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Worksheet Grade 8 Math

Galebrook Lead Sampling: Which Rules Are Functions?

A grade 8 worksheet using a fictional lead sampling program, where students decide which pairings assign exactly one output to each input.

Objectives

  1. Students will decide whether a table, a mapping, or a set of ordered pairs is a function.
  2. Students will name the repeated input that keeps a relation from being a function.
  3. Students will evaluate a function rule for a given input and solve for the input that gives a stated output.

Vocabulary

Service line
A service line is the small pipe that connects the water main under the street to the plumbing inside one home or building. It is the last leg of the trip water makes to your tap.

Also: function, input, output, ordered pair, flushing

Teacher overview

The lead action level is 15 parts per billion, dropping to 10 parts per billion in 2027 under the Lead and Copper Rule Improvements, so utilities keep careful sampling records tied to bottles, addresses, and pipe materials. The fictional Galebrook Water Department keeps six such records, and only some of them pair each input with exactly one output. That distinction is the core of the grade 8 function definition, and a sampling program makes it concrete, because an address that carries two different results is an ordinary record-keeping problem for a utility. The error to watch for is the belief that two inputs sharing an output breaks the function rule; only a repeated input with two different outputs does.

Galebrook Lead Sampling: Which Rules Are Functions? — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

In our scenario, all six records below come from the fictional Galebrook Water Department. Lead results are given in parts per billion.

Relation A pairs each bottle with its result.

Bottle ID Result
B-101 3
B-102 6
B-103 3
B-104 11
B-105 2

Relation B pairs each address with a result.

Address Result
118 Ashgrit Street 4
214 Ashgrit Street 9
118 Ashgrit Street 12
307 Frostwick Lane 5

Relation C uses the Relation A data with the result as the input and the bottle ID as the output.

Relation D is the rule L = 22 - 3f, where f is the number of minutes a tap is flushed and L is the lead result.

Relation E is this set of ordered pairs, written as flush minutes and result: (0, 22), (1, 19), (2, 16), (2, 9), (4, 10).

Relation F pairs each service line material with a number of homes.

Material Homes
Copper 412
Galvanized steel 148
Lead 96
Plastic 244
  1. In Relation A, is each bottle ID paired with exactly one result? Answer yes or no.
  2. Two bottles in Relation A share the result 3. Does that keep Relation A from being a function? Explain.
  3. Is Relation B a function? Name the input that breaks the rule.
  4. Is Relation C a function? Explain using one input.
  5. Using Relation D, find L when f is 4.
  6. Using Relation D, find L when f is 6.
  7. Using Relation D, find f when L is 7.
  8. Using Relation D, find L when f is 0.
  9. Explain what your answer to item 8 means for a tap that has not been flushed at all.
  10. Is Relation E a function? Name the input that appears twice.
  11. In Relation F, find the output paired with the input "lead."
  12. Is Relation F a function? Explain.
  13. The department's rule is that each home is sampled once per year. Explain why that rule makes the address-to-result pairing a function within a single year.

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: Yes. Each of the five bottle IDs appears one time in the table, so each input is paired with exactly one output.
  2. Answer: No, it does not. The function rule limits how many outputs one input may have, not how many inputs may share an output. Bottles B-101 and B-103 are different inputs.
  3. Answer: No, Relation B is not a function. The address 118 Ashgrit Street appears twice, once with a result of 4 and once with a result of 12, so that input has two different outputs.
  4. Answer: No, Relation C is not a function. The input 3 pairs with both B-101 and B-103, which gives one input two different outputs.
  5. Answer: 10 parts per billion. Substitute f = 4 into L = 22 - 3f to get 22 - 3(4), which is 22 - 12, or 10.
  6. Answer: 4 parts per billion. Substitute f = 6 into L = 22 - 3f to get 22 - 3(6), which is 22 - 18, or 4.
  7. Answer: 5 minutes. Set 22 - 3f = 7. Subtract 22 from both sides to get -3f = -15, then divide both sides by -3 to get f = 5 minutes.
  8. Answer: 22 parts per billion. Substitute f = 0 into L = 22 - 3f to get 22 - 0, which is 22.
  9. Answer: The value 22 is the starting output of the rule, so it describes the water that has been sitting in the household pipes. Flushing has not yet lowered the result, which is why the first draw is the highest value the rule produces.
  10. Answer: No, Relation E is not a function. The input 2 appears twice, paired once with 16 and once with 9, so one input has two different outputs.
  11. Answer: 96 homes. The input "lead" appears once in the table and is paired with 96 homes.
  12. Answer: Yes, Relation F is a function. Each of the four materials appears one time, so every input is paired with exactly one output.
  13. Answer: A once-per-year rule means each address can supply only one result in that year, so no address is ever paired with two different outputs. The pairing then meets the definition of a function for that year, though combining several years would break it.

Differentiation

Support

Give these students a decision card that reads "look for the same input twice with different outputs" and a highlighter for marking repeated inputs in the fictional Galebrook tables. Assign items 1, 3, 5, 8, 11, and 12, and provide the substitution L = 22 - 3f with one worked example already filled in.

Extension

Ask students to add one more row to Relation F that keeps it a function, and one more row that breaks it, then to explain the difference in one sentence each. Have them also write a rule in the form L = a - bf that gives a result of 6 after 4 minutes of flushing and state the values of a and b they chose.

ELL supports

Pre-teach input, output, service line, and flushing with a labeled drawing of a tap, a sample bottle, and the pipe leading to the street. Post an arrow diagram showing one input with one arrow and one input with two arrows, labeled function and not a function. Provide the sentence frame "Relation ___ is not a function because the input ___ has two outputs" for the fictional Galebrook items.

Extensions

Have students collect a small classroom relation of their own, such as each student's locker number paired with a floor, and decide whether it is a function. Then ask them to reverse the two columns and decide again, and to write two sentences explaining why reversing a relation from the fictional Galebrook records can change the answer.