Classroom Activities
Lesson: Number Tricks
Objectives
-To explore numerical expressions and to discover the freedom to choose any number to plug into a number trick.
-To show that the original number can be easily guessed based on the result if you know what was the recipe for computation in sufficiently simple form.
-To emphasize a number trick as an expression and make a clear connection to an expression as a recipe for computation.
-To increase comfort with the equals sign in the problem and what that means.
-To display there are infinite results to some number tricks.
-To display the different types of number tricks and their results.
-To learn how to translate computation in words to computation with mathematical symbols (i.e. parentheses, operations and variables).
Procedure
The lesson begins with a number trick (Appendix C, number tricks problem 1). The teacher asks students to choose any number and do the performed steps that are asked and the students are asked to record their results. The results are recorded on the board, and the teacher begins to guess what different students starting numbers were using the results. Students are asked to try and brainstorm ways in which the teacher was able to guess the numbers so easily. Another similar number trick is used (Appendix C, number tricks, problem 2) and the same process is repeated, results and process are discussed as a class. Finally a different kind of number trick is presented in which all students get the same result (Appendix C, number tricks, problem 3). This second type of number trick is discussed as a class and the students are asked why the result was always the same. Teacher and students come up with a list of ways in which to guess the result of a number trick.
Now as a class we discuss what can represent any chosen number (a variable) and how we can use a variable to write the steps in the number trick as an expression. As we are writing a mathematical statement to represent the steps, the idea of an expression as a recipe for computation is emphasized. Students will then be asked to write a number trick using words, as done in the previous problems and then translate them into mathematical expressions.
Lesson: Evaluating Expressions
Objectives
-To see the connections between an expanded expression and its simplified form.
-To understand the idea of equivalence and the equals sign in this particular context.
-To evaluate the expanded and simplified form at any given value in order to check simplification.
Procedure
Students are given a long and complicated expression in expanded form (Appendix C, evaluating/simplifying, 2-5) and asked to evaluate these expressions at different values. Then the students are presented with the simplified versions of the expanded form and asked to evaluate those expressions at the same values. In groups students discuss results and are asked "Do you think the results will be equivalent for any value of x, why or why not?" The groups share their answers to the questions and the class debriefs why it works and how they think we can transform the expanded form to look like the simplified form. The idea of equivalence should be discussed here and this particular context of the equals sign should be emphasized. Students should be able to explain what equivalence is and that we are not looking for a solution. As a class, we will then draft an informal list of different ways we can simplify expressions, prior to the nine rules of arithmetic being presented.
Lesson: Solving Equations
Objectives
-To solve one-step and multi-step equations.
-To be able to justify all steps when solving equations.
-To understand and articulate what the solution to an equation represents.
Procedure
Students start solving one step equations and then multi step equations (Appendix C, Equations, problems 1, 2) and included in solutions are justification for each step using the nine rules of arithmetic (see Strategies) as well as our solving equations properties (equals added to equals and equals multiplied by equals). After students have practiced many problems, we will discuss equations that look different (i.e. equations with variables and constants on both sides and equations that need to be simplified first). In these problems students also need to provide their justification steps. After students are very comfortable with different types of algebraic equations, they will be allowed to solve equations without justification. Finally students will apply their equation solving skills to word problems in which they are required to define variables, set up equations, solve the equations and explain what the answer represents (Appendix C, Equation Word Problems, 1-7)
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