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Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Do Now 1 Name: Date: Do Now 1 – Patterns and Inductive Reasoning Week 1 Packet Page 1 Geometry Level 2 Inductive Reasoning Name: Date: Inductive Reasoning Vocabulary: Inductive Reasoning _____________________________________________ _____________________________________________________________ Conjecture _____________________________________________________ _____________________________________________________________ Counterexample _________________________________________________ _____________________________________________________________ Premise: a reason offered as support for another claim As a class you will watch two clips from the show House. For each scene, identify the premises and conjectures in the table below: Premises Conjecture Scene One Scene Two Week 1 Packet Page 2 Geometry Level 2 Inductive Reasoning Name: Date: Using Inductive Reasoning in Arithmetic Directions: Use the pattern in the examples to complete the conjecture. Counterexamples Directions: Find a counterexample to show that the statement is false. Answer in a complete sentence or draw a figure and explain why it is a counterexample. 3. If the quotient of two numbers is positive, then both numbers must be positive. 4. If a four-sided shape has two sides the same length, then it must be a rectangle. 5. If a four-sided shape has opposite sides that are the same length, then it must be a square. Week 1 Packet Page 3 Geometry Level 2 Inductive Reasoning Name: Date: Using Inductive Reasoning in Geometry For each situation, complete the chart and use inductive reasoning to make a conjecture. 6. Martine drew concurrent lines (lines that all share exactly one point) and determined how many regions they divided her paper into. Figure Number of Lines 1 2 2 4 3 4 5 Number of Regions Complete Martine’s conjecture. If you draw n concurrent lines, it will divide your paper into __________ regions. 7. Geraldo measured the sum of the angles in regular polygons and made the following chart. Figure Number of Sides Angle Sum 3 4 5 180° 360° 540° 6 7 Complete Geraldo’s conjectures: a) To find the number of degrees in a polygon with one more side, _______________ ____________________________________________________________ b) The sum of the angles in a regular polygon with n sides is ___________________ Week 1 Packet Page 4 Homework 1 Name: Date: Geometry Level 2 Homework 1 – Finding Patterns Directions: For problems 1-6, use inductive reasoning to find the next two figures you expect in the pattern. Draw the two next figures. Directions: for problems 7 – 14, use inductive reasoning to find the next two numbers in the pattern. List the next two numbers. Directions: for problem 15, show your work and/or explain how you got your answer for each part of the problem. 15. Clara started a business braiding hair before proms and dances. The table below shows the amount of profit Clara earned after each person paid. Client Number Profit Earned 1 $- 5 2 $10 3 $25 4 $40 5 6 a. Predict the amount of profit Clara will have earned after 5 customers and 6 customers. Show or explain how you got your answer. b. Clara started out by buying a special comb and sanitizer for her hair business when she started. Use the chart to determine what her start-up expenses were. Show or explain how you got your answer. c. Clara predicts that she will earn a total profit of $280 by the end of prom season. How many customers must Clara have in order to make this amount of profit? Show or explain how you got your answer. Week 1 Packet Page 5 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Do Now 2 Name: Date: Do Now 2 – Conjectures and Counterexamples Try some cases, then complete the conjecture. 1. The sum of any three odd numbers is ______________. 2. The difference between and integer and its opposite is _________________. Show the conjecture is false by finding a counterexample. 3. Conjecture: If a positive fraction is multiplies by a positive integer, then the product is greater than the fraction. 4. Conjecture: If all the sides of a figure are the same length, then the figure must be a square. Week 1 Packet Page 6 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Undefined Terms - Notes Name: Date: Undefined Terms – Guided Notes In geometry, there are three terms/figures that are considered “undefined” because they cannot be explained based on other terms. Definitions: Point: Description: Naming Conventions: Figure: Example(s): Line Description: Naming Conventions: Figure: Example(s): Week 1 Packet Page 7 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Undefined Terms - Notes Name: Date: Plane Description: Naming Conventions: Figure: Example(s): All of the other terms that we use in geometry can be defined based on these terms. Here are a few of the first ones. Collinear Defintion: Naming Conventions: Figure: Example(s): Week 1 Packet Page 8 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Undefined Terms - Notes Name: Date: Coplanar Points Definition: Naming Conventions: Figure: Example(s): Coplanar Lines Definition: Naming Conventions: Figure: Example(s): Week 1 Packet Page 9 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Undefined Terms - Notes Name: Date: Examples: Week 1 Packet Page 10 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Undefined Terms - Notes Name: Date: Modeling Collinear Points and Coplanar Points and Lines Your teacher will give you and your partner a few sticky notes to make points with. Place the “points” somewhere visible within the classroom. After all the points have been placed, walk around the room and answer the following questions. 1. Points _______, _________, and __________ are collinear because ______________________________________________________. 2. Points _______, _________, and __________ are not collinear because _______________________________________________________. 3. Points _______, _________, __________ and __________ are not coplanar because __________________________________________________. 4. Points _______, _________, ___________ and __________ are coplanar because __________________________________________________. 5. Lines _________, ___________, and __________ are coplanar because _______________________________________________________. 6. Lines _________, ___________, and __________ are not coplanar because ________________________________________________________. Week 1 Packet Page 11 Geometry Level 2 Homework 3 Name: Date: Homework 3 – Points, Lines and Planes Directions: Copy and complete each sentence. Include the answer in your answer column. Directions: Name each of the figures described. Be sure to include the correct symbols. Directions: In exercises 10 – 12, sketch and label each figure. 10. Sketch 𝐸𝐹 11. Sketch 𝐺𝐻 12. Sketch 𝐽𝐾 Directions: In exercises 13 – 14, a) name three points that are collinear and b) name three points that are not collinear. Directions: for problem 16, show your work and/or explain how you got your answer for each part of the problem. 16. Danny sketched a figure that included points A, B, C, D, and E. In Danny’s figure, no three points were collinear. a. Sketch a figure that could be Danny’s figure. b. Name each of the lines that are formed by the pairs of points in Danny’s figure. c. Explain how you know that you named all of the possible lines in your answer to part b. Week 1 Packet Page 12 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Do Now 3 Name: Date: Do Now 3 – Points, Lines, and Planes Directions: the figure at right to answer questions 9 – 13. 9. Name the points on plane S. 10. Name two lines. 11. Name the plane that contains point D. 12. Name three collinear points. Week 1 Packet Page 13 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Chapter 1 Quiz 1 Review Name: Date: Chapter 1 Quiz 1 Review Classwork Students will be able to (SWBAT): • • • Find patterns and use them to make predictions. Use inductive reasoning to make conjectures. Use undefined terms and defined terms to describe figures. Content Review: Week 1 Packet Page 14 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Chapter 1 Quiz 1 Review Name: Date: Quiz Practice: Directions: Write or draw the next two terms in each sequence. 1. 2. 3. 4. Directions: Show that each conjecture is false by drawing a counterexample. 5. If three lines lie in the same plane, then they intersect in at least one point. 6. Points A, G, and N are collinear. If AG = 7 inches and GN = 5 inches, then AN = 12 inches. Week 1 Packet Page 15 Geometry Level 2 Ms. Sheppard-Brick 617-596-4133 Chapter 1 Quiz 1 Review Name: Date: Directions: Use the diagram at right to answer the following questions. 7. Name three collinear points 8. Name three non-collinear points Possible answer D 9. Name four coplanar points 10. Name four non-coplanar points Sample a , B, and D Sample answer: J, D, P, and B 11. Name a line that intersects 𝐶𝐷. 12. Name the intersection of 𝐽𝐾and plane ℛ. Directions: for problem 13, show your work and/or explain how you got your answer for each part of the problem. 13. Amari drew the figures below on paper. Jenisteen measured each angle with a protractor. They added the measures of each pair of angles to form a conjecture. 49° 131° 87° 93° 105° 75° a. Complete the conjecture that they wrote “If two adjacent angles form a line, …” b. Amari then drew one more figure. Jenisteen measured only one of the angles. Use your conjecture to find the measure of the other angle. Show or explain how you got your answer. 115° Week 1 Packet Page 16 Geometry Level 2 Homework 4 Name: Date: Homework 4 – Review For Chapter 1 Quiz 1 THIS IS A TWO PAGE ASSIGNMENT Directions: Use inductive reasoning to find the next two numbers in each pattern. (TB 1.1) 1) 2) 3) 4) ! ! ! ! 1, 10, 100, 1000, ?, ? 7, 3, -1, -5, -9, -13, ?, ? 1, 1, 2, 3, 5, 8, 13, ?, ? 32, 30, 26, 20, 12, ?, ? 5) , , , , ?, ? ! ! ! ! 6) 1, 3, 6, 10, 15, 21, ? , ? 7) 1, 4, 9, 16, 25, 36, ? , ? 8) 1, 2, 4, 8, 16, 32, ? , ? Directions: Use inductive reasoning to draw the next figure in each pattern. (TB 1.1) 9) 10) Directions: Use the examples to complete the conjecture. (TB 1.2) 11) Conjecture: The product of 5 and any even number is divisible by __?___. Examples: 12) Conjecture: The square of an even number is __?__. Examples: Directions: Give a counterexample in words, numbers, or a sketch to disprove each conjecture. (TB 1.2) 13) Conjecture: The difference of any two even numbers is positive. 14) Conjecture: If two circles touch each other, then one circle is inside the other circle. Directions: Use the diagram at right for problems 15 – 20 15) Name three points that are collinear. 16) Name four points that are coplanar. 17) Name two lines that are coplanar. 18) Name three points that are not collinear. 19) Name four points that are not coplanar. 20) Name two lines that are not coplanar. OVER è OVER è OVER è Week 1 Packet Page 17 Geometry Level 2 Homework 4 Name: Date: Directions: for problems 21 and 22, show your work and/or explain how you got your answer for each part of the problem. 21) Olivia makes the following conjecture: “The square of a number is greater than the number.” a) Give two examples that support Olivia’s conjecture. b) Sam believes that the conjecture is false. He points out that 1! = 1, and 1≯ 1 (1 is not greater than 1). Find another counterexample to Olivia’s conjecture. c) Rewrite the conjecture to make it true. 22) The table below shows figures composed of circles. The number of circles in each figure and the diameter of each circle in each figure follow a pattern, as shown. a. What is the number of circles in figure 5? Show or explain how you got your answer. b. What is the diameter, in inches, of each circle in figure 5? Show or explain how you got your answer. c. What is the ratio of the number of circles in figure 6 to the number of circles in figure 7? Show or explain how you got your answer. d. Write an algebraic expression that could be used to determine the number of circles in figure n. e. Write an algebraic expression that could be used to determine the diameter, in inches, of each circle in figure n. Week 1 Packet Page 18