Geometry: All-In-One Answers Version A Name_____________________________________ Class____________________________ Date________________ Name_____________________________________ Class____________________________ Date ________________ Lesson 1-1 Patterns and Inductive Reasoning 3 Finding a Counterexample Find a counterexample for the conjecture. Since 32 ϩ 42 ϭ 52, the sum of the squares of two consecutive numbers is Lesson Objective NAEP 2005 Strand: Geometry the square of the next consecutive number. 1 Use inductive reasoning to make Topic: Mathematical Reasoning ϩ 5 conjectures Begin with 4. The next consecutive number is 4 1 . Local Standards: ____________________________________ The sum of the squares of these two consecutive numbers is 42 ϩ 52 ϭ 16 ϩ 25 41 . Vocabulary. The conjecture is that 41 is the square of the next consecutive number. Inductive reasoning is reasoning based on patterns you observe. The square of the next consecutive number is (5 ϩ 1)2 ϭ 62 36 . So, since 42 ϩ 52 and 62 are not the same, this conjecture is false . A conjecture is a conclusion you reach using inductive reasoning. 4 Applying Conjectures to Business The price of overnight shipping was $8.00 in A counterexample is an example for which the conjecture is incorrect. 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. Write the data in a table. Find a pattern. 2000 2001 2002 Examples. $8.00 $9.50 $11.00 1 Finding and Using a Pattern Find a pattern for the sequence. 384 192 96 48 Use the pattern to find the next two terms in the sequence. Each year the price increased by $1.50 . A possible conjecture is that 384, 192, 96, 48,... the price in 2003 will increase by $1.50 . If so, the price in 2003 would ϬϬϬ2 2 2 Each term is half the preceding term. The next two be $11.00 ϩ $ 1.50 ϭ $.12.50 terms are 48 Ϭϭ224and 24 Ϭϭ2 12 . Quick Check. 2 Using Inductive Reasoning Make a conjecture about the sum of the cubes of the first 25 counting numbers. 1. Find the next two terms in each sequence. Find the first few sums. Notice that each sum is a perfect square and that a. 1,2,4,7,11,16,22,29 ,37 ,... the perfect squares form a pattern. 13 ϭ 1 ϭϭ12 12 b. Monday,Tuesday,Wednesday,Thursday , Friday ,... 13 ϩ 23 ϭ 9 ϭϭ32 (1 ϩ 2)2 3 3 3 2 2 1 ϩϩ2 3 ϭ 36 ϭ 6 ϭ (1 ϩ 2 ϩ 3) c. 3 3 3 3 2 2 1 ϩϩϩ2 3 4 ϭ 100 ϭ 10 ϭ (1 ϩ 2 ϩ 3 ϩ 4) , ,... 13 ϩϩϩϩϭ23 33 43 53 225ϭ 152 ϭ (1 ϩ 2 ϩ 3 ϩ 4 ϩ 5)2 Answers may vary. Sample: The sum of the first two cubes equals the square of the sum of the first two counting numbers. The sum of the first three cubes equals the © Pearson Education, Inc., publishing as Pearson Prentice Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. square of the sum of the first three counting numbers. This pattern continues for the fourth and fifth rows. So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 ϩ 2 ϩ 3 ϩ . ϩ 25)2. 2 3 Geometry Lesson 1-1 Daily Notetaking Guide Daily Notetaking Guide Geometry Lesson 1-1 Name_____________________________________ Class____________________________ Date ________________ Name_____________________________________ Class____________________________ Date ________________ 2. Make a conjecture about the sum of the first 35 odd numbers. Use your Lesson 1-2 Drawings, Nets, and Other Models calculator to verify your conjecture. Lesson Objectives 1 ϭϭ1 12 NAEP 2005 Strand: Geometry 1 Make isometric and orthographic Topic: Dimension and Shape 1 ϩ 3 ϭϭ4 22 drawings 2 Draw nets for three-dimensional Local Standards: ____________________________________ 13ϩ ϩ 5 ϭϭ9 32 figures 1 ϩϩϩ3 5 7 ϭϭ16 42 Vocabulary. 1 ϩϩϩ3 5 7 ϩ 9 ϭϭ25 52 An isometric drawing of a three-dimensional object shows a corner view of the figure drawn on isometric dot paper. The sum of the first 35 odd numbers is 352, or 1225. An orthographic drawing is the top view, front view, and right-side view of a three-dimensional figure. A foundation drawing shows the base of a structure and the height of each part. 3. Finding a Counterexample Find a counterexample for the conjecture. Some products of 5 and other numbers are shown in the table. A net is a two-dimensional pattern you can fold to form a three-dimensional figure. 5 ϫ 7 ϭ 35 5 ϫ 13 ϭ 65 5 ϫ 3 ϭ 15 5 ϫ 9 ϭ 45 5 ϫ 11 ϭ 55 5 ϫ 25 ϭ 125 Therefore, the product of 5 and any positive integer ends in 5. Examples. Isometric Drawing Since 5 ϫ 2 ϭ 10, not all products of 5 end in 5. 1 Make an isometric drawing of (But they all either end in 5 or 0.) the cube structure at right. An isometric drawing shows three sides of a figure from a corner view. Hidden edges are not shown, so all edges are solid lines. 4. Suppose the price of two-day shipping was $6.00 in 2000, $7.00 in 2001, and $8.00 in 2002. Make a conjecture about the price in 2003. Each year the price increased by $1.00. A possible conjecture is that the price in 2003 will increase by $1.00. If so, the price in 2003 would be $8.00 ϩ $1.00 ϭ $9.00. © Pearson Education, Inc., publishing as Pearson Prentice Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. © Pearson Education, Inc., publishing as Pearson Prentice Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. 4 5 Geometry Lesson 1-1 Daily Notetaking Guide Daily Notetaking Guide Geometry Lesson 1-2 1 All-In-One Answers Version A Geometry Geometry: All-In-One Answers Version A (continued) Name_____________________________________ Class____________________________ Date ________________ Name_____________________________________ Class____________________________ Date ________________ 2 Orthographic Drawing Make an orthographic drawing of the 4 Identifying Solids from Nets Is the pattern a net for a cube? isometric drawing at right. If so, name two letters that will be opposite faces. A Orthographic drawings flatten the depth of a figure. An orthographic The pattern is a net because you can Front BDEF drawing shows three views. Because Right fold it to form a cube. Fold squares A and C up to form the back no edge of the isometric drawing is hidden in the top, front, and front of the cube. Fold D up to form a side. Fold E over to C and right views, all lines are solid. form the top. Fold F down to form another side. After the net is folded, the following pairs of letters are on opposite faces: A and C are the back and front faces. B and E are the bottomand top faces. DFand are the right and left side faces. Front Top Right 5 Drawing a Net Draw a net for the figure with a square base 3 Foundation Drawing Make a foundation drawing for the and four isosceles triangle faces. Label the net with its dimensions. 10 cm isometric drawing. Think of the sides of the square base as hinges, and “unfold” the To make a foundation drawing, use the top view of the orthographic drawing. Front figure at these edges to form a net. The base of each of the four Right isosceles triangle faces is a side of the square . Write in the 8 cm known dimensions. Top 0 cm Because the top view is formed from 3 squares, show 3 squares in the 8 cm 1 foundation drawing. Identify the square that represents the tallest part. Write the number 2 in the back square to indicate that the back section is 2 cubes high. Write the number 1 in each of the two front squares to indicate Quick Check. that each front section is 1 cube high. 1. Make an isometric drawing of the cube structure below. 2 1 1 Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. Right Front 6 7 Geometry Lesson 1-2 Daily Notetaking Guide Daily Notetaking Guide Geometry Lesson 1-2 Name_____________________________________ Class____________________________ Date ________________ Name_____________________________________ Class____________________________ Date ________________ 2. Make an orthographic drawing from this isometric drawing. 5. The drawing shows one possible net for the Graham Crackers box. 14 cm 7 cm Front Right 20 cm GRAHAM GRAHAM 20 cm Front Top Right CRACKERS CRACKERS 7 cm 14 cm Draw a different net for this box. Show the dimensions in your diagram. Answers may vary. Example: 14 cm 3. a. How many cubes would you use to make the structure in Example 3? 4 20 cm b. Critical Thinking Which drawing did you use to answer part (a), the foundation drawing or the isometric drawing? Explain. 7 cm Answers may vary. Sample: the foundation drawing; you can just add the three numbers. 4. Sketch the three-dimensional figure that corresponds to the net. © Pearson Education, Inc., publishing as Pearson Prentice Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. Hall. Prentice Education, Inc., publishing as Pearson © Pearson All rights reserved. © Pearson Education, Inc., publishing as Pearson Prentice Hall.