# The California Frog-Jumping Contest

## Algebra

**is one of five units in the**

*The California Frog-Jumping Contest: Algebra**Contexts for Learning Mathematics’ Investigating Fractions, Decimals, and Percents*(4–6)

*The Celebrated Jumping Frog of Calaveras County*—to develop equivalence and its use in solving algebraic problems. The context of a frog jumping along a track is used to foster number line representations in which students solve for an unknown amount, which is usually the length of a frog jump. Equivalent sequences of jumps

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#### Full Description

**is one of five units in the**

*The California Frog-Jumping Contest: Algebra**Contexts for Learning Mathematics’ Investigating Fractions, Decimals, and Percents*(4–6)

*The Celebrated Jumping Frog of Calaveras County*—to develop equivalence and its use in solving algebraic problems. The context of a frog jumping along a track is used to foster number line representations in which students solve for an unknown amount, which is usually the length of a frog jump. Equivalent sequences of jumps are represented naturally on a double number line by having them start and end at the same location, with one expression shown on top of the line and the other shown underneath the line. The representation can then be used as a tool for solving the problem.

*Adding It Up*(National Research Council 2001, 264), illustrates typical difficulties students may have. Known as the reversal error, it is illustrated by work on the following problem: At a certain university, there are six times as many students as professors. Using S for the number of students and P for the number of professors, write an equation that gives the relation between the number of students and the number of professors. A majority of students, ranging from first-year algebra students to college freshmen, wrote the equation 6S=P. Apparently they used 6 as an adjective and S as a noun, following the natural language in the problem. However, they needed to multiply the number of professors by 6 to find the number of students. The correct response is 6P=S. Because learning to write algebraic expressions is so difficult, we don’t push symbolizing early in this unit. The representation of the number line is used to fix students’ attention on the distinction between the lengths of jumps and the number of jumps. Once this is set, students can begin symbolizing in problems like this in a meaningful way. The unit ends with the students constructing more formal algebraic notation as they develop methods to simplify their earlier representations.