Lesson plan Grade 6 Math

Sedgely's PFAS Results: Modeling Parts per Trillion

A 50-minute grade 6 math lesson building a thousand-by-thousand ratio ladder so students can compare four wells' PFAS results to the federal limit.

Grade: Grade 6
Subject: Math
Material: Lesson plan
Time: 50 min
Prep: Low prep
Materials: A jar of about 1,000 small beads with one bead of a different color, Chart paper, Markers, Calculators, Printed Student Handout, Index cards

Standards

6.RP.A.3 Common Core Math

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations, including finding a percent of a quantity and converting measurement units.

How this material aligns: Handout items 2 and 5 ask what percent of the federal limit a well result represents, and item 4 expresses a drop as a percent of the original result, the percent reasoning this standard requires.

All materials for 6.RP.A.3 →

6.RP.A.1 Common Core Math

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, 'The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'

How this material aligns: The bead ladder has students describe one marked bead among a thousand as a ratio of one to one thousand, then extend that ratio down the scale in handout items 6 through 8.

All materials for 6.RP.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will describe the relationship between parts per thousand, million, billion, and trillion.
Students will express a water test result as a percent of a regulatory limit.
Students will explain why a very small concentration can still exceed a limit.

Vocabulary

Parts per trillion: A unit for measuring extremely small amounts of a chemical in water. PFAS limits are set in parts per trillion because these chemicals matter even in tiny doses.

Also: concentration, limit, percent

Teacher overview

PFAS are a family of manufactured chemicals that turn up in some drinking water sources, and the federal maximum contaminant level for two of them, PFOA and PFOS, is 4.0 parts per trillion each. Numbers that small are hard to picture, so this lesson builds a ladder from parts per thousand down to parts per trillion using a jar of beads, then applies it to four well results in the town of Sedgely, which is fictional. The misconception the lesson addresses is the assumption that a number below 10 must mean a small amount of contamination; students often treat a result of 6.4 as harmless because 6.4 is a small number, without comparing it to the limit. Water systems must meet this limit by April 2029. In May 2026 EPA proposed letting eligible systems apply for a two-year exemption, to April 2031, which would not change the limit itself; the proposal was not yet final as of September 2026.

Procedure

Hook (5 min): Hold up a jar holding about 1,000 beads with exactly one bead in a different color. Ask students to name the fraction and the percent that single bead represents, then tell them that the unit of the limit they will work with today, one part per trillion, is a billion times smaller than the one-in-a-thousand share that single bead shows.

Direct instruction (12 min): Build the ratio ladder on chart paper in four rows: one part per thousand, one part per million, one part per billion, one part per trillion. Between each pair of rows write the phrase "one thousand times smaller," and record how many items a jar would need for a single marked item to represent that share. Introduce the federal limit of 4.0 parts per trillion for PFOA, and note that the compliance date is April 2029, while EPA has proposed letting eligible systems apply for an exemption to April 2031.

Guided practice (13 min): Distribute the Student Handout and work items 1 and 2 with the class. Model item 2 as a division of the well result by the limit, then convert the quotient to a percent. Ask a student to explain why a result above the limit must give a percent above 100.

Independent practice (12 min): Students complete handout items 3 through 9 on their own or in pairs. Circulate and check that any student who writes a percent above 100 can say in words what that means about the well.

Closure and exit ticket (8 min): Collect the handouts and pose one question on an index card: a well tests at 5.0 parts per trillion against a limit of 4.0, so is the result above or below the limit, and what percent of the limit is it? Scan for the reasoning that the well is above the limit at 125 percent. If fewer than three quarters of the cards show the division against the limit, reteach item 2 at the start of the next class.

Sedgely's PFAS Results: Modeling Parts per Trillion — Grade 6 Math

Name: ____________________________ Date: ______________

Student Handout

The town of Sedgely, a fictional town, tested four wells for PFOA. The federal limit is 4.0 parts per trillion. Well 1 measured 2.0 parts per trillion, Well 2 measured 6.4, Well 3 measured 4.0, and Well 4 measured 1.2.

Which of the four wells has a result above the federal limit of 4.0 parts per trillion?
Well 2 measured 6.4 parts per trillion. What percent of the 4.0 limit is that result?
How many parts per trillion must Well 2 drop to reach the 4.0 limit exactly?
The drop you found in item 3 is what percent of Well 2's original result of 6.4?
Well 1 measured 2.0 parts per trillion. What percent of the 4.0 limit is that result?
One part per thousand is how many times larger than one part per million?
One part per million is how many times larger than one part per trillion?
A jar of 1,000 beads with one marked bead shows one part per thousand. How many beads would the jar need for one marked bead to show one part per million?
Explain in one or two sentences why an amount as small as 4 parts per trillion can still be worth setting a limit on.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

Answer: Well 2.
Well 2 measured 6.4 parts per trillion, which is above 4.0. Well 3 sits exactly at the limit, and Wells 1 and 4 are below it.
Answer: 160 percent.
6.4 divided by 4.0 equals 1.6, which is 160 percent of the limit.
Answer: 2.4 parts per trillion.
6.4 minus 4.0 equals 2.4 parts per trillion.
Answer: 37.5 percent.
2.4 divided by 6.4 equals 0.375, which is 37.5 percent.
Answer: 50 percent.
2.0 divided by 4.0 equals 0.5, which is 50 percent of the limit.
Answer: 1,000 times.
A million is a thousand thousands, so one part per thousand is one thousand times larger than one part per million.
Answer: 1,000,000 times.
Million to billion is one thousand times, and billion to trillion is another thousand times, so the two steps together give one million times.
Answer: 1,000,000 beads.
One part per million means one marked item among a million items.
Answer: Because these chemicals can affect health even in very small amounts, and people drink the water every day for years. Accept any answer noting that people drink water every day for years, that the number looks small only because the amount of water it is measured against is so large, or that health agencies set the limit after studying long-term exposure.

Differentiation

Support

Give students the ladder from the direct instruction as a printed reference with the four rows already labeled, and provide a worked model of item 2 using a different result, such as 8.0 compared with a limit of 4.0. Allow calculators for every division, and pair students for items 6 through 8.

Extension

Ask students to find the percent of the limit for a fifth well that measures 3.6 parts per trillion, then to determine how much that result could rise before it crosses the limit, stated both as an amount and as a percent increase.

ELL supports

Post the four ladder rows with a picture beside each: one marked bead in a small jar, then in a large jar, then in a barrel, then in a tank. Provide the sentence frame "Well ___ measured ___, which is ___ percent of the limit, so it is above or below the limit" for items 1 through 5.

Extensions

Students can read about the treatment methods a utility uses when a well exceeds a limit, then rank the four Sedgely wells in the order they would treat them and defend that ranking with the percents they calculated.