Parentheses in Number Sentences


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1 Parentheses in Number Sentences Objective To review the use of parentheses. 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 Identify and write sentences that model number stories. [Patterns, Functions, and Algebra Goal ] Solve problems involving parentheses and nested parentheses. Insert parentheses in order to make true number sentences. Key Activities Students use parentheses in number sentences involving more than one operation. They translate number stories into number sentences, inserting parentheses in the proper places, and determine whether number sentences containing parentheses are true or false Playing Name That Number Student Reference Book, p. 35 per partnership: 1 complete deck of number cards (from the Everything Math Deck, if available) Students apply number properties, equivalent names, arithmetic operations, and basic facts. Math Boxes 7 4 Math Journal, p. 1 Students practice and maintain skills through Math Box problems. Study Link 7 4 Math Masters, p. 197 Students practice and maintain skills through Study Link activities. READINESS Reviewing Parentheses in Number Sentences Math Masters, p. 19 Students insert parentheses to make number sentences true. ENRICHMENT Describing Dot Patterns with Number Models Math Masters, p. 199 Students partition dot grids and use the patterns to write number models. EXTRA PRACTICE 5Minute Math 5Minute Math, pp. 77 and 7 Students practice using grouping symbols. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). Key Vocabulary expression ambiguous nested parentheses Materials Math Journal, pp. 19 and 0 Study Link 7 3 Math Masters, p. 414 slate Advance Preparation Teacher s Reference Manual, Grades 4 6 pp. 9, 99, 10, 103,, 9 Lesson
2 Getting Started Mental Math and Reflexes Use your slate procedures for problems such as the following: = 470, = 356, = 4, = 4, = = = = = = 70, = = 79,040 Math Message Complete Problems 1 and at the top of journal page 19. Study Link 7 3 FollowUp Have partners compare answers and resolve differences. 1 Teaching the Lesson Math Message FollowUp (Math Journal, p. 19) WHOLECLASS Date 7 4 Math Message 1. Write each statement as an expression. Parentheses and Number Stories a. Add and 3, and then multiply by 5. ( + 3) 5 b. Add to the product of 3 and 5. + (3 5) c. 10 times the difference of 7 subtracted from. 10 (  7) d. The product of 10 and, minus 7. (10 )  7. Insert parentheses to rewrite the following problem to make four different true sentences. 6 4 / =? Sample answers: ((6 4)) ) / = 11 6 (4 ) / = 6 (6 4) ( / ) = 3 6 (4 ( / )) = 1 Draw a line to match each number story with the expression that fits it. 3. Story 1 Tom s Total Number of Bottles of Juice Tom had 4 bottles of juice. (4 + 3) 6 He went shopping and bought 3 sixpacks of bottles of juice. Story Tom had 4 sixpacks of bottles of juice. 4 + (3 6) He went shopping and bought 3 more sixpacks of bottles of juice. Algebraic Thinking Discuss students answers. Ask: What do parentheses mean in number sentences? Operations inside parentheses are done first. Write the statements from Problem 1 on the board with their correct expressions, but without parentheses. Guide students to see that without the mathematical punctuation of parentheses, number expressions can take on different values depending on the order in which the operations are performed. Without parentheses, the expression is said to be ambiguous because it has more than one possible meaning. NOTE An expression is a group of mathematical symbols (numbers, operation signs, variables, grouping symbols) that represents a number or can represent a number if values are assigned to any variables it contains. A number sentence is made up of at least two numbers or expressions separated by a relation symbol such as =, >, or <. Example: The number sentence / = n includes the expression /, the variable n, and the equal symbol =. Ask volunteers to share their answers to Problem and explain the steps they used to solve the problem. As the students explain the steps, list them on the board. For example: Steps Numerical Expression Multiply 6 times Subtract from the result. (6 4)  Divide that result by. ((6 4)  ) / Math Journal, p _EMCS_S_MJ_G5_U07_ indd 19 3/1/11 5:6 PM 55 Unit 7 Exponents and Negative Numbers
3 Point out that when two or more sets of parentheses are used in the same expression, the operation inside the inner parentheses is done first. In ((6 4)  ) /, 6 times 4 is the operation in the inner parentheses. The operation in the outer parentheses is done next, followed by all the remaining operations. Parentheses inside parentheses are referred to as nested parentheses. Date Parentheses and Number Stories continued 4. Story 1 Number of Cookies Alice Ate Alice ate 3 cookies before going to a 3 (45 / 5) party. At the party, Alice and 4 friends ate equal shares of 45 cookies. Story There was a full bag with 45 cookies (45 3) / 5 and an opened bag with 3 cookies. Alice and 4 friends ate equal shares of all these cookies. 5. Story 1 Number of Cookies Baked Links to the Future This lesson stresses the use of parentheses to create unambiguous expressions. Lesson 75 will introduce the conventional order of operations, which often reduces or eliminates the need for grouping symbols in many number sentences. Matching Number Stories to Appropriate Expressions (Math Journal, pp. 19 and 0) PROBLEM SOLVING Mr. Chung baked 5 batches of cookies. 15 (4 5) Each of the first 4 batches contained 15 cookies. The final batch contained only 5 cookies. Story In the morning, Mr. Chung baked (4 15) 5 4 batches of 15 cookies each. In the afternoon, he baked 5 more batches of 15 cookies each. 6. A grocery store received a shipment of 10 cases of apple juice. Each case contained 4 sixpacks of cans. After inspection, the store found that 9 cans were damaged. Write an expression that represents the number of undamaged cans. Sample answer: (10 (4 6)) 9 Ask students to read the two number stories in Problem 3 on journal page 19 and match each story with an expression. Ask volunteers to explain their choices. Story 1 goes with the second expression since 3 must be multiplied by 6 to find the total number of cans in 3 sixpacks. Story goes with the first expression since stands for the total number of sixpacks. Assign journal page 0. When most students are done, bring the class together and go over the answers. Have students explain their reasons for matching a given number story with a particular expression. Math Journal, p. 0 Ongoing Assessment: Exit Slip Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414) to assess students facility with writing expressions containing parentheses to represent a number story. Have students explain how they used parentheses in Problem 6 on journal page 0 to write the expression for the total number of undamaged cans. Students are making adequate progress if they refer to the use of nested parentheses to identify the total number of undamaged cans. Ongoing Learning & Practice Playing Name That Number (Student Reference Book, p. 35) Students practice applying number properties, equivalent names, arithmetic operations, and basic facts by playing Name That Name That Number Materials 1 complete deck of number cards Players or 3 Skill Naming numbers with expressions Object of the game To collect the most cards. Directions 1. Shuffle the deck and deal 5 cards to each player. Place the remaining cards numberside down on the table between the players. Turn over the top card and place it beside the deck. This is the target number for the round.. Players try to match the target number by adding, subtracting, multiplying, or dividing the numbers on as many of their cards as possible. A card may only be used once. 3. Players write their solutions on a sheet of paper. When players have written their best solutions: Each player sets aside the cards they used to match the target number. Each player replaces the cards they set aside by drawing new cards from the top of the deck. The old target number is placed on the bottom of the deck. A new target number is turned over, and another round is played. 4. Play continues until there are not enough cards left to replace all of the players cards. The player who has set aside the most cards wins the game. Target number: 16 Player 1 s cards: 7 Some possible solutions: (3 cards used) 7 * (4 cards used) / (all 5 cards used) The player sets aside the cards used to make a solution and draws the same number of cards from the top of the deck. Student Reference Book, p Games Lesson
4 Date Math Boxes Number. Encourage students to extend the game by using the cards in their hands to form exponents and/or fractions. 1. Measure the length and width of each of the following objects to the nearest half inch. Answers vary for b d. 13 a. journal cover b. desktop 10 7 length 1 in. width in. length in. width in. c. index card d. (your choice) length in. width in. length in. width in. Math Boxes 7 4 (Math Journal, p. 1) INDEPENDENT. a. Make a stemandleaf plot of the handspan measures in Ms. Grip s fifthgrade class. 163, 179, 170, 165, 1, 157, 154, 165, 170, 175, 16, 15, 15, 170, 165, 154 Stems Leaves (100s and 10s) (1s) Measure M to the nearest degree. M measures about 167. M Math Journal, p b. Find the following landmarks for the data. Median: 165 Minimum: 154 Range: 31 Mode(s): 165, Calculate the sale price. Regular Price Discount $.99 0% $ % $9.00 0% $9.99 0% Sale Price $7.19 $.99 $71.0 $ Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 7 and 76. The skill in Problem 4 previews Unit content. Writing/Reasoning Have students write a response to the following: Use your solution for Problem to explain how to read a stemandleaf plot. Sample answer: In this problem, the stems are the hundreds and tens digits for each number, and the leaves are the ones digits. The first number on this stemandleaf plot has 1 in the hundreds place, 5 in the tens place, and 4 in the ones place. It is read one hundred fiftyfour. Study Link 7 4 (Math Masters, p. 197) INDEPENDENT Home Connection Students insert parentheses to make number sentences true. In several cases, students will need to insert nested parentheses. 3 Differentiation Options Name Date STUDY LINK Using Parentheses Make each sentence true by inserting parentheses. ( ) ( ) Study Link Master 1. = 3 4 / 1. 3 = / / 4. Write seven names for. Use only numbers less than 10, and use at least three different operations in each name. Use parentheses. Follow the directions in Problem 7 to fill in the last two rows. Make each sentence true by inserting parentheses. Reminder: When you have a pair of parentheses inside another pair, the parentheses are called nested parentheses. Example: ((5 6) ) / 4 3 Sample answers for Problems 1 6: ( ) ( ) ( ) (9 1) (1 / 1) (9 5) ( ) (0.5 / 1) (.5 3) ( ) / (6 4) (6. / ) (7.1) ( 1 ) (9 5) (7 9) 5 (4 (3 3)) 1 (( 1) 4 9) 1 READINESS Reviewing Parentheses in Number Sentences (Math Masters, p. 19) Min To explore the use of parentheses in number sentences, have students insert parentheses to make true sentences. Students compare the use of commas in text sentences to the use of parentheses in number sentences. When students have finished the page, have them share why they think parentheses are important. (( ) ) (( ) ) / 6. 3/ 1 7. Add two names to your namecollection box in Problem 4. Use nested parentheses. Practice Find the number that each variable represents (1 a) a 1 4 1, or ( p) d d, or y 6 y 0 5 p 1 1 Math Masters, p Unit 7 Exponents and Negative Numbers
5 ENRICHMENT INDEPENDENT Describing Dot Patterns with Number Models (Math Masters, p. 199) Min To apply students understanding of parentheses, have them write number sentences to describe dot patterns. Students partition a dot grid and write number sentences to model the indicated number patterns. EXTRA PRACTICE 5Minute Math SMALLGROUP 5 15 Min To offer students more experience with grouping symbols, see 5Minute Math, pages 77 and 7. Name Date Reviewing Parentheses 1. Read the following sentence. Mary Grace the lizard ate three crickets. This sentence could have multiple meanings. 1. The speaker is telling someone named Mary Grace that the lizard ate three crickets.. The lizard, named Mary Grace, ate three crickets. 3. The speaker is telling someone named Mary that the lizard, named Grace, ate three crickets. Without commas, it s hard to tell which meaning was intended. Write the number of the meaning next to each sentence below. 1 3 a. Mary Grace, the lizard, ate three crickets. b. Mary Grace, the lizard ate three crickets. c. Mary, Grace the lizard, ate three crickets. By adding commas, the meaning of a sentence becomes clear. In number sentences, parentheses are used to indicate what to calculate first.. Insert parentheses in each sentence to make the sentence true. a b c. 7 / Insert parentheses in the expressions below, and find their solutions. a b Teaching Master 3 (4 7) 33 6 (9 5) 51 7 / (4 5) 6 9 (7 5) 4 31; 7 (5 4) 7 (6 9) 3 5; 6 (9 3) 9 Planning Ahead In Lesson 76, you will need newspapers and magazines that contain line graphs. Math Masters, p. 19 Teaching Master Name Date Describing Dot Patterns The total dots in this dot array can be found by using patterns. Here is one way to find the total: ((3 3) (4 3) 4) Use shape outlines or colors to identify a pattern on this dot array. Write a number model for your pattern. Then write a number story that matches your number model. Sample answers: Number model: (11 11) (4 (5 4)) (1 4) (4 4) 65 Number story: Sample answer: In a large city school, the fifth graders fill the auditorium in colorful choir robes. Students wearing purple robes stand in 11 rows of 11 students each. Four groups in gold robes stand in 4 rows of 5 students each. There are 4 groups of 1 students in black robes and 4 groups of 4 students in green robes. How many students are in the auditorium? Math Masters, p. 199 Lesson
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