why do we use letters in algebra
It may help you to read first An equation says that two things are equal. It will have an equals sign = like this:
That equation says: what is on the left (x + 2) is equal to what is on the right (6) So an equation is like a statement this equals that So people can talk about equations, there are names for different parts (better than saying that thingy there! ), and all its parts: A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y. A number on its own is called a Constant. A Coefficient is a number used to multiply a variable ( 4x means 4 times x, so 4 Variables on their own (without a number next to them) actually have a coefficient of 1 ( x is really 1x Sometimes a coefficient is a letter like a or b An Operator is a symbol (such as +, etc) that shows an operation (ie we want to do something with the values).
A Term is either a single number or a variable, or numbers and variables multiplied together. An Expression So, now we can say things like that expression has only two terms, or the second term is a constant, or even are you sure the coefficient is really 4? A can have constants, variables and the exponents 0,1,2,3,. But it never has division by a variable. Monomial, Binomial, Trinomial There are special names for polynomials with 1, 2 or 3 terms: are terms whose variables (and their ) are the same. In other words, terms that are like each other. (Note: the coefficients Research around the world has shown that students draw on what they know of other symbol systems when they are trying to understand symbols in algebra. students drawing analogies with other symbol systems (e. g. letters in algebra are abbreviations or initials for things) misleading teaching materials (e. g. fruit salad algebra and variations) students trying to write mathematical ideas that cannot easily be written in algebra (e. g. trying to say in symbols that when x increases by 1, y increases by 4).
A that compared students, progress in algebra in different schools revealed that misinterpretation of algebraic letters as abbreviated words or labels for objects was a persistent difficulty in schools where teaching materials explicitly presented letters as abbreviated words (e. g. , c could stand for a cat so 5 c could mean 5 cats).
The study suggests that вthe letter as an abbreviated wordв can be a particularly difficult misconception to eliminate, if it has been caused by certain teaching materials or teachersв explanations. In contrast, teachers seemed to be able to correct the naГve misinterpretation that the value of letters was related to an alphabetic code quite easily, by explicitly pointing it out. This misconception is often evident in puzzling numerical answers to algebraic questions (e. g. occasionally we see students who evaluate 2 n as 28 because n is the 14th letter of the alphabet). Not all misconceptions are deep. Students experience further difficulties when numbers are combined with pronumerals. In other symbol systems, such as chemical formulae, Roman numerals or indeed in our own number system, putting together two or more symbols implies addition in some way.
For example 35 = 30 + 5, XVI = X + V + I, CO means a molecule comprising 1 atom of carbon and 1 atom of oxygen. Over generalising Roman numerals, a few students will write h 10 instead of h + 10 and write 1 y instead of y In algebra, putting symbols together does not mean addition, but multiplication instead. So students need to learn a new set of rules that are not intuitive 3 + x is not written as 3 x. Instead 3 x means 3 Г x and this in turn means x + x + x, just as 3 Г 4 = 4 + 4 + 4. MacGregor, M. Stacey, K. (1997). Studentsв understanding of algebraic notation. Educational Studies in Mathematics, 33, 1 в 19.
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