Math Forum - Problems Library - Middle School, Arithmetic/Early Mathematics

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Middle School: Arithmetic/Early Mathematics: Number and Operations

At the middle-school level students study the meanings of operations and how operations relate to each other, and learn to compute fluently and make reasonable estimates. They deepen their understandings of fractions, decimals, and percents, and learn to understand ratio and proportion using methods such as scaling and finding equivalent ratios. Students work with large numbers, exponents, and scientific notation, and practice using factors, multiples, prime factorization, and relatively prime numbers.

Many of the Middle School Problems of the Week allow students to exercise these skills, and address the NCTM Number and Operations Standard for Grades 6-8. Below are some of them.

For background information elsewhere on our site, explore the Middle School area of the Ask Dr. Math archives. To find relevant sites on the Web, browse and search Arithmetic/Early Mathematics in our Internet Mathematics Library; to find middle-school sites, go to the bottom of the page, set the searcher for middle school (6-8), and press the Search button.


Access to these problems requires a Membership.

Accuracy Please - Judy Ann Brown
Pre-Algebra, difficulty level 3. Considering precision, accuracy, and appropriateness in finding the volume of a cone. ... more>>

Back-to-School Shopping - Judy Ann Brown
Math Fundamentals, difficulty level 2. Mrs. Bender noticed that the total of the items her children purchased was $155. Can you determine the cost of one pair of jeans? ... more>>

Baseball Attendance Figures - Judy Ann Brown
Pre-Algebra, difficulty level 3. Find the difference between the greatest possible number of people at the stadium and the least possible number of paid admissions to the game. ... more>>

The Big Picture! - Judy Ann Brown
Pre-Algebra, difficulty level 3. A picture is enlarged to four times its area. ... more>>

Building Fences - Judy Ann Brown
Pre-Algebra, difficulty level 3. Plan a garden so as to maximize the area surrounded by a 36-yard fence. ... more>>

Canadian Soccer - Judy Ann Brown
Pre-Algebra, difficulty level 2. Find the distance a player runs to complete three "Double V" laps. ... more>>

Can You Find My Numbers? - Roya Salehi
Pre-Algebra, difficulty level 3. I have eight numbers in my list. Use the clues to find my numbers. ... more>>

Car Rental Quandary - Judy Ann Brown
Pre-Algebra, difficulty level 3. Compare the car rental deals using the given information. ... more>>

Coin Count - Roya Salehi and Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the given information to determine the number of nickels and dimes in Lisa's bank. ... more>>

The Complex Apartment Complex - Judy Ann Brown
Pre-Algebra, difficulty level 2. Find the number of apartments in a new building if each floor has twice as many apartments as the floor above. ... more>>

Distance Driving - Judy Ann Brown
Pre-Algebra, difficulty level 3. Write an expression that will help you determine how long it will take the car to travel any given number of kilometers. ... more>>

Downhill All the Way - Judy Ann Brown
Pre-Algebra, difficulty level 2. Starting at 10,000 feet above sea level, descending a 6% grade followed by a 4% grade will put a car how many feet above sea level? ... more>>

Do you listen to WMTH or WALG? - Judy Ann Brown
Pre-Algebra, difficulty level 4. Each week listeners switch radio stations. Predict the long-term distribution of listeners for each station. ... more>>

Duke is Missing - Judy Ann Brown
Pre-Algebra, difficulty level 3. Duke's head is 6 inches long, his tail is as long as his head plus half the length of his body, and his body is as long as the head and tail combined. How long is Duke? ... more>>

Even Number Puzzle - Judy Ann Brown
Pre-Algebra, difficulty level 3. How many even four-digit whole numbers exist such that the thousands digit is odd, and all four digits are different? ... more>>

Flipping Over Pennies - Judy Ann Brown
Pre-Algebra, difficulty level 3. How many equally likely outcomes are there when flipping 10 coins? ... more>>

Frequent Flier Figuring - Judy Ann Brown
Math Fundamentals, difficulty level 2. Use Web resources to help plan the shortest route from Allentown, Pennsylvania, to New Orleans, Louisiana. ... more>>

The Graduating Class - Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the information about students with different grades to determine the number of students in Mrs. Smith's class. ... more>>

Guess My Number If You Can! - Judy Ann Brown
Pre-Algebra, difficulty level 4. Use division clues to reveal the secret number. ... more>>

Happy New Year! - Judy Ann Brown
Pre-Algebra, difficulty level 4. A puzzle for the new year: how is the grid designed? ... more>>

He Ate It All! - Mr. Marthinsen's Geometry Class
Pre-Algebra, difficulty level 3. The professor ate an entire pizza in 28 minutes. Figure out the size of the pizza. ... more>>

A Holiday Gift for Grandmother Brown - Judy Ann Brown
Pre-Algebra, difficulty level 3. Determine the cost of the dishwasher and the contribution from each of five children. ... more>>

Homerun Percentages - Judy Ann Brown
Pre-Algebra, difficulty level 3. Baseball box scores give us a chance to practice percent skills. ... more>>

How Old is Ms. Nelson? - Judy Ann Brown
Pre-Algebra, difficulty level 2. Ms. Nelson claims that she is three times as old as her little sister. However, six years ago she was five times as old as her sister was. How old is Ms. Nelson? ... more>>

How Wide Is My Garden? - Judy Ann Brown
Pre-Algebra, difficulty level 3. A scale drawing is used to determine the width and area of a garden. ... more>>

I Get A Kick Out Of Soccer! - Judy Ann Brown
Math Fundamentals, difficulty level 3. How many laps of the soccer field would a team member need to complete to run more than a mile? ... more>>

Inching Along Down the Football Field - Terrel Trotter
Math Fundamentals, difficulty level 3. Fred put his football down on a goal line of the 100-yard field. He then moved it forward half the field distance, over and over again. When did the distance first become less than one inch? ... more>>

Lost in Des Moines - Judy Ann Brown
Pre-Algebra, difficulty level 2. Calculate the speed of a car using given information. ... more>>

Luscious Lemonade - Judy Ann Brown
Math Fundamentals, difficulty level 3. The label on the lemon concentrate says that I need to mix 2 cups of concentrate with 3 cups of water. How much lemon concentrate should I buy to make 25 gallons of lemonade? ... more>>

The Magic of Number 6174 - Suzanne Alejandre
Pre-Algebra, difficulty level 4. Have fun exploring this number magic. ... more>>

Making Money - Alice Miller
Pre-Algebra, difficulty level 3. How much money would you have by December 31, 1999, if you put 10 cents in the special "math" bank on January 1, 1999? ... more>>

Martian Holiday - Judy Ann Brown
Middle School, difficulty level 2. On January 1, 2190 a space ship leaves Earth. On what Earth date will the space ship return home? ... more>>

Math Club Lollipop Profits - Judy Ann Brown
Math Fundamentals, difficulty level 3. The Math Club plans to sell lollipops. Help them to calculate their profit. ... more>>

The Most Appealing Nose Contest - Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the given information to determine the amount of money raised for the Humane Society by the Student Council. ... more>>

Mr. Carmel's Party Plan - Judy Ann Brown
Pre-Algebra, difficulty level 2. Help Mr. Carmel's class earn an ice cream party by counting the number of possible sundaes. ... more>>

The Name Game - Judy Ann Brown
Pre-Algebra, difficulty level 3. The first student says his/her own name; the second student repeats the first student's name and adds his/her own name... How many names are said if the numbers of students in my math classes on the first day are 27, 28, 32, 31, and 30? ... more>>

The Nutty Squirrel - Alice Miller
Pre-Algebra, difficulty level 3. How high could a human jump if she could jump with the power of a squirrel? ... more>>

Open and Shut Case - Judy Ann Brown
Pre-Algebra, difficulty level 2. Find the locker combination using mathematical clues. ... more>>

Palindrome Day - Judy Ann Brown
Pre-Algebra, difficulty level 2. How many Palindrome Days are there in a year? ... more>>

Peanut's Back Yard - Jonathan Bauer
Pre-Algebra, difficulty level 3. Peanut, the overexcited dog, makes 40 laps around a stake and travels a total distance of 1 mile. If his bowl of water is 20 feet away from the stake, can he get a drink? ... more>>

Pi Day - Let's Eat Cake! - Judy Ann Brown
Pre-Algebra, difficulty level 3. In honor of Pi Day we will be comparing the volumes of slices taken from round and rectangular cakes. ... more>>

Pizza Pie - Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the given information to decide how many slices of the small pizza would be equal to or greater than 3 slices from the large pizza. ... more>>

Pouring the Punch - Neil Bress
Pre-Algebra, difficulty level 3. The Refreshments Committee needs to pour exactly four liters of eggnog using only three-liter and five-liter measures. ... more>>

Power Outage - Jackson Geometry Class
Pre-Algebra, difficulty level 3. The power went out. Use the current reading on the clocks to determine when the power went out, and for how long. ... more>>

Price Fluidity - Judy Ann Brown
Pre-Algebra, difficulty level 2. Compare the per-gallon cost of eleven different liquids, and put them in order from least to most expensive. ... more>>

Rounding Mountains - Judy Ann Brown
Math Fundamentals, difficulty level 2. Use estimated figures to determine the maximum and minimum differences between the heights of two mountains. ... more>>

Runsums, Part 1 - Ron Knott
Pre-Algebra, difficulty level 3. By spotting the pattern, tell me the smallest number that is a sum of a run of one hundred consecutive numbers. ... more>>

Runsums, Part 2 - Ron Knott
Pre-Algebra, difficulty level 3. Explore number patterns and report on what you have learned. ... more>>

Santa's Little Helpers - Judy Ann Brown
Pre-Algebra, difficulty level 3. Help Santa to solve the puzzle created by the clever elves. ... more>>

Seating Arrangements - Judy Ann Brown
Pre-Algebra, difficulty level 3. Four new students arrive in Mrs. Brown's homeroom. After a few minutes she turns to them and says, "There are 1680 ways that you can sit in the empty seats." How many empty seats are there in the room? ... more>>

Seating Dilemma - Judy Ann Brown
Pre-Algebra, difficulty level 2. Count the number of ways to seat people, if some are math teachers who cannot be next to each other. ... more>>

Sequence of Exercise - Judy Ann Brown
Middle School, difficulty level 2. Each day Chris and Jody jog three blocks farther than they did the day before. How many days have Chris and Jody been jogging? ... more>>

Shine On Harvest Moon - Judy Ann Brown
Pre-Algebra, difficulty level 2. Find the area of the largest circle that can be cut from a nine-inch by twelve-inch sheet of paper. ... more>>

Shopping Spree - Judy Ann Brown
Middle School, difficulty level 1. Jamie is shopping for holiday presents. How many $9.49 shirts can she buy? How much change will she have left? ... more>>

The Silver Mile - Judy Ann Brown
Pre-Algebra, difficulty level 3. Determine the total value of the coins in a silver mile. ... more>>

Simple Arithmetic - Judy Ann Brown
Middle School, difficulty level 2. Titles can be deceiving - this is a simple arithmetic problem but you need to find the numbers. ... more>>

A Simple Reversal - Judy Ann Brown
Math Fundamentals, difficulty level 1. The product of 3 and 24 is 72. The sum of 3 and 24 is 27, which is the reverse of 72. Can you find other pairs of numbers such that their product is the reverse of their sum? ... more>>

Snow Day Phone Chain - Judy Ann Brown
Pre-Algebra, difficulty level 4. Use the given information to find out how many phone calls were made. ... more>>

Spring Garden Planning - Judy Ann Brown
Pre-Algebra, difficulty level 4. Find the area as well as the length and width of my fractional garden. ... more>>

Square Dancing - Judy Ann Brown
Pre-Algebra, difficulty level 2. The sum of each couple's numbers is a perfect square. Use other information in the problem to find the number of Ashley's partner. ... more>>

Temperature Change - Judy Ann Brown
Pre-Algebra, difficulty level 2. Use the given information to find the temperature. ... more>>

Three-dimensional Television - Mr. Marthinsen's Geometry Class
Pre-Algebra, difficulty level 3. Help my friend find which TV will fit snugly into her TV nook. ... more>>

The Tower of Hanoi Puzzle - Roya Salehi
Pre-Algebra, difficulty level 4. Tower of Hanoi: What is the minimum number of moves it will take to transfer 25 disks from one post to another? ... more>>

Treasure Hunt Math - Judy Ann Brown
Pre-Algebra, difficulty level 3. Use clues to find the mystery palindrome. ... more>>

Valley Springs Snow Cream - Kelly Crisp
Pre-Algebra, difficulty level 3. Compare the volume of a sphere, cone, and cylinder using ice cream. ... more>>

Where Does Mrs. Darien Live? - Judy Ann Brown
Middle School, difficulty level 2. Use clues to help Mrs. Darien's students find her house number. ... more>>

Wilda's Swamp Water - Judy Ann Brown
Pre-Algebra, difficulty level 3. Measure 8 quibits of swamp water using only 6-quibit and 10-quibit containers. ... more>>

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