### Purpose

To develop thinking skills. To learn to see numerical relationships and how to solve complex problems by manipulating objects and solving equations.

### Context

Students' beliefs and understanding of mathematical inquiry remain relatively unclear throughout their academic lives. Some of the misconceptions that students carry are: there is only one correct way to solve any math problem; mathematics problems have only one correct answer; mathematics is done by individuals in isolation; mathematical problems can be solved quickly or not at all; and mathematical problems and their solutions do not have to make sense. (*Benchmarks for Science Literacy*, p. 334.)

Because of this, students limit their mathematical behavior. It is important, therefore, that students be exposed to a wide array of concrete representations to help develop a foundation for the higher abstract ideas associated with mathematical inquiry. In this lesson, students will work with manipulatives (beans) to create and solve problems, some of which have more than one correct answer.

### Motivation

To begin the bean exploration, have students work in pairs to sort several beans into different piles according to a rule they make up.

Ask each group the following questions:

- How many piles of beans did you make?
- How would you describe each of the piles you have made?
- What was your rule?

Have students group the beans according to a different rule and ask the same questions. They can repeat this several times, creating as many different rules as possible to sort the beans.

There are many different ways to sort beans and other items. By challenging students to sort the same items using different categories, the students develop thinking skills by looking at the same problem in different ways. Such thinking skills are necessary for students to understand and analyze mathematical situations using algebraic symbols and solving equations, as they’re required to do in the following activity.

If your students need more practice with sorting before continuing with this lesson, they could do the Flood! game on the Between the Lions (PBS Kids) website. In this activity, books float by in groups of five, but each shelf only holds three books. To fill the shelves, students need to choose three books whose titles share a common theme.

### Development

Distribute the Bag the Beans student sheet and have students work in pairs to pack eight bean bags following the rules outlined on the student sheet. Have them record the number of beans for each bag on the student sheet.

While students are working on this activity, ask questions such as the following:

- Can you set up a ratio or an equation to help you determine the answer?
- Why do some problems have one correct answer while other problems may have more than one correct answer?

This activity demonstrates that when students solve problems using manipulatives, the solution almost reveals itself. In addition, students develop confidence in their answers even when they differ from those of their neighbors.

### Assessment

Assess student understanding by checking their answers on the Bag the Beans student sheet. (See the Bag the Beans teacher sheet for answers.)

In addition, have each student make up at least one new rule for filling the bags, and have them give their rules to others to solve. In order to address the benchmark idea, “Results should always be judged by whether they make sense and are useful,” it will be important for students to reflect on and evaluate their rules.

### Extensions

You could use beans as counters in the classroom. For example, you could plan your next class party by solving problems with beans, such as how many bottles of juice will be needed if you use one bottle for every four people.

For another Science NetLinks lesson that allows students to explore with manipulatives, see What’s in a Shape?