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## Memorization of Multiplication Tables

• Because calculator use is widespread, even many bright students have failed to memorize their multiplication tables. Unfortunately, this knowledge is essential if student are to succeed in algebra. During a pre-algebra course, students must not only learn that 8 x 2 = 16. They must also know the different ways of obtaining a product of 16 such as by multiplying 1 x 16, 2 x 8, and 4 x 4. After negative numbers are introduced, they must also realize that a product of 16 may be obtained by multiplying (- 1) x (- 16) or (- 2) x (- 8) or (- 4) x (- 4). Such knowledge is essential for students to be able to solve quadratic equations or perform arithmetic operations on rational expressions.

## Add, Subtract, Multiply and Divide Fractions

• One of the most basic concepts in Algebra 1, that of slope, is a fraction produced by dividing how fast a line goes up by how fast it goes across. This concept is often expressed as rise/run. Algebra also requires students to add, subtract, multiply and divide rational expressions, which are fractions with variables in them. Students cannot master these techniques without the ability to perform arithmetic operations on ordinary fractions.

## Positive and Negative Numbers

• Learning how to work with negative numbers requires memorizing a set of rules and procedures. The rules are not difficult but they do require practice. Students must learn, for example, that the sum of two positive numbers is always positive while the sum of two negative numbers is always negative. The difficulty arises when students attempt to add a positive and a negative number. In this case, students are confused to learn that they must "subtract" to obtain the correct answer to an addition problem. Students often find it helpful to work with checkers of different colors and to be told that checkers of opposite colors cancel each other out. For example, 5 + (- 3) could be represented as five black checkers and three ed checkers. Each red checker cancels a black checker. Only two black checkers will be left. Arithmetically, we can express this as subtraction.

Students also find the idea of subtracting negative numbers to be problematic. In this case, students find it helpful to be told that subtraction is the same thing as addition of the opposite. If you are using checkers as a visual aid, replace the color of checkers being subtracted with checkers of the opposite color and tell the student to add. If the checkers are all of the same color, they are added. If they are different colors, use cancellation.

These hands-on activities give students a visual, intuitive sense of what is happening and allow them to memorize the rules with more ease.

## Mastery of Graphing Skills

• The most important subject in Algebra 1 is lines: how to find equations for them and how to graph them. A student with substandard graphing skills will struggle throughout the entire course. At a minimum, pre-algebra students should know how to plot a point when given an ordered pair representing the position of that point on a graph. They should also be able to reverse the process and find the ordered pair of a point when given its graph.

Students with limited spatial skills may find this skill difficult and will benefit from extra practice. It is often helpful to help the students read the kinds of maps laid out in grids with letters and numbers used to locate places on the maps. Give them a map and ask them what city or cities are located at the intersection of D and 4, for example. Reverse the process and ask what coordinates they would use to help a friend locate their hometown on the map.

## Basic Equation Solving

• Too often, teachers spend large amounts of time teaching the mechanics of equation solving but fail to emphasize the ultimate reason why students are performing all of these algebraic manipulations. Solving algebraic equations is the process of using what is known to find what is unknown.

A teacher or parent can illustrate this by using a concrete problem. Suppose, she says, that she spent \$10 at the store and bought four items. She remembers that three of the items cost \$1, \$3 and \$4 dollars but doesn't remember what the fourth item costs. How could she find that answer? Most students should be able to tell her to subtract the costs of the items she knows from 10 to find the price of the unknown object. This simple exercise illustrates the heart of algebraic equation solving. It is a simple matter from that point to call the unknown x and write the equation x + 1 + 3 + 4 = 10 and to show why, algebraically, that the cost of the unknown item is \$2.

These simple, concrete problems make mathematics less intimidating because they teach students that they already have the reasoning ability to solve problems. They simply need to learn how to apply the kinds of thinking they already do automatically in "real life" to algebra equations.