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Algebra is sometimes referred to as generalized arithmetic because it formalizes arithmetic relationships. Its power lies in the ways it allows us to represent relationships among quantities, to describe properties of operations (such as commutative and distributive), and to describe patterns. Basic algebra rules provide for manipulating symbols, such as simplifying and translating algebraic expressions and then solving for an unknown. Reasoning algebraically is essential to the study of other domains of mathematics, including algebraic geometry, measurement, probability, and data analyses. The algebraic reasoning is also needed in the sciences, the social sciences, and the arts. Earlier, algebra instruction was reserved for older students and focused primarily on the manipulation of algebra symbols and the solutions to basic algebra problems, but today, realizing its significance, elementary algebra is introduced to younger kids with more focus on the "big ideas" of algebra and on reasoning algebraically.
There are three big ideas of algebra to tackle. Although these big ideas of abstract algebra are explained individually, they often appear in simple algebra problems in combination.
Big Idea 1: Variables
A variable can be an object, a geometric shape, or a letter that represents a number of things. Variables are used in three ways- to represent unknowns, to represent quantities that vary, and to generalize properties.
Unknowns are variables that don't vary! That is, they have fixed values. Variables with fixed values are used extensively in equations in the early grades, for example,3 + 2 = n and t + 3 = 7. Kids learn that the same letters or shapes in equations represent the same values.
Following is a fun activity for kids to demonstrate the same:
Children are shown two pictures. In one picture, there are two identical toy cars with a price tag of $2 for both toys. The second picture shows the same toy car and a toy dog with a price tag of$5 for both toys. This problem is their introduction to solving algebraic equations with two unknowns; the two equations are car + car = $2 and car + dog = $5. The unknowns are the price of the toy car and the price of the toy dog. Children will learn in grade 5 how to solve algebra word problems that involve first writing algebraic expressions to represent the mathematical relationships and then solving algebraic expressions for the values of the variables.
In some types of algebra equations, the variables are not fixed. The variables can assume different values. Children can be introduced to the concept of variables as 'representing varying quantities as early as the preschool level and then how to solve algebra problems in later years.
One of the best math aids to illustrate this is the following math activity:
Children are presented with two cups (call them a and b) and 10 chips. Then they are asked to find all different ways in which they can place the chips into the cups. This problem can be represented algebraically by writing a + b = 10. A and b vary because they can have lots of different values. For example, if a = 0, then b = 10; if a = 1, then b = 9; if a = 2, then b = 8; and so on. Not only do children learn that there are 11 different pairs of values that solve the equation, but they also learn that once they choose a value for a, the value of b is determined. Similarly, if they select a value for b, then the value of a is determined.
Variables that take on different values are found in all algebra formulas. Consider the formula for the perimeter of a square. In the formula p = 4s, where s is the side of the square. Now the value of s may be different for different squares.
Children are introduced to special relationships that exist among sets of numbers under specific operations. Children are expected to learn these facts so that the computations are simplified.
If students learn that 3 + 4 = 7, then they also know that 4 + 3 = 7, because for the addition of whole numbers, when you change the order of the addends, the sum remains the same. This rule, the commutative property of addition of whole numbers, holds true for all pairs of whole numbers, so it can be represented using variables as a + b = b + a.
Using variables provides a shorthand way of describing relationships that apply to all numbers in a set.
Big Idea 2: Patterns, Relations, and Functions
Children should investigate numerical and geometric patterns and express them mathematically in words or symbols. They should analyze the structure of the pattern and how it grows or changes, organize this information systematically, and use their analysis to develop generalizations about the mathematical relationships in the pattern.
There are two different types of patterns: those that repeat and those that grow or shrink. A repeating pattern is one in which a group of elements repeats whereas a growing or shrinking pattern is one in which elements don't repeat. Rather, there is a change between successive elements, and every element in the pattern is related to the preceding element in the pattern in the same way.
An example of a growing pattern is 1,3,5,7,9, … each element in the pattern is two more than the preceding element.
An example of a shrinking pattern is 100, 90, 80, 70, ... Each element in the pattern is 10 less than the preceding element. Growing and shrinking patterns lead to generalizations and to then representation of the generalizations using variables. Children learn to describe growing and shrinking patterns in words and to predict what comes next.
The same can be taught through a fun activity for kids. Show them the following image:
Children should be encouraged to explain these patterns verbally and to make predictions about what will happen if the sequence is continued.
Fun math activities for kids like these give opportunity to engage them in thinking about how to articulate and express a generalization—"how can we talk about how this pattern works for a square of any size?" grade 4 kids should be able to predict the next element in a sequence by examining a specific set of examples. By the end of fifth grade, students should be able to generalize by reasoning about the structure of the pattern.
In intermediate algebra, kids can also investigate properties like commutativity, associativity, and distributivity of multiplication over addition. Is 3x5 the same as 5x3? Area model can help children discover these questions. For example:
Another example is to find out if 8x14 can be decomposed into 8x10 + 8x4
Another mathematics activity that can be done with kids is to show the child a pattern of constructions made from cubes like this:
In this math problem, the pattern in figure 1 shows one cube on top and one cube at the bottom, figure 2 shows one cube on top and two cubes at the bottom, figure 3 shows one cube on top and three cubes at the bottom, and so on. The child might describe the pattern by saying, "the number of cubes on the bottom is the same as the number of the figure. There is always one cube on top."
Children can use symbols to write the function that shows the relationship between the total number of cubes (let's call it t) and the figure number (let's call it n). The function would be t = n + 1.
Children also identify patterns when they complete function tables. They learn that for every input there is exactly one output. In pre algebra, kids complete function tables when the rule is given, such as "add 3." but in later grades, they not only complete function tables when the rule is given, but they also have to collect data, organize the data in a table, figure out the function, and record the function using symbols.
Big Idea 3: Proportions and Proportional Reasoning
A proportion is a special type of function in which the relationship between quantities is multiplicative. Reasoning about quantities that are related by multiplication is referred to as proportional reasoning. Proportional reasoning is encountered by kids in early years. For example, in the study of place value, students learn that 1 ten is equal to 10 ones, so they figure that 2 tens (2 x 10) is equal to 20 ones, 3 tens (3 x 10) is equal to 30 ones-each regrouping involves multiplication.
In the study of measurement, when children know that there are 4 cups in 1 quart, they can reason that there must be 8 cups (2 x 4) in 2 quarts, and 12 cups (3 x 4) in 3 quarts. In this process, the child is reasoning proportionally. When students construct equivalent fractions and when they use the key on a map to figure out actual distances between cities, they are reasoning proportionally.
In grade 4, basic algebraic ideas should emerge and be investigated because children—
