Overview:
There are many ways to solve math problems. Some can be solved more easily by drawing a diagram, making a chart or graph of all the values, or looking for a pattern. Another strategy to use is to solve a simpler problem, then use information from the steps and the solution to solve another problem.
What Is Estimation?
One of the ways simpler problems are solved every day is by rounding numbers and estimating. Suppose you want to buy a book that costs $45.00, but you will need to pay tax of 8.8%. Most people cannot figure what 8.8 % of $45.00 without a calculator, but 10% is an easier problem. By estimating 10% of $45.00 at 4.50 , the final cost of the book will be somewhere around $49.50, which is close enough to know whether there’s enough money to make the purchase.
What About Smaller Pieces?
Many times problems can be made simpler by breaking them into smaller pieces. A complex geometric figure can be broken up into its component rectangles, squares, and triangles, and the area of each part calculated, then added together for the area of the entire figure. An algebra problem with many variables can be solved for one variable at a time, and the like variables can be grouped together.
What About Familiar Operations?
In addition, one of the ways to make math problems simpler is to break the problem into a series of familiar operations. For example, when solving the equation 2(c+1) = 8c – 22, the first step is to remove the grouping, in a familiar step of multiplying 2(c + 1) = 2c + 2. The next step is a familiar step of subtraction so that the variables are all on the same side of the equation, so that 2c – 2c + 2 = 8c – 2c – 22, or 2 = 6c – 22. Then the next step is moving the numbers so that the variables are on one side of the equation and the numbers are on the other side, as 2 + 22 = 6c -22 + 22. If 24 = 6c, then a last familiar step: 24/6 = 6c/6 or c = 4.
What About Simpler Patterns?
Suppose the problem were to find the sum of the first 100 natural numbers, starting with 1 + 2 + 3…98 + 99 + 100. It would take a long time, even with a calculator, and there’s always a possibility of a mistake in addition. There’s a simpler way to break it up into smaller pieces, use familiar operations, and use a simpler pattern. One small piece of the problem is to add the first 10 numbers:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
There is a pattern to it, but in a different order: 1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11, 4 + 7 = 11, and 6 + 5 = 11. In other words the sum of the first 10 numbers is 5 (11) or 55. The pattern to the first 100 numbers (still doable, but more time-consuming) is (1 + 100) + (2 +99) + (3 + 98) and so on, or 101 times 50, or 5050.
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