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Lesson plan Grade 8 Math

Ostermere Lab Logbook: Deciding What Counts as a Function

A 45-minute grade 8 lesson in which students test relations from a fictional plant's lab logbook against the definition of a function and find a duplicated entry.

Objectives

  1. Students will decide whether a relation given as a table, as ordered pairs, or as a graph is a function.
  2. Students will explain why one input paired with two different outputs breaks the definition of a function.
  3. Students will use the definition of a function to locate a duplicated entry in a data table.

Vocabulary

Key words: function, input, output, ordered pair, vertical line test

Teacher overview

The laboratory at a drinking water plant keeps a logbook in which every sample gets a number, a time, and a set of test results, and the plant relies on being able to look up one sample and get one answer. That expectation is exactly the definition of a function, which makes a lab logbook a useful place for students to meet the definition rather than a page of abstract ordered pairs. This lesson uses records from the fictional Ostermere water plant, and it ends with a page where the same sample number appears twice with two different results, which students identify as a recording error precisely because the relation is not a function.

Procedure

Hook (5 min): Write two lines from a fictional Ostermere logbook on the board: sample 12 with a chlorine result of 1.0, and sample 12 with a chlorine result of 1.4. Ask the class what an operator should do with those two lines. Collect answers, then say that the mathematical name for the problem is that the relation is not a function.

Direct instruction (10 min): State the definition: a function assigns exactly one output to each input. Model it with a small table from the fictional plant in which the input is the sample time and the output is the turbidity reading. Show that repeated outputs are allowed by pairing two different times with the same reading, and that repeated inputs with different outputs are not allowed. Draw a graph of the same data and demonstrate the vertical line test, connecting it back to the definition rather than presenting it as a separate rule.

Guided practice (12 min): As a class, work through two relations from the fictional Ostermere logbook. The first pairs basin numbers with detention times and is a function. The second pairs detention times with basin numbers and is not, because two basins share the same detention time.
Ask the check-for-understanding question: "Which side of the pair is allowed to repeat, the input or the output?" Expect students to answer the output. If more than a quarter of the class answers the input, stop and redo the mapping diagram before releasing students to the handout.

Independent practice (13 min): Students complete the Student Handout, deciding whether each relation is a function and writing a reason for each decision.

Closure and exit ticket (5 min): On an index card, students write one sentence explaining why a logbook page with the same sample number twice, showing two different results, cannot be a function. Collect the cards and look for the phrase "one input with two outputs" or an equivalent statement in students' own words.

Ostermere Lab Logbook: Deciding What Counts as a Function — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

Every table and record below comes from the fictional Ostermere water plant. For each item, decide whether the relation is a function and write one sentence giving your reason.

  1. The plant recorded turbidity four times in one morning: 6:00 gave 0.18 NTU, 8:00 gave 0.15 NTU, 10:00 gave 0.18 NTU, and 12:00 gave 0.17 NTU. Is turbidity a function of the time of day?
  2. Use the same four records: 6:00 gave 0.18 NTU, 8:00 gave 0.15 NTU, 10:00 gave 0.18 NTU, and 12:00 gave 0.17 NTU. Is the time of day a function of the turbidity reading?
  3. A page of the logbook lists these pairs of (sample number, chlorine result): (1, 1.2 mg/L), (2, 0.9 mg/L), (3, 1.1 mg/L), and (4, 0.9 mg/L). Is the chlorine result a function of the sample number?
  4. A different page lists these pairs of (sample number, chlorine result): (12, 1.0 mg/L), (13, 0.8 mg/L), and (12, 1.4 mg/L). Is the chlorine result a function of the sample number?
  5. Look again at the page in item 4, which lists sample 12 twice with two different results. What does that tell an operator about the record?
  6. A graph shows tank depth on the vertical axis and time on the horizontal axis. At 9:00 the graph shows two separate points, one at 18 feet and one at 22 feet. Is depth a function of time on that graph?
  7. A mapping diagram sends basin 1 to 2 hours, basin 2 to 3 hours, and basin 3 to 2 hours. Is the detention time a function of the basin number?
  8. A student says that a function must give a different output for every input. Explain what is wrong with that statement, and use item 7 in your explanation.

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, it is a function. Each of the four times appears once and is paired with exactly one reading, so no input has two outputs. The reading 0.18 NTU appears twice, but a repeated output does not break the definition.
  2. Answer: No, it is not a function. The reading 0.18 NTU is paired with both 6:00 and 10:00, so that one input has two different outputs.
  3. Answer: Yes, it is a function. Sample numbers 1, 2, 3, and 4 each appear once, so each input has exactly one output. Samples 2 and 4 share the result 0.9 mg/L, and a repeated output is allowed.
  4. Answer: No, it is not a function. Sample 12 is paired with 1.0 mg/L and also with 1.4 mg/L, so a single input has two different outputs.
  5. Answer: The record contains an error. One of the two lines for sample 12 was probably mislabeled or copied from another sample, because a single sample cannot have two different chlorine results at the same time. The lab would recheck the bench sheet before reporting.
  6. Answer: No, it is not a function. The single input of 9:00 is paired with a depth of 18 feet and a depth of 22 feet, and a vertical line drawn at 9:00 crosses the graph twice.
  7. Answer: Yes, it is a function. Basins 1, 2, and 3 each appear once as an input, so each has exactly one detention time. Basins 1 and 3 share the output of 2 hours, which is allowed.
  8. Answer: The statement is wrong because outputs may repeat. The definition only requires that each input has exactly one output. Item 7 shows this directly, since basin 1 and basin 3 both have a detention time of 2 hours and the relation is still a function.

Differentiation

Support

Provide a two-column organizer labeled "input" and "output" and have students copy each relation into it before deciding, so the repeated value is visible in a column rather than buried in a sentence. Offer the sentence frame "This is not a function because the input ___ has the outputs ___ and ___."

Extension

Ask students to write two of their own logbook relations for a fictional plant, one that is a function and one that is not, exchange them with a partner, and defend their judgments. Then have them explain why reversing the inputs and outputs of a function does not always produce another function.

ELL supports

Introduce "input" and "output" with a physical demonstration using a labeled box: a card goes in and exactly one card comes out. Keep a word wall showing "input," "output," "ordered pair," and "function," each paired with a small mapping diagram, and allow students to answer orally before writing.

Extensions

Have students look at a public water quality report from any utility and identify which columns serve as inputs and which as outputs, then state one relation from the report that is a function and explain how they know.