Inequalities. Assessment Management. Dividing Decimals by Decimals: Part 1

Transcription

1 Inequalities Objectives To find and represent all the values that make an inequality in one variable true; and to represent real-world situations with inequalities. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Translate between word and number sentences. [Patterns, Functions, and Algebra Goal 1] Identify relation symbols in number sentences. [Patterns, Functions, and Algebra Goal ] Determine whether inequalities are true or false. [Patterns, Functions, and Algebra Goal ] Recognize that most inequalities have an infinite number of solutions. [Patterns, Functions, and Algebra Goal ] Key Activities Students extend their work with equations to finding solution sets of inequalities. They practice solving inequalities by playing Solution Search. Students write and graph inequalities that represent real-world situations. Ongoing Assessment: Informing Instruction See page 599. Ongoing Assessment: Recognizing Student Achievement Use journal page 44. [Patterns, Functions, and Algebra Goal ] Key Vocabulary inequality solution set Materials Math Journal, pp B Student Reference Book, pp. 44 and 33 Study Link 6 11 per group: Math Masters, p. 473, complete deck of number cards (the Everything Math Deck, if available) scissors Dividing Decimals by Decimals: Part 1 Math Journal, pp. 4 and 43 Students solve division problems with decimal divisors by renaming the dividend and divisor. Math Boxes 6 1 Math Journal, p. 45 straightedge Students practice and maintain skills through Math Box problems. Study Link 6 1 Math Masters, p. 10 Students practice and maintain skills through Study Link activities. READINESS Reviewing Relation Symbols and Inequalities Math Masters, p. 11 Students review relation symbols and properties of number sentences. ENRICHMENT Graphing Compound Inequalities Math Masters, p. 1 Students graph compound inequalities and write inequalities to describe graphs. ENRICHMENT Playing Solution Search with Student-Created Cards index cards Students play Solution Search with self-created cards. Advance Preparation Plan to spend days on this lesson. For Part 1, make a copy of Math Masters, page 473 for every three or four students. Have students cut apart the game cards. Teacher s Reference Manual, Grades 4 6 pp Unit 6 Number Systems and Algebra Concepts

2 Getting Started Mathematical Practices SMP1, SMP, SMP4, SMP6 Content Standards 6.NS.3, 6.EE.5, 6.EE.6, 6.EE.8 Mental Math and Reflexes Students estimate products and quotients by showing thumbs-up for an answer greater than 1 and thumbsdown for an answer less than 1. Suggestions: thumbs-down 1 1_ thumbs-up 9_ 8 1_ 4 thumbs-down 1_ 3_ 8 thumbs-up thumbs-down thumbs-up If time permits, compare the dividend and quotient when the divisor is less than 1 and when it is greater than 1. Math Message Tell whether each inequality is true or false false true 3. 54_ 9 > 7 false true Work with a partner to write an inequality that is neither true nor false. x + 3 < 6 Study Link 6 11 Follow-Up Go over the answers with the class. Ask volunteers to share solution strategies. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Algebraic Thinking Problems 1 4 of the Math Message review the relation symbols discussed in Lesson 6-7. Briefly go over the answers. Then ask students to share the inequalities they wrote. Some may think the task is impossible. Others may extend the concept of open sentences to include inequalities and use a variable. For example, the inequality x + 3 < 6 is neither true nor false until a value is assigned to x. If x =, then the inequality is true; if x = 3, then the inequality is false. Introducing Solution Sets to Inequalities (Student Reference Book, p. 44) ELL WHOLE-CLASS DISCUSSION Algebraic Thinking During the discussion, write the key ideas along with examples on the board to support English language learners. While most equations students have solved so far have had only one solution, many inequalities have an infinite number of solutions. To demonstrate this point, write x < on the board. Have volunteers help you generate a list of possible solutions. Include rational and irrational numbers in this list (for example, 1.835, -1 3_ 5, -π, - _, 0. 3 ). When students recognize that listing all solutions of x < is not viable, discuss the following options for representing all the values that make an inequality true: Lesson

3 Describe the solution set of an inequality in words. For example: The solution set of x < consists of all real numbers less than. Graph the solution set of an inequality on a number line. Include the following steps in your discussion about graphing the solution set of x <. Step 1: On a number line, draw an open circle at. An open circle shows that is not a solution but numbers close to are Step : Plot three solutions of x < on the number line Step 3: Start at the open circle and draw a line that is thick enough to cover the three plotted solutions. Draw an arrowhead to show that there are an infinite number of solutions. Then use one additional solution to check your graph. Date 6 1 Inequalities 1. Name two solutions of each inequality. 10, 1 56, 60, 1 a. 15 > r b. 8 < m c. t 56 d p e. 1_ > y f. w > -3 7 g. 6.5 > 3 d 0, 1 h. g < 0.5. Name two numbers that are not solutions of each inequality. a. (7 + 3) q > 40 1, 0.5 b. 1_ + 1_ 4 < t 10, 100 c. y d. 6 g > 1 3. Describe the solution set of each inequality. Sample answers: 9, 1 4, 5-1, 0.4, Sample answers 0, - 3, For example, -4 7_ is included in the graph and -47_ is less 8 8 than. So, the graph is correct. If an inequality contains the relation symbol <, >, or, the endpoint of the graph is shown with an open circle, indicating that the circled point is not part of the solution set. If an inequality contains the relation symbol or, the endpoint of the graph is shown with a closed, or filled-in circle, indicating that the circled point is part of the solution set. Have students work in pairs to name three solutions of an inequality and then graph that inequality on a number line. Suggestions: y -3 Sample solutions: -3, -1, m - > 0 Sample solutions:.5, 3, 5 Example: t + 5 < 8 Solution set: All numbers less than 3 a. 8 - y > 3 Solution set b. 4b 8 Solution set 4. Graph the solution set of each inequality. a. x < 5 All numbers less than 5 All numbers greater than or equal to n 5 Sample solutions: -, 4.9, b. 6 > b c. 1 1_ h Math Journal, p Before assigning independent journal practice, work with students to solve the Check Your Understanding problems at the bottom of page 44 of the Student Reference Book. 598 Unit 6 Number Systems and Algebra Concepts

4 Ongoing Assessment: Informing Instruction Some students may recognize that for inequalities such as x < and y -3, the direction of the relation symbol and the arrowhead on the graph are the same. Encourage students to use more reliable methods for deciding which part of the number line they should include in their graphs. Solving Inequalities (Math Journal, p. 44) PARTNER PROBLEM SOLVING Algebraic Thinking Circulate and assist as students work in pairs to complete the problems on the journal page. Remind students to use a solution to check each graph. Ongoing Assessment: Recognizing Student Achievement Journal Page 44 Problems 1 and Use journal page 44, Problems 1 and to assess students ability to determine whether inequalities are true or false. Students are making adequate progress if they are able to complete Problems 1 and. Some students may be able to describe the solution sets in Problem 3. [Patterns, Functions, and Algebra Goal ] Links to the Future Graphing solution sets of inequalities with one variable is not a Grade 6 Goal. The graphing activities in this lesson provide students with the skills they will need in future algebra courses. Playing Solution Search (Student Reference Book, p. 33; Math Masters, p. 473) SMALL-GROUP Algebraic Thinking Go over the directions for Solution Search. You might want to play a few rounds of the game with three volunteers while the rest of the class observes. Use this opportunity to answer any questions regarding rules of play. Then students play Solution Search in groups of three or four. Using Inequalities to Describe Real-World Situations WHOLE-CLASS DISCUSSION Present this situation to students: An amusement park s rules state that a guest must be at least 4 feet tall to ride a roller coaster. Ask students to generate a list of possible heights of guests who could ride the roller coaster. Sample answers: 4 feet; 5 feet 1_ inches; 6 feet 1.75 inches After students realize that they cannot list all the possible heights, ask them to describe all the possible heights in words. All heights greater than or equal to 4 feet Lesson

5 Date 6 1 Inequalities in Real-World Contexts Each situation below can be described by an inequality. Follow the steps to identify an appropriate inequality and draw a graph that fits the situation. 1. A city ordinance says that fences can have a maximum height of 6 feet. a. List several allowable fence heights. Sample answers: 3 feet; 5 feet 9 inches; 4 and a half feet b. Describe in words the set of all allowable fence heights. All positive heights that are 6 feet or less c. Use an inequality or two inequalities to describe the set of allowable heights. h 6 and h > 0 d. Graph the set of all allowable heights on the number line below e. Do the values represented on the graph make sense in the situation? Explain your answer. Yes. A height could be any positive number, but negative heights or a height of zero do not make sense. It is unlikely anyone would have a fence that is 0.1 feet high, but since we don t know the minimum, it makes sense to end the graph at 0.. A CD player can hold up to 8 CDs. a. List several possible numbers of CDs in the player. Sample answers: 8 CDs, 5 CDs, 0 CDs b. Describe in words the set of possible numbers of CDs in the player. Any whole number between and including 0 and 8 CDs c. Use an inequality or two inequalities to describe the possible numbers of CDs. n 0 and n 8 d. Graph the set of all possible numbers of CDs on the number line below e. Do the values represented on the graph make sense in the situation? Explain your answer. Yes. You can t have a negative number of CDs or part of a CD, so I only showed dots on 0 and positive whole numbers. Math Journal, p. 44A Date 6 1 Inequalities in Real-World Contexts continued 3. The manager of an apartment building turns on the heat when the outside temperature drops below 50 F. a. List several possible outside temperatures when the heat is on. Sample answers: 45 F; 15 F; 10 F b. Describe in words the set of outside temperatures when the heat is on. All temperatures less than 50 F c. Use an inequality or two inequalities to describe the possible outside temperatures when the heat is on. t < 50 d. Graph the set of all possible temperatures on the number line below e. Do the values represented on the graph make sense in the situation? Explain your answer. Not all of the values make sense. You can have negative temperatures and parts of degrees, but the temperature will never be 1,000 F. Since I don t know the minimum, this is the best graph I can draw. 4. a. Describe your own real-world situation that can be modeled by an inequality. Answers vary. b. List several possible values for your situation. Answers vary. c. Describe the set of possible values in words and with an inequality or two. Answers vary. d. Graph the set of all the possible values on the number line below. Answers vary. e. Do the values represented on the graph make sense in the situation? Explain your answer. Answers vary. Next, ask students to use an inequality to represent all the possible heights and have a volunteer graph the inequality. h 4 feet The graph should look like the number line below Ask: Do all the solutions to this inequality make sense in this situation? No. For example, a person cannot be 0 feet tall. Explain to students that it is important to always keep the context in mind when using inequalities to represent real-world situations. In this example, it is not possible to say exactly what the maximum height of a person is, so the best we can do is leave the graph as it is. However, it is important to acknowledge that the inequality is not a perfect model. Tell students that in other situations, it is sometimes possible to draw a graph that better represents the real-world situation. Outline the process that students just followed to describe a real-world situation with an inequality. List several possible values that fit the situation. Describe the set of all possible values in words. Represent all the possible values with an inequality or inequalities. Graph the possible values on a number line. Check by asking whether the solutions on the graph make sense in the real-world situation. Next, present the following situation: A restaurant allows children under the age of 10 to eat from the kids menu. Guide students through the process above. Ask students to list several possible ages of children who can eat from the kids menu. Sample answers: 3 years; 4 and a half years; 5 years and 9 months; years, 6 months, and 8 days Next, ask students to describe all the possible ages in words. All ages less than 10 years old Ask students how they might describe the solution set using an inequality. Encourage students to keep the context in mind. Ask: Can a child have an age that is not a whole number of years? Yes. A child can be part of a year old. Can a child have a negative age? No. Age is positive. If students suggest a < 10, ask them whether this accurately represents the fact that age cannot be negative. If nobody mentions it, suggest that students might use two inequalities to represent the possible ages: a < 10 and a > 0. NOTE The restaurant situation could also be represented by the compound inequality 0 < a < 10. Acknowledge students who point this out, but do not insist that they write this inequality. Keep the focus of this activity on writing simple inequalities and drawing graphs that are appropriate for the situation. See the Enrichment activity in Part 3 for work with compound inequalities. Math Journal, p. 44B 599A Unit 6 Number Systems and Algebra Concepts

6 Ask a volunteer to graph the inequalities a < 10 and a > 0 together on one number line. The graph should show only the numbers that satisfy both inequalities, as shown on the number line below Ask students whether the ages represented on the graph make sense in the real-world context. Yes. The graph shows all ages between 0 and 10 years. Present one more situation to students: You may have no more than 6 books checked out of the library at one time. Again, guide students through the process above to help them describe the possible values in this situation. List several possible numbers of books. Sample answers: 6, 3, 0, Describe the possible numbers of books in words. All whole numbers between and including 0 and 6. You cannot check out a part of a book, nor can you check out a negative number of books. Describe the solution set with an inequality or inequalities. b 6; b 0 Graph the inequalities on a number line. Ask students how they would indicate on the graph that only whole numbers are included. If no one suggests it, show students how to draw dots at each possible value, as shown below. Date 6 1 Dividing Decimals by Decimals: Part 1 When you multiply the numerator and denominator of a fraction by the same nonzero number, you rename the fraction without changing its value. For example: , so In general, you can think of a fraction as a division problem. The fraction 3 5 equals 3 5, or 53. The numerator of the fraction is the dividend; the denominator is the divisor. As you do with a fraction, you can multiply the dividend and divisor by the same number without changing the value of the quotient. Study the patterns in the table below. Fraction Division Problem Quotient What do you notice about the quotient when you multiply the dividend and the divisor by the same number? The quotient stays the same. Rename each division problem so the divisor is a whole number. Then solve the equivalent problem using partial-quotients division or another method. Example: , Math Journal, p Check that the values represented on the graph make sense in the real-world context. Yes. It shows 0, 1,, 3, 4, 5, or 6 books could be checked out from the library. Graphing Inequalities in Real-World Contexts (Math Journal, pp. 44A and 44B) PARTNER Date 6 1 Dividing Decimals by Decimals continued Rename each division problem so the divisor is a whole number. Then solve the equivalent problem using partial-quotients division or another method. 149, ,40 Circulate and assist as students work in pairs to complete the journal pages. Encourage partnerships to discuss how the graph will look before starting to draw each graph., , , Math Journal, p. 43 Lesson B

7 Date 6 1 Math Boxes 1. Find the solution to each equation. a. y y b. 45 3n 45 n c. 7p 19 p 55 p d. 7 4w 10w w 3. Complete the table. Then graph the data and connect the points. (hr) Distance (mi) Rebecca walks at an average speed of 3 1 miles per hour. Rule: Distance 3 1 mph number of hours (h) (3 1 º h) Solve the pan-balance problem One weighs as much as 1 marbles. Distance (mi) Rebecca s Walks Ongoing Learning & Practice Dividing Decimals by Decimals: Part 1 (Math Journal, pp. 4 and 43) INDEPENDENT Students rename decimal-number divisors as whole-number divisors while maintaining the value of the quotient. They solve equivalent problems using partial-quotients division or method of their choice. Students complete Part in Lesson (hr) 4. The spinner at the right is divided into 5 equal parts. Suppose you spin this spinner 75 times. About how many times would you expect the spinner to land on a prime number? 45 times Math Journal, p Serena keeps 4 stuffed animals lined up on a shelf over her bed. How many different arrangements of the stuffed animals are possible? (Hint: Label the animals A, B, C, and D. Then make a list.) 4 arrangements 156 Math Boxes 6 1 (Math Journal, p. 45) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson The skills in Problems 4 and 5 preview Unit 7 content. Study Link 6 1 (Math Masters, p. 10) INDEPENDENT Home Connection Students review some of the major concepts of the unit. They name solutions and then graph solution sets of inequalities. Study Link Master 3 Differentiation Options Name Date STUDY LINK 6 1 Review 1. Write a number sentence for each word sentence. Word sentence a. 15 is not equal to 3 times 7. b. 5 more than a number is 75. c. 13 more than 9 divided by 9 is less than or equal to 14.. Insert parentheses to make each equation true. Number sentence 15 3 º 7 x READINESS Reviewing Relation Symbols and Inequalities (Math Masters, p. 11) PARTNER 5 15 Min a. 00 (4 º 5) 10 b. 16 (5 3) 1 3. Use the order of operations to evaluate each expression. a. 5 º 6 8 º b. 0 8 c º d. 4 (4 º ) 3 º 4. Solve each equation. a. 3x 5 5x 3 b. (4y 5) y 9 Solution Solution 5. Name three solutions of the inequality. Then graph the solution set. a. f 3 x f 3, f 1, f 18 y To provide experience identifying relation symbols and properties of number sentences, have students work with a partner to complete Math Masters, page 11. This activity is preparation for finding and graphing solution sets of inequalities. Practice $.5 1 Estimate Quotient Sample estimates given. $0.5 $ Estimate Quotient ,70 Estimate Quotient 781 Math Masters, p Unit 6 Number Systems and Algebra Concepts

8 Teaching Master ENRICHMENT INDEPENDENT Graphing Compound Inequalities (Math Masters, p. 1) Min To further explore compound inequalities, students graph the solution sets of compound inequalities and write compound inequalities to describe graphs. For Problem 4, remind students that all square roots have a positive and a negative value; that is, _ 9 = 3 or -3, because 3 3 = 9 and -3-3 = 9. Students can use the word and when writing the inequalities for Problems 5 7. For example, the graph in Problem 5 represents the solution set x -8 and x 5. Name Date 6 1 Reviewing Relation Symbols and Inequalities 1. Translate between word and number sentences. Word sentence a. 7 9 is greater than 3. Number sentence b c. 0 is greater than or equal to 5 less than 5 squared. d. The product of 4 and 19 is less than 80. e. 6 plus a number y is greater than 8. f. 19 is not equal to 54 divided by 3. is less than or equal to the quotient of a number x and 17.. Indicate whether each inequality is true or false. false false true a. 5 º 4 0 b. (7 3) º 6 60 c. 54 / 9 7 d º 5 e f. 18 º y 8 false true true x Are the inequalities and equivalent? Explain. Yes. Both the symbols and the sides of the inequality have been reversed, so the inequalities are equivalent. ENRICHMENT SMALL-GROUP Playing Solution Search with Student-Created Cards Min Math Masters, p. 11 To extend Solution Search, students create sets of Solution Search cards and play the game using these cards. Teaching Master Name Date 6 1 Graphing Compound Inequalities Graph all solutions of each inequality x < 8 (Hint: -1 x < 8 means x -1 and x < 8. For a number to be a solution, it must make both number sentences true.). 3 < y m - 4. x 9 Write an inequality for each graph. 5. x -8 and x 5 OR -8 x 5 6. x -7 and x < 1 OR -7 x < 1 7. x < 1 and x > 1 OR x 1 Math Masters, p. 1 Lesson