<p> Dots and Patterns</p><p>Dots and Patterns</p><p>Description Students will be asked to continue certain patterns, write generalized equations for those patterns, and make certain inferences based off of observations. </p><p>Objectives (Lessons to be learned)  Pattern recognition  Pattern equation writing  The relationship between explicit and recursive equations</p><p>Sunshine State Standards/Benchmarks  MA.D.1.4.1 I can describe, analyze, and generalize relationships, patterns, and functions using words, symbols, variables, tables, and graphs.  MA.D.1.4.2 I can determine the impact when changing parameters of given functions.  MA.E.2.4.1 I can represent real-world problem situations using finite graphs, matrices, sequences, series, and recursive relations.</p><p>Bodies of Knowledge (Approved September 2007)  MA.912.A.10.1 I can use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, working backwards, and creating a table.  MA.912.D.1.1 I can use recursive and iterative thinking to solve problems, including identification of patterns, population growth and decline, and compound interest.  MA.912.D.1.2 I can use finite differences to solve problems to find explicit formulas for recurrence relations.</p><p>Relevance Programming is highly dependent on iterative loops. These loops act just like the dot patterns seen in this exercise. </p><p>Discrete Mathematics Dots and Patterns</p><p>Learning Challenges Inquiry Questions  Can you write an equation to help you find later iterations of the pattern?  Did you notice some similarities between the problems on the last two pages? Conclusion Statement  The last problem is an easy way to add up all the numbers between 1 and n The “Aha!” Moments  When they realize equation writing makes the patterns problems very easy  They realize that the last problem is an easy way to sum 1 to n</p><p>Tools Needed  Pen and paper</p><p>Discrete Mathematics Dots and Patterns</p><p>Assignment The first four V-patterns are given below.</p><p>#1 #2 #3 #4</p><p>1. Draw the next two V-patterns in the sequence.</p><p>2. Which V-pattern will have 31 dots? Explain how you know.</p><p>3. How many dots will the 10th V-pattern have? Explain.</p><p>4. How many dots will the 100th V-pattern have? Explain.</p><p>5. Is it possible to make a V-pattern with 24 dots? With 35,778 dots? Explain.</p><p>6. How many dots will be in the 101st V-pattern? Did you use the same strategy or a different strategy than you used in problem #4 to solve this task?</p><p>Discrete Mathematics Dots and Patterns</p><p>The first four dot patterns are given below.</p><p>#1 #2 #3 #4</p><p>1. Draw the next two dot patterns in the sequence.</p><p>2. Which dot pattern will have 41 dots? Explain how you know.</p><p>3. Is it possible to make a dot pattern with 37 dots? With 4081 dots? Explain.</p><p>4. How many dots will the 10th dot pattern have? 100th? Explain.</p><p>5. Write a formula that will summarize how to find the number of dots in the Pth dot pattern. Explain what each of your letters stands for in your formula.</p><p>6. Share your formula with a partner or another group and see if they can use it to find the number of dots in the 130th dot pattern.</p><p>Discrete Mathematics Dots and Patterns</p><p>The tile patterns below are made of white and gray tiles. Gray tiles cost $2.00 each. White tiles cost $1.00 each.</p><p>#1 #2 #3 #4</p><p>1. Draw the 5th and the 8th tile pattern.</p><p>2. How much would the white tiles cost for the 5th, 6th, 7th and 12th patterns? Explain.</p><p>3. How much would the gray tiles cost for the 5th, 6th, 7th and 12th patterns? Explain.</p><p>4. How much would the white tiles cost for the 100th pattern? How much would the gray tiles cost for the 100th pattern?</p><p>5. Come up with a way to describe the number of gray tiles there will be in any pattern # in the sequence. Record your description so another group can understand.</p><p>Discrete Mathematics Dots and Patterns</p><p>Each shape below is made with beams (------) and bolts (dots).</p><p>#1 #2 #3 #4</p><p>1. How would you change pattern #4 to create pattern #5 in the sequence? How many more bolts were needed? How many more beams were needed?</p><p>2. How many beams are in the 25th pattern in the sequence? How many bolts are in the 25th pattern?</p><p>3. Write a formula for the number of beams.</p><p>4. Write a formula for the number of bolts.</p><p>5. A company has 301 bolts. What is the largest pattern that they can make?</p><p>Discrete Mathematics Dots and Patterns</p><p>ANOTHER PATTERN:</p><p>1. Draw pattern #5. Make a sketch of pattern #P.</p><p>2. Without going one-by-one, how would you figure out the number of white circles in pattern number 50?</p><p>3. Write a formula with P standing for the pattern number # of white circles= Total # of circles=</p><p>ANOTHER PATTERN:</p><p>#1 #2 #3 #4</p><p>1. Draw pattern #5. Make a sketch of pattern #P.</p><p>2. Without going one-by-one, how would you figure out the number of white circles in pattern number 50?</p><p>3. Write a formula with P standing for the pattern number</p><p>Discrete Mathematics Dots and Patterns</p><p>#1 #2 #3 #4</p><p>1. Draw pattern #5.</p><p>2. Draw pattern #P.</p><p>3. If your pattern number is P, how do you figure out the number of white tiles in the frame?</p><p>4. Without going one-by-one, how would you figure out the number of gray tiles in pattern #50?</p><p>5. If you know that the pattern number is P, write a formula for:</p><p>Total # of gray tiles= Total # of white tiles= Total number of tiles=</p><p>Discrete Mathematics Dots and Patterns</p><p>Find the sum total of 1 through 5.</p><p>1+2+3+4+5=???</p><p>Now do it for 1 to 5 AND 5 to 1.</p><p>1+2+3+4+5 + 5+4+3+2+1 ???????</p><p>Now try to sum 1 to 100, and 100 to 1 like you did in the previous exercise</p><p>1 + 2 +…….….99+100 + 100+99…………2 + 1 ???????????</p><p>Finally, sum 1 to 100.</p><p>1+2+……..99+100=???</p><p>Discrete Mathematics Dots and Patterns</p><p>1. The first four dot patterns are given below.</p><p>#1 #2 #3 #4</p><p>Figure out a formula for the dot pattern sequence.</p><p>2. The first four dot patterns are given below.</p><p>#1 #2 #3 #4</p><p>Figure out a formula for the dot pattern sequence.</p><p>Discrete Mathematics</p>