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Algebraic Expressions Objective To introduce the use of algebraic expressions to represent situations and describe rules. www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Key Concepts and Skills • Identify and use patterns in tables to solve problems. [Patterns, Functions, and Algebra Goal 1] • Write algebraic expressions to model rules. [Patterns, Functions, and Algebra Goal 1] • Use variables to write number models that describe situations. [Patterns, Functions, and Algebra Goal 2] Key Activities Students complete statements in which the variable stands for an unknown quantity. They state the rule for “What’s My Rule?” tables in words and with an algebraic expression. Ongoing Assessment: Recognizing Student Achievement Use journal page 341. [Patterns, Functions, and Algebra Goal 2] Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice 1 2 4 3 Playing Name That Number Curriculum Focal Points Interactive Teacher’s Lesson Guide Differentiation Options READINESS Student Reference Book, p. 325 per partnership: 1 complete deck of number cards (the Everything Math Deck, if available) Students apply number properties, equivalent names, arithmetic operations, and basic facts. Exploring “What’s My Rule?” Tables Math Boxes 10 3 Student Reference Book, p. 218 Students choose variables to write algebraic expressions. Math Journal 2, p. 344 compass Geometry Template or ruler Students practice and maintain skills through Math Box problems. Study Link 10 3 Math Masters, p. 299 Students practice and maintain skills through Study Link activities. Key Vocabulary algebraic expression Math Masters, p. 300 Students use patterns in tables to solve problems. EXTRA PRACTICE Writing Algebraic Expressions ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students define and illustrate the term algebraic expression. ENRICHMENT Analyzing Patterns and Relationships Math Masters, p. 300A Students analyze patterns and relationships. Materials Math Journal 2, pp. 341–343 Student Reference Book, p. 218 Study Link 102 Class Data Pad slate Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 278–289 Lesson 10 3 797 Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP6, SMP7, SMP8 Content Standards Getting Started 5.OA.1, 5.OA.2, 5.OA.3, 5.NBT.2, 5.NBT.5, 5.MD.5a, 5.MD.5b Mental Math and Reflexes Math Message Ava, Joe, and Maria are 5th graders. Ava is 1 centimeter taller than Joe, and Joe is 2 centimeters taller than Maria. Make a table of 4 possible heights for Ava, Joe, and Maria. Students solve extended multiplication and division facts problems involving powers of 10. Write the problems on the board or Class Data Pad. 6 ∗ 102 = 600 0.56 ÷ 102 = 0.0056 0.254 ∗ 103 = 254 36.5 ÷ 103 = 0.0365 7.538 ∗ 10 = 753.8 8 ÷ 102 = 0.08 2 Heights Ava Joe Maria Study Link 10 2 Follow-Up 4.3 ∗ 103 = 4,300 Have partners compare answers and resolve differences. 7.6 ÷ 10 = 0.76 24 ∗ 107 = 240,000,000 1 Teaching the Lesson ▶ Math Message Follow-Up WHOLE-CLASS DISCUSSION Algebraic Thinking On the Class Data Pad, draw and label the table for the Math Message. Ask students to share heights listed in their tables. Record them on the Class Data Pad table. Ask: How tall is Ava? Expect that students will respond with one of the heights from the table. Ask them to explain their reasoning. Sample answer: Ava’s height is 1 centimeter more than Joe’s. How did you determine Joe’s height? Sample answer: Joe is 2 cm taller than Maria. How tall is Maria? Sample answer: Maria’s height isn’t given, so I picked a likely height for a fifth grader. Student Page Emphasize that Ava’s height depends on Joe’s height, and Joe’s depends on Maria’s. We only know that Ava is 1 cm taller than Joe, and that Joe is 2 cm taller than Maria. If Maria could be any height, then there would be an infinite number of possibilities for Joe’s and Ava’s heights. If Maria is 147.5 cm tall, then Joe’s height is 149.5 cm, and Ava’s height is 150.5 cm. If Maria is 1.48 m tall, then Joe is 1.50 m tall, and Ava is 1.51 m tall. Algebra Algebraic Expressions Variables can be used to express relationships between quantities. Claude earns $6 an hour. Use a variable to express the relationship between Claude’s earnings and the amount of time worked. If you use the variable H to stand for the number of hours Claude worked, you can write his pay as H ∗ 6. H ∗ 6 is an example of an algebraic expression. An algebraic expression uses operation symbols (+, -, ∗, , and so on) to combine variables and numbers. Write the statement as an algebraic expression. Statement Marshall is 5 years older than Carol. Algebraic Expression If Carol is C years old, then Marshall’s age in years is C + 5. Evaluating Expressions Some algebraic expressions: 2-x m∗m C+5 6∗H (C + 5) (6 ∗ H) Ask: How does Ava’s height compare with Maria’s height? Ava is 3 cm taller than Maria. If Maria is 151.5 cm tall, how tall is Ava? 154.5 cm If Ava is 150.2 cm tall, how tall is Maria? 147.2 cm Other expressions that are not algebraic: 7+5 6 ∗ 11 (7 + 5) (6 ∗ 11) To evaluate something is to find out what it is worth. To evaluate an algebraic expression, first replace each variable with its value. ▶ Introducing Algebraic Evaluate each algebraic expression. 6∗H x∗x∗x If H = 3, then 6 ∗ H is 6 ∗ 3, or 18. If x = 3, then x ∗ x ∗ x is 3 ∗ 3 ∗ 3, or 27. Expressions WHOLE-CLASS DISCUSSION ELL (Student Reference Book, p. 218) Write an algebraic expression for each situation using the suggested variable. 2. Toni runs 2 miles every day. How 1. Alan is A inches tall. If Barbara is 3 inches shorter than Alan, what is many miles will she run in D days? Barbara’s height in inches? Algebraic Thinking On the board or Class Data Pad, make a table of just Maria’s and Joe’s heights. Point out that it is similar to a “What’s My Rule?” table. Label Maria’s Height in and Joe’s Height out. Ask: What is the rule for this table? out = in + 2 What is the value of each expression when k = 4? 3. k + 2 4. k ∗ k 5. k 2 6. k 2 + k - 2 Check your answers on page 440. Student Reference Book, p. 218 215_234_EMCS_S_SRB_G5_ALG_576515.indd 218 798 Unit 10 3/8/11 5:09 PM Using Data; Algebra Concepts and Skills Ask volunteers to represent Joe’s height using an algebraic expression. Let M represent Maria’s height in inches. Then M + 2 represents Joe’s height in inches. Add M and M + 2 to the column headings in the table on the Class Data Pad. Review and discuss Student Reference Book, page 218. Have students study the examples of expressions that are algebraic and those that are not algebraic. Ask students to explain how the examples are similar and how they are different. Expressions use operation symbols (+, -, ∗, ÷) to combine numbers, but algebraic expressions combine variables and numbers. Tell students that it is important to remember the following points. To support English language learners, write the statements on the board: A situation can often be represented in several ways: in words, in a table, or in symbols. Algebraic expressions use variables and other symbols to represent situations. To evaluate an algebraic expression means to substitute values for the variable(s) and calculate the result. Ask students to propose algebraic expressions to fit simple situations. To support English language learners, write the situations and respective expressions on the board. For example: Sue weighs 10 pounds less than Jamal. If J = Jamal’s weight, then J - 10 represents Sue’s weight. Isaac collected twice as many cans as Alex. If A = the number of cans Alex collected, then 2 ∗ A, or 2A, represents the number of cans Isaac collected. There are half as many problems in today’s assignment as there were in yesterday’s. If y = the number of problems in y 1 y, _ 1 ∗ y, or _ yesterday’s assignment, then there are _ problems 2 2 2 in today’s assignment. Pose the following problems. Ask students to write an algebraic expression for each problem on their slates. ● Six times the sum of 9 and some number 6 (9 + n) ● 10 times the product of a number and 6 10 (n ∗ 6) ● Triple the sum of a number and 20 3 (n + 20) ● 10 less than a number n - 10 ● 7 less than the product of a number and 6 n ∗ 6 - 7 Algebraic expressions can be combined with relation symbols (=, <, >, and so on) to make number sentences. For example, x + 2 = 15, or 3y + 7 < 100. Ask volunteers for the name of number sentences that contain algebraic expressions. Algebraic equations Links to the Future Pictures, diagrams, and graphs are important ways to represent situations, and they are discussed throughout Everyday Mathematics. Graphs in an algebra context are discussed in Lesson 10-4. Lesson 10 3 799 Student Page Date Time LESSON Algebraic Expressions 10 3 䉬 Complete each statement below with an algebraic expression, using the suggested variable. The first problem has been done for you. 1. First Hometown Bank If Leon gets a raise of $5 per week, then his salary is S ⫹ $5 Beth 141 Pay to the order of: Leon Amount: S dollars The Boss When most students have finished, discuss students’ answers, and point out that there are often several ways to write an algebraic expression. The answer to Problem 7 can be written 10 + (5 ∗ D), 5D + 10, or 10 + 5D. The answer to Problem 8 can be written 1 ∗ X, _ 1 X, or _ X. _ 3 3 3 Leon’s salary is S dollars per week. . If Ali’s grandfather is 50 years older than Ali, then Ali is G ⫺ 50 4. PROBLEM PRO P RO R OBL BLE B LE L LEM EM SO S SOLVING OL O LV VIN IIN NG Algebraic Thinking Go over Problem 1 on journal page 341. Partners then complete the statements on journal pages 341 and 342. . Kesia’s allowance is D dollars. 3. (Math Journal 2, pp. 341 and 342) PARTNER ACTIVITY If Beth’s allowance is $2.50 more than Kesia’s, then Beth’s allowance is D ⫹ $2.50 2. ▶ Writing Algebraic Expressions 夹 years old. Ali’s grandfather is G years old. Ali Ask students for the algebraic equation they would write to represent Problem 1. B = D + $2.50 Have students choose and write the algebraic equation for two problems. They should write their equations in the space beneath the problem answer line. Seven baskets of potatoes weigh 7 ⴱ P, or 7P A basket of potatoes weighs P pounds. pounds. Ongoing Assessment: Recognizing Student Achievement Math Journal 2, p. 341 Journal Page 341 Use journal page 341 to assess students’ ability to write algebraic expressions that model situations. Students are making adequate progress if they correctly identify and write the expressions for Problems 3 and 4. Some students will correctly write the algebraic equations for the two problems of their choice. [Patterns, Functions, and Algebra Goal 2] ▶ Expressing a Rule as an PARTNER ACTIVITY Algebraic Expression (Math Journal 2, p. 343) Algebraic Thinking Have students complete journal page 343. Everyday Mathematics students are very familiar with “What’s My Rule?” tables because of their experiences with them starting in first grade. Use the Readiness activity in Part 3 with students who do not understand “What’s My Rule?” tables. Student Page Date Time LESSON Algebraic Expressions 10 3 5. If a submarine dives 150 feet, then it will be traveling at a depth of X + 150 continued X ft feet. A submarine is traveling at a depth of X feet. 6. _1 * A, _1 A, or _A 5 7. 5 5 ft 2. The gym floor has an area of A square feet. ▶ Playing Name That Number Sports Heroes Students practice applying number properties, equivalent names, arithmetic operations, and basic facts by playing Name That Number. Encourage students to find number sentences that use all 5 numbers and to use numbers as exponents or to form fractions. A library charges 10 cents for each overdue book. It adds an additional charge of 5 cents per day for each overdue book. 2 If Kevin spends _ 3 of his allowance on a book, then he has _1 * X, _1 X, or _X 3 3 3 dollars left. Kevin’s allowance is X dollars. Math Journal 2, p. 342 EM3MJ2_G5_U10_333-368.indd 342 800 Unit 10 PARTNER ACTIVITY (Student Reference Book, p. 325) The charge for a book that is D days overdue is 10 + (5 * D), 5D + 10, cents. or 10 + 5D 8. 2 Ongoing Learning & Practice The floor is divided into 5 equal-size areas for gym classes. Each class has a playing area of 4/1/10 1:04 PM Using Data; Algebra Concepts and Skills Student Page ▶ Math Boxes 10 3 INDEPENDENT ACTIVITY (Math Journal 2, p. 344) Date Time LESSON “What’s My Rule?” 10 3 1. a. State in words the rule for the “What’s My Rule?” table at the right. X Y 5 1 Subtract 4 from X. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-1. The skill in Problem 4 previews Unit 11 content. Writing/Reasoning Have students write a response to the following: Explain the strategy and reasoning you would use to solve Problem 3b with the standard multiplication algorithm. Answers vary. b. INDEPENDENT ACTIVITY (Math Masters, p. 299) 1 -3 2 -2 Q Z 1 3 Y= 4 -X 3. a. 3 5 -4 -2 -3 -1 -2.5 -0.5 g 1 _ t Circle the number sentence that describes the rule. 1 ∗ Z=_ 2Q 1 Z=2∗Q State in words the rule for the “What’s My Rule?” table at the right. Multiply g by 4. 2 2 0 0 2.5 10 1 _ 1 4 5 20 Circle the number sentence that describes the rule. g=2∗t Y=X-4 State in words the rule for the “What’s My Rule?” table at the right. Z=Q+2 b. ▶ Study Link 10 3 -5 Add 2 to Q. b. Writing/Reasoning Have students write a response to the following: Explain how you found the volume for Problem 5. I knew the base (B) of the prism was 42 cm2 and the height (h) of the prism was 3 cm. I used the formula B ∗ h to calculate the volume (V ): 42 ∗ 3 = 126 cm3. 0 -1 Circle the number sentence that describes the rule. Y=X/5 2. a. 4 t=2∗g t=4∗g Math Journal 2, p. 343 333-368_EMCS_S_G5_MJ2_U10_576434.indd 343 2/22/11 5:21 PM Home Connection Students complete statements with algebraic expressions. They write the rule and identify the related number sentence for “What’s My Rule?” tables. 3 Differentiation Options READINESS ▶ Exploring “What’s My Rule?” PARTNER ACTIVITY 5–15 Min Tables (Math Masters, p. 300) Algebraic Thinking To provide experience with using patterns in tables to solve problems, have students make rules and then complete the related table. Partners work together to complete each of their Math Masters pages. When students have finished, discuss the rules and tables they made. Ask partners to share what they think is important to remember when solving “What’s My Rule?” tables. Sample answers: If you know the in value, follow the rule to find the out value. If you know the out value, do the opposite of the rule to find the in value. Student Page Date Time LESSON 10 3 1. Math Boxes Identify the point named by each ordered number pair. a. (0,4) b. (3,3) c. (5,4) d. (4,0) y 5 B A D C D B 4 A 3 2 1 0 C 0 1 2 3 4 x 5 208 2. Add or subtract. a. b. c. d. e. 3. 10 -8 + (-17) = -25 0 -12 – (-12) = 0 -45 + 45 = -31 - 14 = -45 20 + (-10) = Multiply. Use the algorithm of your choice. Show your work. a. 43 * 78 3,354 b. 19 * 86 c. 1,634 79 * 42 3,318 92 4. a. Draw a circle that has a diameter of 4 centimeters. 19 5. The rectangular prism below has a volume 153 164 of 126 cm. 3 (unit) 3 cm Area of base 42 cm 2 Write a number model for the formula. b. The radius of the circle is 2 cm . 42 ∗ 3 = 126 cm3 197 Math Journal 2, p. 344 333-368_EMCS_S_MJ2_G5_U10_576434.indd 344 3/22/11 12:43 PM Lesson 10 3 801 Study Link Master Name Date STUDY LINK Time EXTRA PRACTICE Writing Algebraic Expressions 10 3 Complete each statement below with an algebraic expression, using the suggested variable. 1. ▶ Writing Algebraic Expressions Augusto Algebraic Thinking Students complete the Check Your Understanding problems on Student Reference Book, page 218. Ask students to write algebraic expressions for the first two problems. Lamont and Mario harvested L + M carrots. carrots. Rhasheema and Alexis have a lemonade stand at their school fair. They promise to donate one-fourth of the remaining money (m) after they repay the school for lemons (l ) and sugar (s). So the girls donate _1 ∗ (m - (l + s)), or _1 (m - (l + s)) 4 4 dollars. 3. a. State in words the rule for the “What’s My Rule?” table at the right. Multiply N by 3 and add 5. b. 4. a. N Q 2 11 4 17 Circle the number sentence that describes the rule. 6 23 Q = (3 + N) ∗ 5 8 29 10 35 Q = 3 ∗ (N + 5) Q = 3N + 5 State in words the rule for the “What’s My Rule?” table at the right. E R 7 57 Multiply E by 6 and add 15. b. Circle the number sentence that describes the rule. R = E ∗ 6 ∗ 15 R = (E ∗ 6) + 15 10 75 31 201 R = E ∗ 15 + 6 3 33 108 663 ELL SUPPORT ▶ Building a Math Word Bank 576 120 _37 384 ∗ 1.5 = 6. 50.3 ∗ 89 = 7. 843 _ = 8. 70.4 / 8 = 7 5–15 Min To provide language support for algebra concepts, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the term algebraic expression, draw pictures relating to the term, and write other related words. See the Differentiation Handbook for more information. 4,476.7 8.8 Math Masters, p. 299 294-322_439_EMCS_B_MM_G5_U10_576973.indd 299 SMALL-GROUP ACTIVITY (Differentiation Handbook, p. 142) Practice 5. 5–15 Min (Student Reference Book, p. 218) Lamont, Augusto, and Mario grow carrots in three garden plots. Augusto harvests two times as many carrots as the total number of carrots that Lamont and Mario harvest. So Augusto harvests 2 ∗ (L + M ), or 2(L + M ) 2. 218 231 232 INDEPENDENT ACTIVITY 2/23/11 4:21 PM ENRICHMENT ▶ Analyzing Patterns and INDEPENDENT ACTIVITY 5–15 Min Relationships (Math Masters, p. 300A) To extend students’ understanding of rules and patterns, students use two sets of rules to generate two tables of values. They form ordered pairs consisting of corresponding terms from two patterns, and graph the ordered pairs on a coordinate plane. Students then use the tables of values, rules, ordered pairs, or graphs to identify a relationship between corresponding terms from each pattern. Teaching Master Name Date LESSON Time Patterns and Relationships 10 3 A car is traveling at a given speed over a stretch of highway. You can find the distance the car travels by multiplying its speed by the amount of time it travels. Car A travels at a speed of 30 miles per hour (mph). Car B travels at 60 miles per hour. Complete the tables to find the distance each car travels for the given times. Car A Car B Speed: 30 mph Speed: 60 mph Time (hr) Distance (mi) Time (hr) Distance (mi) 0 0 0 1 30 1 2 60 90 120 2 0 60 120 180 240 3 4 2. For each car, write the rule that is used to find the distance. Car A: Car B: 3. Use the tables to write a set of ordered pairs in the form (Time, Distance) for each car. Then graph the data and connect the points for each car. Label each graph. (0,0) (0,0) (1,30) (1,60) (2,60) (3,90) (2,120) (3,180) (4,120) (4, 240) 240 210 180 rB Car B 270 150 120 90 A Car A Multiply each hour by 60. Ca Multiply each hour by 30. 60 ar 4 30 C 3 Distance (miles) 1. 0 0 1 2 3 4 Time (hours) 4. As the amount of time increases, explain how the distance Car B travels compares with the distance Car A travels? Sample answer: For each hour, Car B travels twice as far as Car A. This is because Car B is traveling twice as fast. Math Masters, p. 300A 300A-300B_EMCS_B_MM_G5_U10_576973.indd 300A 802 Unit 10 3/22/11 9:37 AM Using Data; Algebra Concepts and Skills Name Date LESSON Time Patterns and Relationships 10 3 A car is traveling at a given speed over a stretch of highway. You can find the distance the car travels by multiplying its speed by the amount of time it travels. Car A travels at a speed of 30 miles per hour (mph). Car B travels at 60 miles per hour. Complete the tables to find the distance each car travels for the given times. Car A Car B Speed: 30 mph Speed: 60 mph Time (hr) Distance (mi) Time (hr) 0 0 0 1 30 1 2 2 3 3 4 4 Distance (mi) 2. For each car, write the rule that is used to find the distance. Car A: Car B: 3. Use the tables to write a set of ordered pairs in the form (Time, Distance) for each car. Then graph the data and connect the points for each car. Label each graph. Copyright © Wright Group/McGraw-Hill Car A Car B (0,0) (1,30) 270 240 210 Distance (miles) 1. 180 150 120 90 60 30 0 0 1 2 3 4 Time (hours) 4. As the amount of time increases, explain how the distance Car B travels compares with the distance Car A travels? 300A