2.  Symbolic manipulation with artificial applications …  With little or no connection to the real world…

3.  Structure in our number system and methods to compute can be generalized  Symbolism, involving equations and variables is used to express generalizations from arithmetic  Variables are symbols that take the place of numbers or ranges of numbers  Patterns can be recognized, extended and generalized

4.  Functions are relationships- members of one set with members of another set  Functional relationships can be represented in graphs, symbolic equations, tables and words  Each representation of a function provides a different view of the same relationship

5.  Not a single idea but it consists of different forms of thought and an understanding of symbolism

6.  One of the most important symbols in arithmetic  The “=“ sign is a poorly understood symbol  Students believe that the problem is on the left side of the equal sign and the answer is on the right  It is all right to remind students that the equal sign means the same as

7.  Why is it so important that students understand the equal sign?  1. It is important for students to understand the relationships in our number system  For example… 6 X 7 = 5 X 7 + 7

8.  A number can be expressed as a sum  6 = 1 + 5  The distributive property allows us to multiply the parts separately  (1 + 5) x 7 = (1 x 7) + (5 x 7)  Using further properties this can then be rewritten as 7 + (5 x 7)  or 5 x 7 + 7

9.  2. When students have difficulty with the meaning of the equal sign, they often have difficulty when encountered in algebraic equations  5x – 24 = 81 would require that the students see what is on each side of the equal sign as equivalent  It is not possible to “do” the left hand side  But if both sides are the same then they will remain the same when 24 is added to each side

10.  Powerful, allow for generalizations  Variables used as an Unknown Value  Variables used as Quantities that Vary  Solving Equations or Inequalities

11.  Student first experience variables used as symbols that stand for an unknown value ?

12.  Different symbols or variables in a single equation may have different values  a + 6 = 10 – b, one solution could be a = 3 and b = 1  Another solution for a and b is 2  Many students think that if two variables are different then their values must be different, a misconception….

13.  The use of two or three variables is a forerunner of variables that are used to describe functions such as  y =mx +b or y=3x – 5  For situations, context becomes very important