Zigzag Paths and Binary Strings Counting, Pascal s Triangle, and Combinations Part I

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1 Zigzag Paths and Binary Strings Counting, Pascal s Triangle, and Combinations Part I LAUNCH In this task, you will learn about a special triangular array of numbers called Pascal s triangle. You will explore the triangle using zigzag paths and binary strings. In the companion Part II task, you will learn more about counting, subsets, and combinations in Pascal s triangle. Suppose a frog hops downward from START through the triangular array of lily pads below. The frog can jump only to lily pads that are on the row just below it to the right or to the left, so the frog hops downward right or left in a zigzag fashion. You can code its path as a binary string of 0s and 1s. Use a 0 to denote a hop downward and to the left from your perspective and a 1 to denote a hop downward and to the right. For example, a frog can use any of four zigzag paths to hop from START to location F. These paths are recorded in the figure at location F as four binary strings: 0111, 1011, 1101, and You will analyze this figure by working through the problems below. Record your work on separate sheets of paper as needed. START 0 1 <--- row one A 11 B 001 C D E F G 1. The figure shows all the binary strings for rows 1 and 2. That is, for each location in rows 1 and 2, the binary strings representing all paths from START to the location have been recorded at that location. (Row 1 is the first row after START, row 2 is the second row after START, and so on.) Consider row 3. a. Trace the path represented by 010. b. Use a different color to trace a path from START to location C. Represent the path as a binary string. c. Record all the binary strings for each location in row 3. d. Why do all of the binary strings in row 3 have exactly three digits? e. List all the strings in row 3. List them in systematic order, and describe the order that you use. NCTM Adapted from Navigating through Discrete Mathematics in Grades 6-12 Page 1 of 5

2 EXPLORE 2. Explain why there is just one string in each outside location of the triangle. 3. In row 4, consider locations D and F. a. Do you think that the number of strings at location D will be the same as the number of strings at location F? Explain your thinking. b. List the binary strings for all zigzag paths from START to location D. c. Compare the strings for location D with those for location F. What do they have in common? How do they differ? Explain. 4. Consider all the locations in row 4. a. How many strings would you need to code all zigzag paths to E? Write down all the strings. b. Find all the binary strings for each location in row 4 (you have already found many of the locations, so just complete row 4). Describe some patterns you see in the strings in row Consider the locations E, F, and G. a. Strings for location G represent zigzag paths from START to G. Find several strings for location G. b. Study the zigzag paths to G in the triangle. To zig and zag from START to G, a path must travel through E or F. Use the strings in locations E and F to determine all the strings for G. Explain your method. NCTM Adapted from Navigating through Discrete Mathematics in Grades 6-12 Page 2 of 5

3 So far you have been finding and describing the binary strings for each location in the triangle. Next, consider the number of strings at each location. That is, as you work on the following problems, look for answers to this question: How many zigzag paths are there from START to each location? The how many paths question can be answered with the help of the array below. It is known as Pascal s triangle. The numbers in the array are calculated using the following rules: i. Row one consists of two 1s. ii. Each row begins and ends with a 1. iii. To compute any other number in any row, add the two numbers that are just above to the left and just above to the right. (This is the addition rule for Pascal s triangle.) S 1 1 row one Complete row 10 of Pascal s triangle. 7. Compare Pascal s triangle to the triangle of lily pad locations, as follows. a. In Problem 4, you found all the binary strings for each location in row 4 of the lily pad triangle. Compare the binary strings with the numbers in row 4 of Pascal s triangle. Describe the relationship between row 4 of the lily-pad triangle and row 4 of Pascal s triangle. b. In Problem 5, you considered how you can determine the strings for location G from the strings for E and F. Underline the entries in Pascal s triangle that correspond to locations E, F, and G. Describe how the three numbers you have underlined relate to the numbers of strings in locations E, F, and G. The addition rule for Pascal s triangle (see rule iii above), describes how the three numbers in locations E, F, and G are related. Explain why this addition relationship makes sense in terms of paths to E, F, and G in the lily pad triangle. NCTM Adapted from Navigating through Discrete Mathematics in Grades 6-12 Page 3 of 5

4 c. Explain the addition rule for Pascal s triangle in terms of zigzag paths. (For the addition rule, see rule iii for creating Pascal s triangle, above.) d. State the connection between the number of binary strings in each location of the lily pad triangle and the corresponding numbers in Pascal s triangle. 8. Consider zigzag paths from the apex (S) of the triangle to the underlined 35. a. A binary string that describes one such path is Sketch this path. b. Write the binary strings for two other paths from S to the underlined 35. c. What is the length of each binary string that represents a path from S to the underlined 35? What does the length of the string represent? d. How many 1s are in each string corresponding to the underlined 35? How many 0s are in each string corresponding to the underlined 35? Explain the number of 1s and 0s in terms of zigzag paths. e. Consider all zigzag paths of length 7. If you assume that all paths are equally likely, what is the probability that such a path stops at the underlined 35? 9. Consider zigzag paths to locations in row 9. a. What is common to every path from S to the underlined 84 in the triangle? b. What is the length of any path from S to any location in row 9? c. How many nine-step paths from S contain four downward steps to the right? Explain how you determined your answer. d. How many strings of length 9 contain three digits that are 1s? Explain your answer in terms of zigzag paths. List two such strings and mark the paths they represent in the triangle. 10. Pascal s triangle has left-right symmetry. Explain the reasons for this left-right symmetry in terms of zigzag paths. NCTM Adapted from Navigating through Discrete Mathematics in Grades 6-12 Page 4 of 5

5 SUMMARIZE Pascal's triangle is named for Blaise Pascal ( ), who studied and wrote about the patterns it contains. Pascal was an influential French mathematician and philosopher who contributed to many areas of mathematics. However, the triangle was partially described in China beginning 500 years earlier, in the 11th century, by a Chinese mathematician named Jia Xian and later by Yang Hui (13 th century) and Chu Shih-Chieh (14 th century), and was also known to Islamic mathematicians al-karaji (11 th century) and al-kashi (15 th century) and the famous Persian poet Omar Khayyam, who was also a mathematician and astronomer. In China, Pascal's triangle is called Jia s triangle and sometimes Yang Hui s triangle, since Yang s work, although building on Jia's initial discoveries, was evidently more widely known. In this task you explored zigzag paths, binary strings, and Pascal s triangle. Review and summarize what you learned by answering the following questions. a. Describe how a zigzag path is coded as a binary string. Give an example. b. What does the length of a binary string tell you about its location in the triangular array? c. What does the number of 1s in a binary string tell you about the zigzag path that it encodes? d. Describe the relationship between the numbers in Pascal s triangle and zigzag paths. e. Explain the addition rule for Pascal s triangle in terms of zigzag paths. So far, you have represented and explored Pascal s triangle using zigzag paths and binary strings. In the companion Part II task, you will investigate the triangle using subsets and combinations. In the process, you will learn more about Pascal s triangle and many important ideas of systematic counting associated with the triangle. NCTM Adapted from Navigating through Discrete Mathematics in Grades 6-12 Page 5 of 5