Mathematics Success Level H

Transcription

1 T393 [OBJECTIVE] The student will solve two-step inequalities and graph the solutions on number lines. [MATERIALS] Student pages S132 S140 Transparencies T372 from Lesson 15, T405, T407, T409, T411, T413, T415 [ESSENTIAL QUESTIONS] 1. How does the solution of the inequality 2x x + 3 < 7 differ from the solution of the equation 2x x + 3 = 7? 2. Which operation(s) do we undo first in two-step inequalities? 3. How do you make an inequality a true statement when multiplying or dividing by a negative number? [WORDS FOR WORD WALL] inequality, inverse operation(s), isolate the variable, less than, greater than, inequality symbols (<, >,, ), number line [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Algebraic Formula, Graph, Verbal Description, Graphic Organizer [WARM-UP] (5 minutes IP, WG, I) S132 (Answers on T404.) Have students turn to S132 in their books to begin the Warm-Up. Students will solve one-step inequalities. Monitor students to see if any of them need help during the Warm-Up. Give students 4 minutes to complete the problems and then spend 1 minute reviewing the answers as a class. {Algebraic Formula} [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night.

2 T394 Mathematics Success Level H [LESSON] (50 60 minutes M, GP, IP, WG, CP) SOLVE Problem (2 minutes GP, WG) T405, S133 (Answers on T406). Have students turn to S133 in their books, and place T405 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to solve two-step inequalities. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description} Inequality Symbols (3 minutes M, GP, WG) T372 from Lesson 15 Place T372 from Lesson 15 on the overhead. Use the following activity to review comparing integers and to review the four inequality symbols (<, >,, ). {Verbal Description, Graphic Organizer, Algebraic Formula, Graph} Inequalities Step 1: Point out the location of zero on the number line. Tell students that the numbers to the right of zero are positive values and the values to the left of zero on the number line are negative values. Step 2: Explain to students that they can compare 4 and 0 using the mathematical symbol meaning greater than (>). On T372 write 4 is greater than 0 and then write the number sentence using the greater-than sign (>). 4 is greater than 0 4 > 0 Step 3: Explain to students that they can compare - 12 and 0 using the mathematical symbol meaning less than (<). On T372, write - 12 is less than 0 and then write the number sentence using the less-than sign (<) is less than 0-12 < 0 Step 4: Show students the inequality x 6. This means that x is less than or equal to 6. Show students the inequality x 3. This means that x is greater than or equal to 3.

3 T395 Two-Step Inequalities Introduction (8 minutes M, GP, WG) T405, S133 (Answers on T406.) Introducing Two-Step Inequalities Step 1: Have students look at Problem 1 on T405 (S133). Ask students what type of equation they are looking at (two-step equation). Ask, What is a two-step equation? (an equation that requires more than one operation to solve) Tell students that when solving two-step equations they will need to follow the order of operations (PEMDAS) in reverse order. Usually, multiplication and division are completed before addition and subtraction. However, in two-step equations, students will add or subtract before multiplying or dividing. Step 2: Work through the two-step equation on the overhead as students work in their books. Review the process of isolating the variable and balancing the equation. The answer for the equation in Problem 1 is x = 5. After completing the equation, complete the check. Step 3: Have students look at the problem in the second column, which is 2x 3 < 7. Ask students how this problem is different from the equation? (It has an inequality symbol instead of an equals sign.) Explain to students that inequalities can be solved using the same process as solving equations. Ask students what two things needed to happen each time they solved an equation (isolate the variable and balance the equation). Tell students that in solving inequalities they will also need to isolate the variable and that whatever they do to one side of the inequality they must also do to the other.

4 T396 Mathematics Success Level H Step 4: Ask students what step needs to be done first in the inequality. (Add 3 to both sides.) Model how to add 3 to both sides as students do the same. Ask students if the variable is isolated. (No.) Ask students what needs to be done to isolate the variable. (Divide both sides of the inequality by 2.) Model how to divide both sides by 2 as students do the same. Ask students if the variable is now isolated. (Yes.) Ask, Have we performed the same operations on both sides of the inequality? (Yes.) Step 5: Ask students what the value of x is in the inequality (x x < 5). 2x x 3 < x 2 < 10 2 x < 5 Step 6: Tell students that they can check the answer for an inequality using the same process as the one they used to check the answer for an equation. Have students look back at the equation in Problem 1. Ask students how many values there were for the variable x (1). Explain that in an inequality, there are often several values that will make the statement true. Have students look at the inequality they just solved. Tell students that the solution for the inequality was x < 5. Therefore, any value that is less than 5 should work when substituted back into the original inequality. Step 7: Model for students how to substitute the value of 1 back into the original inequality. Have students complete the check on S133 as you model on T405. Explain that the value of 1, which is less than 5, makes this a true statement because - 1 is less than 7. 2(1) 3 < < 7-1 < 7 True Step 8: Choose a value that is greater than 5, such as 7, to try in the original inequality. Explain that any value greater than 5 will make the inequality not true. 2(7) 3 < < 7 11 < 7 Not a true statement because 11 is not less than 7.

5 T397 Step 9: Have students look at the number line in the third column. Ask them what the solution was for the inequality (x x < 5). Have them write the solution in the third column on S133 as you model on T405. Step 10: Tell students that because there is more than one value that can be a solution to the inequality, the solution can be graphed using a number line. Step 11: Tell students that they will use the solution of the inequality (x < 5) to determine how to number the number line and how to graph the solution. Explain that because the solution contains a positive 5, students can place a 5 in the middle of the number line and label values to the left and right of the 5. Step 12: Model for students how to begin graphing the inequality by drawing an open circle above the 5 as they work on S133. Ask students if 5 is a solution to the inequality. (No, because x is less than 5.) Tell students that the circle above the 5 is open because 5 is not included in the solution. Step 13: Tell students that they now must decide which direction the arrow should point on the number line. Explain that because the solution set includes all the values less than 5, the arrow should point to the left. Model for students how to draw the arrow extended from the circle above the 5. Tell students that the arrowhead at the end of the line shows that the values for the solution will continue to negative infinity. Step 14: Use the steps above to model solving the equation and inequality in Problem 2.

6 T398 Mathematics Success Level H Two-Step Inequalities with Multiplication (8 minutes M, GP, IP, CP, WG) T407, S134 (Answers on T408.) 4 Minutes M, WG: Have students turn to S134 in their books, and place T407 on the overhead. Use the following activity to help students solve two-step inequalities with multiplication. Two-Step Multiplication Inequalities Step 1: Direct students attention to Problem 1. Ask students what they need to do first to solve the inequality. (Subtract 1 from both sides.) Model how to subtract 1 from both sides as students do the same in their books. 3x x + 1 < x x < 9 Step 2: Ask students if the variable is isolated. (No.) Ask, What inverse operation must be used to isolate the variable? (Division, because it is a multiplication problem.) Model dividing both sides by 3. 3x 3 < 9 3 x < 3 Step 3: Model for students how to check the problem by choosing a value for x that is less than 3. For this problem, substitute 2 back into the original inequality. 3(2) + 1 < < 10 7 < 10 True Step 4: Have students look at the number line in the third column. Ask them what the solution was for the inequality (x x < 3). Have them write the solution in the third column on S134 as you model on T407. Step 5: Model for students how to graph the solution of the inequality in the third column.

7 T399 3 minutes IP, CP: Have students work in partners to complete Problems 2 4 on S134. Students should solve each inequality, check its solution, and then graph the solution on the number line in the third column. {Algebraic Formula, Graphic Organizer, Graph} 1 minute WG: Have students come back together as a class and share their results. Solve Two-Step Division Inequalities (8 minutes M, GP, WG, IP, CP) T409, S135 (Answers on T410.) 4 minute M, WG: Have students turn to S135 in their books, and place T409 on the overhead. Use the following activity to help students solve two-step inequalities with division. Two-Step Inequalities with Division Step 1: Direct students attention to Problem 5. Point out that the division problem is written as a fraction with the fraction bar representing the division symbol. Ask students what they need to do first to solve the inequality. (Subtract 2 from both sides.) Model how to subtract 2 from both sides as students do the same in their books. x x 4 7 Step 2: Ask students if the variable is isolated. (No.) Ask, What operation must be used to isolate the variable? (Multiplication, because it is a division problem.) Model multiplying both sides by 4. (4) x 4 7(4) x 28 Step 3: Model for students how to check the problem by choosing a value for x that is greater than or equal to 28. For this problem, substitute 32 back into the original inequality True

8 T400 Mathematics Success Level H Step 4: Have students look at the number line in the third column. Ask them what the solution was for the inequality (x x 28). Have them write the solution in the third column on S135 as you model on T409. Step 5: Model for students how to graph the solution of the inequality in the third column. 3 minutes IP, CP: Have students work in partners to complete Problems 6 8 on S135. {Graphic Organizer, Algebraic Formula, Graph} 1 minute WG: Have students come back together as a class and share their results. Inequalities Multiplication and Division with Negative Numbers (4 minutes M, GP, WG) T411, S136 (Answers on T412.) Have students turn to S136 in their books, and place T411 on the overhead. Use the following activity to help students solve inequalities with negative numbers. Inequalities with Negative Numbers Step 1: Have students look at Problem 1. Ask students if this statement is true or false (true). Record. Step 2: In the third column, show how to multiply both sides by negative 2. Ask students if the statement 14 < - 18 is true or false (false). Record. Ask, What can be done to make the statement true? (Flip the inequality symbol.) Step 3: Model for students how to complete statements about what they did to make the inequality true. Step 4: Have students look at Problem 2. Ask students if this statement is true or false (true). Record. Step 5: In the third column, show how to divide both sides by negative 2. Ask students if the statement - 7 > - 1 is true or false (false). Ask, What can be done to make the statement true? (Flip the inequality symbol.) Step 6: Model for students how to complete statements about what they did to make the inequality true.

9 T401 Solve and Graph Inequalities Multiply and Divide with Negative Numbers (6 minutes M, GP, WG) T411, S136 (Answers on T412.) Use the following activity to help students solve and graph inequalities that include negative numbers with multiplication and division. Two-Step Inequalities with Negative Numbers x, Step 1: Have students look at Problem 3. Ask students what they need to do first to isolate the variable. (Add 4 to both sides.) Model how to add 4 to both sides of the inequality. Ask students if the variable is isolated. (No.) Ask, What operation needs to be done to isolate the variable? (division) Remind students that when they divide or multiply by a negative integer, they need to flip the inequality sign to make the inequality true. Model how to divide both sides of the inequality by x x x x - 10 Step 2: In the second column, model how to check the answer by substituting in a solution into the original inequality. Ask students to explain what they did during the process of solving the inequality. (Possible answer: Since they had to divide by a negative to solve the inequality, they flipped the symbol to make the inequality true.) Record. - 2(4) Step 3: In the third column, write the solution to the inequality and then model for students how to graph the solution on the number line. Review with students how to set up the number line using the solution as the middle value on the number line. Step 4: Have students look at Problem 4. Ask students what they need to do to isolate the variable. (Subtract 8 from both sides and then multiply both sides by negative 8.) Model how to isolate the variable as students do the same in their books. Remind students that when they divide or multiply by a negative integer, they need to flip the inequality sign to make the inequality true.

10 T402 Mathematics Success Level H x < ( - 8) x - 8 < - 11( - 8) x > 88 Step 5: In the second column, model how to check the answer by substituting in a solution into the original inequality. Ask students to explain what they did during the process of solving the inequality. (Possible answer: Since they had to multiply by a negative to solve the inequality, they flipped the symbol to make the inequality true.) Record < < < - 3 Step 6: In the third column, write the solution to the inequality and then model for students how to graph the solution on the number line. Review with students how to set up the number line using the solution as the middle value on the number line. Solve and Graph Inequalities (6 minutes IP, CP, WG) T413, S137 (Answers on T414.) 5 minutes IP, CP: Have students work in partners to solve and graph the inequalities for Problems 1 4 on S137. Remind students that when they are dividing or multiplying by a negative number, they will have to flip the inequality symbol at the end to make the inequality true. {Graphic Organizer, Algebraic Formula, Graph} 1 minute WG: Have students come back together as a class and share their results. {Graphic Organizer, Algebraic Formula, Graph}

11 T403 SOLVE Problem (3 minutes GP, IP) T415, S138 (Answers on T416.) Have students turn to S138 in their books, and place T415 on the overhead. Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (They will solve a two-step inequality.) {SOLVE, Verbal Description, Algebraic Formula} If time permits (10 minutes I, IP) S139 (Answers on T417.) Have students complete Problems 1 8 on S139 independently. {Graphic Organizer, Graph, Algebraic Formula} [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. How does the solution of the inequality 2x x + 3 < 7 differ from the solution of the equation 2x + 3 = 7? (An equation has only one solution and an inequality has many solutions). Which operation(s) do we undo first in two-step inequalities? (Addition or subtraction) How do you make an inequality a true statement when multiplying or dividing by a negative number? (When multiplying or dividing by a negative value, flip the inequality sign to make the solution true.) [HOMEWORK] Assign S140 for homework. (Answers on T418.) [QUIZ ANSWERS] T419 T420 The quiz can be used at any time as extra homework or to see how students did on two-step inequalities.