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Math problem set Grade 8 Math

Sandgate Aquifer Storage: Scientific Notation Comparisons

A grade 8 problem set where students compare a fictional aquifer's stored water, yearly pumping, and recharge in scientific notation and say how much larger each is.

Objectives

  1. Students will divide two numbers written in scientific notation to say how many times larger one is.
  2. Students will rewrite a product or difference so the result is in correct scientific notation.
  3. Students will convert a large water volume into acre-feet and into a number of households using given conversion factors.

Vocabulary

Acre-foot
The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.
Recharge
Recharge is water soaking down through the ground to refill an aquifer, the way rain refills an underground sponge. Aquifers stay healthy only if recharge keeps up with pumping.

Also: scientific notation, power of ten

Teacher overview

Groundwater numbers run across many orders of magnitude at once: the water stored underground is measured in hundreds of billions of gallons, the pumping that draws on it in hundreds of millions, and the rain that recharges it in tens of millions. This set uses a fictional aquifer under the town of Sandgate to make those comparisons manageable, since a quotient of two numbers in scientific notation answers the question of how many times larger one is without any long division. The error to watch for is an answer left as 0.5 times 10 to the third power, which is a correct value written in incorrect form and should be rewritten as 5 times 10 squared.

Sandgate Aquifer Storage: Scientific Notation Comparisons — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

The fictional Sandgate aquifer holds about 4 × 10^11 gallons of stored water. The town of Sandgate pumps about 8 × 10^8 gallons from it each year. Rain and snowmelt recharge the aquifer with about 4 × 10^7 gallons each year. Write every answer in correct scientific notation unless the problem asks for a plain number.

  1. How many times larger is the water stored in the aquifer than one year of the town's pumping? Give a plain number.
  2. The town pumps 8 × 10^8 gallons each year. Write the total for 10 years in scientific notation.
  3. A neighboring aquifer holds 2 × 10^13 gallons. How many times larger is that than the Sandgate aquifer's 4 × 10^11 gallons? Give a plain number.
  4. The town pumps 8 × 10^8 gallons each year and the aquifer recharges by 4 × 10^7 gallons each year. How many times larger is the pumping than the recharge? Give a plain number.
  5. Subtract the yearly recharge of 4 × 10^7 gallons from the yearly pumping of 8 × 10^8 gallons to find the net loss for one year. Write the answer in scientific notation.
  6. The aquifer loses a net 7.6 × 10^8 gallons each year. How many years would it take to lose 3.8 × 10^10 gallons at that rate?
  7. One acre-foot equals 325,851 gallons. Write that number in scientific notation, rounded so the first factor has two decimal places.
  8. Use 3.26 × 10^5 gallons per acre-foot. About how many acre-feet is the aquifer's 4 × 10^11 gallons? Round the first factor to one decimal place.
  9. An average US household uses about 300 gallons of water per day. Write one year of that use, at 365 days, in scientific notation, rounded so the first factor has two decimal places.
  10. Use 1.1 × 10^5 gallons per household each year. About how many households could the town's yearly pumping of 8 × 10^8 gallons supply? Round to a single digit times a power of ten.

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: 500 times. Divide 4 × 10^11 by 8 × 10^8. Divide the first factors, 4 divided by 8, to get 0.5, and subtract the exponents, 11 minus 8, to get 10^3. That gives 0.5 × 10^3, which is rewritten as 5 × 10^2, or 500.
  2. Answer: 8 × 10^9 gallons. Multiply 8 × 10^8 by 10. The first factor stays 8 and the exponent rises by 1, from 8 to 9.
  3. Answer: 50 times. Divide 2 × 10^13 by 4 × 10^11. Divide 2 by 4 to get 0.5, and subtract the exponents, 13 minus 11, to get 10^2. That gives 0.5 × 10^2, which is rewritten as 5 × 10^1, or 50.
  4. Answer: 20 times. Divide 8 × 10^8 by 4 × 10^7. Divide 8 by 4 to get 2, and subtract the exponents, 8 minus 7, to get 10^1. The result is 2 × 10^1, or 20.
  5. Answer: 7.6 × 10^8 gallons. Rewrite 4 × 10^7 as 0.4 × 10^8 so both terms share the same power of ten. Then 8 × 10^8 minus 0.4 × 10^8 equals 7.6 × 10^8 gallons.
  6. Answer: 50 years. Divide 3.8 × 10^10 by 7.6 × 10^8. Divide 3.8 by 7.6 to get 0.5, and subtract the exponents, 10 minus 8, to get 10^2. That gives 0.5 × 10^2, which is 5 × 10^1, or 50 years.
  7. Answer: 3.26 × 10^5 gallons. Move the decimal point five places to the left, from 325,851 to 3.25851, so the power of ten is 5. Rounding 3.25851 to two decimal places gives 3.26.
  8. Answer: 1.2 × 10^6 acre-feet. Divide 4 × 10^11 by 3.26 × 10^5. Divide 4 by 3.26 to get about 1.227, and subtract the exponents, 11 minus 5, to get 10^6. Rounding the first factor to one decimal place gives 1.2 × 10^6 acre-feet.
  9. Answer: 1.10 × 10^5 gallons. Multiply 300 gallons per day by 365 days to get 109,500 gallons.
    Move the decimal point five places to the left to get 1.095 × 10^5, and round the first factor to 1.10.
  10. Answer: 7 × 10^3 households. Divide 8 × 10^8 by 1.1 × 10^5. Divide 8 by 1.1 to get about 7.27, and subtract the exponents, 8 minus 5, to get 10^3. Rounded to a single digit, that is 7 × 10^3, or about 7,000 households.

Differentiation

Support

Give these students a worked model of one division in scientific notation, with the first factors and the powers of ten circled in different colors, and a reminder card that reads "divide the front numbers, subtract the exponents." Mark items 1 through 5 as the required set and allow a calculator for every step.

Extension

Ask students to find how many years of the town's current pumping the aquifer could supply if the recharge stopped entirely, then compare that with the answer to item 6 and explain in a paragraph why the two numbers are so different. Have them state the assumption each calculation makes.

ELL supports

Post a place-value strip showing 10^3 through 10^12 with the plain word for each, such as thousand, million, and billion, so students can name the size of every answer aloud. Provide the sentence frame "The aquifer holds ___ times as much water as the town pumps in one year."

Extensions

Have students find the population of their own town and estimate its yearly household water use in scientific notation, using the 300 gallons per day figure and an assumed household size, then compare that estimate with the Sandgate town figure and explain which parts of the estimate are the least certain.