Some Mental Computation techniques
Mental computation is the process of finding an exact answer to a computation without the use of any other computational aid.
Here are a few common mental computation techniquesName / What / When / How
Count on and Count Back / Count up or down by place value. For example, 352 – 3 would be calculated, “352, 351, 350. / Use this technique when the number to be added or subtracted is 1, 2, 3; 10, 20, 30; 100, 200, 300; and so on. / Begin by saying the larger number. Count on to add or back to subtract.
Choose Compatible Numbers / Select pairs of compatible numbers (numbers that are easy to compute mentally) to perform the computation. / Use this technique if one or more pairs of numbers can be easily added, subtracted, multiplied, r divided, or if they can produce multiples of 10, 100, or other numbers that make calculations easy. / First look for pairs of numbers that are easy to calculate. Perform these calculations first. Then look for other numbe combinations that can be calculated easily.
Left to Right / Break apart the numbers into their place values and perform you work from right to left. / Use this technique when one of the numbers is a single or when most digit-by-digit computations are simple. / Think about each number in its expanded form. Do the calculations for the largest place values on down to the smallest place values. Now combine your answers to each of the smaller computations.
345+130232422121÷11Use Compensation / Substitute a compatible number for one of the numbers so that you can complete the computation mentally. / Use this technique when a calculations can be chosen that is close to the original one and that is easy to do mentally. / Change the original calculation to one that is easy to do mentally. (Try not to change more than one number.) Keep track of how you adjusted the original calculation, and find the answer to the original calculation.
Equal Additions / Since the difference between two numbers does not change if the same number is added to both of the original numbers, select an addend that will transform the subtraction into a recognizable difference. / Use this when one of the numbers in a subtraction calculation can be changed to make the resulting computation easy to do mentally. / Identify a number that can be added to one of the numbers in the original calculation to give a new computation that is easy to do. Then add this number to both numbers in the original problem.
65+38196908 - 39
3493÷7145 – 771456 - 397
More mental mathematics – this time for comparing the magnitude of rational numbers.
Key Ideas: Every rational number may be expressed in the form , where a may be any integer, and b may be any integer except the integer 0. The rational numbers may be viewed in two interrelated ways:
(Equations) The set of rational numbers in the smallest possible set of numbers that are necessary for solving all possible equations of the form bx+a=0, where a and b are integers, b≠0. (This point of view dominates high school mathematics or college algebra courses.)
(Ratios) The set of rational numbers is the set of all possible ratios. Ratios come in two forms: Part-to-Whole and Part-to-Part. (In elementary curriculum, the ratio view dominates.) Our focus is on the Part-to-Whole representation.
When I write the ratio , a reasonable question to ask is, “ of what?” In order to relate rational number to a whole number of objects, we must define a context in which a “whole” or “unit” is clearly identifiable. Is the unit a loaf of bread? Is it a one dollar bill? Is it the distance to the nearest mile from here to San Francisco? Is it the number of teaspoons in a cup? The understanding of the number changes with respect to the context. Thus when working with rational numbers, it is quite important to clearly identify the whole (unit) you are considering.
Comparing Fractions – How can you determine the largest fraction between two given fractions?
Using Benchmarks to Estimate
1.How do you know when a fraction is close to 1?
2.Which of the fractions above are close to 1? Among these, which are greater than 1?
3.How do you know when a fraction is close to ?
4.Which of the fractions above are close to ? Among these, which are greater than ?
5.How do you know when a fraction is close to 0?
6.Which of the fractions above are close to 0?
7.Which fraction in each pair listed below is larger? Do not find a common denominator! Use the benchmarks 1 and to reason through each comparison. Is one fraction more than 1 while the other is less than 1? Is the fraction more than while the other is less than ? Explain your thinking for each comparison.
a. or d. or
b. or e. or
c. or f. or
Using Reasoning to Compare Fractions
Many fractions can be compared by simply reasoning from a conceptual knowledge (a geometric understanding) of fractions rather than by using a process to find a common denominator or by using cross multiplication. Use the hints given to develop a thinking strategy based on conceptual knowledge for comparing each pair of fractions.
1.The fractions and have the same size parts (the same denominator). Which is larger and why?
2.The fractions and both have the same number of parts (the same numerator). Which is larger and why?
3.Both of the fractions and are one fractional part away from being a whole. Which is larger and why?
4.Both of the fractions and are one fractional part away from . Which is larger and why?
5.Determine which fraction in each pair is larger by reasoning from your conceptual knowledge of fractions. Explain your thinking for each comparison.