MAT 152_SAS#01.pdf

MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class numbe...

MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Lesson Title: The Nature of Mathematics: Articulating Materials: the Importance of Mathematics in One’s Life; Student activity sheets Recognizing Patterns in Nature and Configurations in the World References: https://www.niu.edu/mathmatters/ev Learning Targets: eryday-life/index.shtml At the end of the module, students will be able to: https://lifehacks.io/reasons-whymath- 1. Identify the Fibonacci numbers in nature and art. is-important/ 2. Draw a spiral using the Fibonacci sequence. https://www.youtube.com/watch?v=m e6Dnl2DOtM Productivity Tip: “Use an online or physical calendar or planner to organize your time ”. A. CONNECT A.1 LESSON PREVIEW/REVIEW Activity A.1 Study the picture and answer the following questions briefly. 1. What do you notice about this construction of the nautilus shell? _________________________________________________________ _________________________________________________________ 2. What other things in nature (plants, animals, etc.) you observe this kind of pattern? _________________________________________________________ _________________________________________________________ B. COACH B.1 Content Notes Topic: Recognizing Patterns in Nature and Configurations in the World Reading Comprehension Strategy: Main Ideas and Supporting Details. In this lesson, you are going to read to learn about Recognizing Patterns in Nature and Configurations in the World. To help us understand what we read, we will use the comprehension strategy of finding the main ideas and the supporting details. This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Instructions: In the last lesson, I showed you the two steps to finding the main idea and supporting details. Let’s do this together. What was the first step? - - - - - -You’re right! Step one: Read and highlight information. Let’s do that now. Read carefully and highlight the important ideas. I’ll do some, you do the rest! Look at my notes! A Fractal is a detailed Main Idea: A fractal is a complex pattern that is self-similar across different pattern that looks scales and repeats itself over time. similar at any scale and repeats itself over time. It shows simple shapes multiplying Supporting detail 1: Supporting detail 2: Supporting detail 3: Each snowflake has a The branching pattern Fern leaves, the over time yet unique pattern and self- of trees, the smaller smaller leaflets maintaining the same similarity. branches replicate the mirroring the shape of pattern. We can find structure of larger the entire leaf. fractals in nature in branches. snowflakes, tree branching, lightning, and ferns. A Spiral is a curved Main Idea: A spiral is a curved pattern with circular shapes revolving around a pattern that focuses central point. on a center point and a series of circular shapes that revolve Supporting detail 1: Supporting detail 2: Supporting detail 3: around it. We can find spirals in nature in Pinecone scales form Many shells, like those In hurricanes, the pinecones, some spirals that follow the of snails and nautilus, spinning makes shells, pineapples, Fibonacci sequence, have spiral shapes bands of clouds that and hurricanes. making them efficient where the shell grows spiral out from the and beautiful. larger but retains its center. It's kind of like shape. how water goes down a drain in a spiral. This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ A Voronoi pattern is a Main Idea: Voronoi pattern is like drawing borders around things based on how way of dividing space close they are to each other. into regions based on how close things are to each other. It helps show how nature likes Supporting detail 1: Supporting detail 2: Supporting detail 3: to be efficient, meaning it likes to use the least Voronoi patterns are like Voronoi is like a puzzle, Bees make amount of space and drawing lines on a map maximizing coverage honeycombs with cells energy. You can see to show where things efficiently. that fit together like a these patterns in are close together. This puzzle, helping them different parts of nature. For example, helps us see how nature save space and the spots on a giraffe's tries to use space and organize their hive. skin look like pieces of energy wisely. a puzzle that fit together perfectly. Honeycombs made by bees are another example, where the hexagon shapes use the least material to store the most honey. Fractals Spirals Voronoi This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ FIBONACCI SEQUENCE Main Idea: Fibonacci sequence is a numerical pattern found in nature and The Fibonacci mathematics, where each number is the sum of the two preceding ones. sequence, named after the Italian mathematician Leonardo Fibonacci of Pisa, who introduced it Supporting detail 1: Supporting detail 2: Supporting detail 3: in 1202, creates a mathematical pattern Introduced by Italian Fibonacci numbers It shows how found in nature. It uses Mathematician help us understand mathematical patterns simple equations to Leonardo of Pisa in how things in nature are reflected in the produce Fibonacci 1202. grow with patterns. beauty of our world. numbers, which can describe many of the complex spiral growth patterns seen in nature. Nonetheless, studying the Fibonacci sequence offers us an opportunity to explore the interplay between mathematics, nature, and art in a meaningful and engaging way. Rule of a Fibonacci Sequence The Fibonacci Sequence can be written as a "Rule". The terms are numbered from 0 onwards like this: n= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14... xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377... From the table above, the nth term is the Fibonacci number xn, or when n = 6 , x6 = 8; n = 7 , x7 = 13. The Rule can be written as: xn = xn-1 + xn-2 xn-1 is the previous term or the Fibonacci number corresponding to n−1 xn-2 is the term or the Fibonacci number corresponding to n−2 Example: Find the 8th term of the Fibonacci sequence: x8 = x7 + x6 = 13 + 8 = 21. This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Done! Finding the main idea and supporting details in what we read will help us understand the content better. This reading strategy can be used in any subject! It takes practice to do this well. Keep trying! The more you practice finding main ideas and supporting details, the better you'll get at it. Strategy Review: 1. What is the title of the text we read? __________________________________________ 2. What reading comprehension strategy did we use? ______________________________ 3. What are the two steps in this strategy? _______________________________________ 4. How many main ideas were in the text? _______________________________________ 5. How many supporting details were there in the text? _____________________________ B.2 Skill-building Activities Activity B.2.1 Read the problem carefully and then answer the questions provided. 1. A certain man put a pair of rabbits in a place by a wall. Started with just one pair of rabbits, a baby boy rabbit, and a baby girl rabbit. They were fully grown after one month and in the next month two more baby rabbits (again a boy and a girl) were born. The next month these babies were fully grown, and the first pair had two more baby rabbits (again, a boy and a girl). Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature. (Note: Given no rabbits die in the entire time). How many rabbits will be produced in the bottom of the figure? __________________ *It is an 800-year-old problem created by Fibonacci that led him to study the sequence of numbers. (Liber abbaci, pp. 283-284) This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ 2. Can you identify how Fibonacci numbers are used in Pascal’s Triangle? 3. Tree branching also makes use of the Fibonacci sequence. Draw a sample to show it. Activity B.2.2. Drawing the Fibonacci Spiral Using the graphing paper, you will now draw a spiral. You need to estimate which square you will start (not in the middle). You may also extend your graphing paper to make it bigger if you have a bigger space in your work area. Step 1. Draw a square (S) that measures one centimeter on a side. You can just estimate the measure if you do not have a ruler. Step 2. Draw a second square to the right (R) of the first square (S) that measures one centimeter on a side. Step 3. Draw a third square above (A) the 2 squares previously drawn that measure two centimeters on a side. Step 4. Draw a fourth square to the left (L) of the other three squares that measure three centimeters on a side. Step 5. Draw a fifth square below (B) the other four squares that measure five centimeters on a side. Step 6. Continue this pattern (R-A-L-B) until you have filled up the graphing paper with the same number of squares as a side for the next set of squares. Step 7. To draw the spiral, you need to draw an arc starting on the inside of the initial square and have it pass from one corner to the next so that it is continuously passing each new square from corner to corner. This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ After creating the spiral, explain what pattern is used to find the remaining numbers for the sequence. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ C. CHECK C.1 Check for Understanding Multiple Choice: Encircle the letter of the correct answer. 1. A fractal is a _____ pattern repeated over and over. a. complicated b. complex c. simple d. chaotic 2. Which of the following is an example of fractal patterns found in nature. a. Bumblebees b. Snowflakes c. Dolphins d. Rocks 3. What is the rule for this pattern? 32, 36, 40, 44 a. add 4 b. multiply 2 c. subtract 4 d. divide by 2 4. What are the three next numbers in the pattern? 5, 10, 15, 20, 25, ____, ____, ____ a. 30, 35, 40 b. 65, 35, 116 c. 26, 27, 28 d. 20, 15, 10 5. Which pattern follows the given rule? Rule: Start with 4 and multiply by 3 each time until there are 4 numbers. a. 4, 12, 38, 114 b. 4, 12, 36, 108 c. 4, 7, 10, 13 d. 4, 7, 11, 14 6. The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13... These numbers are named after Fibonacci, whose real name is. a. Leonardo da Vinci b. Leonardo da Pisa c. Leonardo Dicaprio d. Leonardo da Euler 7. Fibonacci spirals (generated by drawing a quarter-circle in each box, where a larger box lays adjacent to a smaller one, and the lengths of these boxes are Fibonacci numbers) are claimed to appear in the arrangements and patterns of fruits, vegetables, pinecones, seed heads and shells. a. True b. False This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ 8. Which of the following is NOT an example of Fibonacci numbers found in nature? a. spirals on a sunflower b. pinecone spiral c. the number of petals on a daisy c. a mountain range 9. What is a spiral? a. curve that starts at a center point and moves away from the center. b. repetition of straight-line shapes c. start from a single point and grow outward in many directions. d. occurs when an organism has no right or left side. 10. What is a pattern? a. movement of energy from one place to another b. curve that starts at a center point c. a group of recurring objects or shapes d. starts from a single point and grows outward in many directions. D. CONCLUDE D.1 Frequently Asked Questions 1) How does the Fibonacci Sequence work? Each number in the sequence is found by adding the two numbers before it. For example, 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, and so on. 2) How do plants and animals use the Fibonacci Sequence? Plants and animals use Fibonacci patterns to optimize growth, space utilization, and resource distribution. For example, leaves on a stem arrange themselves to maximize exposure to sunlight. 3) How can I use the Fibonacci Sequence to create art or design? You can incorporate Fibonacci patterns into your artwork or design by using the sequence to determine proportions, spacing, and composition, creating visually harmonious and engaging results. 3) Where are patterns used in real life? Examples of natural patterns include waves, cracks, or lightning. Man -made patterns are often used in design and can be abstract, such as those used in mathematics, science, and language. In architecture and art, patterns can be used to create visual effects for the observer. This document is the property of PHINMA EDUCATION  MAT 152: Mathematics in the Modern World Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ D.2 Thinking about learning This time let’s end the module activities by answering the following questions about your learning experience. 1. What felt confusing about what you learned today? Why? ______________________________________________________________________________ ______________________________________________________________________________ 2. What motivated you to finish the lesson today? ______________________________________________________________________________ ______________________________________________________________________________ This document is the property of PHINMA EDUCATION