SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS

12 SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS INSTRUCTIONAL ACTIVITY Lesson 2 LEARNING GOAL Students will solve linear equations using concrete and semi-concrete models, algebraic procedures, and the concept of balance to determine and explain the number of solutions. The critical outcome of this activity is for students to solve linear equations in any form to determine how many solutions (if any) the equation has. In addition, students should explain what the solution means in terms of the variable value in the original equation. Students should continue justifying their work using the properties of equality. PRIMARY ACTIVITY Students will solve linear equations to determine the number of solutions and explain the following concepts: Conditional equation Contradiction (equation) Identity (equation) OTHER VOCABULARY Students will need to know Variable Solution Equation Conditional Contradiction Identity Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Distributive Property of Equality

13 u u Commutative Property of Equality Associative Property of Equality MATERIALS u u u u Materials to make a balance (binder clip, ruler, yarn/string, cups/bowls, and a pencil/pen); see FIGURE 1 Brown paper bags Items to use on balances (counters, nickels, pennies, paper clips, number cubes, pencils, etc.) INSTRUCTIONAL ACTIVITY STUDENT HANDOUT Figure 2. Example balance IMPLEMENTATION The focus of this activity is to identify and describe conditional, contradiction, and identity equations. Students will discover scenarios that indicate a contradiction or identity, first on a balance (concrete representation), then algebraically. Students should continue using Properties of Equality to justify their work. Begin the activity with a review of the previous lesson. Any equation with a solution is appropriate for an example. Solve the equation with students, asking for justification for each operation. Ask students to explain what the solution means once a value has been determined. (They should indicate that the number is the value that, when substituted for the variable, makes the equation true.) Explain to students that this is a conditional equation.

14 GUIDING QUESTIONS Determine if the student can SOLVE LINEAR EQUATIONS IN 1 VARIABLE WITH RATIONAL OR REAL NUMBER COEFFICIENTS: What are you trying to find by solving this equation? How would you start solving for the variable? Why? Describe each step you would take to solve for the variable. Which Properties of Equality justify each step you described? Determine if the student can EXPLAIN CONDITIONAL EQUATION: Does a conditional (linear) equation have a solution? If so, how many? What does the solution mean in terms of the original equation? The activity s introduction can be represented using physical objects or an online version (http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2. html) of a comparable situation. Activities with Algebra Tiles are also appropriate. Model the idea of a variable representing an unknown quantity by writing an x or another variable on the outside of brown paper bags filled with an equal number of counters or similar items. Write an equation such as 2x + 3 = 2x + 3 to introduce an identity equation, and model this on the balance. (This is an identity; therefore, the value of the variable can be any number.) Explain that the equation will be balanced if one side of the balance has the two bags with the unknown number of counters and three individual counters and the other side of the balance has two bags with the unknown number of counters and three individual counters. Direct students to construct a semi-concrete model of the equation on paper (in the INSTRUCTIONAL ACTIVITY STUDENT HANDOUT). An appropriate model is shown in the following figure.

15 GUIDING QUESTIONS Elicit student thinking: What do you notice about the expressions on each side of the balance? Determine if the student can REPRESENT EXPRESSIONS WITH NUMBERS AND/OR VARIABLES: How can you express 2x in your drawing? How can you express +3? Determine if the student can EXPLAIN EQUATION: What do you know about the expressions on either side of the equal sign? Begin solving as you would a conditional equation. Ask students what they would like to do to isolate one paper bag. Students may request to remove three counters or two variables from each side of the equation first (order does not matter). Eventually, all numbers and variables will subtract to zero and the balance (or circles in student drawings) will be empty. Emphasize that because the result is a true and balanced statement (e.g., 0 = 0), the variable can equal any real number. An example of the complete problem is shown in the following table.

16 SEMI-CONCRETE REPRESENTATION PROPERTY OF EQUALITY ALGEBRAIC REPRESENTATION Given Equation 2x + 3 = 2x + 3 Subtraction Property of Equality -2x -2x 3 = 3 Subtraction Property of Equality -3-3 0 = 0 x can equal any real number Students work should be complete and thorough as in the previous lesson. Show students that substituting various numbers (e.g., 5, 0, 8) will always produce true statements in the original equation; therefore, x can equal any real number. Model the equation on the balance by using any number of counters as the variable value. Emphasize this type of equation is an identity. Connect the definition of the word identity and the appearance of an identity equation (e.g., 2x + 3 = 2x + 3 or any expression equal to itself). Encourage students to discuss the connection in pairs and then report back to the class.

17 GUIDING QUESTIONS Elicit student thinking: How can we find out what x represents in this equation? Determine if the student can EXPLAIN THE STEPS TO SOLVE AN EQUATION: How do you decide what should be removed from each side of the balance? Determine if the student can EXPLAIN IDENTITY EQUATION: Does an identity (linear) equation have a solution? If so, how many? What does the solution mean in terms of the original equation? Repeat the activity by having students model the solving process with a second equation with drawings of the concrete representation, algebraic representation, and corresponding properties of equality. An example [3(x + 2) = 3x + 6] is included in the INSTRUCTIONAL ACTIVITY STUDENT HANDOUT. The following practice examples can be added. Those including large numbers, negative numbers, and fractions or decimals are more difficult to model concretely. NOTE: Combining like terms is allowed based on the Distributive Property of Equality and the Symmetric Property of Equality. Therefore, the Property of Equality that can be used as justification is Distributive Property of Equality. 6(x + 2) = 3(2x + 4) 1 + x + 4x = 5x 3 + 4 1 1 x + 2 + x = x + 9 2 2 7

18 GUIDING QUESTIONS Elicit student thinking: How would you approach this problem? What do you notice first? Determine if the student can USE PROPERTIES OF OPERATIONS TO GENERATE EQUIVALENT EXPRESSIONS: When you see multiplication of a quantity in parentheses, which property allows you to simplify? How does 3(2x + 4) (or a similar expression) simplify? What expression is equivalent to 1 + x + 4x? Are 1 + 5x and 5x + 1 equivalent expressions? Determine if the student can EXPLAIN IDENTITY EQUATION: Can you determine the number of solutions before you finish solving the equation? If so, what do you look for? What does it mean to say x can equal any real number in the equation? Write an equation such as 2x + 3 = 2x + 5 to introduce a contradiction equation and model this on the balance. (This is a contradiction; therefore, there is no real number that, when substituted for x, will balance the scale.) Model this equation on a balance and have students construct an equivalent model of the equation on paper (in the INSTRUCTIONAL ACTIVITY STUDENT HANDOUT). Ask students if they notice anything different about this equation from the beginning. They may state that it isn t balanced or note that the expressions both have the same variable term (2x) but different constants.

19 GUIDING QUESTIONS Elicit student thinking: How does this equation compare to the last one we saw? What is the same? What is different? Determine if the student can REPRESENT EXPRESSIONS WITH NUMBERS AND/OR VARIABLES: How can you express 2x in your drawing? How can you express +3 or +5? Determine if the student can EXPLAIN EQUATION: What should be true about the expressions on either side of the equal sign? Ask students what they would like to do to isolate one paper bag. Students may request to remove three counters or two variables from each side of the equation first (order does not matter). Eventually, all numbers and variables will subtract to zero on one side of the balance (or circles in student drawings) and there will be two counters left on the other side. Emphasize that because the result is not a true (or balanced) statement, there is no real number, substituted for x, that makes the equation true. An example of the complete problem is shown in the following table.

20 SEMI-CONCRETE REPRESENTATION PROPERTY OF EQUALITY ALGEBRAIC REPRESENTATION Given Equation 2x + 3 = 2x + 5 Subtraction Property of Equality -2x -2x 3 = 5 Subtraction Property of Equality -3-5 0 = 2 No real number, substituted for x, makes the equation true Students work should be complete and thorough as in the previous lesson. Show students that substituting various numbers (e.g., 5, 0, -8) will not produce true statements in the original equation; therefore, no real number, substituted for x can make the equation true. Model this on the balance by using any number of counters as the variable value (the scale will never be balanced). Emphasize this type of equation is a contradiction. Connect the definition of the word contradiction and the appearance of the initial equation (expressions with the same variable term but different constants). Encourage students to discuss the connection in pairs and then report back to the class.

21 GUIDING QUESTIONS Determine if the student can EXPLAIN THE STEPS TO SOLVE AN EQUATION: How do you know how much must be removed from each side of the balance? Determine if the student can EXPLAIN CONTRADICTION EQUATION: Does a contradiction (linear) equation have a solution? If so, how many? What does this mean in terms of the original equation? Repeat the activity by having students model the solving process with a second equation with drawings of the concrete representation, algebraic representation, and the corresponding properties of equality. An example [4x + 5 = 2(2x + 1)] is included in the INSTRUCTIONAL ACTIVITY STUDENT HANDOUT. The following practice examples can be added. Those including large numbers, negative numbers, and fractions or decimals are more difficult to model concretely. NOTE: Combining like terms is allowed based on the Distributive Property of Equality and the Symmetric Property of Equality. Therefore, the Property of Equality that can be used as justification is Distributive Property of Equality. 8(x 1) = 2(4x + 5) -3 + 7x x = 6x 3 + 6 1 5 x + 9 + x = 3x + 4 x 3 3

22 GUIDING QUESTIONS Determine if the student can USE PROPERTIES OF OPERATIONS TO GENERATE EQUIVALENT EXPRESSIONS: When you see multiplication of a quantity in parentheses, which property allows you to simplify? How does 8(x 1) (or a similar expression) simplify? What expression is equivalent to -3 + 7x x? Are -3 + 6x and 6x 3 equivalent expressions? Determine if the student can EXPLAIN CONTRADICTION EQUATION: Can you determine the number of solutions before you finish solving the equation? If so, what do you look for? What does it mean about the value of the variable in the original equation to say no real number, substituted for x, makes the equation true? At the end of the activity, Write an equation (conditional, identity, or contradiction) in any form. Ask students to determine what step can occur first (as well as the Property of Equality they will use) and use the idea of balance to equally adjust both sides. This process should continue until the variable is isolated (if possible). Ask students what the solution is and what it means about the variable value in the original equation. Repeat this activity with a few different equations, ensuring that equations in several forms and with different types of solutions are represented.

Name SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS Lesson 2 1) 2x + 3 = 2x + 3 SEMI-CONCRETE REPRESENTATION PROPERTY OF EQUALITY ALGEBRAIC REPRESENTATION Copyright 2015 by The University of Kansas 9