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Using Variables to Describe Number Patterns Objectives To describe general number patterns in words; and to t write special cases for general number patterns. www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice Key Concepts and Skills Math Boxes 3 1 • Apply a general pattern to find 10% of a number. Math Journal 1, p. 85 Students practice and maintain skills through Math Box problems. [Number and Numeration Goal 2] • Extend numeric patterns. [Patterns, Functions, and Algebra Goal 1] • Write a number sentence containing a variable to describe a general pattern. [Patterns, Functions, and Algebra Goal 1] • Apply general patterns to explore multiplicative and additive inverses. [Patterns, Functions, and Algebra Goal 4] Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 2. [Patterns, Functions, and Algebra Goal 1] READINESS Connecting General Number Patterns and Special Cases Math Masters, p. 71 Students use the “What’s My Rule?” routine to relate general number patterns and special cases. ENRICHMENT Math Masters, p. 74 Students practice and maintain skills through Study Link activities. Math Masters, pp. 72 and 73 Students describe patterns and relationships among triangular numbers, square numbers, and rectangular numbers. Ongoing Assessment: Informing Instruction See page 182. Key Vocabulary general pattern variable special case Materials Math Journal 1, pp. 82–84 Student Reference Book, p. 105 calculator Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 278–289 Variables, Formulas, and Graphs Differentiation Options Exploring Number Patterns Students describe a general number pattern in words and write examples or special cases of it. They are given special cases for a general pattern and describe it with a number sentence having one variable. Unit 3 Interactive Teacher’s Lesson Guide Study Link 3 1 Key Activities 180 Curriculum Focal Points EXTRA PRACTICE 5-Minute Math 5-Minute Math™, pp. 157, 240, and 242 Students write rules to describe numeric patterns. Mathematical Practices SMP2, SMP3, SMP5, SMP6, SMP7, SMP8 Content Standards Getting Started 6.NS.6a, 6.NS.6c, 6.EE.2, 6.EE.2a, 6.EE.2b, 6.EE.2c, 6.EE.6 Mental Math and Reflexes Math Message Students rename mixed numbers and whole numbers as fractions. Suggestions: 5 1 _ 33 1 _ 1 _ 22 1_ 8_ 3_ 1. Write the number that is the opposite of a. 15 -15 b. -8 8 c. π -π d. 5.25 -5.25 2. Add. a. 5 + -5 0 b. -2.6 + 2.6 0 c. -(-11) + (-11) 0 d. y + (-y) 0 4 4 8 2 _ 2_ 3 3 4 4 5 5_ 1 7 7 64 64 _ 1 1 Teaching the Lesson ▶ Math Message Follow-Up Interactive whiteboard-ready ePresentations are available at www.everydaymathonline.com to help you teach the lesson. WHOLE-CLASS DISCUSSION Draw a number line on the board. As students share their answers, plot the numbers and their opposites on the number line. Ask: What do you notice about the positions of a number and its opposite in relationship to 0? They are on opposite sides of 0 on the number line and are each the same distance from 0 on the number line. What does this tell you about the opposite of 0? 0 is its own opposite. Is any other number its own opposite? no What is the sum of any number and its opposite? 0 ▶ Describing General Number 7 7 e - _8 _8 PROBLEM BL BLE LEM LE LEM SOLVING LVIN V NG Patterns with Variables Adjusting the Activity To review the concept of opposites, use the number line to model the three properties of opposites at the top of Student Reference Book, page 105. AUDITORY KINESTHETIC TACTILE VISUAL PARTNER ACTIVITY ELL (Math Journal 1, pp. 82 and 83) Algebraic Thinking Read and discuss the text at the top of journal page 82. Have students complete Problem 1 and discuss their solutions. Be sure to cover the following points: Rules that describe patterns are sometimes called general patterns. A general pattern may be described in words. Student Page Date Time LESSON 3 1 Patterns and Variables 103 Study the number sentences at the right. All three sentences show the same general pattern. 10 10% of 50 100 50 10 10% of 200 100 200 This general pattern may be described in words: To find 10% 10 1 of a number, multiply the number by 100 (or 0.10, or 10 ). 10 10% of 8 100 8 The pattern may also be described by a number sentence 0 that contains a variable: 10% of n 1 n. 1 00 A variable is a symbol, such as n, x, A, or . A variable can stand for any one of many possible numeric values in a number sentence. 10 10 Number sentences like 10% of 50 100 50 and 10% of 200 100 200 are 0 examples, or special cases, for the general pattern described by 10% of n 1 n. 100 A general numeric pattern may be described with symbols, at least one of which represents a number. Symbols that represent numbers are called variables. To write a special case for a general pattern, replace the variable with a number. Example: General pattern Special case 1. A variable can have any one of many possible numeric values. A common misunderstanding of variables is that a variable always stands for one particular number. 10 10% of n n 1 00 10 10% of 35 35 1 00 Here are 3 special cases for a general pattern. 10 10 1 725 725 1 2 1 2 1 1 Sample answers: a. Describe the pattern in words. Any number divided by itself equals 1. b. 2. Give 2 other special cases for the pattern. 9 9 1 Here are 3 special cases for another general pattern. 15 (15) 0 50 0 500 3 (3) 0 1 4 1 1 (4) 0 Sample answers: a. Describe the pattern in words. Any number added to its opposite equals zero. b. Give 2 other special cases for the pattern. 0.75 (0.75) 0 100 (100) 0 Math Journal 1, p. 82 Lesson 3 1 181 Student Page Date Time LESSON Patterns and Variables 3 1 3. 103 A spider has 8 legs. The general pattern is: s spiders have s 8 legs. Write 2 special cases for the general pattern. 10 8 80 legs a. 4. continued Sample answers: 22 8 176 legs b. There are many ways to describe the same pattern using n = 1, _ b = 1, and = 1 all describe variables. For example, _ n b the pattern in Problem 1 on journal page 82. Study the following special cases for a general pattern. Sample answers: 6 The value of 6 quarters is 4 dollars. 10 The value of 10 quarters is 4 dollars. 33 The value of 33 quarters is 4 dollars. Describe the general pattern in words. a. The value of n quarters is n4 dollars. b. Give 2 other special cases for the pattern. The value of The value of To support English language learners, discuss the everyday meaning of variable and of special case, as well as their meanings in this context. 15 quarters is 145 dollars. 0 dollars. 100 quarters is 10 4 Have partners complete journal pages 82 and 83. They may use calculators. Sample answers: Write 3 special cases for each general pattern. 5. pp2p 6. c 1c 1 7. p p (3 p) 5 p 8. s 2 s (s 1) s 4 4 2 4 1.8 1.8 2 1.8 20 20 2 20 2 2 (3 2) 5 2 3.8 3.8 (3 3.8) 5 3.8 16 16 (3 16) 5 16 When a particular number is substituted for the variable in a general pattern, the result is called an example, or a special case, of the general pattern. 1 2 1 33 1 6.4 2 1 33 1 6.4 1 NOTE There are no actual calculations required on these pages, but some students may want to verify that a particular number sentence is true. This may ) . involve the use of parentheses keys ( 52 5 (5 1) 5 102 10 (10 1) 10 92 9 (9 1) 9 When most students have completed the journal pages, ask volunteers to share their solutions. Have students describe each general pattern in their own words before giving special cases for the pattern. There is often more than one way to describe a pattern in words. Two ways of n = describing the general pattern _ 1 follow. n Math Journal 1, p. 83 1. If the numerator and denominator of a fraction are the same number (except 0), the fraction is equivalent to 1. NOTE General patterns may be described in words or by an open number sentence. Whenever Everyday Mathematics asks students to write a general pattern, they should write a number sentence that describes the pattern, unless the directions specifically state that they are to describe it in words. Time LESSON 3 1 Writing General Patterns 103 Following is a method for finding the general pattern for a group of special cases. 8 /1 8 0.3 / 1 0.3 Solution Strategy Step 1 Write everything that is the same for all of the special cases. Use blanks for the parts that change. / 1 Each special case has division by 1 and an equal sign. Step 2 Fill in the blanks. Each special case has a different number, but the number is the same for both blanks, so use the same variable in both blanks. N /1 N , or x Write a general pattern for each group of 3 special cases. 1. 6 10 6 1 1 0 1 2 General pattern x 1x General pattern T 00 General pattern c cats have c 4 legs 00 5 cats have 5 4 legs. 6 6 62 1 2 2 1 1 2 (2) 0.7 0.7 (0.7)2 General pattern 2 Math Journal 1, p. 84 182 Unit 3 with Number Sentences (Math Journal 1, p. 84) Algebraic Thinking As a class, read and discuss the problem and solution strategy at the top of journal page 84. Ask students to decide which parts stay the same in all of the special cases and which parts change from case to case. Circulate and assist as needed. When most students have completed the page, ask volunteers to share their solutions. Ongoing Assessment: Informing Instruction 1 cat has 1 4 legs. 2 cats have 2 4 legs. 4. /1 , or Sample answers: 600 78.7 0 0 3. x 18 1 18 2.75 1 2.75 2. /1 PARTNER ACTIVITY 12.5 / 1 12.5 Example: Write the general pattern for the special cases at the right. Possible solutions: A general pattern that is written with a variable looks the same as any of the special cases for that pattern. The only difference is that the variable has been replaced by a specific value. (See margin.) ▶ Describing General Patterns Student Page Date 2. If a nonzero number is divided by itself, the result is equal to 1. Variables, Formulas, and Graphs Watch for students who do not recognize what the special cases have in common. Some students may benefit from circling all the numbers and symbols that stay the same from one special case to the next. Student Page Date 2 Ongoing Learning & Practice Time LESSON Math Boxes 3 1 1. Complete the “What’s My Rule?” table. 2. Rule: Subtract 1.32 ▶ Math Boxes 3 1 INDEPENDENT ACTIVITY in out 8 6.68 0.83 0.48 2.15 1.8 (Math Journal 1, p. 85) 4.89 7.33 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-3. The skills in Problems 5 and 6 preview Unit 4 content. Writing/Reasoning Have students write a response to the following: Explain how you know where to place the decimal point in the quotient of Problem 4. Sample answer: I estimated. 90 divided by 6 is 15, so 93.6 divided by 6 is about 15. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 2 3. 3.57 6.01 32 33 253 103 4. Divide. 6冄9 9 苶3 苶.6 苶 200,000 200 thousand, or 0.2 million b. 16,900,000,000 c. 58,400,000,000,000 16.9 billion 58.4 trillion 5. Use Math Boxes, Problem 2 to assess students’ ability to write special cases for a general pattern. Students are making adequate progress if they are able to write three special cases. Some students may recognize the general pattern as illustrative of the Distributive Property of Multiplication over Addition. a (b + c) (a b) + (a c) Sample answers: 1 (2 3)(1 2) (1 3) 5 (10 5) (5 10) (5 5) 7 (20 9)(7 20) (7 9) Write each number in number-and-word notation. a. Write 3 special cases for the general pattern. 93.6 6 Add or subtract. 5 1 4 7 a. 7 7 5 7 2 9 b. 9 9 10 3 7 c. 11 11 11 1 5 6 d. 1 6. 3 a. 10 83 b. 100 4 c. 15 19 d. 20 6 15.6 42–45 Compare each pair of fractions. Write < or >. 5 10 81 100 5 15 20 20 83 75 Math Journal 1, p. 85 [Patterns, Functions, and Algebra Goal 1] ▶ Study Link 3 1 INDEPENDENT ACTIVITY (Math Masters, p. 74) Home Connection Students practice describing general patterns with words and number sentences having one variable. They write special cases for general patterns. Teaching Master Study Link Master Name Date STUDY LINK 7 8 43 0 43 36.09 0 36.09 (2 24) 24 3 24 0.5 0.25 0.25 0.5 For each set of special cases, write a general pattern. 5. Sample answers: 7 7 0.1 6. 10 52 53 55 3 3 0.1 132 133 135 4 0.1 4 10 in 13 12 11 16 21 7 20 25 60 20 105 110 300 100 2. out 4 Rule: Divide the in number by 3. You are writing special cases for a general number pattern when you complete a “What’s My Rule?” table. s 0.1 Rule: Add the opposite of the number. (x x 0) out in out 0 8 1 25 0 0 0 9 1 1 1 53 m0 1 Rule: Divide by the number. (y y 1) 3 7 1 (2)0 1 s 10 1 4 100 Use the values from the table above to write special cases for the following general number patterns: x x 0. y y 1. Special cases Special cases Practice Sample answers: Complete. 3 10. 4 out 8 in 20 1 1460 1 10 x2 x3 x5 103, 253 Complete. 100 0.25 0.25 100 1 7. 10 in Rule: Add 5 to the in number. (2 10) 10 3 10 s 0.25 0.25 s 32 33 35 General Patterns and Special Cases You are describing a general number pattern for a special case when you write a rule for a “What’s My Rule?” table. Write a rule for each table shown below. Sample answers: (2 m) m 3 m Time 8 52 0 52 For each general pattern, give 2 special cases. 4. 103 7 0 Give 2 other special cases for the pattern. b. 3. Describe the general pattern in words. Sample answers: The sum of any number and 0 is equal to the original number. a. 2. 3 1 1. Following are 3 special cases representing a general pattern. 17 0 17 Date LESSON Variables in Number Patterns 31 1. Name Time 10 100 75 100 0.10 0.75 1 8. 4 4 11. 5 25 — — 0. — 25 1 9. 5 80 7 12. 10 100 80 100 Math Masters, p. 74 0. 20 100 70 100 0.20 0. 70 Example: 3 3 0 Example: 8 8 1 25 25 0 7 7 0 53 53 0 9 9 1 1 1 1 4 4 100 100 1 Math Masters, p. 71 Lesson 3 1 183 Teaching Master Name Date LESSON Time 3 Differentiation Options Number Patterns 31 Triangular, square, and rectangular numbers are examples of number patterns that can be shown by geometric arrangements of dots. Study the number patterns shown below. Triangular Numbers Square Numbers READINESS 1st 2nd 3rd 4th 1st 2nd 3rd 4th Rectangular Numbers ▶ Connecting General Number INDEPENDENT ACTIVITY 5–15 Min Patterns and Special Cases 1st 1. 2nd 3rd (Math Masters, p. 71) 4th Use the number patterns to complete the table. Number of Dots in Arrangement 2. 1st 2nd 3rd 4th Triangular Number 1 3 6 10 Square Number 1 4 9 16 25 36 49 64 81 100 Rectangular Number 2 6 12 20 30 42 56 72 90 110 What is the 11th triangular number? 5th 6th 7th 8th 9th 10th 15 21 28 36 45 55 To provide experience with algebraic notation, have students use the “What’s My Rule?” routine. By completing “What’s My Rule?” tables, some students may more easily make the connection between general number patterns and special cases in Part 1 of this lesson. 66 How does the 11th triangular number compare to the 10th triangular number? It is 11 more than the 10th triangular number. ENRICHMENT ▶ Exploring Number Patterns Math Masters, p. 72 PARTNER ACTIVITY 5–15 Min (Math Masters, pp. 72 and 73) To apply students’ understanding of general patterns, have them work with a partner to discover some relationships among figurate numbers—special numbers associated with geometric figures. For example, the sum of two successive triangular numbers is a square number; and the sum of a rectangular number and its corresponding square number is a triangular number. EXTRA PRACTICE ▶ 5-Minute Math Teaching Master Name Date LESSON 31 3. Number Patterns continued Describe what you notice about the sum of 2 triangular numbers that are next to each other in the table. Add the second square number and the second rectangular number; the third square number and the third rectangular number. What do you notice about the sum of a square number and its corresponding rectangular number? The sum is a triangular number. 5. Describe any other patterns you notice. Sample answer: Rectangular numbers are twice the corresponding triangular numbers. 6. You can write triangular numbers as the sum of 4 triangular numbers when repetitions are allowed. For example: 6 1 1 1 3 Find 3 other triangular numbers that can be written as sums of exactly 4 triangular numbers. Sample answers: 10 1 3 3 3 15 3 21 3 3 6 3 6 6 6 Math Masters, p. 73 184 Unit 3 5–15 Min Time Sample answer: The sum is a square number. 4. SMALL-GROUP ACTIVITY Variables, Formulas, and Graphs To offer more practice extending and describing numeric patterns, see 5-Minute Math, pages 157, 240, and 242.