Algorithms and Arithmetic in Everyday Mathematics
Everyday Mathematics Parent Handbook
Algorithms and Arithmetic in
An algorithm is a set of rules for solving a math problem which, if done properly, will give a correct answer each time.
Algorithms generally involve repeating a series of steps over and over, as in the borrowing and carrying algorithms and in the long multiplication and division algorithms. The Everyday Mathematics program includes a variety of suggested algorithms for addition, subtraction, multiplication and division. Current research indicates a number of good reasons for this primarily, that students learn more about numbers, operations, and place value when they explore math using different methods.
Arithmetic computations are generally performed in one of three ways: (1) mentally, (2) with paper and pencil, or (3) with a machine, e.g. calculator or abacus. The method chosen depends on the purpose of the calculation. If we need rapid, precise calculations, we would choose a machine. If we need a quick, ballpark estimate or if the numbers are easy, we would do a mental computation.
The learning of the algorithms of arithmetic has been, until recently, the core of mathematics programs in elementary schools. There were good reasons for this. It was necessary that students have reliable, accurate methods to do arithmetic by hand, for everyday life, business, and to support further study in mathematics and science. Todays society demands more from its citizens than knowledge of basic arithmetic skills. Our students are confronted with a world in which mathematical proficiency is essential for success. There is general agreement among mathematics educators that drill on paper/pencil algorithms should receive less emphasis, and that more emphasis be placed on areas like geometry, measurement, data analysis, probability and problem solving, and that students be introduced to these subjects using realistic problem contexts. The use of technology, including calculators, does not diminish the need for basic knowledge, but does provide children with opportunities to explore and expand their problem solving capabilities beyond what their pencil-and-paper arithmetic skills may allow.
Sample Algorithms: Below are examples of a few procedures that have come from childrens mental arithmetic efforts. Each is a legitimate algorithm, that is, a set of rules that if properly followed yields a correct result. As parents, you need to be accepting and encouraging when your children attempt these computational procedures. As they experiment and share their solution strategies, please allow their ideas to flourish. If you are not comfortable with the vocabulary of arithmetic, you may want to review the glossary entries for addition, subtraction, multiplication and division before reading the sample algorithms.
1. Left-to-right Algorithm
2. Partial-Sums Algorithm
3. Rename-Addends Algorithm (Opposite Change)
If a number is added to one of the addends and the same number is subtracted from the other addend, the result remains the same. The purpose is to rename the addends so that one of the addends ends in zeros.
4. Counting-On Algorithm
1. Add-Up Algorithm
2. Left-to-Right Algorithm
3. Rename Subtrahend Algorithm (also called Same Change)
If the same number is added to or subtracted from both the minuend (top number) and subtrahend (bottom number), the result remains the same. The purpose is to rename both the minuend and the subtra-hend so that the subtrahend ends in zero.
This type of solution method shows a strong ability to hold and manipulate numbers mentally.
4. Two Unusual Algorithms
In Third Grade Everyday Mathematics, a partial-products algorithm is the initial approach to solving multiplication problems with formal paper-and-pencil proce-dures. This algorithm is done from left to right, so that the largest partial product is calculated first. As with left-to-right algorithms for addition, this encourages quick estimates of the magnitude of products without neces-sarily finishing the procedure to find exact answers. To use this algorithm efficiently, students need to be very good at multiplying multiples of 10, 100, and 1000. The fourth-grade program contains a good deal of practice and review of these skills, which also serve very well in making ballpark estimates in problems that involve multiplication or division, and introduces the * as a symbol of multiplication.
1. Partial-Product Algorithm
2. Modified Standard U.S. Algorithms
A Division Algorithm
The key question to be answered in many problems is, How many of these are in that, or How many n's are in m? This can be expressed as division: m divided by n, or m/n.
One way to solve division problems is to use an algorithm that begins with a series of at least/less than estimates of how many ns are in m. You check each estimate. If you have not taken out enough ns from the ms, take out some more; when you have taken out all there are, add the interim estimates.
You would record 10 as your first estimate and remove (subtract) ten 12s from 158, leaving 38. The next question is, How many 12s are in the remaining 38? You might know the answer right away (since three 12s are 36), or you might sneak up on it: More than 1, more than 2, a little more than 3, but not as many as 4. Taking out three 12s leaves 2, which is less than 12, so you can stop estimating.
To obtain the final result, you would add all of your estimates (10 + 3 = 13) and note what, if anything, is left over (2). There is a total of thirteen 12s in 158; 2 is left over. The quotient is 13, and the remainder is 2.
The examples show one method of recording the steps in the algorithm.
One advantage of this algorithm is that students can use numbers that are easy for them to work with. Students who are good estimators and confident of their extended multiplication facts will need to make only a few estimates to arrive at a quotient, while others will be more comfortable taking smaller steps. More important than the course a student follows is that the student understands how and why this algorithm works and can use it to get an accurate answer.
A student with good number sense might answer, At least one-tenth, since 0.1 * 12 is 1.2, but less than two-tenths, since 0.2 * 2 = 2.4. The answer then could be l3.1 (12s) in 158, and a little bit left over.
The question behind this algorithm, How many of these are in that? also serves well for estimates where the information is given in scientific notation (see glossary). The uses of this algorithm with problems that involve scientific notation or decimal information will be explored briefly in grades 5 and 6, mainly to build number sense and understanding of the meanings of division.
An algorithm is any series of steps which, if followed properly, always yield a correct result. There are many ways to add, subtract, divide, and multiply that meet this definition. Your child will learn to compute accurately and quickly.
Children gain valuable confidence and insight when permitted to explore algorithms of their own invention. A given child may be more comfortable with this way or that. A given approach may be more useful for this problem or that one.
Although you probably learned only one or two algorithms for each kind of arithmetic, it is important that you support your childs use of many. In fact, if you closely observe your own computations in a variety of real-life settings counting change, making estimates, balancing your checkbook, etc. you will probably find that you use different algorithms at different times, and some of them are probably your own inventions.
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