Water for Schools
Menu

Lesson plan Grade 8 Math

Havenscroft Hydrant Flushing: Comparing Rates in Three Forms

A 50-minute grade 8 lesson where students find three fictional flushing crews' gallons-per-minute rates from a table, a graph, and a written rule, then rank them.

Objectives

  1. Students will find the unit rate of a proportional relationship shown as a table, as a graph, and as a written rule.
  2. Students will compare three proportional relationships given in different forms and rank them by rate.
  3. Students will explain why a steady flushing rate produces a straight line through the origin.

Vocabulary

Key words: unit rate, slope, proportional relationship, origin

Teacher overview

Field crews at the fictional Havenscroft Water Department flush hydrants every spring to clear sediment from the mains, and each crew records how many gallons it has released and how long the hydrant has been open. This lesson gives three crews' records in three different forms, so students find a unit rate in a table, in two points on a graph, and in a written sentence, and then compare the three. The misconception this lesson is built to correct is the habit of ranking crews by the total gallons released rather than by the rate: a crew that releases more gallons over a longer time is not necessarily working faster, and students have to divide before they can compare.

Procedure

Hook (5 min): Tell the class that three fictional Havenscroft crews flushed hydrants on the same morning. Crew Ash released 780 gallons, Crew Birch released 350 gallons, and Crew Cedar released 290 gallons. Ask which crew had the fastest hydrant. Take answers without comment, then reveal that the three crews worked for 12, 5, and 5 minutes. Ask whether anyone wants to change an answer.

Direct instruction (12 min): Model the calculation with a fourth crew that is not on the handout. Crew Dell released 480 gallons in 8 minutes. Write the ordered pairs (0, 0) and (8, 480), find the slope as 480 divided by 8, and label the result 60 gallons per minute. Draw the line through the origin and label the point (1, 60) to show what the unit rate looks like on the graph. Write the rule as g equals 60 times m, where g is gallons and m is minutes. Point out that all three forms carry the same number.

Guided practice (13 min): Put a table for Crew Ash on the board with the pairs (0, 0), (4, 260), (8, 520), and (12, 780). Have students find the rate two ways, once from the first pair of rows and once from the last pair of rows, and confirm they match. Ask the check-for-understanding question: "If a crew's table shows 0 gallons at 0 minutes, what does that tell you about the graph?" Expect students to say the line passes through the origin. If fewer than three-quarters of the class can answer, reteach the origin point before moving on.

Independent practice (15 min): Students complete the Student Handout, working through all ten items and showing the division for each rate.

Closure and exit ticket (5 min): On an index card, students answer this question: a crew releases 405 gallons in 9 minutes, so what is the crew's rate in gallons per minute? Collect the cards and check for the answer of 45 gallons per minute. If more than a third of the class multiplies instead of dividing, reteach the unit rate step at the start of the next session.

Havenscroft Hydrant Flushing: Comparing Rates in Three Forms — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

All of the crews and numbers below belong to the fictional Havenscroft Water Department. Show your division for every rate you find.

  1. Crew Ash recorded these pairs of (minutes, gallons): (0, 0), (4, 260), (8, 520), and (12, 780). What is Crew Ash's rate in gallons per minute?
  2. Crew Birch's record is graphed as a straight line through the points (0, 0) and (5, 350), where the first number is minutes and the second is gallons. What is the slope of that line in gallons per minute?
  3. Crew Cedar released 290 gallons in 5 minutes at a steady rate. What is Crew Cedar's rate in gallons per minute?
  4. Write an equation for the number of gallons g that Crew Cedar releases in m minutes.
  5. Crew Ash flushes at 65 gallons per minute, Crew Birch at 70 gallons per minute, and Crew Cedar at 58 gallons per minute. Which crew has the greatest rate?
  6. Crew Ash flushes at 65 gallons per minute. How many gallons does Crew Ash release in 20 minutes?
  7. Crew Cedar flushes at 58 gallons per minute and must release 1,450 gallons. How many minutes does that take?
  8. Crew Birch flushes at 70 gallons per minute. How many gallons does Crew Birch release in 12 minutes?
  9. Crew Ash releases 910 gallons in 14 minutes and Crew Birch releases 840 gallons in 12 minutes. How many more gallons does Crew Ash release?
  10. Each crew's graph is a straight line that passes through the point (0, 0). Explain why the line starts at the origin.

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: 65 gallons per minute. From (4, 260), the rate is 260 divided by 4, which is 65.
    Checking with the last two rows, 780 minus 520 is 260 gallons over 12 minus 8, or 4 minutes, which is again 65 gallons per minute.
  2. Answer: 70 gallons per minute. The slope is the change in gallons over the change in minutes, or 350 minus 0 divided by 5 minus 0, which is 350 divided by 5, or 70.
  3. Answer: 58 gallons per minute. Divide 290 gallons by 5 minutes to get 58.
  4. Answer: g = 58m. The relationship is proportional because 0 minutes gives 0 gallons, so the rule is the rate 58 multiplied by the number of minutes m.
  5. Answer: Crew Birch. At 70 gallons per minute, Crew Birch has a greater rate than Crew Ash at 65 and Crew Cedar at 58, so Crew Birch's line is the steepest of the three.
  6. Answer: 1,300 gallons. Multiply the rate by the time: 65 times 20 equals 1,300 gallons.
  7. Answer: 25 minutes. Divide the total by the rate: 1,450 divided by 58 equals 25 minutes.
  8. Answer: 840 gallons. Multiply 70 gallons per minute by 12 minutes to get 840 gallons.
  9. Answer: 70 gallons. Crew Ash releases 910 gallons and Crew Birch releases 840 gallons, and 910 minus 840 equals 70 gallons. Note that Crew Ash releases more in total even though Crew Birch has the faster rate, because Crew Ash worked two minutes longer.
  10. Answer: Because no water has been released before the hydrant opens. At 0 minutes the crew has released 0 gallons, so the ordered pair (0, 0) is on every crew's graph, and a steady rate keeps the graph a straight line from that point.

Differentiation

Support

Give these students a partially completed table for each crew with the division already set up, so they only carry out the arithmetic rather than deciding which two numbers to divide. Allow a calculator throughout, and provide a labeled model of the Crew Dell example from the direct instruction step to keep on the desk as a reference.

Extension

Ask students to write the equation for all three crews, graph the three lines on one set of axes, and determine how many minutes Crew Cedar would need to release as much water as Crew Birch releases in 20 minutes. Have them explain what the answer says about the two slopes.

ELL supports

Pre-teach "rate," "per minute," and "origin" with a simple demonstration: fill a measuring cup for ten seconds and record the amount, then ask students to say the rate aloud in a full sentence. Provide the sentence frame "Crew ___ releases ___ gallons every minute, so its rate is ___ gallons per minute" for items 1 through 3.

Extensions

If your water utility publishes a hydrant flushing schedule, have students find how many hydrants are flushed in one week and estimate the total water released using a rate from this lesson, stating clearly which numbers are estimates and which come from the published schedule.