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Lesson 53 TAKS Grade 9 Objective 10 (8.16)(A) Making Conjectures A conjecture, or statement predicting what is true in certain cases, is often a part of mathematical problems. To make conjectures, you will need information that can be confirmed as either true or false. New Vocabulary • conjecture • sequences Using Patterns to Make Conjectures Patterns that contain repeating sequences of numbers, shapes, or colors can be used for conjectures. Note what features in the sequence are repeated, and how often they are repeated. From this information, you can predict where these repetitions will occur at a point in the sequence that is not given explicitly. Sometimes the repetition is not in the form of a recurring number or figure, but in a procedure. In the sequence 2, 4, 6, 8, 10, … , no number is repeated. Instead, the adding of 2 to each number to produce the next is repeated. This repetition allows you to make the conjecture that each number in the sequence will be 2 larger than the previous number. Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. EXAMPLE 1 What conjecture can you make about the following sequence of numbers? 1, 1, 2, 3, 5, 8, 13, 21, 34, … Step 1 Determine the pattern in the given sequence. The third number, 2, equals the sum of the first two numbers, 1 and 1. The fourth number, 3, equals the sum of the second and third numbers, 1 and 2. The fifth number, 5, equals the sum of the third and fourth numbers, 2 and 3. Step 2 Make your conjecture based on the pattern. After the first two numbers in the sequence, each number in the sequence equals the sum of the previous two numbers. Step 3 Test your conjecture using the given information. The seventh number is 13. The eighth number is 21, The ninth number, 34, should equal 13 21. This is the case, which supports the conjecture. Be sure that you have recognized the pattern that is taking place in the sequence, so that you can accurately predict where the repetitions will occur at a later point in the sequence. Quick Check 1 1a. What conjecture can you make about the following sequence of numbers? 1, 5, 9, 13, 17, 21, … TAKS Review and Preparation Workbook 1b. What conjecture can you make about the following sequence of numbers? 1, 4, 2, 4, 3, 4, … LESSON 53 ■ Making Conjectures 157 TAKS Objective 10 (8.16)(A) LESSON 53 Using Examples or Non-Examples to Make Conjectures Sometimes sets of examples have a common property about which you can make a conjecture. Sometimes a conjecture can even be made about a property that is not a part of the examples. These “non-examples” indicate what the conjecture is. Start by looking at all of the examples to see what they have in common. These are factors you can ignore. Look for patterns in how you get from one example to the next. Use these patterns to make a conjecture about the next example. In the event that there is something the examples are all missing, a conjecture can still be made from this set of non-examples. If there is enough information to make a conjecture, then there is enough information to check it. EXAMPLE 2 A group of students were given 20 problems to solve in different amounts of time. The number of mistakes made by students for each block of time is shown in the scatterplot below. What conjecture can you make about the data? 6 5 4 3 2 1 0 5 10 15 20 25 Time (minutes) 30 Step 1 Examine the information for common features or patterns. The number of mistakes decreases as the time for solving the problems increases. Step 2 Make your conjecture. The greater the time spent on each problem, the fewer mistakes that are made. Step 3 Test your conjecture using the given information. The line that follows the trend of the data has a negative slope. Therefore, as the number in the domain (amount of time per problem) increases, the number in the range (number of mistakes made) decreases. Quick Check 2 2a. What conjecture would you make if the data in Example 2 formed a horizontal line? 2b. What conjecture would you make if the data in Example 2 followed no particular trend? 158 LESSON 53 ■ Making Conjectures TAKS Review and Preparation Workbook Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Number of Mistakes Number of Mistakes vs. Time Name__________________________Class____________Date________ 1 The figures below have a repeating pattern. ... 1 2 3 4 5 6 7 8 ... Which of the following will appear as the 18th figure in the pattern? A C B D 4 Five balls are chosen such that the radius of each ball is as follows: 2 centimeters, 4 centimeters, 8 centimeters, 16 centimeters, and 32 centimeters. Assuming that each ball is a perfect sphere, which of the following statements about the four largest balls is correct? F The volume of any ball is 8 times greater than the volume of the next smallest ball. G The volume of any ball is 4 times greater than the volume of the next smallest ball. H The volume of any ball is 3 times greater than the volume of the next smallest ball. J The volume of any ball is 2 times greater than the volume of the next smallest ball. 2 Which statement is a correct conjecture about the following numerical sequence? 1, 1, 1,… 1, 14, 19, 16 25 36 Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. F Each number in the sequence equals one half the previous number (12n). 5 Angela is comparing the prices of different shoes. She notes the prices for two sizes and two styles for each of three brands. Angela’s findings are listed in the table below. G Each number in the sequence equals one third the previous number (13n). H Each number in the sequence equals the reciprocal of the square of the number 12 . (n ) J Each number in the sequence equals the reciprocal of the cube of the number 13 . (n ) 3 The arrangement of colored lights along the edge of a carnival booth is as follows: blue, green, yellow, and purple. If the sequence starts with a blue light, what color will the 32nd light in the sequence be? Size 6 dress Size 8 dress Size 6 loafers Size 8 loafers Brand A $22 $22 $25 $25 Brand B $16 $16 $11 $11 Brand C $19 $19 $14 $14 Which conjecture follows from the data in the table? A The cost of all shoes increases with shoe size. B The cost of shoes does not depend on the brand. C The cost of shoes is always highest for brand A and lowest for brand B. D The cost of loafers is always less than the cost of dress shoes. A blue B green C yellow D purple TAKS Review and Preparation Workbook LESSON 53 ■ Making Conjectures 159