## Example 2

### Midpoint on a Number Line (Formula Development)

The following problem should be given in advance of the midpoint formula so that students can invent the formula. It becomes less abstract, contextually richer, and more accessible if students work in small groups using a physical number line (e.g., a meter stick or a yard stick) in the solution process. Freudenthal (1991, pp. 36-37) discusses typical student behaviors, ways to guide students' reinvention of the formula, and possible extensions for this problem type.

Problem. Locate each of the following midpoints:

a. Locate the midpoint between 18 cm and 76 cm on a meter stick. Describe the mathematics used to help calculate the midpoint.

b. Locate the midpoint between 15 cm and 18 cm on a meter stick. Describe the mathematics used to help calculate the midpoint.

c. Locate the midpoint between 5 in. and 24 3/4 in. on a yard stick. Describe the mathematics used to help calculate the midpoint.

d. Locate the midpoint between the numbers x and y. Describe the procedure used to determine the midpoint expression.

Extension. This problem can be extended to having students develop an algebraic expression for a point that is any fractional distance from x to y.