1. Algebra

2. What Algebra consisted of? • Symbolic manipulation with artificial applications … • With little or no connection to the real world…

3. Big Ideas • 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. Big Ideas • 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. Algebraic Thinking • Not a single idea but it consists of different forms of thought and an understanding of symbolism

6. The Meaning of the Equal Sign • 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. The Meaning of the Equal Sign • 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. Basic Ideas in Arithmetic • 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. The Meaning of the Equal Sign • 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. Variables in Equations • Powerful, allow for generalizations • Variables used as an Unknown Value • Variables used as Quantities that Vary • Solving Equations or Inequalities

11. Variables used as an Unknown Value • Student first experience variables used as symbols that stand for an unknown value ?

12. Variables Used as Quantities That Vary • 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 bis 2 • Many students think that if two variables are different then their values must be different, a misconception….

13. Variables Used as Quantities That Vary • 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

14. Solving Equations or Inequalities