Discovering. Algebra. An Investigative Approach. Condensed Lessons for Make-up Work

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1 Discovering Algebra An Investigative Approach Condensed Lessons for Make-up Work

2 CONDENSED L E S S O N 0. The Same yet Smaller Previous In this lesson you will apply a recursive rule to create a fractal design use fraction operations to calculate partial areas of fractal designs review the order of operations Investigation: Connect the Dots You can create a fractal design by repeatedly applying a recursive rule to change a figure. First, you apply the rule to the starting, or Stage 0, figure to create the Stage figure. Then, you apply it to the Stage figure to create the Stage 2 figure, to the Stage 2 figure to create a Stage figure, and so on. Usually, parts of later stages of a fractal design look like Stage. This feature is called self-similarity. Steps 5 The diagram on page 2 of your book shows Stages 0 of a fractal design created by using the rule Connect the midpoints of the sides of each upward-pointing triangle. If you continued applying this rule forever, you would get a design called the Sierpiński triangle. On the Connect the Dots worksheet, create Stage of the fractal design by following the directions in Step. As you work, think about the patterns from one stage to the next. Your completed Stage triangle should look like the figure on page 9 of your book. Steps 6 0 Suppose the area of the Stage 0 triangle is. The area of the smallest triangle at Stage is, and the combined area of the upward-pointing triangles is or. The area of the smallest triangle at Stage 2 is or 6. Because there are 9 smallest upward-pointing 9 triangles, their combined area is. 6 Stage At Stage, there are 27 (9 ) smallest upwardpointing triangles, each with area 6 or 6. The combined area of these triangles is At Stage, there are 8 (27 ) smallest upward-pointing triangles, each with area 6 or 256. The combined area is Steps 2 The combined areas of triangles at different stages can be found. Here is an example. Suppose the area of the Stage 0 triangle is 8. Then the combined area of one smallest triangle at Stage, three at Stage 2, and two at Stage is Stage 2 6 (continued) 2002 Key Curriculum Press Discovering Algebra Condensed Lessons

3 Previous Lesson 0. The Same yet Smaller (continued) You might want to check your work on a calculator. Calculator Note 0A shows you how to set your graphing calculator to give answers in fraction form. When you evaluate an expression, be sure to follow the order of operations: simplify all expressions inside parentheses, then evaluate all powers, then multiply and divide from left to right, then add and subtract from left to right. EXAMPLE The diagram on page 5 of your book shows three stages of a different fractal design. a. If the area of the Stage 0 triangle is, what is the combined area of one smallest triangle at Stage and four smallest triangles at Stage 2? b. If the area of the Stage 0 triangle is 8, what is the combined area of the shaded triangles? Solution a. The area of the smallest triangle at Stage is 9. The area of the smallest triangle at Stage 2 is 9 9 or 8. Now find the combined area of one smallest Stage triangle 9 and four smallest Stage 2 triangles The combined area is 8. Multiply by before you do any adding. Rewrite the fractions with a common denominator. Add the numerators. b. The area of the Stage 0 triangle is 8, so you can find the combined area of the shaded triangles by multiplying each area by Or you can multiply the result from part a by The combined area is Discovering Algebra Condensed Lessons 2002 Key Curriculum Press

4 CONDENSED L E S S O N 0.2 More and More Previous In this lesson you will look for patterns in the way a fractal grows use patterns to make predictions use exponents to represent repeated multiplication Investigation: How Many? The number of new upward-pointing triangles grows with each stage of the Sierpiński design. If you look closely, you will see a pattern you can use to predict the number of new triangles at any stage. Stage 0 Stage Stage 2 Stage Stage If you count the new upward-pointing triangles at Stages 0, you will get the results shown in this table. Notice that the number of new upward-pointing triangles at each stage is times the number in the previous stage. Stage Number of new number upward-pointing triangles ( is times.) (9 is times.) (27 is times 9.) (8 is times 27.) Continuing this pattern, the number of new triangles at Stage 5 is 8 or 2, the number at Stage 6 is 2 or 729, and the number at Stage 7 is 729 or 287. To find the number of new upward-pointing triangles at Stage 5, you can continue multiplying by to find the number at Stage 8, Stage 9, Stage 0, and so on up to Stage 5. Or you can use this pattern: Number at Stage (one ) Number at Stage 2 Number at Stage Number at Stage... (product of two s) (product of three s) (product of four s) (continued) 2002 Key Curriculum Press Discovering Algebra Condensed Lessons

5 Previous Lesson 0.2 More and More (continued) Continuing this pattern, the number of new triangles at Stage 5 is the product of fifteen s:,8,907. EXAMPLE Describe how the number of white squares is growing in this fractal design. Stage 0 Stage Stage 2 Stage Solution Study how the design changes from one stage to the next. The recursive rule is At each stage, create a -by- checkerboard pattern (with white squares in the corners) in each white square from the previous stage. Notice that each checkerboard pattern contains five white squares. Stage has five white squares. At Stage 2, five white squares are created in each of the white squares from Stage, for a total of 5 5 or 25 white squares. At Stage, five white squares are created in each of the 25 squares from Stage 2, for a total of 5 25 or 25 white squares. You can show these results in a table. Number of white squares Stage Repeated Exponent number Total multiplication form or In the last column, the small, raised numbers, called exponents, tell you how many 5 s are multiplied together. Notice that, for each stage, the exponent is equal to the stage number. You can use this idea to find the number of white squares at any stage. For example, the number of white squares at Stage is 5 or 625, and the number at Stage 0 is 5 0 or 9,765,625. To see if this pattern works for Stage 0, enter 5 0 into your calculator. (See Calculator Note 0B to learn how to enter exponents.) The result is, which is the number of white squares at Stage 0. This fits the pattern. Discovering Algebra Condensed Lessons 2002 Key Curriculum Press

6 CONDENSED L E S S O N 0. Shorter yet Longer Previous In this lesson you will find a recursive rule for generating a fractal design use fractions to express the lengths of segments in various stages of the fractal design use exponents to describe a rule for finding the total length of any stage of the fractal design use a calculator to convert fractions to decimals Investigation: How Long Is This Fractal? The Koch curve is a fractal created from line segments. Think about how the curve changes from one stage to the next. Stage 0 Stage Stage 2 To discover the recursive rule for creating the design, study what happens from Stage 0 to Stage. One possible recursive rule is To get the next stage, divide each segment from the previous stage into thirds and build a equilateral triangle above the middle third. Then, remove the bottom of each triangle. You can make a table to show how the curve changes from stage to stage. Notice that the number of segments at each stage is times the number in the previous stage and that the length of each segment is the length of the segments at the previous stage. Total length (Number of segments times length of segments) Stage Number of Length of each Decimal form number segments segment Fraction form (rounded) If you continue the patterns in the table, you ll see that Stage has or segments and that the length of each segment is or. So the total length of the Stage figure is, which can be written. This simplifies to 6 2, 7 or about 2.7. Stage (continued) 2002 Key Curriculum Press Discovering Algebra Condensed Lessons 5

7 Previous Lesson 0. Shorter yet Longer (continued) Notice that, when you write the total length of a stage using exponents, the exponent is equal to the stage number. So the total length of the Stage figure is, which 56 simplifies to 2 8, or about.6. EXAMPLE Look at the beginning stages of this fractal: Stage 0 Stage Stage 2 a. Describe the fractal s recursive rule. b. Find its length at Stage 2. c. Write an expression for its length at Stage 2. Solution a. To find the recursive rule, consider how the fractal changes from Stage 0 to Stage. The rule is To get the next stage, divide each segment of the previous stage into thirds. Then, replace the middle third with an M made from segments that are each the length of the segments from the previous stage. b. To find the length of the fractal at Stage 2, first look at its length at Stage. The Stage figure has six segments of length. Each Stage 2 segment is replaced by six new segments, so the Stage 2 figure has 6 6 or 6 2 segments. Each Stage 2 segment is the length of the Stage segments, so each Stage 2 segment has length or 2. The total length at Stage 2 is 6 2 2, which can be written as 6 2 or 2 2. So the total length is. c. At each stage, each segment from the previous stage is replaced with six new segments. The length of each new segment is the length of the segment at the previous stage. By Stage 2, this has been done 2 times. The Stage 2 figure is or 6 2 or 2 2 long. 6 Discovering Algebra Condensed Lessons 2002 Key Curriculum Press

8 CONDENSED L E S S O N 0. Going Somewhere? Previous In this lesson you will review operations with integers use a recursive process to evaluate expressions use a calculator to evaluate expressions identify the attractors of expressions Investigation: A Strange Attraction You have seen how recursion can be used to create fractal designs. In this investigation you look at recursive processes involving numerical expressions. Steps 5 2 For example, consider this expression: Start with any number, put it in the box, and do the computation. For example, here s the result if you start with 0: 2 (0) 0 Now, take the result,, and put it in the box in the original expression. 2 () 6 You can continue this process, each time using your result from the previous stage. Here are the results for the first 0 stages. Steps 6 9 You can draw a number line diagram to show how the value of the expression changes at each stage. This diagram shows the results for Stages Now, try evaluating 2 recursively for a different starting value. When you evaluate 2 recursively, the results get farther and farther apart. For some expressions, no matter what value you start with, the results get closer and closer to a particular number. Original expression: 2 Starting number (at Stage 0): 0 Stage number Expression Result (continued) 2002 Key Curriculum Press Discovering Algebra Condensed Lessons 7

9 Previous Lesson 0. Going Somewhere? (continued) EXAMPLE What happens when you evaluate this expression recursively with different starting numbers? 0.5 Solution Let s see what happens for a couple different starting numbers. In both cases, the results seem to get closer to one number, perhaps 6. If, no matter what starting value you try, the results get closer and closer to 6, then 6 is an attractor for the expression. To check whether 6 is an attractor, use it as the starting number. If the result stays at 6, then it is an attractor. 0.5 (6) 6 Because you get back exactly what you started with, 6 is an attractor for the expression 0.5. Starting number: 0.5 () () (.5) (.75) (5.75) (5.6875) Starting number: (0) (8) (7) (6.5) (6.25) (6.25) As you have seen, not all expressions have an attractor. Other expressions have attractors that are difficult or impossible to find. With practice, you may be able to predict the attractors for some simple expressions without actually doing any computations. 8 Discovering Algebra Condensed Lessons 2002 Key Curriculum Press

10 CONDENSED L E S S O N 0.5 Out of Chaos Previous In this lesson you will review measuring distances and finding fractions of distances use a chaotic process to create a pattern by hand use a chaotic process to create a pattern on a calculator Investigation: A Chaotic Pattern? When you roll a die, the results are random. If you looked at the results of many rolls, you would not expect to see a pattern that would let you predict exactly when and how often a particular number will appear. However, as you will see, sometimes a random process, like rolling a die, can be used to generate an orderly result. Steps 9 Suppose you start with this triangle and choose a starting point anywhere inside the triangle., 2 Starting point, 5, 6 Then you roll a die and mark the point halfway from the starting point to the vertex labeled with the number on the die. For example, if you roll a 2, you would mark the point halfway between the starting point and the top vertex. This point becomes the starting point for the next stage. Starting point for next stage, 2, 5, 6 Roll the die again and mark the point halfway between the new starting point and the vertex labeled with the number on the die., 2 Starting point for next stage, 5, 6 (continued) 2002 Key Curriculum Press Discovering Algebra Condensed Lessons 9

11 Previous Lesson 0.5 Out of Chaos (continued) If you repeat this process many times, an interesting pattern will emerge. Rolling the die and plotting the points by hand is time-consuming. Fortunately, your calculator can do the same process much more quickly. The Chaos program chooses a starting point and follows the procedure described above to plot 000 points. Steps 0 2 Try running the program on your calculator. It will take a while for the calculator to plot all the points, so be patient. When the program is finished, you should see this familiar pattern on the screen. The pattern of points looks like the Sierpiński triangle! Mathematicians use the word chaotic to describe an orderly procedure that produces a random-looking result. Here we started with a random procedure that produces an ordered-looking result. The orderly result is called a strange attractor. No matter what starting point you choose, the points will fall toward the shape. Many fractal designs, like the Sierpiński triangle, are strange attractors. Accurate measurements are essential to seeing a strange attractor form. EXAMPLE Find point F three-fifths of the way from D to E. Give the distance from D to F in centimeters. 2 cm D E Have your ruler handy so that you can check the measurements. Use your calculator to check the computations. Solution Measuring segment DE shows that it is about 2 cm long. Three-fifths of 2 is 5 2, which you can rewrite as 6 5 or 7.2. Place point F 7.2 cm from point D. 7.2 cm D F E 0 Discovering Algebra Condensed Lessons 2002 Key Curriculum Press