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Worksheet Grade 9 Math

Copper Hollow's Chlorine Log: Precision and Significant Figures

A grade 9 worksheet on chlorine dosing at a fictional treatment plant, where students match reported precision to test-kit resolution and significant figures.

Objectives

  1. Students will calculate chlorine demand from a dosage and a residual reading using the formula demand = dosage minus residual.
  2. Students will determine an appropriate level of precision for a calculated result based on the precision of the measuring instruments used.

Vocabulary

Chlorine residual
Chlorine residual is the small amount of chlorine still left in the water after it leaves the treatment plant. It keeps protecting the water from germs all the way to your faucet.

Also: chlorine demand, significant figures, maximum residual disinfectant level

Teacher overview

At the fictional Copper Hollow water treatment plant, a technician applies a measured dose of chlorine to a batch of water and, after a standard contact time, measures how much chlorine remains as a residual. The difference between those two numbers is the chlorine demand, the amount of chlorine that reacted with organic matter and pathogens in the water rather than remaining as protection in the distribution system. This worksheet uses Copper Hollow's daily log to practice that subtraction while asking a second question that operators face constantly: how many decimal places can a calculated result honestly claim?

A common misconception is treating a calculator's full decimal display as automatically correct. If a dosage is only known to the nearest tenth of a milligram per liter, no amount of arithmetic can make a demand value calculated from it precise to the nearest thousandth. The worksheet asks students to match their reported precision to the least precise measurement that went into the calculation, the same habit a lab technician is trained to follow.

Copper Hollow's Chlorine Log: Precision and Significant Figures — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

Use the formula: Chlorine demand (mg/L) = dosage (mg/L) − residual (mg/L).

  1. A technician at the fictional Copper Hollow treatment plant applies a chlorine dose of 3.2 mg/L to a batch of water. After the standard contact time, a lab titration method, which resolves to the nearest 0.01 mg/L, measures a residual of 1.43 mg/L. Calculate the chlorine demand.

  2. The dosage (3.2 mg/L) is known only to the nearest tenth, while the residual (1.43 mg/L) is known to the nearest hundredth. When subtracting two measurements, the result should be reported only to the least precise of the two decimal places. Round your answer from problem 1 to the correct number of decimal places, and state which measurement's precision controlled the rounding.

  3. A field colorimeter at the same plant resolves chlorine readings only to the nearest 0.1 mg/L. If the true residual in the sample is 1.43 mg/L, what value would the colorimeter most likely display?

  4. Recalculate the chlorine demand using the colorimeter's rounded residual reading from problem 3. How does this answer compare to your rounded answer from problem 2?

  5. Compare the residual reading of 1.43 mg/L to the maximum residual disinfectant level of 4.0 mg/L. Is Copper Hollow's treated water within the allowed level? By how much?

  6. Distribution-system operators typically maintain a chlorine residual between 0.2 and 2.0 mg/L once water leaves the plant. Does Copper Hollow's 1.43 mg/L residual fall within that typical range?

  7. The next day, Copper Hollow raises its dose to 3.8 mg/L to handle a period of higher organic load, and the residual after contact time measures 1.10 mg/L (lab titration, resolves to the nearest 0.01 mg/L). Calculate the chlorine demand and round it to the correct number of decimal places.

  8. Compare the rounded chlorine demand values from problem 2 and problem 7. Did the water's chlorine demand increase or decrease from one day to the next, and by how much?

  9. A new technician calculates a chlorine demand using colorimeter readings, which resolve only to the nearest 0.1 mg/L, and reports the result on the log sheet as "1.75000 mg/L." Explain why this reported value claims more precision than the colorimeter can support, and rewrite it at an appropriate level of precision.

  10. In your own words, explain why an operator should never report more decimal places in a calculated result than the least precise measurement used to calculate it.

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: 1.77 mg/L.
    3.2 mg/L − 1.43 mg/L = 1.77 mg/L, calculated exactly before any rounding for precision is applied.

  2. Answer: 1.8 mg/L.
    The dosage (3.2 mg/L) is known to one decimal place, which is less precise than the residual's two decimal places, so the result must be rounded to one decimal place: 1.77 rounds to 1.8 mg/L. The dosage's precision controls the rounding because it is the less precise of the two measurements.

  3. Answer: 1.4 mg/L.
    1.43 mg/L rounds to the nearest tenth as 1.4 mg/L, since 1.43 is closer to 1.4 than to 1.5.

  4. Answer: 1.8 mg/L.
    3.2 mg/L − 1.4 mg/L = 1.8 mg/L. This matches problem 2's correctly rounded answer, showing that once both values are reported at an honest level of precision, the colorimeter-based and titration-based results agree.

  5. Answer: yes, by about 2.57 mg/L.
    4.0 mg/L − 1.43 mg/L = 2.57 mg/L, so Copper Hollow's residual is well within the maximum residual disinfectant level.

  6. Answer: yes.
    1.43 mg/L falls between the typical distribution range of 0.2 mg/L and 2.0 mg/L.

  7. Answer: 2.7 mg/L.
    3.8 mg/L − 1.10 mg/L = 2.70 mg/L exactly, which rounds to 2.7 mg/L at the dosage's one-decimal-place precision.

  8. Answer: increased by 0.9 mg/L.
    The demand rose from 1.8 mg/L (problem 2) to 2.7 mg/L (problem 7), an increase of 0.9 mg/L, consistent with a higher organic load consuming more chlorine.

  9. Answer: the reported value is over-precise; correct it to 1.8 mg/L.
    A colorimeter that resolves only to the nearest 0.1 mg/L cannot support five decimal places of precision in a calculated result. Reporting "1.75000 mg/L" falsely implies a level of precision the instrument never measured. The value should be rounded to one decimal place, such as 1.8 mg/L, to match what the colorimeter can actually distinguish.

  10. Answer: a calculated result cannot be more precise than its least precise input.
    Subtraction and other arithmetic operations cannot manufacture precision that was not present in the original measurements. Reporting extra decimal places suggests a false level of confidence and can mislead anyone reading the log about how carefully the water was actually tested.

Differentiation

Support (below grade level)

Provide a reference card stating the rule "round to the same number of decimal places as the least precise measurement" alongside a worked example using different numbers than problems 1 and 2. Allow a calculator for all subtraction so students can focus on the rounding decision rather than the arithmetic itself.

Extension (above grade level)

Ask students to determine, using only the instrument resolutions given in this worksheet, what would happen to the reportable precision of chlorine demand if the fictional Copper Hollow plant switched to a titration method that resolved to the nearest 0.001 mg/L instead of 0.01 mg/L, while keeping the same 3.2 mg/L dosage. Have them explain whether this upgrade would change the answer to problem 2.

ELL supports

Pre-teach "residual," "dosage," and "demand" with a simple diagram showing chlorine going in (dosage), some being used up (demand), and some remaining (residual). Provide sentence starters for problems 8 through 10, such as "The demand increased because ___" and "This value is too precise because ___."