![]() ![]() “Four is greater than eleven” doesn’t make sense, and it is recognizing that error that gives greater than and less than its instructional power.ĭo you agree that students often have trouble with these particular symbols? What tips do you have for teaching greater than/less than? Come and share in our WeAreTeachers HELPLINE group on Facebook. How will they know if they are reading it correctly? The numbers should be in the correct order (unlike 4 < 11 read as “eleven is greater than four”), and the number sentence should make sense. Then it is a matter of practice with reading the inequalities aloud, to teachers, classroom partners, and parents. Second, students should read the whole inequality, naming numbers and symbols left to right, like they would read any sentence. If they forget which is which, I like to point out that the less than symbol makes an L. This is actually a simple, and more fruitful, switch.įirst, explicitly teach that the symbols have names. Tips for teaching greater than/less than (without the alligator mouth) Taken with a pair of numbers, the greater than and less than symbols form “inequalities,” a fundamental way of explaining the relationship between two numbers. You likely need more context to how large x needs to be for a particular application.We have the opportunity to teach how the language of all math works. The sine approximation is even better, with error on the order of x³.īut if an approximation holds for x ≫ 1, there’s often an implicit asymptotic series in the background, and these are more subtle. That’s the case in our square root approximation above. If an approximation holds for | x| ≪ 1, there’s often an implicit power series in the background, and the error is on the order of x². It’s often harder to tell from context when something is large than when it is small. The error in the example is more than 30,000, but this value is small relative to 10! = 3,628,800. The relative error is small, not the absolute error. Note that the approximation error above is small relative to the exact value. For instance, if n = 10, the approximation above has an error of less than 1%. For example, Stirling’s formula for factorials saysįor n ≫ 1. Sometimes you see something like n ≫ 1 to indicate that n must be large. How small is small enough? The post explains how to know. A lot of people memorize “You can replace sin θ with θ for small angles” without thoroughly understanding what this means. If θ is small, the error in the approximation above is very small.Ī few years I wrote a 700-word blog post unpacking in detail what the previous sentence means. Rather than saying a variable is “small,” we might say it is much less than 1. The ratio b/a = 0.03, and your error should be small relative to 0.03, so the approximation above should be good enough. Suppose you need to know √103 to a couple decimal places. If, in your context, you decide that b/ a is small, the approximation error will be an order of magnitude smaller. So when is | b| much less than a? That’s up to you. ![]() You might see somewhere that for | b| ≪ a, the following approximation holds: All jargon is like this.īelow are some examples of ≪ and ≫ in practice. You have to know the context to understand how to interpret them, but they’re very handy if you are an insider. The symbols ≪ and ≫ can make people uncomfortable because they’re insider jargon. Sometimes you’ll see ≫, or more likely > (two greater than symbols), as slang for “is much better than.” For example, someone might say “prototype > powerpoint” to convey that a working prototype is much better than a PowerPoint pitch deck. ![]() Is 5 much less than 7? It is if you’re describing the height of people in feet, but maybe not in the context of prices of hamburgers in dollars. Here’s a little table showing how to produce the symbols. The symbol ≪ means “much less than, and its counterpart ≫ means “much greater than”. The symbols ≪ and ≫ may be confusing the first time you see them, but they’re very handy. ![]()
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