Writing Original Problems

Students use higher-order thinking skills when they write original problems similar to those they have done in class. Have students proof and solve each other's problems before publishing them on the computer, along with the author's name and the date.


These problems may be used as do-nows, homework assignments, or placed in the math center. Teachers who regularly include student writing find that students are highly motivated to solve problems created by their peers AND teachers develop a larger bank of appropriate word problems over time.


An easy way to start is to give students an open-ended problem to solve.  Then challenge students to write their own version of the  problem to share with their classmates.

Problem Solving: Stoplight

The Stoplights problem challenges students to figure out how many different stoplights they can create with 3 different color linking cubes if each stoplight must have one cube of each color. 


Use linking cubes so that students can assemble all possible stoplights. You may choose to have students use any three different colors so that each student pair has enough linking cubes to assemble all possible combinations using one of each color to create the stoplight.


In these kinds of organized counting problems, it is important to question students about how they know they have found all of the possible combinations.  A first grader once responded that he knew he had them all because each color gets to be on top two times.  What a great insight!


Download the Stoplights problem student handout.

Seussical Combinations

The book states that not one of them is like another. These original Mathwire problems challenge students to use combinations to figure out how many unique fish could be created.

• Challenge younger students to find all of the different combinations for  Seussical Fish using attributes from the story. Making an orderly list is an effective strategy for solving this problem. Other students might opt to draw the solution.
• Older students will enjoy the challenge of the   Fishy Combinations problem which involves many more options. The model solution uses the multiplication principle of counting to easily solve the problem. Students may also use a tree diagram or list all of the possible red fish, then extend this list to the blue and black fish combinations.

Domino Problem Solving

Students will have to really think about dominoes as they solve Garage Sale Dominoes.  A sample solution using an orderly list is included in the pdf file.


After students have solved Garage Sale Dominoes, challenge them to solve More Garage Sale Dominoes in which they must figure out how many different dominoes there are in a complete double-nine set.  A sample solution using an orderly list is included in the pdf file.

Enrichment on the Internet:

• Challenge students to solve Math Forum's Domino Activities.
• Amaze students with your psychic abilities using the Domino Trick.
• Challenge students to solve the Domino Maze from Games magazine.
• Check out examples of Domino Artwork.

Problem Solving: Valentine's Day Treat

Valentine's Day Treat challenges students to list all of the possible sundaes that can be made if students choose one ice cream and one topping from the cafeteria list. Note that students must choose a topping for this problem, otherwise they are simply buying ice cream which doesn't count as a sundae.


This problem may be solved using various methods and is accessible to students of all levels.  Some students may simply list the various combinations.  Other students might elect to use a probability tree.  Others might use a more developed concept of combinations in counting and naming the various combinations.  All are valid methods of solving this problem and reflect students' current understanding of the concept of combinations.


Download  Valentine's Day Treat which includes the student handout and a sample solution.

Gingerbread Man Problem Solving

Students are asked to use words, pictures and numbers to solve these gingerbread problems:

Gingerbread Bake Sale challenges students to figure out how many different gingerbread men Ruby can make for the holiday bake sale.


Gingerbread Man Chains requires students to calculate how many gingerbread men will be needed to create chains across the bulletin board.  Students also need to figure out how many sheets of brown construction paper will be needed for the project.


Creative Extension:  Ask students to write original problems that involve the gingerbread man.  Encourage them to think of measurement and patterns as well as numbers and operations as they create their word problems.  Provide time for students to type in, print out and decorate their problems.  Assemble them into a class Gingerbread Man booklet that may be shared with other classes as well.

Turkeys and Cows

In honor of Thanksgiving, here is a turkey and cow problem, an adaptation of the familiar chicken and cows problems.  In these problems, students are given the number of heads and the number of legs and must figure out how many chickens and cows there are.  Central to the solution of these types of problems is the fact that chickens have two legs and cows have four legs.

The following method is a simple pictorial way to solve these types of problems.  Very young first and second graders have used this method to successfully tackle these problems, itching for more challenges.

PROBLEM:  Farmer Brown counted the turkeys and cows in his barnyard.

• There were 5 heads.
• There were 14 legs.

How many turkeys and how many cows were in Farmer Brown's barnyard?

First, the students concentrate on the number of heads.  In this case there were 5 heads, so the students draw 5 circles to represent the heads.

Second, the students give each head 2 legs to make them turkeys.

Now the student checks the number of heads and legs he has drawn.  The student has 5 heads (correct) and 10 legs (not enough).  Farmer Brown saw 14 legs, so the student needs to add 4 more legs.  He must add 2 legs at a time because a cow has 4 legs.

Again, the student counts and checks his drawing.  He has 5 heads (correct) and 14 legs (correct), so he now labels each drawing either T (turkey) or C (cow).

It is now easy to see that the solution to this problem is that Farmer Brown had 3 turkeys and 2 cows in his barnyard.

Teachers should model this method with students.  It might be very helpful to model this solution on chart paper or chalkboard, so that students may refer back to this model as they solve problems on their own.  Next, let pairs of students work together to solve another problem.  After that, students may work individually or as pairs to solve additional problems involving turkeys and cows.

Download Mathwire's Turkeys and Cows problem set.

CHALLENGE:  Students may be challenged to create their own turkey and cow problems for their classmates to solve.   Each problem should also have a solution attached for easy checking.

Fall Problem Solving

These open-ended assessments require students to apply mathematical concepts and skills to solve problems and explain their thinking using words, pictures and/or numbers.


 Candy Corn presents a triangular numbers problem using a candy corn pattern. Younger students might use candy corn to model the problem. A sample solution shows how older students might use an input-output table to model the pattern and find the solution without the use of manipulatives.


 Annual Fall Parade challenges students to use the triangular pattern to figure out how many students are in the fourth grade. Given the number of full rows, students must apply the pattern and use effective recording (picture, table, etc.) to explain their reasoning.


 Pumpkin Picking is a pattern problem that can be solved using a picture or an input/output table. [contributed by Shannon Collier, Joseph C. Caruso School, Keansburg, NJ]


 Trick or Treat requires students to create a table of values and identify the pattern or rule to solve the problem.


 Grade 2 Halloween Word Problems were written by Tammie Holcombe and Karen Zeigler, second grade teachers at Port Monmouth Road School in Keansburg, NJ.


Younger students will enjoy analyzing and completing  Fall Patterns. They should then classify the pattern and explain their reasoning as part of the class discussion. It is possible that students will see different patterns in some of the items so their explanation and justification are very important mathematical discourse.

Back-to-School Problem Solving

September is a great time for data collection activities as students are naturally curious about their new classmates. Ask questions that require students to analyze data and support their conclusions.

Challenge students to solve some of these problems to practice collecting, organizing and analyzing data in real-life applications. Require students to explain their reasoning and solution to develop mathematical communication skills.

See all Back-to-School Word Problems.

Here's a sample 3rd grade problem that targets student understanding of probability:

Third graders cut apart each of the letters in THIRD GRADERS and place
the squares in a bag. Kevin pulls out one square and looks at the letter.

• What is the probability that the letter he pulls out will be the letter, D?  Explain how you know.
• What is the probability that the letter he pulls out will be S?  Explain.
• On Friday, his teacher says that if anyone can correctly predict which tile he or she will pull out the bag of THIRD GRADERS squares, the class will not have any homework over the weekend.  She asks Kevin's table to predict for the class.  They are nervous because the class is counting on them to say the right letter.
•  Which letter should Kevin’s team predict that they will pull from the bag?
•  Explain how you know that the students should predict this letter.

Note:  It is very important that students explain how they know their answers are correct.  They may use pictures, words and numbers to explain their thinking.

Data Analysis: TV Survey

TV Survey is an open-ended problem that was designed to introduce students to this real-life application of sampling. Because TV is a part of students' daily life, this scenario is accessible to all students. And, because students often exhibit a keen sense of "fairness," they are sometimes disconcerted by the notion that a small group of people get to decide what all of us get to watch on TV.

TV Survey was also designed to address different Bloom levels so that students have to use higher-order thinking skills to analyze, synthesize and evaluate information about the sampling method.

• Download the TV Survey open-ended assessment.

See more Mathwire Sampling Activities.

Problem Solving: Pattern Blocks

Pattern blocks are a familiar manipulative in most elementary classrooms.  Students learn a lot of geometry from informally interacting with pattern blocks.  These problem-solving activities were specifically designed to target important mathematical concepts using these available manipulatives.

• In Pattern Block Symmetry students must create a pattern block design to satisy the given conditions involving line symmetry and the number of different pattern block pieces used.
• Pattern Block Design (Grade 3) requires students to create a design to meet specified criteria.
• Pattern Block Design (Grade 4) requires students to create a design to meet specified criteria.
• Hexagon Dragons challenges students to use an input-output table to record dragon growth, then analyze the pattern and write a rule for dragon growth.
• In Fir Tree, students must use an input-output table to record and analyze data, then write a rule for this growing pattern.
• Symmetric Snowflake challenges students to use pattern blocks to complete the snowflake design so that it has line symmetry.

Problem Solving: Fractions 2

Students sometimes find the set model of fractions harder to understand and use.  These problems were designed to help students concretely work with the set model to solve problems.


Let's look at a sample problem:

Mrs. Puglisi’s third graders visited the park nature center to see the tadpoles and 

frogs.  They counted 20 animals in the tank. 

•  3/4 of the animals had already changed to frogs.  

•  The rest  were still tadpoles. 

•  How many frogs were in the tank? 

•  How many tadpoles were in the tank? 

•  Explain how you know your answers are correct.

Concretely Solving the Problem:  

• Students should begin by counting out 20 manipulative objects they will use to solve the problem.
• Next, students look at the fraction 3/4.  The denominator tells us that the animals were divided into 4 groups.   NOTE:  It is often helpful to use a piece of paper that is folded into quarters so that students may sort the manipulatives evenly into each of the 4 spaces. 
• Students deal out the manipulatives as if they were cards, placing one in each of the 4 spaces and repeating until all of the manipulatives have been distributed.  
• Students check to make sure that each quarter has the same number of animals.
• Now students look back at the fraction 3/4.  The denominator of 4 told students how many groups.  The numerator of 3 in this case tells students how many of the groups had already changed to frogs.  
• Students count the total number of frogs in 3 groups.
• The other group are still tadpoles.  Students count how many animals are still tadpoles.
• Finally, students may use words, pictures and numbers to record their thinking and explain how they knew their answer was correct.

Extension:  After students have mastered this concrete approach to solving fraction problems using the set model, they may move to a paper-and-pencil semi-concrete solution method.

• The student draws a square on his paper. 
• He checks the denominator of the fraction to see how many groups are needed.  The student divides the square into that many identical parts.
• The student then makes dots or tally marks in each section, as above, dividing the total number of objects evenly among the groups.
• The student checks the numerator of the fraction to see how many groups he must label for his answer.

• NOTE:  The beauty of this method is that students may easily use this approach in standardized testing situations.  It's completely portable!
• Students may also simply draw four circles to represent the 4 groups and proceed as above.

Practice Problems:

• Tadpoles and Frogs requires students to use fractions to figure out how many tadpoles and frogs there were in the tank.
• Animal Shelter requires students to use fractions to figure out how many cats and dogs were available for adoption at the animal shelter.
• Ask students to write original problems, peer solve and edit, then add to your classroom collection for future use.  

Problem Solving: Fractions

Problem solving activities help students develop conceptual understanding of fractions as they use appropriate fraction models to think about and solve the different tasks.  These Mathwire open-ended assessments were designed to measure students' conceptual understanding of fractions.


Pattern Block Fraction Design requires students to fill a shape with pattern blocks to create a design that meets certain requirements.   Students must also write a fraction that describes the part of the total design represented by each different color pattern block.


Fraction Game simulates a Fraction War game but students must draw a representation of each fraction and explain who won, based on the drawings.  This assessment was designed for students in an Everyday Math program.  Students may draw any representation that helps them decide which fraction is larger and explain their thinking.


Mrs. Meatball challenges students to solve pizza fraction problems pictorially.  The problems are listed on one page, but students should use lots of room to draw pictures, label and explain how they solved each problem pictorially.  NOTE:   This task was originally designed for a teacher in-service class where participants were not allowed to use any previous fraction knowledge to solve the problems.  They were instructed to use only words, pictures and numbers to solve the problems and explain their thinking.  They were not allowed to use any fraction algorithms, etc.

Problem Solving: Multiplication Arrays

Arrays are one graphical representation of multiplication that students must understand.  These problems were created to provide practice in creating arrays.

• Ants Marching assesses student understanding of the concept of multiplication as arrays.
• Third Grade Parade also assesses student understanding of the concept of multiplication as arrays.
• Cheerleader Competition was designed to assess student understanding of multiplication as an array.

Problem Solving: Number Patterns

Mathematics is full of patterns and mathematicians analyze data, searching for patterns that will allow them to draw conclusions and make hypotheses.

These problems were specifically designed to reinforce common mathematical patterns.  Some patterns are easily identified.  Others are best solved using an input/output table to collect and analyze mathematical patterns.

Grades 1-2:

• Bay Street requires students to analyze the pattern of house numbers on Bay Street. Students write down the pattern they see and use this number pattern to write in the missing house numbers. 
• King Street requires students to analyze the pattern of house numbers on King Street. Students write down the pattern they see and use this number pattern to write in the missing house numbers.  

Grades 3-4:

• Baseball Season is a pattern problem that can be easily solved using a table of values. 
• Anthony's Allowance provides additional pattern practice.  

It is important that students identify the patterns they see, using their own words to describe these patterns.  Supply mathematical terms as you summarize their findings, adding terms such as odd, even, Fibonacci sequence, repeating pattern, growing pattern, etc. Add these terms to the math word wall so that students may consult this list in future problem solving.

Problem Solving: Fractions

Students are challenged to solve these problems by drawing pictures rather than using traditional fraction algorithms.  This is an extremely effective strategy to develop student conceptual understanding of fractions, fractional parts and fraction operations.

• Tadpoles and Frogs requires students to use fractions to figure out how many tadpoles and frogs there were in the tank.
• Animal Shelter requires students to use fractions to figure out how many cats and dogs were available for adoption at the animal shelter.
• Fraction Game simulates a Fraction War game but students must draw a representation of each fraction and explain who won, based on the drawings.
• High Number Game requires that students apply their knowledge of fractions, decimals and percents to decide whose number card is highest.
• Pattern Block Fraction Design requires students to use pattern blocks to fill the shape so that it has a line of symmetry. They must then write a fraction for each color pattern block used in the design.
• The Adventures of Mrs. Meatball problem set challenges students to draw pictures to solve each problem and prove their answers. 
• Monkey Business challenges students to work backwards to solve this fraction division problem. 
• Bake Sale challenges students to work backwards to solve another fraction division problem involving disappearing cookies.

More Organized Counting

Houses in a Row

This problem challenges students to figure out how many different color combinations are possible for a row of houses, if no two houses can be painted exactly alike. Use large color squares and triangles to model this problem for students as an introduction to combinations.

Download Houses in a Row student worksheet and solution.

NOTE:  Have students explain their thinking and how they organized their information.  It is also important to ask students how they knew that they had found all of the possible combinations.

Problem Solving: Organized Counting

Stoplights challenges students to figure out how many different stoplights they can create with 3 different color linking cubes if each stoplight must have one cube of each color. Use linking cubes so that students can assemble all possible stoplights. 

NOTE:  You may choose to have students use any three different colors so that each student pair has enough linking cubes to assemble all possible combinations using one of each color to create the stoplight. 

Discussion:  In these types of problems, it is important that students know when they have found all of the possible combinations.  Be sure to ask students to explain how they know they have all of the combinations.  How did they know when they could stop looking?

Try to let students work on the problem without organizing their thinking.  Let pairs of students work together.  Ask students to explain their method and their thinking.  Peer explanations are so powerful in problem solving.  

You may be surprised at how students organize the challenge.  I remember a first grader telling me that he knew when he had them all because each color got to be on top twice.  He simply switched the bottom two cubes, then let another color be on top, etc.  To him, it was like being in the front of the line.  It was such a succinct explanation from a young learner.  Amazing!

Download Stoplights which includes the student handout and a brief explanation.  Be sure to make extra copies available to students,  so they aren't led into believing they must find 6, which is the correct answer. 

Suggestion:  To keep this task truly open-ended, hand out blank paper and have students draw their answers.  This method does not prompt students and also mimics state testing.

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Problem Solving: Working Backwards

Bake Sale challenges students to work backwards to figure out how many cookies the girls actually baked for the school bake sale.  They left the cookies on cooling racks, but each got up at different times throughout the night, ate some cookies, then packaged some others to take home.  Lo and behold there are only 10 cookies left in the morning.  Each girl admits eating a few during the night but there are a lot of cookies missing.  Students must work backwards to figure out how many cookies they originally baked, then decide how many cookies each girl got and whether or not the cookies were fairly split.

Download the Bake Sale problem with possible solution.

Problem Solving: Remainders in Division

Top of the Morning to You!

This problem challenges students to identify the 100th and 1000th letter that will be printed on the electronic signboard. To solve this, students will grapple with a division problem, figuring out how many letters are in the message and how many times the message will repeat to identify the 100th letter.  This is a good introductory problem because there are no remainders, making this a straightforward division problem. Nonetheless, students must explain their thinking and the process they used to solve the problem, so it's effective practice for state testing.

•  Download the  Top of the Morning problem set with possible solution.