Beginning Algebra: Concepts and Problems
Nowadays, children begin learning algebraic concepts in kindergarten through addition and subtraction equations. A summary of beginning algebra concepts is presented here, with example problems and solutions.
Beginning Algebraic Concepts and Problems
Addition and Subtraction Equations
In kindergarten, children are exposed to linear equations for addition and subtraction. They also learn to add and subtract for numbers 0-5. In first grade, they begin doing more complex problems, using numbers up to 100. To familiarize students with equations, ask questions such as:
- Which of these is not correct?: 5 = 5, 8 = 9 -1, 6 + 2 = 4 + 3, 4 + 3 = 3 + 4; Answer: 6 + 2 = 4 + 3
At first, the numbers in equations are all constants. For a challenge, an unknown answer may be represented with a question mark, a box or a similar symbol.
- Examples:
5 + 4 = ? --> 5 + 4 = 9
7 - 6 = ? --> 7 - 6 = 1
Examples of more complex problems that are presented later are:
- x + 4 = 12 --> x = 8
r - 3 = 6 - 2 --> r = 6 - 2 + 3 --> r = 7
Multiplication and Division Equations
Third graders learn two more operations: multiplication and division. Students may use visuals, such as models or drawings, to solve for unknown quantities. Problems common for third grade include:
- 25 ÷ y = 5 --> 5y = 25 --> 5 = 5. The problem may also be written as 25/y = 5.
8 x a = 16 (may also be written as 8a = 16) --> a = 16/2 --> a = 8
More complex problems include combinations of the four operations: addition, subtraction, multiplication and division.
- Examples:
2x + 5 = 9 --> 2x = 9 - 5 --> x = 4/2 --> x = 2
8a - 2 = 50 --> 8a - 50 - 2 --> a = 48/8 --> a = 6
7r/4 = 7/2 --> 7r = 7/2 x 4 --> 7r = 28/2 --> 7r = 14 --> r = 2
Order of Operations
At first, students learn to do problems where the answer will be the same no matter in what order the operations are performed. In sixth grade, students generate equivalent expressions by using the properties of operations. This sometimes requires the use of parentheses in an equation.
- Examples:
4(3 + x) is the same as 12 + 4x
15x + 30y may also be expressed as 5(3x + 6y)
a + a + a + a is the same as 4a
The actual order of operations is presented later. This order is:
- Do operations that appear in parentheses. If there is a set of parentheses inside another set of parentheses, do the innermost one first.
- Do all multiplication and division problems, starting at the left and moving to right.
- Do all addition and subtraction problems, again starting at the left and going to the right.
- Examples:
x = 20 ÷ (4 - (10 - 8)) --> x = 20 ÷ (4 - (2)) --> x = 20 ÷ 2 --> x = 10
y = (6 - 4) * 49 ÷ 7 --> y = 2 x 49 ÷ 7 --> y = 98 ÷ 7 --> y = 14
b = 45/{8(5 - 4) - 3} --> b = 45/{8(1) - 3} --> b = 45/5 --> b = 9
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