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.