Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics and art wikipedia , lookup
Mathematics and architecture wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Large numbers wikipedia , lookup
Bra–ket notation wikipedia , lookup
Recurrence relation wikipedia , lookup
Collatz conjecture wikipedia , lookup
Moiré pattern wikipedia , lookup
Hyperreal number wikipedia , lookup
Line (geometry) wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Unit 1, Activity 1, Extending Number and Picture Patterns Geometry Blackline Masters, Geometry Page 1-1 Unit 1, Activity 1, Extending Number and Picture Patterns Date ___________ Name ________________________ Extending Patterns and Sequences When presented with a sequence and asked to find the next term, inductive reasoning is applied. Analyze the specific examples provided, determine a pattern, and then find the missing term. Making a prediction about missing terms is called making a conjecture. Examples: For each of the following, write the next two terms and describe the pattern. 1) 2, 4, 6, 8, 10, … _____, _____ 2) -1, 0, 1, 2, 3, … _____, _____ 3) 4, 7, 10, 13, 16, … _____, _____ 4) 1, 4, 9, 16, 25, … _____, _____ 5) 1, 3, 6, 10, 15, … _____. _____ 6) 1, 3, 7, 15, 31, 63, … _____, _____ 7) 1, 1, 2, 3, 5, 8, … _____, _____ 8) 3, 5, 9, 15, 23, … _____, _____ Inductive reasoning can also be used to find missing terms in sequences and patterns dealing with pictures. Draw the next two figures for each of the following and describe the pattern. 9) 10) 11) Blackline Masters, Geometry Page 1-1 Unit 1, Activity 1, Extending Number and Picture Patterns with Answers Date ___________ Name ________________________ Extending Patterns and Sequences When presented with a sequence and asked to find the next term, inductive reasoning is applied. Analyze the specific examples provided, determine a pattern, and then find the missing term. Making a prediction about missing terms is called making a conjecture. Examples: For each of the following, write the next two terms and describe the pattern. 1) 2, 4, 6, 8, 10, … __12_, _14__ even numbers or +2 2) -1, 0, 1, 2, 3, … __4__, __5__ add 1 to each 3) 4, 7, 10, 13, 16, … _19_, _22_ add 3 4) 1, 4, 9, 16, 25, … __36_, __49_ perfect squares 5) 1, 3, 6, 10, 15, … __21_. _28__ add 2, then 3, then 4, etc. 6) 1, 3, 7, 15, 31, 63, … _127_, _255_ add 2, then 4, then 8, then 16, etc. 7) 1, 1, 2, 3, 5, 8, … __13_, _21__ add the preceding two terms Fibonacci Sequence 8) 3, 5, 9, 15, 23, … _33__, _45__ add 2, then 4, then 6, then 8, etc. Inductive reasoning can also be used to find missing terms in sequences and patterns dealing with pictures. Draw the next two figures for each of the following and describe the pattern. 9) The student should draw a shaded triangle, then an unshaded square. 10) The student should draw two shaded pentagons. 11) The student should draw a circle with an inscribed pentagon. The points on the circles increase by one in each picture, which are connected to make polygons. Blackline Masters, Geometry Page 1-2 Unit 1, Activity 1, Linear or Non-linear “Tis Linear or Not linear; That is the Question” Directions: Decide whether each of the given rules, sequences, or tables represents a linear or non-linear pattern. Place a check under the column which corresponds to your decision. Be prepared to explain why you made your decision. Is the given pattern Linear 1) 2,5,8,11,14,... 2) 1, 2, 4,8,16,... 3) 3n 1 4) x 1 2 3 4 5 y 18 15 12 9 ? 5) n2 1 6) 15, 10, 6, 3, 1,... 7) 8) x 1 2 3 4 5 y 100 50 25 12.5 ? Non-linear n4 2 Blackline Masters, Geometry Page 1-3 Unit 1, Activity 1, Linear or Non-linear with Answers “Tis Linear or Not linear; That is the Question” Directions: Decide whether each of the given rules, sequences, or tables represents a linear or non-linear pattern. Place a check under the column which corresponds to your decision. Be prepared to explain why you made your decision. Is the given pattern Linear Non-linear 1) 2,5,8,11,14,... 2) 1, 2, 4,8,16,... 3) 3n 1 4) x 1 2 3 4 5 y 18 15 12 9 ? 5) n2 1 6) 15, 10, 6, 3, 1,... 7) 8) x 1 2 3 4 5 y 100 50 25 12.5 ? n4 2 Blackline Masters, Geometry Page 1-4 Unit 1, Activity 1, Using Rules to Generate a Sequence Linear versus Non-linear Relationships Linear data are data that ____________________________ Consider a few different patterns. 1) Term n Value n-3 2) 3) 4) 5) 2 -1 3 0 4 5 6 7 8 Term n 1 Value 2n+3 5 2 7 3 9 4 5 6 7 8 Term n 1 Value 3n+1 4 2 7 3 10 4 5 6 7 8 Term n Value n2 1 1 2 4 3 9 4 5 6 7 8 Term n Value n3 1 1 2 8 3 27 4 5 6 7 8 Blackline Masters, Geometry 1 -2 Page 1-5 Unit 1, Activity 1, Using Rules to Generate a Sequence Questions to answer: 6) Which patterns had common differences (the same number added over and over)? Does that number appear in the rule? 7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern were rewritten in this form, how should m be interpreted? Graph each of the sequences above on a sheet of graph paper to determine if they are linear or not linear. 8) Which sequences produced a line? What did these sequences have in common? 9) Which sequences did not produce a line? What did these sequences have in common? 10) Write a conjecture about all linear relationships and all non-linear relationships based on your examples above. Are the following sequences linear or non-linear? 11) -1.5, -1, -0.5, 0, 0.5, … 12) 4, 10, 18, 28, 40, … 13) 2, 1, 2/3, ½, 2/5, … 14) 1, 4, 7, 10, 13, … Blackline Masters, Geometry Page 1-6 Unit 1, Activity 1, Using Rules to Generate a Sequence with Answers Linear versus Non-linear relationships Linear data are data that _forms a line when graphed__ Consider a few different patterns. 1) Term n Value n-3 1 -2 2 -1 3 0 4 1 5 2 6 3 7 4 8 5 4 11 5 13 6 15 7 17 8 19 4 13 5 16 6 19 7 22 8 25 4 16 5 25 6 36 7 49 8 64 6 216 7 343 8 512 Difference between the terms is 1 2) Term n 1 Value 2n+3 5 2 7 3 9 Difference between the terms is 2 3) Term n 1 Value 3n+1 4 2 7 3 10 Difference between the terms is 3 4) Term n Value n2 1 1 2 4 3 9 There is no common difference between terms 5) Term n Value n3 1 1 2 8 3 27 4 64 5 125 There is no common difference between terms Questions to answer: Blackline Masters, Geometry Page 1-7 Unit 1, Activity 1, Using Rules to Generate a Sequence with Answers 6) Which patterns had common differences (the same number added over and over)? Does that number appear in the rule? Patterns 1, 2, and 3 had common differences. These numbers are the coefficients of n. 7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern were rewritten in this form, how should m be interpreted? The m stands for slope. If I rewrote the rule in the slope-intercept form it would tell me the slope of the line which is the rate of change—how much y changes when x changes. Graph each of the sequences above on a sheet of graph paper to determine if they are linear or non-linear. 8) Which sequences produced a line? What did these sequences have in common? Patterns 1, 2, and 3; each of these patterns had a common difference which is the coefficient of n. 9) Which sequences did not produce a line? What did these sequences have in common? Patterns 4 and 5; these patterns did not have a common difference. 10) Write a conjecture about all linear relationships and all non-linear relationships based on your examples above. Patterns which represent linear relationships will have a common difference between terms. Patterns which are non-linear will not have a common difference between terms. Are the following sequences linear or non-linear? 11) -1.5, -1, -0.5, 0, 0.5, … Linear 13) 2, 1, 2/3, ½, 2/5, … Non-linear Blackline Masters, Geometry 12) 4, 10, 18, 28, 40, … Non-linear 14) 1, 4, 7, 10, 13, … Linear Page 1-8 Unit 1, Activity 2, Generating the nth Term for Picture Patterns Date ___________ Name ________________________ Directions: Find the indicated term for each of the patterns below. 1) How many sides will the 15th term have? 2) What will the 23rd figure look like? 3) What is the 50th term of the sequence above? 4) What is the 103rd term of the sequence? Blackline Masters, Geometry Page 1-9 Unit 1, Activity 2, Generating the nth Term for Picture Patterns with Answers Date ___________ Name ________________________ Directions: Find the indicated term for each of the patterns below. 1) How many sides will the 15th term have? Solution: n + 2; 17 sides Add two to the figure number, to determine the number of sides. For example, the 3rd figure has 5 sides. 2) What will the 23rd figure look like? Solution: Since the pattern repeats after four figures, students should realize that every term that is a multiple of four will look like the fourth figure. The nearest multiple to 23 is 20; the students should then continue the pattern—it is the 3rd figure. 3) What is the 50th term of the sequence above? Solution: The shapes repeat after 3 terms so 48 is the closest multiple of 3 to 50, so the shape is a square. The square is not shaded because the even terms are not shaded. 4) What is the 103rd term of the sequence? Solution: The pattern repeats after five terms. The 100 th term is the fifth figure, so the 103rd term is the third figure. Blackline Masters, Geometry Page 1-10 Unit 1, Activity 3, Square Figurate Numbers Date _____________ Name ________________________ Square Numbers Consider the following sequence: 1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a square? Blackline Masters, Geometry Page 1-11 Unit 1, Activity 3, Square Figurate Numbers with Answers Date _____________ Name ________________________ Square Numbers Consider the following sequence: 1) What is the number pattern? 1, 4, 9, 16, 25 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term? Formula: n 2 ; the 25th term is 625. 4) How does each number relate to the area of a square? The area of a square is s 2 where s is the measure of the side. In each of the squares, the measure of the sides are the same, and they increase by one each time. Therefore the area is 22, 32, 42, … n 2 . Blackline Masters, Geometry Page 1-12 Unit 1, Activity 3, Rectangular Figurate Numbers Date _____________ Name ________________________ Rectangular Numbers Consider the following: 1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a rectangle? Blackline Masters, Geometry Page 1-13 Unit 1, Activity 3, Rectangular Figurate Numbers with Answers Date _____________ Name ________________________ Rectangular Numbers Consider the following: 1) What is the number pattern? 2, 6, 12, 20, 30 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term? Formula: n 2 n or n n 1 ; the 25th term is 650. 4) How does each number relate to the area of a rectangle? Each rectangle has a height the same as the figure number and a base which is one greater than the height; therefore, the number of dots needed for any figure is the same as the area of the rectangle, n(n+1), where n is the height and the base is one more than the height. Blackline Masters, Geometry Page 1-14 Unit 1, Activity 3, Triangular Figurate Numbers Date _____________ Name ________________________ Triangular Numbers Consider the following: 1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a triangle? Blackline Masters, Geometry Page 1-15 Unit 1, Activity 3, Triangular Figurate Numbers Date _____________ Name ________________________ Triangular Numbers Consider the following: 1) What is the number pattern? 1, 3, 6, 10, 15 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term? Formula: n n 1 n2 n or or 0.5n2 0.5n ; the 25th term is 325. 2 2 4) How does each number relate to the area of a triangle? 1 The area of a triangle is half the area of a rectangle, A bh , so if we take the formula 2 for rectangular numbers, we can divide it by 2 to get the area of a triangle with the same base as its corresponding rectangle. Blackline Masters, Geometry Page 1-16