Interactive lesson includes instant self-check, tutorials, and activities. Recognizing and Extending Patterns

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1 - -. Plan What You ll Learn OJETIVE OJETIVE To continue a number pattern To write a rule for a number pattern... nd Why To find the number of tiles in a pattern, as in Example OJETIVE ontinuing a Number Pattern heck Skills You ll Need For help, go to Lesson -3. Order each set of decimals from least to greatest. 6. See margin , 0.005, , 3., , 9.6, , 6.0, , 0.0, , 0.4, 0.35 New Vocabulary term conjecture Interactive lesson includes instant self-check, tutorials, and activities. Investigation: The first three designs in a pattern are shown at the right.. ontinue the pattern. Sketch the fourth and fifth designs on grid paper. See margin.. How many squares are in the fourth design? The fifth design? 3 squares; 7 squares 3. opy and complete the table. Recognizing and Extending Patterns Lesson Preview heck Skills You ll Need Ordering ecimals Lesson -3: Example 4. Extra Practice p. 64. Lesson Resources Optional Materials graph paper Teaching Resources Practice, Reteaching, Enrichment Reaching ll Students Practice Workbook - Spanish Practice Workbook - Guided Problem Solving - Presentation ssistant Plus! Transparencies heck Skills You ll Need - Problem of the ay - dditional Examples - Student Edition nswers - Lesson Quiz - PH Presentation Pro -ROM - esign Number Number of Squares 5 omputer Test Generator 4. nswers may vary. Sample: Multiply (0 ), or 9, by 4, and then add Reasoning escribe how you will find the tenth design in the pattern. Technology Resource Pro -ROM omputer Test Generator PH Presentation Pro -ROM Reading Math Read the three dots in the pattern, 4, 7, 0, cas and so on. INSTNT HEK SYSTEM The numbers, 4, 7, 0, cform a number pattern. Each number in the pattern is a term. For example, the third term in this pattern is 7. The three dots after the number 0 tell you that the pattern continues beyond what is shown. Ongoing ssessment and Intervention efore the Lesson iagnose prerequisite skills using: heck Skills You ll Need uring the Lesson Monitor progress using: heck Understanding dditional Examples Test Prep - 63 fter the Lesson ssess knowledge using: Lesson Quiz omputer Test Generator Student Site Teacher Web ode: aak-5500 lgebra Readiness Puzzles 45 Self-grading Lesson Quiz PH SuccessNet Teacher enter Lesson Planner Resources Plus 63

2 . Teach Math ackground Patterns are the cornerstone of mathematics. In fact, mathematics has been described as the search for pattern. In this lesson, students learn to recognize, extend, and describe addition/subtraction patterns and multiplication/division patterns. In an addition/subtraction pattern, each term is derived from the one before it by consistently adding or subtracting the same amount. Formally, this type of pattern is called an arithmetic sequence. In a multiplication/division pattern, each term is derived from the one before it by consistently multiplying or dividing by the same amount. Patterns of this type are called geometric sequences. Teaching Notes English Learners When students hear the word pattern, they might think of a repeating pattern, the type of pattern they see on rugs, quilts, and other decorative items; Help students understand that the word pattern here refers to a rule that is repeated again and again. Investigation (Optional) Lead students to see the following relationships: Each design in the geometric pattern has four more squares than the preceding design. Each term in the number pattern is four greater than the term that precedes it. EXMPLE Teaching Tip fter discussing Example, ask: How would the pattern be different if the first term were? Each term would increase by ; the pattern would be, 5, 8,, 4,.... OJETIVE heck Understanding a. 4, 5 b. 4, 6 c. 57, 64 Writing a Rule for a Number Pattern 64 hapter Patterns and Variables conjecture is a prediction about what may happen. You can use the terms you know to make a conjecture about how a pattern will continue. EXMPLE Finding Number Patterns ecorating Jacob is creating a pattern of colored tiles for a wall in his bathroom. The first four designs are shown below. How many squares will be in the fifth and sixth designs? ount the squares in each design. There are, 4, 7, and 0 squares. Notice that each design has three more squares than the one before it. 3 3 dd 3 to 0 to get the fifth term., 4, 7, 0, 3, 6,... d dd 3 to 3 to get the sixth term. So, the fifth and sixth designs will have 3 and 6 squares. Write the next two terms in each number pattern. a.,,, 3, c b. 56, 48, 40, 3, c c. 9, 36, 43, 50, c d. Reasoning In Example, the eighth design will go all the way across Jacob s bathroom wall. How many tiles will be in the eighth design? tiles One way you can describe a number pattern is to give the first term and the rule. rule is an explanation of how you go from one term to the next. EXMPLE Writing Number Patterns From Rules Write the first six terms in the number pattern described by this rule: Start with and multiply by The first term is. T,, 4, 8, 6, 3 Real World Multiply each term by d to find the next term. heck Understanding Write the first six terms in each number pattern. a. Start with 90 and subtract 5 b. Start with and multiply by 3 90, 75, 60, 45, 30, 5, 3, 9, 7, 8, 43 c. Start with 7 and add 9 7, 36, 55, 74, 93, Reaching ll Students 64 elow Level Have students practice skip-counting by, by 3, by 4, and so on., 4, 6, 8, 0,...; 3, 6, 9,, 5,...; 4, 8,,6, 0,...; and so on dvanced Learners Have students create original geometric patterns that represent number patterns, such as the designs in the Investigation and in Example. English Learners See note on page 64. Inclusion See note on page 65.

3 3 EXMPLE Writing a Rule Patterns Write the next three terms in each pattern. Then write a rule for the pattern. a. 53, 49, 45, 4, , 49, 45, 4, 37, 33, 9 The rule is start with 53 and subtract 4 b..5, 4.5, 3.5, 40.5, , 4.5, 3.5, 40.5,.5, 364.5,,093.5 The rule is start with.5 and multiply by 3 To get from one term to the next, subtract 4. To get from one term to the next, multiply by 3. heck Understanding 3 Write the next three terms and write a rule to describe each number pattern. See margin. a., 7, 49, 343, c b. 0.0, 8.8, 7.6, c c. 56, 8, 64, c EXMPLE Teaching Tip fter discussing Example, ask: How would the pattern be different if the first term were? It is the same pattern, but without the term ; the pattern would be, 4, 8, 6, 3, EXMPLE Inclusion Stress the importance of identifying the two parts of the rule: the first term, and the explanation of how you go from one term to the next. sk: When you write the rule for the pattern in part a, why is it not enough to say, Subtract 4 from a term to get the next term.? Sample: That sentence by itself could describe a different pattern, such as 0, 6,, 8,.... dditional Examples epartures Red Train :5 P.M. :5 P.M. :5 P.M. 3:5 P.M. EXERISES Practice by Example lue Train :7 P.M. :0 P.M. :47 P.M. :3 P.M. Example (page 64) Example (page 64) Example 3 (page 65) Write the next two terms in each number pattern.., 6, 0, 4, c 8,. 99, 88, 77, 66, c 55, 44 3., 3, 9, 7, c 8, 43 4.,.4,.8,., c.6, 3.0 Write the first six terms in each number pattern. 5. Start with 7 and add 4 6. Start with 5 and divide by For more practice, see Extra Practice. 7,, 5, 9, 3, 7,... 5, 56, 8, 64, 3, 6 Write the next three terms and write a rule for each number pattern See margin. 7., 5, 5, 5, c 8., 0, 50, 50, c 9. 6,000,000; 600,000; 60,000; c 0. $.85, $5.70, $8.55, c. Schedules The schedule shows the departure times for two trains. Predict the remaining departure times before 6 P.M. for each train. a. The lue Train. b. The Red Train. 4:5, 5:5 3:7, 4:0, 4:47, 5: Write the next two terms in this number pattern. 5,, 9, 6,... 33, 40 Write the first six terms in the number pattern described by this rule: Start with 47 and subtract 3 47, 44, 4, 38, 35, 3 Write the next three terms and write a rule to describe this number pattern..3, 4.4, 6.5, 8.6, ,.8, 4.9; rule: Start with.3 and add. losure What is a number pattern? Sample: a set of numbers in which each number is related to the next number by a rule What is a term of a pattern? one of the numbers in the pattern How do you describe a number pattern? Give the first term and the rule. heck Understanding 3a.,40, 6,807, 7,649; the first term is ; multiply each term by 7 to get the next term. b. 6.4, 5., 4; the first term is 0.0; subtract. from a term to get the next term. c. 3, 6, 8; the first term is 56; divide each term by to get the next term. Exercises 7. 65; 3,5; 5,65; the first term is ; multiply a term by 5 to get the next term. 8.,50; 6,50; 3,50; the first term is ; multiply a term by 5 to get the next term. 9. 6,000; 600, 60; the first term is 6,000,000, divide a term by 0 to get the next term. 0. $.40, $4.5, $7.0; the first term is $.85; add $.85 to a term to get the next term. 65

4 3. Practice ssignment Guide Objective ore 4, 4 Extension 3 Objective ore 5, 5 9 Extension 30 Test Prep 3 36 Mixed Review 37 4 Error Prevention! Exercises 0 Have students softly repeat the operation they use for each successive term. Practice - Sketch the next two designs in each pattern... Write the next three terms in each pattern., 3, 5, 9, , 5, 7, 9, 4. 34, 3, 8, 5, 6, 486,,458 44, 5, 60 5., 6, 8, 54, 6., 0, 8, 36, 50, 49, 48 7,, , 53, 5, 5, 8. 7, 8, 0, 3, Find the next three terms and write a rule to describe each number pattern. 9. 4, 7, 0, 3, 9, 9, 9 0., 4, 8, 6, 9, 9, 9 6, 9, ; start with the number 3, 64, 8; start with the number 4 and add 3 and multiply by. 9, 9, 39, 49, 9, 9, 9. 8,, 4, 7, 9, 9, 9 59, 69, 79; start with the number 0, 3, 6; start with the number 9 and add 0 8 and add , 5, 5, 05, 9, 9, , 0, 0, 40, 9, 9, 9 95, 85, 75; start with the 80, 60, 30; start with the number 35 and subtract 0 number 5 and multiply by 5. Write the first five terms in a number pattern starting with the number 6. Write the rule that describes your pattern. heck students answers. Find the missing term. 6. 7,, 63, 9, , 7, 9, 5, , 3, 3, 9, , 0, 80, 30,, nswers may vary. Sample: 480, 40, 0, 60, ; divide by Write the fifth term of each number pattern.., 8, 64, 5, c 4, , 5,, 9, c , 7, 34, 5, c ,.,, 0, c,00 6., 8, 4, 0, c , 89, 78, 67, c stronomy Edmond Halley (656-74) first saw the comet named for him in 68. He correctly predicted that it would return about every 76 years. 986 a. ased on Halley s calculations, when did the comet last appear? b. When is the comet expected to return next? bout how old will you be when that happens? 06; check students work. c. Number Sense id Halley see the comet a second time? Explain. No. The next appearance after 68 was 758, 6 years after his death. Find the missing term ,, 63, 7, , 7, 88, 44,, 6. 7, 80, 44, , 0., 0.04, 7, ,.4, 7, 9.6, , 9, 83, 75 Writing in Math 5. Write a number pattern and its rule. 6. usiness dry cleaner charges $5.00 to clean one item. He offers to GPS clean a second item for $4.50, and a third item for $4.00. a. If he continues to subtract $.50 for each additional item, how much will it cost to clean six items? $.50 b. If the pattern continues, which item will be cleaned for free? the th item 7. omputers The ifference Engine is a computer designed by harles abbage (79-87). If you feed it a list of numbers, it will look for a pattern and continue the list, if possible. Why do you think the computer was given this name? The computer looked at the difference between terms in a list to compute a pattern. Geometry raw the next design in each pattern. Reteaching Find the next three numbers in the pattern. 3, 9, 5,,?,?,? Look at how the second number can be found from the first. 3, 9, 5, or 3, 9, 5, Look at how the third number can be found from the second. 3, 9, 5, or 3, 9, 5, is not Try adding 6 to the third number. 3, 9, 5, Now you can write a rule to describe the pattern. The rule is Start with the number 3 and add 6 The next three numbers in the pattern are 7, 33, and 39. 3, 9, 5,, 7, 33, Find the next three numbers and write a rule for each number pattern , 5, 8,,,,. 3, 6,, 4,,, Start with and add 3 Start with 3 and multiply by , 8, 7, 36,,, 4. 64, 56, 48, 40,,, Start with 9 and add 9 Start with 64 and subtract 8 56,04, 4, , 4, 6, 64,, 6. 75, 70, 65, 60,,, Start with and multiply by 4 Start with 75 and subtract , 8, 7, 63,,, 8. 4, 8, 6, 3,,, Start with 90 and subtract 9 Start with 4 and multiply by 66 hapter Patterns and Variables GPS Use the Guided Problem Solving worksheet with Exercise 6. 66

5 E Leap Year Test Prep Reading omprehension It takes about days for Earth to orbit the sun. Four years of days is about three years with 365 days each, plus a fourth year with 366 days. This fourth year is known as a leap year. Multiple hoice Take It to the NET Online lesson quiz at Web ode: aaa-00 Mixed Review 30. The number pattern below is called a Fibonacci sequence. Write a rule to describe this number pattern. See margin.,,, 3, 5, 8, 3,, 34, 55, Stretch Your Thinking How many different straight lines can you draw through any two of the points at the right? 0 lines Read the article below and answer Exercises 3 and 33. In a leap year, February has 9 days. leap year is any year that can be divided by 4, except century years, such as 500. Only century years that can be divided evenly by 400, such as 600, are leap years. 3. Which years between 007 and 030 will be leap years? 008, 0, 06, 00, 04, Will 00 be a leap year? Explain. No; 00 is not divisible by What is the value of the missing term? 0, 94, 86, 7, 70, 6, I 35. What is the fifth term in the number pattern? ; 0; 400; 8,000; c F. 8,400 G. 6,000 H. 80,000 I. 60, What is the fifth term in the number pattern? 3, 7,, 5, c E 4. ssess Lesson Quiz - Write the next two terms and write a rule for each number pattern.. 3, 9, 5,,... 7, 33; Start with 3 and add 6. 7, 4, 8, 56,..., 4; Start with 7 and multiply by , 0.8, 0.4, 5.,....6,.3; Start with 4.6 and divide by lternative ssessment Each student in a pair writes a number pattern of four terms that can be extended for at least two more terms, using one operation. Partners exchange papers and write the next two terms of the pattern. Then each student writes a six-term pattern and erases one of the numbers. Partners find the missing term in each other s pattern. Test Prep Resources For additional practice with a variety of test item formats: Test-Prep, p. 3 Test-Taking Strategies, p. 09 Test-Taking Strategies With Transparencies Enrichment - Patterns in Numbers Lesson -5 First estimate then find each sum or difference about 3; 3. about ;.06 about 49; about ;.7 about 4.5; 4.3 about ; Number patterns are like secret codes. Once you understand what the pattern is, you have the ability to extend the pattern. reate some number patterns and then share them with others in your class. See if you can unlock the codes of your classmates number patterns.. The number pattern,, 4, 3, 6, 5... is a two-addition pattern.the pattern is +3, +9, +3, reate another pattern that is a two-addition pattern. Show at least ten numbers in your pattern. Sample answer: 0, ;,, 3, 3, 5, 5, 7, 7, 9, 9. What is the number pattern in this sequence, 50, 60, 55, 65, 60, 70...? reate another sequence of numbers using the same operations. Show at least ten numbers in your pattern. Sample answer: 0, 5; 0, 0, 5, 5, 0, 30, 5, 35, 30, You can also use multiplication and division to create patterns. reate a pattern in which all numbers will be even numbers. Use at least three operations. Show at least ten numbers in your pattern. Sample answer: ; 4, ;, 4, 8, 4, 8,, 6,, 6, The first term is ; the second term is ; add the two previous terms to get the next term. 4. pattern needs to contain sufficient numbers to establish the pattern. Find at least three ways that these numbers could start a pattern:, 4...Then find the next five numbers in each pattern. Sample answer:, ;, 4, 3, 5, 4, 6, 5;, 3;, 4, 7, 4, 7, 34, 37;, ;, 4, 8, 0, 0,, a. How many squares will be in the next shape below? 5 b. raw the arrangement. 67