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Lesson plan Grade 4 Math

Barrowgate's Mill Row Water Main: Pipe Sections and Remainders

A grade 4 lesson in the fictional town of Barrowgate where students divide a water main's length by pipe-section length and decide what each remainder means.

Objectives

  1. Students will divide a four-digit length by a one-digit section length and identify the quotient and the remainder.
  2. Students will explain what a remainder means when whole pipe sections, whole workdays, or whole truck loads are being counted.
  3. Students will multiply a four-digit length by a one-digit factor to compare two water mains.

Vocabulary

Water main
A big underground pipe that carries drinking water beneath the streets to a neighborhood. Smaller service lines branch off it to reach each building.

Also: pipe section, remainder, quotient, distribution

Teacher overview

The fictional town of Barrowgate is replacing the old water main that runs under Mill Row, and the replacement pipe arrives on a truck in straight sections of one fixed length. Before the crew orders anything, someone has to divide the length of the street by the length of one section and then decide what to do with whatever is left over. The misconception this lesson is built to correct is the habit of reporting only the quotient and dropping the remainder, so that a class divides 1,236 feet by 8 feet, writes 154, and leaves the last 4 feet of Mill Row with no pipe under it. The same judgment shows up again when the crew counts workdays and truck loads, and each time the leftover forces one more of whatever is being counted.

Procedure

Hook (5 min): Hold up a length of adding-machine tape or string cut to 8 units and a longer piece representing a street. Ask how many of the short pieces it would take to cover the long one, and write two or three student estimates on the board without comment. Tell the class that Barrowgate's crew has to answer this exact question before ordering pipe.

Direct instruction (12 min): Introduce the scenario. In the town of Barrowgate, the Mill Row water main is 1,236 feet long, and each new pipe section is 8 feet long. Model the division 1,236 divided by 8 on chart paper using place value: 8 goes into 12 hundreds once with 4 hundreds left, 8 goes into 43 tens five times with 3 tens left, and 8 goes into 36 ones four times with 4 ones left. Write the result as 154 sections with 4 feet remaining, then draw the last stretch of Mill Row on the board with a 4-foot gap and ask what the crew would find in the street if they ordered only 154 sections.

Guided practice (13 min): Work one parallel problem together on chart paper. A different street in Barrowgate needs 1,150 feet of main, and the sections are still 8 feet long. Have pairs find the quotient and remainder, then ask each pair to say out loud, in a full sentence, how many sections must be ordered and why. For the teacher's reference, 1,150 divided by 8 is 143 with a remainder of 6, so 144 sections must be ordered. Use a quick thumbs-up poll: if fewer than three-quarters of the class adds one to the quotient, work a second parallel problem before moving on.

Independent practice (15 min): Students complete the seven questions on the student handout on their own, using the standard division algorithm or an area model. Circulate and ask individual students what their remainder stands for in that particular question, since the remainder means feet in one item and sections in another. Question 4 divides by a two-digit number, so expect students to reason from multiplication facts there rather than from the standard algorithm.

Closure / exit ticket (5 min): Collect the handouts, then ask every student to write one sentence on an index card finishing this stem: "A remainder matters in this problem because ___." Scan the cards for students who describe the remainder as leftover work still to be done rather than as a number to discard.

Barrowgate's Mill Row Water Main: Pipe Sections and Remainders — Grade 4 Math
Name: ____________________________ Date: ______________

Student Handout

Barrowgate is replacing the water main under Mill Row. Use the numbers given in each question.

  1. The Mill Row water main is 1,236 feet long. Each new pipe section is 8 feet long. How many whole 8-foot sections fit along the 1,236 feet?

  2. The Mill Row main is 1,236 feet long. The crew has laid 154 whole sections that are 8 feet long each. How many feet of Mill Row still have no new pipe under them?

  3. The Mill Row main is 1,236 feet long, and each section is 8 feet long. Every foot of the street must be covered, and a section cannot be built out of scraps. How many sections must the crew order in all?

  4. The crew lays 14 sections each workday. How many workdays does the crew need to lay 155 sections?

  5. One delivery truck carries 9 pipe sections in a load. How many loads are needed to deliver 155 sections?

  6. The old main under Ferry Street is 3 times as long as the 1,236-foot Mill Row main. How long is the Ferry Street main?

  7. Look back at question 3. Explain in one or two sentences why the crew must order 155 sections instead of 154.

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: 154 sections.
    1,236 divided by 8. Eight goes into 12 hundreds 1 time with 4 hundreds left over, into 43 tens 5 times with 3 tens left over, and into 36 ones 4 times with 4 ones left over. The quotient is 154 and the remainder is 4. Check: 154 times 8 equals 1,232 feet.

  2. Answer: 4 feet.
    The 154 whole sections cover 154 times 8, which is 1,232 feet. Subtract that from the street length: 1,236 minus 1,232 equals 4 feet with no pipe under them. This is the same 4 that appeared as the remainder in question 1.

  3. Answer: 155 sections.
    The 154 whole sections leave 4 feet uncovered, and those 4 feet still need pipe. One more section covers them, so the crew orders 154 plus 1, or 155 sections. Part of that last section will be cut off and set aside.

  4. Answer: 12 workdays.
    155 divided by 14 is 11 with a remainder of 1, because 14 times 11 equals 154. Eleven full days leave 1 section still to lay, and that section needs a twelfth day. Accept 11 remainder 1 as work shown, but the answer to the question asked is 12 days.

  5. Answer: 18 loads.
    155 divided by 9 is 17 with a remainder of 2, because 9 times 17 equals 153. Seventeen full loads leave 2 sections behind, and those 2 sections need an eighteenth trip.

  6. Answer: 3,708 feet.
    1,236 times 3. Multiply by place value: 1,000 times 3 equals 3,000, 200 times 3 equals 600, 30 times 3 equals 90, and 6 times 3 equals 18. Adding gives 3,000 plus 600 plus 90 plus 18, or 3,708 feet.

  7. Answer: The leftover 4 feet still needs pipe.
    Accept any explanation stating that dividing leaves 4 feet uncovered and that a partial stretch of street still has to be piped, so the crew rounds the section count up rather than dropping the remainder.

Differentiation

Support (below grade level)

Give these students a version of the handout in which questions 1 and 2 are already set up as a labeled division frame, with the dividend and divisor written in place and boxes waiting for the quotient and the remainder. Provide unit cubes or paper strips cut to 8 units so students can build the last part of Mill Row and see the 4-foot gap physically before they write about it. Allow the numbers in question 6 to be multiplied with an area model rather than the standard algorithm.

Extension (above grade level)

Ask students to plan the delivery schedule for the whole job in the town of Barrowgate: if the truck brings 9 sections per load and the crew lays 14 sections per day, how many loads must arrive before the crew can finish a full day of work without running out of pipe, and on which day would the crew first stand idle if only one load arrived each morning? Students should justify their answer with division and a written explanation of every remainder they meet.

ELL supports

Pre-teach "main," "section," and "remainder" with a labeled drawing showing a long street, one short piece of pipe, and a small leftover gap at the end. Keep the drawing posted during independent practice. Provide sentence frames for questions 3 and 7: "The crew must order ___ sections because ___ feet are still uncovered." Allow students to answer question 7 orally to the teacher or a partner before writing.

Extensions

Invite a distribution operator or public works crew leader from a nearby utility to describe how a real main replacement is measured and ordered, and ask them to bring a short offcut of pipe. Students can then remeasure the classroom scenario using the actual section length the visitor names, keeping every step of the division and every remainder decision the same.