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A nautilus shell repeats the same shape on a larger and larger scale from its tip to its opening. You yourself repeat many activities each day. These three examples all involve an iterative process. Iteration is a process of repeating the same procedure over and over. The following activities demonstrate this process.
You yourself repeat many activities each day. These three examples all involve an iterative process. Iteration is a process of repeating the same procedure over and over. The following activities demonstrate this process. Work with a partner to develop an algorithm to sort the members of your class in order of their birthdays. An algorithm is a procedure or set of rules for solving a problem. Select two people and compare their birthdays.
Rank the person with the later birthday second. Now, compare the next persons birthday with the last ranked birthday. Rank the later birthday of those two last. Describe the continuing process you will use to nd the classmate with the latest birthday. Describe the process you would use to nd the person with the second latest birthday. With whom do you stop comparing? Describe a process to rank all the remaining members of your class by their birthdays.
Illustrate your process with a diagram. The process you described is an iterative process because it involves repeating the same set of steps throughout the algorithm. Computers can easily be programmed to sort data using this process. Using isometric dot paper, draw a large equilateral triangle with side lengths of 32 units.
Divide this equilateral triangle into four smaller equilateral triangles. Shade the middle triangle. What fraction of the original triangle is shaded? For each of the unshaded triangles, repeat this process. For each of the unshaded triangles, repeat this process again. What fraction of the original triangle is shaded now?
Predict the fraction of the original triangle that would be shaded for the fourth and fth steps in this iterative process. Predict the fraction of the original triangle that would be shaded if this iterative process continued indenitely. Method 2: The Geometers Sketchpad 1. Open a new sketch and a new script. Position both windows side by side. Click on REC in the script window. In the sketch window, construct a triangle.
Shift-click on each side of the triangle. Shift-click on one vertex and the two adjacent midpoints. Choose Loop in your script. Repeat step 5 for the other two vertices.
Shift-click on the three midpoints. From the Display menu, choose Hide Midpoints. Stop your script. Open a new sketch. Construct a new triangle. Mark the three vertices. Play your script at a recursion depth of at least 3. You may increase the speed by clicking on Fast. Experiment with recursion scripts to design patterns with repeating shapes.
It is an example of a fractal, a geometric gure that is generally created using an iterative process. One part of the process is that fractals are made of selfsimilar shapes. As the shapes become smaller and smaller, they keep the same geometrical characteristics as the original larger shape. Fractal geometry is a very rich area of study.
Fractals can be used to model plants, trees, economies, or the honeycomb pattern in human bones. Choose an interesting fractal and describe how it is self-similar. Example 1 Modelling With a Fractal Fractals can model the branching of a tree.
Describe the algorithm used to model the tree shown. Solution Begin with a 1-unit segment. Branch off at 60 with two segments, each one half the length of the previous branch. Repeat this process for a total of three iterations.
Arrow diagrams can illustrate iterations. Such diagrams show the sequence of steps in the process. Example 2 The Water Cycle Illustrate the water cycle using an arrow diagram.
Although the timing of the precipitation can vary, the cycle will repeat itself indenitely.
How many different outcomes of the series are possible? Each set of branches represents a new game in the playoff round. As soon as one team wins three games, the playoff round ends, so the branch representing that sequence also stops.
Iterative processes can create fractals. A process that repeats itself can be illustrated using arrows and loops. A tree diagram can illustrate all the possible outcomes of a repeated process involving two or more choices at each step. For recursive functions, the rst step is calculated using initial or seed values, then each successive term is calculated using the result of the preceding step.
Communicate Your Understanding 1. Describe how fractals have been used to model the fern leaf shown on the right. Describe your daily routine as an iterative process. Practise A 1. Which of the following involve an iterative 2. The diagram below illustrates the carbon- oxygen cycle.
Draw arrows to show the gains and losses of carbon dioxide. Draw a tree diagram representing the 8. Application In the Swedish playoffs of eight players in a singles tennis tournament. The tree diagram should show the winner of each game continuing to the next round until a champion is decided.
Apply, Solve, Communicate B 4. Draw a diagram to represent the food chain. Communication Describe how the tracing mathematician Helge von Koch developed a fractal based on an equilateral triangle. Using either paper and pencil or a drawing program, such as The Geometers Sketchpad, draw a large equilateral triangle and trisect each side.
Replace each middle segment with two segments the same length as the middle segment, forming an equilateral triangle with the base removed, as shown below. Illustrate your description with a sketch.
In the rst investigation, on page 6, you developed a sort algorithm in which new data were compared to the lowest ranked birthday until the latest birthday was found.
Then, the second latest, third latest, and so on were found in the same manner. If you are using a computer program, continue this process for at least two more iterations. Application Sierpinskis carpet is similar to number of segments after n iterations? The rst two terms of a sequence are given Sierpinskis triangle, except that it begins with a square. This square is divided into nine smaller squares and the middle one is shaded. Use paper and pencil or a drawing program to construct Sierpinskis carpet to at least three stages.
Predict what fraction of the original square will be shaded after n stages. Determine the next four terms in the sequence. Each of the following fractal trees has a a Select a starting point near the centre of different algorithm. Assume that each tree begins with a segment 1 unit long.
Assign the numbers 1 to 4 to the directions north, south, east, or west in any order. Now, generate random whole numbers between 1 and 4 using a die, coin, or graphing calculator.
Draw successive line segments one unit long in the directions corresponding to the random numbers until you reach an edge of the paper. Repeat part a using this extra rule. After completing the walk, trade drawings with a classmate and see if you can deduce the rules for each others walk. Describe an aspect of chaos theory that interests you. Are all spirals within a tree self-similar? Use the given values for t1 to nd the successive terms of the following recursive formulas.
Continue until a pattern appears. Describe the pattern and make a prediction for the value of the nth term. Draw the tree using paper and pencil or a drawing program. A random walk can illustrate such chaotic outcomes. Draw an angle and, on the x-axis, label the point cos , 0 as P1. Draw a new circle, centred at P1 , with radius of cos.
Continue this iterative process. Evaluate 8!
Communication Music can be written using considered an iterative process. Write a formula for the product of the rst n even positive integers. Explain why your formula is correct.
Look up this type of music in a library or on the Internet. What characteristics does fractal music have? Computers use binary base 2 code to represent numbers as a series of ones and zeros. Base 10 0 1 2 3 4 Binary 0 1 10 11 a Describe an algorithm for converting product of integers from base 10 to binary.
Research the recycling process for a material such as newspaper, aluminum, or glass and illustrate the process with an arrow diagram. List every computer program you can think of that can be used to manage data.
Sort the programs into categories, such as word-processors and spreadsheets. Indicate the types of data each category of software would be best suited to handle. List the advantages and disadvantages of each category of software. Decide which of the programs on your list would be best for storing and accessing the lists you have just made.
Most ofce and business software manage data of some kind. Schedulers and organizers manage lists of appointments and contacts.
E-mail programs allow you to store, access, and sort your messages. Word-processors help you manage your documents and often have sort and outline functions for organizing data within a document. Although designed primarily for managing nancial information, spreadsheets can perform calculations related to the management and analysis of a wide variety of data. Most of these programs can easily transfer data to other applications.
Database programs, such as Microsoft Access and Corel Paradox, are powerful tools for handling large numbers of records. These programs produce relational databases, ones in which different sets of records can be linked and sorted in complex ways based on the data contained in the records.
For example, many organizations use a relational database to generate a monthly mailing of reminder letters to people whose memberships are about to expire.
However, these complex relational database programs are difcult to learn and can be frustrating to use until you are thoroughly familiar with how they work. Partly for this reason, there are thousands of simpler database programs designed for specic types of data, such as book indexes or family trees.
Of particular interest for this course are programs that can do statistical analysis of data. Such programs range from modest but useful freeware to major data-analysis packages costing thousands of dollars. To demonstrate statistical software, some examples in this book have alternative solutions that use FathomTM, a statistical software package specically designed for use in schools.
Data management programs can perform complex calculations and link, search, sort, and graph data. The examples in this section use a spreadsheet to illustrate these operations. A spreadsheet is software that arranges data in rows and columns. For basic spreadsheet instructions, please refer to the spreadsheet section of Appendix B. If you are not already familiar with spreadsheets, you may nd it helpful to try each of the examples yourself before answering the Practise questions at the end of the section.
Formulas and Functions A formula entered in a spreadsheet cell can perform calculations based on values or formulas contained in other cells. Formulas retrieve data from other cells by using cell references to indicate the rows and columns where the data are located.
In both Microsoft Excel and Corel Quattro Pro, it is good practice to begin a formula with an equals sign. Although not always necessary, the equals sign ensures that a formula is calculated rather than being interpreted as text. Built-in formulas are called functions. In Corel Quattro Pro, precede a function with an symbol. Example 1 Using Formulas and Functions The rst three columns of the spreadsheet on the right list a students marks on tests and assignments for the rst half of a course.
Determine the percent mark for each test or assignment and calculate an overall midterm mark. Use the SUM function to nd totals for columns B and C, and then convert to percent in cell D12 to nd the midterm mark. Relative and Absolute Cell References Spreadsheets automatically adjust cell references whenever cells are copied, moved, or sorted. For example, if you copy a SUM function, used to calculate the sum of cells A3 to E3, from cell F3 to cell F4, the spreadsheet will change the cell references in the copy to A4 and E4.
Thus, the value in cell F4 will be the sum of those in cells A4 to E4, rather than being the same as the value in F3. Because the cell references are relative to a location, this automatic adjustment is known as relative cell referencing. If the formula references need to be kept exactly as written, use absolute cell referencing. Enter dollar signs before the row and column references to block automatic adjustment of the references. Fill and Series Features When a formula or function is to be copied to several adjoining cells, as for the percent calculations in Example 1, you can use the Fill feature instead of Copy.
Click once on the cell to be copied, then click and drag across or down through the range of cells into which the formula is to be copied. Use a spreadsheet to produce a conversion table for temperatures from 1C to 15C. Solution Enter 1 into cell E2 and 2 into cell E3. Use the Fill feature to put the numbers 3 through 15 into cells E4 to E Then, use the Fill feature to copy the formula into cells F3 through F Note that the values in these cells show that the cell references in the formulas did change when copied.
These changes are an example of relative cell referencing. Charting Another important feature of spreadsheets is the ability to display numerical data in the form of charts or graphs, thereby making the data easier to understand. The rst step is to select the range of cells to be graphed. To find more information and specific links for each chapter, click on the underlined words or phrases below. Functions 11 is broken down into the following units: Nelson Functions 11 Corrections Chapter 1 Chapter Review 11 This is a present value question.
This course introduces the mathematical concept of the function by extending students' experien Ms Ma's Advanced Functions class - 1. Chapter 2: Dynamics Chapter 9. Tues - Inverse Functions pgs. Chapter 8: Financial Applications Compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
Solutions Manual Appendix mcv4u Class 9, pmp exam guide, life orientation question paper grade 8, microeconomic theory basic principles and extensions 11th edition answers, excpt exam study guide, Murder Past Due Cat In The Stacks 1 Miranda James, free papers on culture, nelson functions 11 chapter 8 solutions, june Here you can download file Nelson Advanced Functions This course requires that you are very comfortable with the material you learned in 2D.
Functions 11 Nelson. Chapter 8. Unit Breakdown. Chapter 5: Trigonometric Ratios. Relationships between Points, Lines, Planes. Chapter 3: Quadratic Functions.
Chapter 5. Nelson Functions 11 Solutions Manual Chapter 1 are becoming more and more widespread as the most viable form of literary media today. Nelson functions eleven. Make sure you write down questions to bri There was a problem previewing this document. Scanning available upon request. Corrections to textbook solutions. Student Achievement Levels.
This page contains links for each chapter's "go to" activities, as well as links to help you with the chapter and unit reviews. Compound Interest vs Simple Interest: Applet; 8. Textbook Answer Corrections. If you have questions on the homework, please see me Mr. Equations of Lines and Planes.
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