Math problem set Grade 6 Math

Pikewell Farm's Shallow Well: Dividing Fractions

A grade 6 problem set where students divide fractions and mixed numbers to plan watering from a shallow farm well that refills three quarters of a foot per hour.

Grade: Grade 6
Subject: Math
Material: Math problem set
Time: 40 min
Prep: No prep
Materials: None

Standards

6.NS.A.1 Common Core Math

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

How this material aligns: Problems 1 through 6 are word problems requiring quotients of fractions, such as 4 and one half divided by three quarters and 8 divided by two thirds.

All materials for 6.NS.A.1 →

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: Problems 7 and 8 ask what percent of the well's daily refill the family uses, which is percent-of-a-quantity reasoning in a real setting.

All materials for 6.RP.A.3 →

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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 divide a mixed number by a fraction to solve a water planning problem.
Students will explain why dividing by a fraction less than 1 gives a larger quotient.
Students will express daily water use as a percent of the well's daily refill.

Vocabulary

Key words: dug well, refill rate, quotient, mixed number

Teacher overview

A shallow dug well fills back up slowly from the soil around it, so a family that draws water faster than the well refills has to plan the day around that rate. Pikewell Farm is a fictional homestead whose well refills three quarters of a foot each hour, and every problem here comes from that single well, the garden it waters, and the farm's storage tank. The misconception this set is designed to correct is the belief that dividing always makes a number smaller; problems 1, 2, and 9 return to the same comparison so students see that dividing by three quarters produces a larger result than the number they started with.

Pikewell Farm's Shallow Well: Dividing Fractions — Grade 6 Math

Name: ____________________________ Date: ______________

Problems

Pikewell Farm draws all of its water from a shallow dug well. The well refills three quarters of a foot of water each hour. Use that rate and the numbers given in each problem.

The well is drawn down 4 and one half feet. At three quarters of a foot per hour, how many hours does it take to refill?
A watering can holds three quarters of a gallon. How many full cans can be poured from 7 and one half gallons?
A drip bucket lets out one eighth of a gallon each minute. How many minutes does it take to let out 2 and one half gallons?
Each row of the garden needs two thirds of a gallon. How many rows can 8 gallons water?
How many one sixth feet are in 1 and one half feet?
Pumping 10 gallons lowers the farm's storage tank by one sixth of a foot. How many gallons must be pumped to lower the tank 1 and one half feet?
The well refills 60 gallons in a day, and the family uses 45 gallons in a day. What percent of the daily refill does the family use?
The family cuts its use to 36 gallons a day. What percent of the 60-gallon daily refill is that?
Explain why dividing 7 and one half by three quarters gives an answer larger than 7 and one half.

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: 6 hours.
4 and one half equals 9/2, and three quarters equals 3/4. Dividing gives 9/2 times 4/3, which equals 36/6, or 6 hours.
Answer: 10 cans.
7 and one half equals 15/2. Dividing by 3/4 gives 15/2 times 4/3, which equals 60/6, or 10 full cans.
Answer: 20 minutes.
2 and one half equals 5/2. Dividing by 1/8 gives 5/2 times 8/1, which equals 40/2, or 20 minutes.
Answer: 12 rows.
8 divided by 2/3 equals 8 times 3/2, which equals 24/2, or 12 rows.
Answer: 9.
1 and one half equals 3/2. Dividing by 1/6 gives 3/2 times 6/1, which equals 18/2, or 9 groups of one sixth of a foot.
Answer: 90 gallons.
From problem 5 there are 9 groups of one sixth of a foot in 1 and one half feet. Each group takes 10 gallons, so 9 times 10 equals 90 gallons.
Answer: 75 percent.
45 divided by 60 equals 0.75, which is 75 percent of the daily refill.
Answer: 60 percent.
36 divided by 60 equals 0.60, which is 60 percent of the daily refill.
Answer: Because three quarters of a gallon is smaller than one whole gallon.
Dividing asks how many groups of the divisor fit inside the number. Since each group is less than 1, more than 7 and one half groups fit, so the quotient of 10 is larger than the 7 and one half the problem started with. Dividing by a number greater than 1 would give a smaller result.

Differentiation

Support

Provide a fraction strip diagram showing 7 and one half divided into three-quarter pieces so students can count the ten groups before computing. Rewrite each mixed number as an improper fraction alongside the problem, and let students keep a written reminder of the rule to multiply by the reciprocal.

Extension

Ask students to find how many hours the family could water at 5 gallons per hour while using no more than half of the well's 60-gallon daily refill, and to show the division that supports the answer.

ELL supports

Give a picture card set showing a well, a watering can, and a garden row, each labeled with its fraction. Provide the sentence frame "There are ___ groups of ___ in ___, so the answer is ___" for problems 1 through 5, and allow problem 9 to be answered with a drawing plus one sentence.

Extensions

Students can build a simple table showing the well's water level hour by hour across a day in which the family draws water twice, once in the morning and once in the evening. Reading the table shows why refill rate, and not just total gallons, decides when the well runs short.