size is written as a single digit times a power of ten, and a more negative exponent means a smaller number.
  8. Answer: 3 times the limit. Divide 12 by 4 to get 3, so the test result is three times the federal limit of 4.0 parts per trillion.
  9. Answer: 4 x 10^-12. Four parts per trillion is 4 times one part per trillion, and one part per trillion is 1 times 10 to the negative twelfth power.
  10. Answer: 8 x 10^-9. Eight parts per billion is 8 times one part per billion, and one part per billion is 1 times 10 to the negative ninth power.
  11. Answer: 2 x 10^3 times. Divide 8 x 10^-9 by 4 x 10^-12. Dividing 8 by 4 gives 2, and subtracting the exponents gives 10 to the third power, so the result is 2,000 times.
  12. Answer: 100 elements. Divide 9 x 10^6 gallons by 30 days to get 3 x 10^5 gallons per day.
    Then divide 3 x 10^5 by 3 x 10^3 to get 10 to the second power, which is 100 elements.
  13. Answer: Powers of ten split each comparison into two easy steps, one with the single digits and one with the exponents. Counting decimal places by hand in 0.0004 and 0.001 invites a lost zero, while subtracting exponents does not.

Differentiation

Support

Give these students a place-value strip that runs from 10 to the third power down to 10 to the negative twelfth power, with the decimal form written under each entry. Assign items 1, 2, 3, 5, 7, and 9 from the fictional Pennifold lab sheet, and allow the decimal form as an accepted answer alongside the power of ten.

Extension

Ask students to find how many membrane pores, laid side by side, would span one human hair, using the sizes on the lab sheet, and to write the result, rounded to one digit, as a single digit times a power of ten. Then have them explain why the answer is not simply the sum of the two exponents.

ELL supports

Pre-teach micrometer, pore, and element with a labeled drawing showing a membrane sheet, its openings, and a spiral element, and keep the pair "how many times as large" next to a division symbol on the board. Provide the sentence frame for item 13: "Powers of ten help because ___, while decimals ___." The fictional Pennifold table can be read aloud in pairs before any writing.

Extensions

Have students research one everyday object whose size they can measure in millimeters, convert that measurement to micrometers, and place it on the ordered list from item 7 for the fictional Pennifold lab sheet. Ask them to write the size as a single digit times a power of ten and to name which two listed items it falls between.