1. Copyright © Cengage Learning. All rights reserved. CHAPTER 1 Foundations for Learning Mathematics

2. Copyright © Cengage Learning. All rights reserved. SECTION 1.2 Patterns and Communication

3. 3 What Do You Think? What do patterns have to do with learning mathematics? How does the ability to recognize and analyze patterns add to your problem-solving toolbox? Why does the NCTM encourage mathematics classrooms to provide numerous opportunities for communication?

4. 4 Patterns and Communication Patterns exist virtually everywhere in the world. Look at the floor, the walls, and the ceiling of most rooms, and you are sure to see patterns. Look at a wallpaper display in a store and see patterns. Virtually all clothing contains patterns. There are also patterns in weather that help us to make predictions. One aspect common to all patterns, whether geometric or numerical, is that there is repetition.

5. 5 Patterns and Communication When we examine this repetition, we find an underlying structure or organization. Understanding this structure enables us to make generalizations, which in turn enables the users of this knowledge to do more powerful mathematics, to design more powerful machines and equipment, or to make manufactured items more cheaply.

6. 6 Patterns and Communication Recognizing patterns: Being able to use patterns makes the guess–check–revise strategy much more powerful. However, as many of you discovered, this is easier said than done. In order to use a pattern, you have to recognize it. As we found, creating a new column called “Difference” helped to make the pattern more visible. We then analyzed the pattern to see that each time we changed a chicken into a pig, we added two feet. This enabled us to get the solution on the third guess.

7. 7 Investigation A – Sequences and Patterns Let us examine three sequences here to understand how reasoning can help us go beyond simply noticing a pattern to understanding the structure of the sequence. For each of the following sequences, see whether you can determine the next number in the sequence, the 20 th number (term) in the sequence, and then the n th term. Sequence 1: 2, 5, 8, 11, 14,... Sequence 2: 3, 6, 12, 24, 48,... Sequence 3: 4, 9, 19, 39, 79,...

8. 8 Investigation A – Discussion Sequence 1: Let us examine the first sequence: 2, 5, 8, 11, 14,.... It is not terribly difficult to determine that the 6 th term will be 17. One way to determine the 20 th term is simply to continue the sequence until you get to the 20 th term. However, this is tedious. With a bit of analysis, you could realize that you need to add 15 more 3s after the 5 th term, and thus, the value of the 20 th term is 14 + 45.

9. 9 Investigation A – Discussion Using the fact that multiplication is repeated addition, we can represent the terms more economically. This not only saves time but also begins to reveal the mathematical structure of the sequence. cont’d

10. 10 Investigation A – Discussion In the last row of the table, the number shown in bold tells how many 3s are there. Notice that this number is always 1 less than the number representing the position of the term in the sequence. That is, the 4 th term has three 3s, the 5 th term has four 3s, etc. Thus, the n th term must have (n – 1) 3s. cont’d Table 1.5

11. 11 Investigation A – Discussion Sequences like the one above, where the difference between each pair of consecutive terms is constant, are called arithmetic sequences. What is different is the starting number and the common difference. If we represent the starting number by a and the common difference by d, then we can state a rule for finding the n th term of any arithmetic sequence, as shown in the last column of Table 1.6. Table 1.6 cont’d

12. 12 Investigation A – Discussion Sequence 2: Now let’s examine the second sequence: 3, 6, 12, 24, 48,.... You probably realized that the two sequences are similar in that there is a pattern and there is a relationship between each term and the next. In this sequence, the relationship is that each term is twice, or double, the preceding term. So the next number in the sequence is 96. cont’d

13. 13 Investigation A – Discussion In the second row of Table 1.7, I have broken down each term in the sequence so you can see how it came to be. Using the fact that exponentiation is repeated multiplication, we can represent the terms more economically. Table 1.7 cont’d

14. 14 Investigation A – Discussion Sequences like 3, 6, 12, 24, 48,.…… where the relationship between each term and the following term is that they always have the same ratio (in this case 2), are called geometric sequences. If we represent the starting number by a and the common ratio by r, then we can state a rule for finding the n th term of any geometric sequence, as shown in the last column of Table 1.8. Table 1.8 cont’d

15. 15 Investigation A – Discussion Sequence 3: Let us examine the third sequence: 4, 9, 19, 39, 79,.... Some students call this a hybrid sequence in that it is not “just like” either of the ones above. One way of describing what repeats is to look at the relationship between each term and the following term. One way of describing this is that you double each term and then add 1. So the next number in the sequence is 159. cont’d

16. 16 Investigation A – Discussion We can represent this relationship between each term and the following term as 2n + 1. This is nice, but it won’t help us to determine the value of the 20 th term or the n th term. Thus we have to look further. Here is where number sense and intuition come into play in mathematics. You might have realized that, after the first term, all of the terms end in 9. What if we wrote out a similar sequence—one in which each term is 1 more than the terms of our sequence. cont’d

17. 17 Investigation A – Discussion This has been done in the third row of Table 1.9. By breaking each number apart (5, 5  2, 5  2  2, 5  2  2  2, 5  2  2  2  2) or by using exponents (see the fourth and fifth rows of Table 1.9). We can see that the n th term of the original sequence (4, 9, 19, 39,...) is simply 1 less; that is, 5  2 (n – 1) – 1. Table 1.9 cont’d

18. 18 Investigation B – Patterns in Multiplying by 11 Let us investigate what happens when we multiply a number by 11. From examining the first three problems in Table 1.10, can you predict the answer to 53  11? Make your prediction and then multiply 53  11 to check your prediction. Table 1.10

19. 19 Investigation B – Discussion If you had trouble, the algorithm for multiplying a two-digit number by 11 is to add the two digits together, and that number is the middle digit of the product. Test this out for 62  11. When we get to 73  11, the problem becomes more challenging because the sum of the two digits is more than 9. First, we know that the digit in the ones place will stay the same.

20. 20 Investigation B – Discussion Because 3 + 7 = 10, the digit in the tens place will be a zero and the digit in the hundreds place increases by 1, like “carrying.” This is how we get the answer of 803. Using this analysis, predict the product of 75  11. We find the algorithm works: 7 + 5 = 12. We still have 5 in the ones place, 2 in the tens place, and the hundreds place is now one more: 8, giving us the predicted product of 825. cont’d

21. 21 Patterns and Communication Communication: Standard 8: Communication Instructional programs from pre-kindergarten through grade 12 should enable all students to organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely.

22. 22 Patterns and Communication Albert Einstein is said to have remarked that “a description in plain language is a criterion of the degree of understanding that has been reached.” Communication and understanding go hand in hand. “When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others verbally or in writing, they are faced with the task of stating their ideas clearly and convincingly to an audience”.

23. 23 Patterns and Communication Implicit in the previous sentence is that there are two distinct kinds of communication that you need to be aware of as you work in this course. 1. Communicating with yourself about the problem—that is, being able to make sense of your own strategies and solutions. 2. Communicating with other people—sharing your observations and solutions and being able to understand others’ observations and solutions.

24. 24 Learning by Communicating

25. 25 Learning by Communicating There is another reason for emphasizing communication at the beginning of this course. Few people learn best in isolation. The National Training Laboratories in Bethel, Maine, created the pyramid in Figure 1.3 to represent what we know about learning. Retention Rates from Different Ways of Learning Figure 1.3

26. 26 Investigation C – Pascal’s Triangle The pattern below, called Pascal’s triangle, was “discovered” by the mathematician Blaise Pascal (1623–1662). Examine the triangle, play with it, and describe the patterns you see.

27. 27 Investigation C – Pascal’s Triangle You may choose to describe patterns with words, in a table or sequence, or by using mathematical notation. Discussion: Here are some patterns to get you started: 1. Both top diagonals contain only 1s. For example, start at the top and go down the left side. You have only 1s. 2. The counting numbers (1, 2, 3, 4,...) are on the second diagonal. cont’d

28. 28 Investigation C – Discussion 3. The sum of any sequence of counting numbers is found diagonally down from the last number counted. Look at 1, 2, 3, 4. The sum of those four numbers is 10, which is diagonally below 4. Now look at 1, 2, 3, 4, 5, the sum of which is 15, which is diagonally below 5! 4. The figure has symmetry such that each diagonal has a twin. For example, look at the second diagonal going down the right: 1, 2, 3, 4, 5, 6, 7, 8,.... cont’d

29. 29 Investigation C – Discussion 5. Each number (not equal to 1) in the triangle is the sum of the two numbers to its top right and its top left. Look at this visual representation: Choose any number c (not equal to 1) in the triangle. If we let a and b represent the two numbers immediately above c, then the value of c is equal to the value of a + b! 6. Add the numbers in each row in the triangle. cont’d

30. 30 Investigation C – Discussion The sums of the rows make the following sequence: 1, 2, 4, 8, 16. This sequence can be described as the powers of 2. In this case a table is helpful. See Table 1.11. Table 1.11 cont’d

31. 31 Investigation D – Communicating Patterns in a Magic Square Magic squares, triangles, and circles have fascinated human beings for thousands of years. The definition of a magic square is that when you add the numbers in any row, any column, or either diagonal, you get the same number. The square at the left was created by Albrecht Dürer in 1514 in his engraving Melancholia. He created a square in which the year appeared in the bottom row of the square! The magic sum for rows, columns, and diagonals is 34. Now take a closer look at this square.

32. 32 Investigation D – Communicating Patterns in a Magic Square What other patterns do you see in this magic square? What observations do you make? Observation will be defined as something interesting, but not repeating—for example, the 16 numbers in the square are consecutive numbers from 1 to 16. Write down your observations and patterns so that someone reading your description could see what you see. cont’d

33. 33 Investigation D – Discussion Below are some of many pattern and observations. There are actually many more 34s. Here are six more! The sum of the four numbers in the center 2  2 square is 34; the sum of the four corner numbers is also 34; if you divide 4  4 the square into four 2  2 squares, the sum of the numbers in each of the 2  2 squares is also 34. There are 2 odd and 2 even numbers in each row and column. There is at least one prime number in each column.

34. 34 Investigation D – Discussion Middle rows and/or columns can be switched without affecting the square; that is, the sum of all rows, columns, and diagonals will still be 34. If you travel on the bottom-right to top-left diagonal, each number is 3 more than the previous number. Now look at the other diagonal. What do you see there? If you look at the middle two columns, each pair of numbers are consecutive numbers: 3 and 2, 10 and 11, 6 and 7, 15 and 14. Now look at the middle two rows. What do you see there? cont’d

35. 35 Investigation D – Discussion Square every number in the table. The sum of each row is 438 or 310, and the sum of these two numbers is 748. The sum of each column is 378 or 370, and the sum of these two numbers is 748. Add the first and second numbers in each row—the sum is either 16 or 18. Add the third and fourth numbers of each row—the sum is either 16 or 18. You can find a similar pattern by summing the top two numbers and then the bottom two numbers in each column; in this case, the sums alternate between 13 and 21. cont’d