Objectives To describe general number patterns in words; and to write special cases for general number patterns. 1 materials Teaching the Lesson Key Activities 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. Math Journal 1, pp. 82–84 Student Reference Book, p. 105 calculator Key Concepts and Skills • Apply a general pattern to find 10% of a number. [Number and Numeration Goal 2] • Extend numeric patterns. [Patterns, Functions, and Algebra Goal 2] • 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] Key Vocabulary general pattern • variable • special case Ongoing Assessment: Informing Instruction See page 182. 2 materials Ongoing Learning & Practice Students practice and maintain skills through Math Boxes and Study Link activities. Ongoing Assessment: Recognizing Student Achievement Use journal page 85. [Patterns, Functions, and Algebra Goal 1] 3 materials Differentiation Options READINESS Students use the “What’s My Rule?” routine to relate general number patterns and special cases. Math Journal 1, p. 85 Study Link Master (Math Masters, p. 74) calculator ENRICHMENT EXTRA PRACTICE Students describe patterns and relationships among triangular numbers, square numbers, and rectangular numbers. Students write rules to describe numeric patterns. Teaching Masters (Math Masters, pp. 71–73) 5-Minute Math, pp.158, 240, and 242 Technology Assessment Management System Math Boxes, Problem 2 See the iTLG. 180 Unit 3 Variables, Formulas, and Graphs Getting Started Mental Math and Reflexes Math Message Students rename mixed numbers and whole numbers as fractions. Suggestions: 1. Write the number that is the opposite of a. 15 15 b. 8 8 c. d. 0 0 2. Add. a. 5 5 0 b. 2.6 2.6 0 c. (11) (11) 0 d. y (y) 0 1 5 14 4 1 33 8 4 4 1 22 37 7 2 8 2 3 3 5 5 1 64 64 1 e. (x) x 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Have students share their answers. Pose additional problems, if necessary. Suggestions: 7 7 ● What is the opposite of 8? 8 ● What is the sum of any number and its opposite? 0 ELL Adjusting the Activity To review the concept of opposites, have students refer to page 105 in the Student Reference Book. Most students are familiar with opposite in relation to position; as in opposite sides of a rectangle are congruent. Draw a number line on the board and show the positional relationship between a number and its opposite to zero. Model the sum of any number and its opposite on the number line. A U D I T O R Y K I N E S T H E T I C T A C T I L E Student Page V I S U A L Date LESSON 3 1 Describing General Number PARTNER ACTIVITY Patterns with Variables (Math Journal 1, pp. 82 and 83) Time 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. 100 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 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: To write a special case for a general pattern, replace the variable with a number. Example: General pattern 10% of n 10 n Special case 10% of 35 10 35 100 100 1. Here are 3 special cases for a general pattern. Rules that describe patterns are sometimes called general patterns. 10 10 1 725 725 1 2 1 1 1 2 Sample answers: a. Describe the pattern in words. Any number divided by itself equals 1. b. Give 2 other special cases for the pattern. 9 9 A general pattern may be described in words. 1 50 0 500 1 2. Here are 3 special cases A general numeric pattern may be described with symbols, at least one of which represents a number. Symbols that represent numbers are called variables. 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. for another general pattern. 15 (15) 0 3 (3) 0 1 4 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 82 Math Journal 1, p. 82 Lesson 3 1 181 Student Page Date Time LESSON Patterns and Variables 3 1 continued 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. 103 3. A spider has 8 legs. The general pattern is: s spiders have s 8 legs. Sample answers: 22 8 176 legs Write 2 special cases for the general pattern. 10 8 80 legs a. b. There are many ways to describe the same pattern using n b variables. For example, n 1, b 1, and 1 all describe the pattern in Problem 1 on journal page 82. 4. 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. a. Describe the general pattern in words. The value of n quarters is n4 dollars. To support English language learners, discuss the everyday meaning of variable and of special case, as well as their meanings in this context. b. Give 2 other special cases for the pattern. The value of 15 quarters is 145 dollars. 100 The value of 100 quarters is 4 dollars. Sample answers: Write 3 special cases for each general pattern. 5. p p 2 p 6. c 1 c 1 7. p p (3 p) 5 p 8. s s (s 1) s 4 4 2 4 1.8 1.8 2 1.8 20 20 2 20 Have partners complete journal pages 82 and 83. They may use calculators. 1 2 2 1 33 313 1 1 6.4 6.4 1 NOTE There are no actual calculations required on these pages, but some 2 52 5 (5 1) 5 102 10 (10 1) 10 92 9 (9 1) 9 2 2 (3 2) 5 2 3.8 3.8 (3 3.8) 5 3.8 16 16 (3 16) 5 16 students may want to verify that a particular number sentence is true. This may ) . involve the use of parentheses keys ( When most students have completed the journal pages, ask volunteers to share their solutions. 83 Math Journal 1, p. 83 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. Possible solutions: N /1 N , or x /1 x /1 , or Sample answers: 1. 18 1 18 2.75 1 2.75 6 10 6 1 10 2. If a nonzero number is divided by itself, the result is equal to 1. 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.) PARTNER ACTIVITY with Number Sentences (Math Journal 1, p. 84) 12.5 / 1 12.5 Example: Write the general pattern for the special cases at the right. Write a general pattern for each group of 3 special cases. 1. If the numerator and denominator of a fraction are the same number (except 0), the fraction is equivalent to 1. Describing General Patterns Student Page Date 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 n 1 follow. General pattern x 1x General pattern T 00 General pattern c cats have c 4 legs 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. 2. 6 0 0 1 2 00 78.7 0 0 Ongoing Assessment: Informing Instruction 3. 1 cat has 1 4 legs. 2 cats have 2 4 legs. 5 cats have 5 4 legs. 4. 6 6 6 1 2 2 1 1 2 2 (2) 0.7 0.7 (0.7)2 General pattern 2 84 Math Journal 1, p. 84 182 Unit 3 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. Write 3 special cases for the general pattern. 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. Math Boxes Problem 2 3.57 6.01 32 33 253 103 3. Write each number in number-and-word 4. Divide. notation. .6 69 93 a. 200,000 200 thousand, or 0.2 million b. 16,900,000,000 16.9 billion c. 58,400,000,000,000 58.4 trillion 1 4 a. 7 7 42–45 6. Compare each pair of fractions. 5 7 Write < or >. 3 a. 10 5 9 10 3 7 c. 11 11 11 1 5 6 d. 1 15.6 93.6 6 5. Add or subtract. 7 b. 9 Ongoing Assessment: Recognizing Student Achievement 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) 2 9 83 b. 100 4 c. 15 19 d. 20 6 5 10 81 100 5 15 20 20 83 75 85 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. 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 24 100 0.25 0.25 100 0.5 0.25 0.25 0.5 For each set of special cases, write a general pattern. 7 0.1 7 10 52 53 55 3 0.1 3 10 132 133 135 4 0.1 5. Sample answers: 6. 3 10. 4 100 75 1 00 12 11 16 21 7 20 25 60 20 2. 110 300 Add 5 to the in number. Rule: out 4 100 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) m0 1 Rule: Divide by the number. (y y 1) out in out 3 0 8 1 25 0 0 0 9 1 1 1 53 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 Sample answers: Complete. in 13 7 1 (2)0 1 s 10 Practice 10 out 8 in 20 1 1460 1 4 10 x2 x3 x5 1 7. 10 103, 253 Complete. s 0.25 0.25 s 32 33 35 in Rule: (2 10) 10 3 10 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. 105 Sample answers: (2 m) m 3 m Time Write a rule for each table shown below. 52 0 52 For each general pattern, give 2 special cases. 4. 1. 103 7 8 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 Following are 3 special cases representing a general pattern. 17 0 17 Date LESSON Variables in Number Patterns 31 1. Name Time 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 (Math Masters, p. 71) 1st 1. 2nd 3rd 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 15 21 28 36 45 55 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 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 Time continued Describe what you notice about the sum of 2 triangular numbers that are next to each other in the table. Sample answer: The sum is a square number. 4. 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 3 3 6 21 3 6 6 6 Math Masters, p. 73 184 Unit 3 Variables, Formulas, and Graphs SMALL-GROUP ACTIVITY 5–15 Min To offer more practice extending and describing numeric patterns, see 5-Minute Math, pages 157, 240, and 242.

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