GED Math Instruction
by Myrna Manly
This article appeared in the January/February 1988 issue of GED Items (ISSN 0896-0518; Volume 5, No. 1), published by the GED Testing Service of the American Council on Education. Myrna Manly is a mathematics instruction consultant and former Math Test Editor for the Tests of General Educational Development (GED Tests).
GED Math Instruction
The GED math instruction article in the November/December 1987 Items recommended that calculator use be integrated into GED mathematics instruction. Implicit in this recommendation is the mandate to use calculators properly and intelligently. The ability to "bestimate" is the key ingredient that separates an intelligent calculator user from a mindless button pusher. We must, therefore, focus on developing estimation skills in our mathematics instruction.
Prerequisite: single-digit mathematics facts
Students must be able to quickly recall basic mathematics facts before they can estimate. In fact, whenever confronted with a single-digit operation one should do this mentally rather than with a calculator. Expand these single digit operations by showing the patterns that occur when dealing with powers of 10. For example, finding the difference of 600 - 200 is no more difficult than 6 - 2. Use similar examples such as 6 x 7 = 42, 60 x 7 = 420, 60 x 70 = 4200 and 56/8 = 7, 560/8 = 70, 5600/80 = 70 to reveal that single-digit mental mathematics can be a very powerful tool.
Estimate, then calculate
Make estimation an integral part of every computation by determining the range of the answer before you begin the operation. "Where would the value of this answer fall? Would it be less than 1? Greater than 10? Less than 100?" are questions to ask before actually calculating the answer. To achieve more accurate estimations, use a rounding technique that requires both the single-digit skill mentioned above as well as a sense of the range of the answer.
Round with a purpose
Rounding with a purpose in mind requires adjusting the numbers involved to be compatible enough so that you can do the operation mentally. The value you choose to round to is determined by the other number in the problem. For example, if 238 were to be divided by 59, it would be appropriate to estimate it as 240/60 = 5. If that same number 238 were to be divided by 51, one might do better using an estimate of 250/50 = 5. This rounding techniques differs from the traditional method in that there is not one specific correct way to proceed; any way that works is acceptable. Fractions and decimals will also be rounded to familiar quantities for estimating. Consider adding these numbers 13/15 and 5/9. the fraction 13/15 is close to 1 and 5/9 is close to 1/2. One can therefore estimate the sum as being close to 1 and 1/2. Common fraction and decimal equivalents are also helpful to know when performing mental math. For example, if asked to find 18% of 147, being able to recognize that 18% is approximately 1/5 and using 150 rather than 147 makes this simply 1/5 x 150 = 30.
Test for reasonableness
Look at the answer in the context of the problem. Ask if it is reasonable. "Is four feet a reasonable height for a building?" Would we expect a person to be 500 inches tall?" are questions that can be answered with common sense. Encourage students to use their innate commonsense to acquire a number sense.
Practice, practice, practice
It is obvious that students need to build confidence in their number sense before they can be comfortable estimating. Practice is the means to that end. Take every chance to develop this confidence deliberately by using repeated examples structured to show patterns. Analyze answers for their logical connection to the problem.
Upon reflection, we realized that estimating and common sense, combined with basic mathematical knowledge, are the critical elements needed for mathematical literacy in today's world of calculating machines. But is it true that these will be enough to pass the GED Tests when calculators are not allowed? Yes, as long as the examinee is also aware of some mathematical applications and concepts. The following sample item is a good illustration to support this answer. The item has a sketch of a triangle in which two of the angles are labeled 37 and 67. The question asks "How many degrees does the third angle measure?" and gives these alternatives:
- Not enough information is given.
Examinees who are familiar with this concept know that the sum of the angles in a triangle is 180 and know that they must add the two values and then subtract their sum from 180. The examinee who is an estimater would round the values to 40 and 70, add them to get 110, subtract this sum from 180 to get 70, and choose 3. as the only possible correct answer-all without paper and pencil computation.
Items that can be solved like this, combined with the "set-up" items that do not require any calculation, make up 75% of the items on the published practice tests [Official GED Practice Tests, Steck-Vaughn, 8701 North MoPac Expressway, Austin, TX 78759-8364; 1-800-531-5015] . This seems to be a sure indication that time spent learning to estimate will be doubly valuable-both as preparation for the GED Mathematics Test and for the big test itself: Real Life.
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