Mental Mathematics |
TASKS
1. In a 6th grade mathematics class, one of the problems requires the following calculation:15 / 32 ÷3 / 8. One student claims, without using pencil or paper, that the answer is 5 / 4 because 15 ÷ 3 = 5 and 32 ÷ 8 = 4. Is the student's answer correct? How can the student's reasoning be explained?
2. The owners of a manufacturing company would like to be able to promise one-week delivery of special orders. The company is not ready, however, to make this promise to customers. Of the 56 orders filled so far this month, 49 were filled in one week or less and 7 took more than one week to fill, a success rate of 87.5%. During the weekly team meeting, the supervisor wants to know how many subsequent orders must be filled within one week to increase this month's success rate to 90%. Several employees know that this problem can be solved with algebra and scrounge around for paper and pencils. One employee, however, announces that the answer is 14. She describes her reasoning as follows: We want the 7 late orders so far to be only 10% of the total number of orders, which therefore must be 70. Subtracting the 49 on-time and 7 late orders so far yields 14 orders. Is she correct? How can her reasoning be explained?
COMMENTARY. Mental arithmetic can save time, money, and even embarrassment--such as being caught short of cash at the check-out counter. In everyday life, people use mental arithmetic to estimate budgets, shopping bills, tips in restaurants, and taxes. On the job, an employee might be asked to give a quick estimate while talking to a boss or a colleague. Mental arithmetic depends not only upon number facts and estimating skills but also problem solving skills and algebraic reasoning skills because the standard paper-and-pencil algorithms are sometimes hard to do in one's head. As people become more familiar with number facts and gain number sense and symbol sense, their mental arithmetic can become more precise and more flexible. A study of 44 mathematicians' numerical estimation strategies found considerable variation in the strategies used (Dowker, 1992). As many as 23 different strategies were used for the same task!
The value in discussing mental arithmetic in high school is that number sense and symbol sense can build on each other, leading to greater facility with and understanding of both algebra and mental arithmetic. The examples above have been chosen to illustrate this possibility. In beginning algebra classes, students often arrive at answers without using symbolic techniques. Furthermore, as these examples illustrate, informal or non-standard methods can reveal sophisticated algebraic reasoning, even though it might not be expressed symbolically. Rather than disregarding such informal methods as distractions from the goal of teaching standard algebraic methods, teachers can exploit informal methods as sources of meaning for students by establishing connections between informal and formal methods. Such connections can go in two directions. First, an informal method may be expressed symbolically, thereby promoting algebra as a way of expressing ideas about numbers. In this way, symbol sense may build upon number sense. Second, after some manipulation, symbolic expressions may be reinterpreted informally, thereby promoting algebra as a way of thinking about numbers. In this way, number sense may build upon symbol sense.
For experienced users of algebra, some of its power of comes from the fact that symbols may be manipulated without concern for their meaning. Nevertheless, for students and experienced users alike, periodic re-interpretation of symbolic expressions can shed new light on the context, its mathematical representations, or its mathematical structure, potentially leading toward an understanding of algebra that is more flexible for solving familiar and unfamiliar problems.
MATHEMATICAL ANALYSIS. Each task is solved separately below.
1. It is easy to verify that the student's answer is indeed correct:
15 / 32 ÷ 3 / 8 = 15 / 32 * 8 / 3 = 5 / 8 * 3 / 4 * 8 / 3 = 5 / 4
But the calculation does not shed much light on whether the approach makes sense. One way of explaining the approach is to note that if the calculation were15 / 32 ÷ 3 / 32, the answer would be 5, because 15 of anything divided by 3 of the same things is 5. But because we are dividing by 3 / 8, which is 4 times bigger than 3 / 32, the answer should be 1 / 4 of 5, or 5 / 4.
Another way to justify the approach is algebraic. The standard method is as follows:
a / b ÷ c / d = a / b * d /c = ad / bc
The student proposes that the answer is a ÷ c / b ÷ d. Simplifying this expression according to standard algebraic rules shows that the answer is indeed correct:
a / b ÷ c / d = a / c / b / d = a / c ÷ b / d = a / c * d / b = ad / bc
The awkwardness of the calculation, however, suggests that this approach, although correct, might not be efficient if, for example, c is not a factor of a.
2. Other employees might check the employee's answer. With 49 successes and 7 failures so far, and 14 future successes, the success rate will be:
49 + 14 / 49 + 7 + 14 = 63 / 70 = 0.90, or 90%.
So her answer is correct. The deeper question is how to explain the employee's reasoning. Her reasoning is clear and understandable, too, indicating that there is probably a general procedure at work. What, then, is the relationship between her reasoning and the standard algebraic approach?
A typical way to solve the problem that the supervisor poses would be to let x represent the unknown number of subsequent (consecutive) orders to be delivered successfully within one week. Then,
49 + x / 49 + 7 + x = 0.90
But this equation is hard to solve mentally. To generalize this approach, let x be as above, let r represent the desired success rate, let s represent the number of successes so far, and let f represent the number failures so far. Then, as before,
s + x / s + ƒ + x= r (1)
Solving this for x yields (after some manipulation):
x = rs + rƒ s / 1 r (2)
Though it is not immediately clear how to interpret this formula, the denominator, 1 r, gives a clue about the employee's reasoning: if r is the desired success rate, then 1 r is the desired failure rate, and the employee's reasoning began with the failure rate--10% in this case. Equation 1, which is based on successes, has an analog that is based on failures. Table 1 gives the analogous equation and several others that follow algebraically from it, along with explanations of how the equation might be interpreted.
Table 1:
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The last equation in Table 1 is close to the reasoning that the employee described, and is a general procedure that may be done mentally. Thus, algebraic manipulation has provided justification for the employee's mental procedure.
Algebra can also provide suggestions for other mental procedures. Further algebraic manipulation of Equation 2, for example, gives additional possibilities, shown in Table 2.
Table 2:
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The last equation in Table 2 gives another way to think about the original problem. Because the ratio of the desired success rate and failure rate is 90:10 or 9:1, there must be 9 times as many successes as failures. With 7 failures so far, there must be 63 successes, or 14 more than the 49 we already have.
EXTENSIONS. Traditional algebra problems can present similar opportunities for students to generate informal solutions. For example,
Box + Box + Box + Triangle = 47 / Box Triangle = 1
A student might say, "Well, I know from the second equation that Box is one more than Triangle. So I imagined that the Triangle in the first equation was a Box. Then the first equation would be Box + Box + Box + Box = 48. So, Box is 12 and Triangle is 11."
References
- Dowker, A. (1992).
- Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23(1), 45-55.
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