Variables are very important
Numbers, symbols, variables and terms
As soon as letters appear, math becomes complicated for some. Variables and terms are useful "simplification tools" that occur throughout school. How can one learn to deal with algebraic structures - and which ideas are important?
With variables and terms, a piece of "pure" mathematics is finding its way into the classroom: Where previously it was about concrete arithmetic with everyday relevance - be it in factual arithmetic with the basic arithmetic operations, with geometric objects, with the rule of three or in percentages - now the step in takes place abstraction and, for many, school mathematics becomes “lifted and empty”. Variables and terms enable an unambiguous, precise and very brief symbolic representation for (mathematical) facts that would require many sentences on a linguistic level.
What is a now? Basic ideas about variables
One difficulty in dealing with variables lies in the many possibilities that dealing with them offers. Unlike in everyday life, variables are not abbreviations for anything. A variable is ultimately a placeholder for a number, for several numbers, for all numbers or for a term. It is denoted by letters. You can insert values (placement aspect) or calculate with the variables (calculus aspect). Variables are used to describe something (object aspect). The different ideas of the term variable (as well as the different meanings of terms and equations) must be taken into account in the classroom.
- The variable as unknown: The variable stands for a number to be determined. The value for the variable is to be determined in such a way that the equation is fulfilled and a true statement is made. Example: 5a + 3 = 13
- The variable as a general number (or as an indeterminate): This basic concept is used in the general description of a law of calculation or a geometric formula. Any number that makes sense can be used for the variables. Examples: (a + b)2 = a2 + 2ab + b2 A = ½ gh
- The variable as changeable: This basic concept occurs in all functional relationships that can be described by equations. All elements of the definition range can then be used for the function variable, which is usually designated with x. If a variable occurs as a variable, there is usually a second dependent variable in play (called y or f (x) for functions). Examples: f (x) = 3x + 5 or also fa(x) = ax2 (here with parameter a).
Set up, transform and solve equations
When solving word problems, modeling, puzzles - "problems" for short, it often helps to set up equations and solve them. To do this, however, the students must be able to handle terms and equations confidently. And so many difficulties should not be concealed in class either. A classic example: sheep (S) and goats (Z) are standing in a meadow. There are five times as many sheep as goats. What does an equation that describes this situation look like? Often the equation 5S = Z is noted, and so, following the sentence structure, exactly the opposite situation is described: Now there would be five times as many goats as sheep on the meadow. The inequality can therefore only be converted into an equality relation by reversing it: S = 5Z (cf. Schmidt 2009).
The path to setting up and solving equations involves a learning development. First of all, it is a matter of grasping the meaning of the equal sign (it can be a request to calculate, a relation sign or be used as a definition sign). Arithmetic expressions to describe real situations are set up ("Two dogs join three dogs ..."). First ways to determine an unknown are taken, the equivalence in arithmetic expressions is recognized. A term is no longer perceived only as a calculation expression, but term structures and relationships between operations as well as the reverse operation as a solution strategy are recognized. The structure of algebraic expressions is recorded and backward computation is used as a helpful strategy. Finally, equations are solved through systematic transformation ("perform the same operation on both sides") (cf. Stacey 2011), possibly with the help of a substitution. Equations cannot always be solved, they can have exactly one, exactly two, three, ... or any number of solutions.
Among other things, the way from number puzzles to equations has proven successful in class (Hesse 2011): Think of a number, add 3, multiply by 2, subtract 6. Starting from a possibly imaginary number x, a term can be developed that can be simplified - it becomes clear how the imaginary number is quickly "guessed". Here the result mentioned has to be divided by 2.
For a flexible use of equations, changing the representation is always important: Tables, situations, equations and graphs should be assigned to one another or moved from one type of representation to the other.
Barzel, B./Herget, W .: Numbers, symbols, variables - abstract and concrete. Plea for a lively use of terms. In: mathematik lehren, issue 136, pp. 4 - 9.
Barzel, B./Holzäpfel, L .: Structures as the basis of algebra. In: mathematik lehren, issue 202, pp. 2 - 9.
Hesse, D .: Mind reading - no magic. In: mathematik lehren, issue 169, pp. 16-20.
Schmidt, W .: Variables, terms, equations. In: Mathematik 5-10, Issue 6, Friedrich-Verlag, pp. 4 - 5.
Stacey, K .: A journey through the years. From arithmetic expression to solving equations. In: mathematik lehren, issue 169, pp. 6-12.
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