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Math problem set Grade 8 Math

Cranmere Sands Aquifer Storage: Sizing a Winter Injection Plan

A grade 8 set on an aquifer storage project where students write injection volumes in scientific notation, divide powers of ten, and check a pipeline loss rule.

Objectives

  1. Students will write large water volumes in scientific notation and interpret the exponent.
  2. Students will divide numbers written in scientific notation to compare two stored volumes.
  3. Students will convert a stored volume to acre-feet and test it against a stated design rule.

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.

Also: scientific notation, aquifer storage and recovery, injection well

Teacher overview

Cranmere Sands is a fictional coastal town whose water district plans to store treated water in a sand aquifer during the wet winter months and pump it back out in summer, a practice engineers call aquifer storage and recovery. The volumes involved run into the hundreds of millions of gallons, and scientific notation keeps those values manageable. Students write those volumes in scientific notation, divide them to compare the aquifer with the town's old surface reservoir, and finish by testing a pipeline loss against the district's stated rule. The mistake to look for is dividing the coefficients correctly but keeping the larger exponent instead of subtracting the exponents.

Cranmere Sands Aquifer Storage: Sizing a Winter Injection Plan — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

The water district in the town of Cranmere Sands is planning one winter of aquifer storage. Use the values inside each problem.

  1. Over one winter the district plans to inject 420,000,000 gallons of treated water into the aquifer. Write that volume in scientific notation.
  2. The town's old surface reservoir holds 7,000,000 gallons. Write that volume in scientific notation.
  3. How many times as much water is the planned winter injection as the surface reservoir holds? Show the division of the two numbers in scientific notation.
  4. The district has six injection wells, and each one takes 3.5 x 10^5 gallons per day. Write the combined daily injection for all six wells in scientific notation.
  5. Working at that combined daily rate, how many days does the district need to inject 4.2 x 10^8 gallons?
  6. One acre-foot equals 325,851 gallons. Write that number in scientific notation, rounding the first factor to two decimal places.
  7. Using the acre-foot value from problem 6, convert 4.2 x 10^8 gallons to acre-feet. Round your answer to the nearest hundred acre-feet.
  8. In August the town uses 1.4 x 10^6 gallons per day. For how many days could 4.2 x 10^8 gallons of stored water meet that demand on its own?
  9. An engineer measures a loss of 6.3 x 10^4 gallons per day from the injection pipeline. One percent of the combined daily injection is 2.1 x 10^4 gallons. How many times larger than that one-percent amount is the measured loss?
  10. The district's rule is that pipeline losses must stay below 2 percent of the combined daily injection. Does the measured loss meet that rule? Support your answer with the numbers from problems 4 and 9.

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: 4.2 x 10^8 gallons.
    420,000,000 has the decimal point moved 8 places to the left to reach 4.2, so the volume is 4.2 x 10^8 gallons.
  2. Answer: 7 x 10^6 gallons.
    7,000,000 has the decimal point moved 6 places to the left to reach 7, so the volume is 7 x 10^6 gallons.
  3. Answer: 60 times.
    Divide the first factors and subtract the exponents: 4.2 ÷ 7 = 0.6, and 10^8 ÷ 10^6 = 10^2, so the quotient is 0.6 x 10^2 = 60. The winter injection is 60 times the reservoir's volume.
  4. Answer: 2.1 x 10^6 gallons per day.
    6 × 3.5 x 10^5 = 21 x 10^5 gallons per day. Rewriting 21 x 10^5 with one digit before the decimal point gives 2.1 x 10^6 gallons per day.
  5. Answer: 200 days.
    4.2 ÷ 2.1 = 2, and 10^8 ÷ 10^6 = 10^2, so the quotient is 2 x 10^2 = 200 days.
  6. Answer: 3.26 x 10^5 gallons.
    Moving the decimal point 5 places left in 325,851 gives 3.25851, which rounds to 3.26, so the value is 3.26 x 10^5 gallons per acre-foot.
  7. Answer: 1,300 acre-feet.
    4.2 ÷ 3.26 = 1.288, and 10^8 ÷ 10^5 = 10^3, so the quotient is about 1.288 x 10^3, which is roughly 1,288 acre-feet and rounds to 1,300 acre-feet.
  8. Answer: 300 days.
    4.2 ÷ 1.4 = 3, and 10^8 ÷ 10^6 = 10^2, so the stored water covers 3 x 10^2 = 300 days of August demand.
  9. Answer: 3 times.
    6.3 ÷ 2.1 = 3, and both numbers carry 10^4, so the exponents cancel and the loss is 3 times the one-percent amount.
  10. Answer: No.
    The measured loss is 3 times the one-percent amount, which makes it 3 percent of the combined daily injection of 2.1 x 10^6 gallons. Since 3 percent is above the district's limit of 2 percent, the pipeline fails the rule and needs repair before the injection season begins.

Differentiation

Support

Provide a place-value strip showing 10^3 through 10^9 with the matching written-out numbers, so students can check each conversion against a reference instead of counting decimal places from memory. Work problem 3 as a class, writing the coefficient division and the exponent subtraction on separate lines before students continue.

Extension

Ask students to find the smallest number of injection wells the district would need to finish the same 4.2 x 10^8 gallon injection in 150 days rather than 200, and to write both the required daily rate and the well count in scientific notation.

ELL supports

Provide a worked bilingual glossary card for "injection well," "aquifer," "acre-foot," and "scientific notation," each with a small labeled sketch. Allow students to show the division in problems 3, 5, and 8 as two stacked steps, coefficients above and powers of ten below, instead of explaining the process in sentences.