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Lesson plan Grade 6 Math

Pellmore's Valve Crew: Expressions for a Week of Inspections

A grade 6 lesson built on a utility valve crew's weekly count: students write and evaluate expressions with a variable, then solve one-step equations about the work.

Objectives

  1. Students will write an algebraic expression for a repeated daily count plus a starting amount.
  2. Students will evaluate an expression with a variable for two different values of that variable.
  3. Students will solve one-step equations in the forms x plus p equals q and px equals q in context.

Vocabulary

Valve
A valve is a device in a pipe that operators can open or close to control where water flows. Valves let crews shut off one street for repairs without turning off the whole town.

Also: expression, variable, evaluate, equation

Teacher overview

Every buried water main has valves along it, and distribution crews turn each valve open and shut on a regular schedule, often once a year for the most important valves, to be certain it will still close during a break. The fictional Pellmore water district tracks that work as a running count, which is exactly the structure of an expression with one variable: a fixed amount already done plus a steady amount added each day. The misconception this lesson addresses is that 14d means 14 followed by the value of d, so students write 145 when d is 5. The procedure names that error out loud and asks students to read 14d aloud as fourteen times d every time it appears.

Procedure

Hook (5 min): Tell the class that a two-person crew in the town of Pellmore inspected 9 valves on Friday and then started a fresh week. Ask how many will be done after one day, two days, and three days if they inspect 14 a day. Record the three answers as a short table, and ask what a student would write for "any number of days" without knowing the number.

Direct instruction (13 min): Build the expression 14d + 9 from the table, naming d as the number of working days and 14 as the number inspected each day. Read the expression aloud as "fourteen times d, plus nine," and evaluate it for d = 3, showing the substitution line 14 times 3 plus 9 before the arithmetic. Contrast it with the wrong reading, writing 143 on the board and crossing it out. Then set up one equation of the same form, 9 + n = 150, and solve it by subtracting 9 from both sides, which gives n = 141.

Guided practice (10 min): Pairs evaluate 14d + 9 for d = 4 and d = 10, which give 65 and 149, and solve 14d = 70 together, which gives d = 5. Require each pair to write the substitution line or the inverse-operation line before the answer. Check that both pairs at each table agree before they continue.

Independent practice (17 min): Students complete handout questions 1 through 8. Watch for two errors: writing 18 + k instead of 18k in question 6, and dividing by 18 twice in question 8.

Closure (5 min): Exit ticket on a slip of paper: each student writes what d stands for in the Pellmore expression 14d + 9 and evaluates it for d = 2, which is 37. Collect the slips at the door and count how many have both parts right. If fewer than four of every five students have both, open the next class by rebuilding the expression from the table.

Pellmore's Valve Crew: Expressions for a Week of Inspections — Grade 6 Math
Name: ____________________________ Date: ______________

Student Handout

In the fictional Pellmore water district, a two-person crew inspects 14 valves each working day. Before this week began, the crew had already inspected 9 valves. The district owns 289 valves in all, and repair kits cost 18 dollars each. Show your work for every item.

  1. Write an expression for the total number of valves inspected after d working days, counting the 9 finished before the week began.
  2. Evaluate your expression from question 1 for d = 5.
  3. Evaluate your expression from question 1 for d = 12.
  4. The district owns 289 valves, and 9 were inspected before the week began. Solve the equation 9 + n = 289 to find how many valves are still waiting for an inspection.
  5. The crew inspects 14 valves a day and has 280 valves left. Solve the equation 14d = 280 to find how many working days that takes.
  6. Repair kits cost 18 dollars each. Write an expression for the cost, in dollars, of k kits.
  7. Evaluate the expression 18k for k = 12.
  8. The district spent 540 dollars on repair kits at 18 dollars each. Solve the equation 18k = 540 to find how many kits it bought.

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: 14d + 9.
    The crew adds 14 valves for every day worked, so the daily work is 14 times d, and the 9 already finished are added once.
  2. Answer: 79 valves.
    Substitute d = 5 to get 14 times 5 plus 9, which is 70 plus 9, or 79.
  3. Answer: 177 valves.
    Substitute d = 12 to get 14 times 12 plus 9, which is 168 plus 9, or 177.
  4. Answer: 280 valves.
    Subtract 9 from both sides of 9 + n = 289 to get n = 280.
  5. Answer: 20 days.
    Divide both sides of 14d = 280 by 14 to get d = 20.
  6. Answer: 18k.
    Each kit costs 18 dollars, so k kits cost 18 times k. The variable is multiplied, not added.
  7. Answer: $216.
    Substitute k = 12 to get 18 times 12, which is 216 dollars.
  8. Answer: 30 kits.
    Divide both sides of 18k = 540 by 18 to get k = 30.

Differentiation

Support

Give these students the Pellmore table from the hook already filled in for one, two, and three days, so question 1 is describing a pattern they can see rather than inventing one. Provide a substitution frame that reads "14 times ___ plus 9" for questions 2 and 3, and allow a calculator throughout.

Extension

Ask these students to write an expression for the fictional Pellmore district's total weekly cost when the crew is paid a flat 320 dollars a day and uses 2 repair kits a day at 18 dollars each, then evaluate it for a five-day week. They should be able to explain which part of their expression changes when the crew works an extra day.

ELL supports

Pre-teach valve, inspect, and kit with a photograph or a simple sketch of a buried valve and the cover that sits over it in the street. Provide a word bank of the four operation phrases used in the handout, and give this sentence frame for question 1: "The crew does ___ valves each day, so after d days they have done ___, plus the ___ already finished."

Extensions

Invite a public works employee to describe how the local system tracks valve and hydrant maintenance, or bring in a printed maintenance log if a visit is not possible. Students can rewrite the Pellmore expression using the real crew's daily rate, which shows them that the same algebraic structure fits any crew size or schedule.