Unit 1, Activity 1, Extending Number and Picture Patterns Geometry

Transcription

1 Unit 1, Activity 1, Extending Number and Picture Patterns Geometry Blackline Masters, Geometry Page 1-1

2 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

3 Unit 1, Activity 1, Extending Number and Picture Patterns with Answers Extending Patterns and Sequences Date Name 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 2) -1, 0, 1, 2, 3, 4, 5 even numbers or +2 add 1 to each 3) 4, 7, 10, 13, 16, _19_, _22_ 4) 1, 4, 9, 16, 25, 36_, 49_ add 3 perfect squares 5) 1, 3, 6, 10, 15, 21_. _28 6) 1, 3, 7, 15, 31, 63, _127_, _255_ add 2, then 3, then 4, etc. add 2, then 4, then 8, then 16, etc. 7) 1, 1, 2, 3, 5, 8, 13_, _21 8) 3, 5, 9, 15, 23, _33, _45 add the preceding two terms add 2, then 4, then 6, then 8, etc. Fibonacci Sequence 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) The student should draw a shaded triangle, then an unshaded square. 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

4 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 Non-linear 1) 2,5,8,11,14,... 2) 1, 2, 4,8,16,... 3) 3n 1 4) x y ? 5) 2 n 1 6) 15, 10, 6, 3, 1,... 7) x ) n y ? Blackline Masters, Geometry Page 1-3

5 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 y ? 5) 2 n 1 6) 15, 10, 6, 3, 1,... 7) x ) n y ? Blackline Masters, Geometry Page 1-4

6 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 ) Term n Value 2n ) Term n Value 3n ) Term n Value n ) Term n Value n Blackline Masters, Geometry Page 1-5

7 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

8 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 Difference between the terms is 1 2) Term n Value 2n Difference between the terms is 2 3) Term n Value 3n Difference between the terms is 3 4) Term n Value n There is no common difference between terms 5) Term n Value n There is no common difference between terms Questions to answer: Blackline Masters, Geometry Page 1-7

9 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, 12) 4, 10, 18, 28, 40, Linear Non-linear 13) 2, 1, 2 / 3, ½, 2 / 5, 14) 1, 4, 7, 10, 13, Non-linear Linear Blackline Masters, Geometry Page 1-8

10 Unit 1, Activity 2, Generating the n th Term for Picture Patterns Directions: Find the indicated term for each of the patterns below. Date Name 1) How many sides will the 15 th term have? 2) What will the 23 rd figure look like? 3) What is the 50 th term of the sequence above? 4) What is the 103 rd term of the sequence? Blackline Masters, Geometry Page 1-9

11 Unit 1, Activity 2, Generating the n th Term for Picture Patterns with Answers Directions: Find the indicated term for each of the patterns below. Date Name 1) 2) How many sides will the 15 th 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. 3) What will the 23 rd 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 3 rd figure. 4) What is the 50 th 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. What is the 103 rd term of the sequence? Solution: The pattern repeats after five terms. The 100 th term is the fifth figure, so the 103 rd term is the third figure. Blackline Masters, Geometry Page 1-10

12 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 25 th term? 4) How does each number relate to the area of a square? Blackline Masters, Geometry Page 1-11

13 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 25 th term? Formula: 2 n ; the 25 th term is ) How does each number relate to the area of a square? 2 The area of a square is s 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 2 2, 3 2, 4 2, n 2. Blackline Masters, Geometry Page 1-12

14 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 25 th term? 4) How does each number relate to the area of a rectangle? Blackline Masters, Geometry Page 1-13

15 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 25 th term? Formula: 2 n + n or n( n+ 1) ; the 25 th term is ) 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

16 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 25 th term? 4) How does each number relate to the area of a triangle? Blackline Masters, Geometry Page 1-15

17 Unit 1, Activity 3, Triangular Figurate Numbers Date Name Consider the following: Triangular Numbers 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 25 th term? Formula: ( + 1) 2 n + n n n or or n + n; the 25 th term is ) 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

18 Unit 2, Activity 3, Proof Puzzle Please print two copies of each proof one to be cut up and one to be used as an Answer Key. Cut the statements and reasons in the following proofs into strips and put them in envelopes to have the students arrange in the correct order. If students need help identifying the strips as either a statement or reason, put all of the statements from one proof in one envelope and the reasons for the proof in a separate envelope. Label the envelopes Statements Proof # and Reasons Proof #. The statements and reasons are not numbered below, but the order in which they are presented is the order that the students should have when their work is completed. Proof #1 Given: 4( x 2) = 52 Prove: x = 15 Statements 4( x 2) = 52 Given Reasons 4x 8 = 52 Distributive Property 4x 8+ 8 = Addition Property 4x = 60 Simplification 4x = 60 Division Property 4 4 x = 15 Simplification Blackline Masters, Geometry Page 2-1

19 Unit 2, Activity 3, Proof Puzzle Proof #2 Given: ( a) ( a ) Prove: x = = 123 Statements ( a) ( a ) = 123 Given Reasons 6 12a 3a 18 = 123 Distributive Property 12 15a = 123 Simplification 12 15a + 12 = Addition Property 15a = 135 Simplification 15a = x = 9 Division Property Simplification Blackline Masters, Geometry Page 2-2

20 Unit 2, Activity 3, Proof Puzzle Proof #3 Given: 2 x + 6 = 2 5 Prove: x = 2 Statements Reasons 2x + 6 = 2 5 Given 5 2x + 6 = 52 5 ( ) Multiplication Property 2x + 6 = 10 Simplification 2x + 6 6= 10 6 Subtraction Property 2x = 4 Simplification 2x = 4 Division Property 2 2 x = 2 Simplification Blackline Masters, Geometry Page 2-3

21 Unit 2, Activity 3, Proof Puzzle Proof #4 Given: 4 1 a= 7 a 2 2 Prove: a = 1 Statements Reasons 4 1 a= 7 a 2 2 Given a = a 2 2 Multiplication Property 8 a= 7 2a Simplification 8 a 8= 7 2a 8 Subtraction Property a= 2a 1 Simplification a+ 2a= 2a 1+ 2a Addition Property a = 1 Simplification Blackline Masters, Geometry Page 2-4

22 Unit 3, Activity 3, Distance in the Plane Process Guide Name: Date: Section A: Use the Pythagorean Theorem to find the missing side in each right triangle below. Give exact answers (in simplest radical form, if necessary). A. 8 x 15 B b C. d 7 7 Blackline Masters, Geometry Page 3-1

23 Unit 3, Activity 3, Distance in the Plane Process Guide Section B: Find the coordinates of the vertices on each right triangle. Find the lengths of the legs of each right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. Then answer the questions that follow. A D E C B F 1. How do the lengths of the horizontal legs of each triangle relate to distance on a number line? Which operation could you perform on the two x-coordinates that would result in the length of the horizontal legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) 2. How do the lengths of the vertical legs of each triangle relate to distance on a number line? Which operation you could perform on the two y-coordinates that would result in the length of the vertical legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Blackline Masters, Geometry Page 3-2

24 Unit 3, Activity 3, Distance in the Plane Process Guide 3. For each triangle, substitute the number sentence you created in question 1 for a and the number sentence you created in question 2 for b in the Pythagorean Theorem. Then solve for c without simplifying the expression you have for a 2 +b 2. (Hint: ) Section C: Given the right triangle below, use the Pythagorean Theorem to find the hypotenuse of the given triangle using points and. Remember to find the legs of the right triangle just as you did in Section B. Solve the equation you create for c. Your final answer will have all variables and should resemble your final equation from number 3 above. Answer the questions that follow to help guide you through this process. M P N 4. To find the length of, which two coordinates would you need to use? What expression represents the distance from N to P? 5. To find the length of, which two coordinates would you need to use? What expression represents the distance from M to P? 6. To find the length of, use the Pythagorean Theorem and replace a with the expression from question 4 and replace b with the expression from question 5. Solve the equation for c. Remember, your result should have variables and should resemble. Blackline Masters, Geometry Page 3-3

25 Unit 3, Activity 3, Distance in the Plane Process Guide 7. The length of any segment is the between the endpoints. Based on that knowledge, how can the equation created in question 6 be useful? Section D: In Section C, you derived the Distance Formula. Typically, the formula is written as, where d represents the between two points (or the length of a segment that connects the two points). Use the formula you derived to find the length of the segments given on the grid below. Give the exact answer, in simplest radical form if necessary D 6 C 4 2 S F -2 G T 2 2 Formula Investigation: Would the formula d ( x x ) ( y y ) = + also work? What about ( ) ( ), ( ) ( ), or ( ) ( ) d = x x + y y d = x x + y y d = y y + x x? Why or why not? Explain your reasoning. Blackline Masters, Geometry Page 3-4

26 Unit 3, Activity 3, Distance in the Plane Process Guide Section E: Real-World Application Saints quarterback, Drew Brees, threw a pass from the Saints 10-yard line, 12 yards from the Saints sideline. The pass was caught by Marques Colston on the Saints 50-yard line, 40 yards from the same sideline. Brees gets credit for a 40-yard pass, but how much credit should Brees get for the pass? Imagine a grid laid over the representation of the field below with the horizontal axis on the bottom sideline and the vertical axis on left goal line, as shown. Goal Line SAINTS SAINTS Saints Sideline Remember, the intersection of the axes is (0,0) or the. What two ordered pairs would be used to represent the locations of Drew Brees and Marques Colston? (Hint: (yard line, yards from sideline)) How many yards did Brees throw the football? How did you find that distance? How many more yards should he get credit for on this pass? Blackline Masters, Geometry Page 3-5

27 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Name: Date: Section A: Use the Pythagorean Theorem to find the missing side in each right triangle below. Give exact answers (in simplest radical form, if necessary). A. 8 x 15 x = 17 B b C. d 7 7 Blackline Masters, Geometry Page 3-6

28 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Section B: Find the coordinates of the vertices on each right triangle. Find the lengths of the legs of each right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. Then answer the questions that follow. A D E C B F 1. How do the lengths of the horizontal legs of each triangle relate to distance on a number line? Which operation could you perform on the two x-coordinates that would result in the length of the horizontal legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Sample answer: The length of the horizontal legs is the distance from one x-coordinate to the next. This measure can be found by subtracting the x-values. The number sentences would be and. 2. How do the lengths of the vertical legs of each triangle relate to distance on a number line? Which operation you could perform on the two y-coordinates that would result in the length of the vertical legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Sample answer: Even though the segments are vertical, the length of the vertical legs is the distance from one y-coordinate to the next. This measure can be found by subtracting the y- values. The number sentences would be and. Blackline Masters, Geometry Page 3-7

29 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers 3. For each triangle, substitute the number sentence you created in question 1 for a and the number sentence you created in question 2 for b in the Pythagorean Theorem. Then solve for c without simplifying the expression you have for a 2 +b 2. (Hint: ) Triangle ABC: Triangle DEF: Note: students may write the equations without the absolute value which is acceptable. Discussions should be held to help students understand why the absolute value symbols are not needed once the expressions have been squared. Section C: Given the right triangle below, use the Pythagorean Theorem to find the hypotenuse of the given triangle using points and. Remember to find the legs of the right triangle just as you did in Section B. Solve the equation you create for c. Your final answer will have all variables and should resemble your final equation from number 3 above. Answer the questions that follow to help guide you through this process. M P N 4. To find the length of, which two coordinates would you need to use? What expression represents the distance from N to P? ; Note: is also acceptable. 5. To find the length of, which two coordinates would you need to use? What expression represents the distance from M to P? ; Note: is also acceptable. 6. To find the length of, use the Pythagorean Theorem and replace a with the expression from question 4 and replace b with the expression from question 5. Solve the equation for c. Remember, your result should have variables and should resemble. Blackline Masters, Geometry Page 3-8

30 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Note: students may write the equations with or without the absolute value which is acceptable. Discussions should be held to help students eventually understand why the absolute value symbols are not needed once the expressions have been squared. 7. The length of any segment is the distance between the endpoints. Based on that knowledge, how can the equation created in question 6 be useful? The equation in number 6 can be used to find the distance between any two endpoints or the length of segment MN in the triangle above. Section D: In Section C, you derived the Distance Formula. Typically, the formula is written as, where d represents the distance between two points (or the length of a segment that connects the two points). Use the formula you derived to find the length of the segments given on the grid below. Give the exact answer, in simplest radical form if necessary D 6 C 4 2 S F -2 G T Formula Investigation: Would the formula d ( x x ) ( y y ) = + also work? What about ( ) ( ), ( ) ( ), or ( ) ( ) d = x x + y y d = x x + y y d = y y + x x? Why or why not? Explain your reasoning. Yes, all of these formulas work. Students should understand that because the differences are squared, in these cases they can change the order for the subtraction even though subtraction is not a commutative operation. Since addition is commutative, whether they subtract x-values or y- values first does not matter. Make sure students know they should subtract x-values from x- values and y-values from y-values. Blackline Masters, Geometry Page 3-9

31 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Section E: Real-World Application Saints quarterback, Drew Brees, threw a pass from the Saints 10-yard line, 12 yards from the Saints sideline. The pass was caught by Marques Colston on the Saints 50-yard line, 40 yards from the same sideline. Brees gets credit for a 40-yard pass, but how much credit should Brees get for the pass? Imagine a grid laid over the representation of the field below with the horizontal axis on the bottom sideline and the vertical axis on left goal line, as shown. Goal Line SAINTS SAINTS Saints Sideline Remember, the intersection of the axes is (0,0) or the origin. What two ordered pairs would be used to represent the locations of Drew Brees and Marques Colston? (Hint: (yard line, yards from sideline)) Drew Brees: (10, 12); Marques Colston: (50, 40) How many yards did Brees throw the football? How did you find that distance? How many more yards should he get credit for on this pass? The distance is 48.8 yards. The distance was found by using the formula developed in this activity and using the ordered pairs listed above. Brees should get 8.8 yards more credit. Blackline Masters, Geometry Page 3-10

32 Unit 3, Activity 4, Dividing Number Line Segments Name: Date: A B Point D lies on between points A and B and divides into a ratio of 2:3. What is the coordinate of point D? 1. What is the length of? 2. If is divided into a ratio of 2:3, how many equal parts would there be? 3. Find the coordinate of D. 4. Explain in your own words how you found the coordinate of D. Given the number line below, answer the following questions If M is located at -3 and N is located at 4, find P such that is divided into a 1:5 ratio. 6. If X is located at -2 and Y is located at -5, find Z such that is divided into a ratio of 2:1. 7. Given A and C as arbitrary points on the number line (they could be located at any value), develop a formula that could help you find the location of any point that divides the segment into the ratio. Blackline Masters, Geometry Page 3-11

33 Unit 3, Activity 4, Dividing Number Line Segments with Answers Name: Date: A B Point D lies on between points A and B and divides into a ratio of 2:3. What is the coordinate of point D? 1. What is the length of? 10 units 2. If is divided into a ratio of 2:3, how many equal parts would there be? 5 3. Find the coordinate of D. 2 ( 10 ) = 4 5 D is located at coordinate ( 1) = 3 4. Explain in your own words how you found the coordinate of D. First, find the length of segment AB which is 10. Then, multiply 10 by 2/5 because we want D to be 2 of the five equal parts (or 2/5 of the distance) way from A. 2/5 of 10 is 4. Then, add 4 to the coordinate for A which is 3. **Note some students may use the coordinate for B; if that happens they should understand that D is to be 3/5 of the length of segment AB from B and that result (6) should be subtracted from B to make sure it is between A and B. Students should be encouraged to always use the leftmost endpoint. Given the number line below, answer the following questions If M is located at -3 and N is located at 4, find P such that is divided into a 1:5 ratio. P should be located at. 6. If X is located at -2 and Y is located at -5, find Z such that is divided into a ratio of 2:1. Z should be located at Given A and C as arbitrary points on the number line (they could be located at any value), where A is the leftmost point, develop a formula that could help you find the location of any point that divides the segment into the ratio. k1 ( C A) + A k + k 1 2 Blackline Masters, Geometry Page 3-12

34 Unit 3, Activity 7, Parallel Line Facts Use the given diagram to complete the steps and answer the questions. Date Name 1. Draw a line through vertex B so that the line is parallel to AC. Locate one point to the left of B and label it L. Locate one point to the right of B and label it R. 2. Given the diagram above with the parallel line drawn through B, prove that m BAC + m ABC + m ACB = Using the same diagram above, extend AC so that it is a line. Draw two points on AC : one to the left of A labeled D and the other to the right of C and label it E. Using the new diagram, prove that m BAD = m ABC + m ACB. Blackline Masters, Geometry Page 3-13

35 Unit 3, Activity 7, Parallel Line Facts 4. Remember, the area of a triangle can be written as answer the following questions. 1 A = bh. Use the diagram below to 2 a. Using D, draw triangle ADC. b. Choose a point anywhere on BD and label it E. Draw triangle AEC. c. What do you notice about the base of each triangle: ABC, AB ' C, ADC, and AEC? d. What do you notice about the height of each triangle? e. What conjecture can you make about the area of any triangle that would be drawn between these parallel lines if A and C are not moved to different positions? Explain your reasoning. f. Would your conjecture still be true if you were able to choose any three points on the two lines to draw your triangles? Explain your reasoning. Blackline Masters, Geometry Page 3-14

36 Unit 3, Activity 7, Parallel Line Facts with Answers Use the given diagram to complete the steps and answer the questions. Date Name 1. Draw a line through vertex B so that the line is parallel to AC. Locate one point to the left of B and label it L. Locate one point to the right of B and label it R. 2. Given the diagram above with the parallel line drawn through B, prove that m BAC + m ABC + m ACB = 180. Given that BR AC, we know that alternate interior angles are congruent. So, BAC ABL and ACB CBR. By definition of congruence, m BAC = m ABL and m ACB = m CBR. Because ABL, ABC, and CBR are adjacent and form a line, m ABL + m ABC + m CBR = 180. Using the substitution property of equality, we now have m BAC + m ABC + m ACB = Using the same diagram above, extend AC so that it is a line. Draw two points on AC : one to the left of A labeled D and the other to the right of C and label it E. Using the new diagram, prove that m BAD = m ABC + m ACB. Given that BR AC, we know that alternate interior angles are congruent. So, BAD ABR and ACB CBR. By definition of congruence, we also know that m BAD = m ABR and m ACB = m CBR. Using the Angle Addition Postulate, we know m ABR = m ABC + m CBR. Next, using the Substitution Property of Equality, we find m ABR = m ABC + m ACB. Using the Substitution Property of Equality one more time, we get m BAD = m ABC + m ACB. Blackline Masters, Geometry Page 3-15

37 Unit 3, Activity 7, Parallel Line Facts with Answers 4. Remember, the area of a triangle can be written as answer the following questions. 1 A = bh. Use the diagram below to 2 a. Using D, draw triangle ADC. b. Choose a point anywhere on BD and label it E. Draw triangle AEC. c. What do you notice about the base of each triangle: ABC, AB ' C, ADC, and AEC? The base of all four triangles is segment AC. The measure of the base doesn t change. d. What do you notice about the height of each triangle? Since the distance between parallel lines is equal everywhere, the height of all four triangles is the same. e. What conjecture can you make about the area of any triangle that would be drawn between these parallel lines if A and C are not moved to different positions? Explain your reasoning. Since both the base and height of these triangles are the same, they will have the same area. f. Would your conjecture still be true if you were able to choose any three points on the two lines to draw your triangles? Explain your reasoning. No, the conjecture would not necessarily work. If the measure of the base were changed each time, the area of each triangle would also change despite the fact that the height remained the same. Blackline Masters, Geometry Page 3-16

38 Unit 3, General Assessment, Scrapbook Rubric Parallel and Perpendicular Lines Scrapbook CATEGORY 4 points 3 points 2 points 1 point 0 points Score Comments Quantity (24) Minimum of 3 photos per term (Parallel or Perpendicular) Only two photos/ pictures per term Only one photo/picture per term Only one picture to demonstrate both terms No photos or pictures x 6 Quality (24) Title Page (8) Reflection (12) Photos are of excellent quality; clear; description is written clearly Excellent quality; typed; includes project title, date, and class period Typed; tells what the student learned from project; grammatically correct Photos/ pictures are of good quality; description is clear but missing some elements Typed; missing date or class period Typed; 1-2 grammar errors; some evidence of learning Photos/pictures are grainy; term is not clearly depicted in picture; description is vague Handwritten with all information; or typed and missing date and class period Handwritten; 3-4 grammar errors; vague evidence of learning Photos/pictures do not depict term at all; description only gives definition Handwritten and missing date and class period; missing title (typed with all other info) Handwritten; 5-6 grammar errors; little to no evidence of learning No description x 6 given No title page No reflection x 2 x 3 Neatness/ Creativity (8) Timeliness (8) Typed; clean; neatly bound pages; original project title; attractive; etc. Turned in on time Typed; project name not original; some pages loose Turned in one day late Handwritten; project is less than attractive; Turned in two days late Dirty, crumpled pages; if handwritten there are scratchouts or places with liquid paper Turned in three days late Pages are not bound Turned in more than three days late x 2 x 2 Blackline Masters, Geometry Page 3-17

39 Unit 3, Activity 1, Specific Assessment, What s My Line? What s My Line? On the attached page, you have been given a line and a point. Every line can be unique and can have its own unique equation. It is your job to find out as much about the line as possible. Listed below is the information that you must determine about the line. 1. Locate, draw, and label an x-axis and y-axis. 2. Find two points on your line. Label their (x,y) coordinates. Using the two points find the slope of your line. Show your calculations below. Write the slope next to the line as m=. 3. Determine the slope-intercept form of the equation for your line. Show your calculations below. Label your line with the equation. 4. Calculate the x and y-intercepts. Show your calculations below. On your graph, identify and label by giving their (x,y) coordinates. 5. Draw the line that is perpendicular to your line that passes through the point that was given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this equation. Show your calculations below. 6. Draw the line that is parallel to your line that passes through the point that was given on the page. Write the word parallel next to this line. Write the slope-intercept form of the equation for the parallel line and label the line with this equation. Show your calculations below. Blackline Masters, Geometry Page 3-18

40 Unit 3, Activity 1, Specific Assessment, What s My Line? with Answers What s My Line? On the attached page, you have been given a line and a point. Every line can be unique and can have its own unique equation. It is your job to find out as much about the line as possible. Listed below is the information that you must determine about the line. 1. Locate, draw, and label an x-axis and y-axis. Will vary by student 2. Find two points on your line. Label their (x,y) coordinates. Using the two points find the slope of your line. Show your calculations below. Write the slope next to the line as m= Graph A: m = ; Graph B: m = ; Graph C: m = ; Graph D: m = ; Graph E: m = 2 3 Other answers cannot be given since there are no axes on the graphs (students must draw these in on their own wherever they choose). The placement of the axes determines other answers. 3. Determine the slope-intercept form of the equation for your line. Show your calculations below. Label your line with the equation. Answers will depend on where x/y axes are drawn by each student. 4. Calculate the x and y-intercepts. Show your calculations below. On your graph, identify and label by giving their (x,y) coordinates. Answers will depend on where x/y axes are drawn by each student. 5. Draw the line that is perpendicular to your line that passes through the point that was given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this equation. Show your calculations below. 3 Graph A: m = ; Graph B: 2 Graph D: 2 m = ; Graph E: m = m = ; Graph C: 7 5 m = ; 2 6. Draw the line that is parallel to your line that passes through the point that was given on the page. Write the word parallel next to this line. Write the slope-intercept form of the equation for the parallel line and label the line with this equation. Show your calculations below. 2 Graph A: m = ; Graph B: 3 5 Graph D: m = ; Graph E: 2 7 m = ; Graph C: 2 1 m = 3 2 m = ; 5 Blackline Masters, Geometry Page 3-19

41 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph A Blackline Masters, Geometry Page 3-20

42 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph B Blackline Masters, Geometry Page 3-21

43 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph C Blackline Masters, Geometry Page 3-22

44 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph D Blackline Masters, Geometry Page 3-23

45 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph E Blackline Masters, Geometry Page 3-24

46 Unit 3, Activity 1, Specific Assessment, What s My Line? Rubric What s My Line? Rubric Description Points Possible Points Earned I. x-axis and y-axis drawn and labeled 5 II. III. IV. Slope of the line A. 2 points labeled with (x,y) coordinates 2 B. Slope calculated correctly 6 C. Labeled line with slope 1 Equation of the line A. Calculated correctly 6 B. Labeled on graph 1 x and y-intercepts A. Calculated correctly 6 B. Labeled on graph 2 V. Perpendicular Line A. Line drawn correctly 3 B. Slope of line correctly identified 2 C. y-intercept calculated correctly 2 D. Equation written correctly based on calculations shown 2 C. Line labeled with perpendicular and equation 2 VI. VII. Parallel Line A. Line drawn correctly 3 B. Slope of line correctly identified 2 C. y-intercept calculated correctly 2 D. Equation written correctly based on calculations shown 2 C. Line labeled with parallel and equation 2 Following directions A. Cover sheet 3 B. Stapled 3 C. Submitted on or before due date 3 Total 60 Blackline Masters, Geometry Page 3-25

47 Unit 4, Activity 1, Vocabulary Self-Awareness Word/Phrase + Definition/Rule Example transformation pre-image image rigid transformation (rigid motion) non-rigid transformation (non-rigid motion) orientation isometry reflection line of reflection translation rotation center of rotation Blackline Masters, Geometry Page 4-1

48 Unit 4, Activity 1, Vocabulary Self-Awareness degree of rotation clockwise counterclockwise dilation center of dilation scale factor similarity transformation composite transformation glide reflection Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 4-2

49 Unit 4, Activity 1, Vocabulary Self-Awareness with Answers Word/Phrase + Definition/Rule Example transformation pre-image A correspondence between two sets of points such that each point in the pre-image has a unique image and that each point in the image has exactly one pre-image; a change in size, orientation, or position of a figure in space. The original object that is to be transformed. image rigid transformation (rigid motion) non-rigid transformation (non-rigid motion) orientation isometry reflection line of reflection The copy of the object that has been transformed. A transformation that preserves measurements of segments and angles; also called an isometry (see below). A transformation that does not preserve measures of segments and angles; the shape of the pre-image may not be preserved either. The location (position and angle) of an object in space in relation to a set of reference axes. A transformation that preserves measurements and more specifically distances between points; a transformation that preserves distances is also bound to preserve angle measures; a congruence transformation. A transformation in which each point in the pre-image has an image that is the same distance from the line of reflection (see below). For a point on the line of reflection, the image is itself; aka flip. The perpendicular bisector of the segment joining each point (pre-image) and its image. Blackline Masters, Geometry Page 4-3

50 Unit 4, Activity 1, Vocabulary Self-Awareness with Answers translation rotation center of rotation angle (degree) of rotation clockwise counterclockwise dilation center of dilation scale factor A transformation which moves an object a fixed distance in a fixed direction; a composite of two reflections over parallel lines; aka slide. A transformation that turns a figure about a fixed point called the center of rotation; a composite of two reflections over intersecting lines; aka turn. A fixed point about which a figure is rotated; the point where two intersecting lines of rotation meet?? Rays drawn from the center of rotation to a point on the preimage and its image form the angle of rotation (measured in degrees). Rotation of an object to the right indicated by a negative angle of rotation. Rotation of an object to the left indicated by a positive angle of rotation. A transformation that produces an image that is the same shape as the pre-image but is a different size; a stretch or shrink of the pre-image. A fixed point in the plane about which all points are expanded (stretched) or contracted (shrunk). The ratio by which an object is enlarged or reduced; if greater than 1 the image is an enlargement; if between 0 and 1 the dilation is a reduction; if the scale factor equals 1, the figures are congruent. Blackline Masters, Geometry Page 4-4

51 Unit 4, Activity 1, Vocabulary Self-Awareness with Answers similarity transformation composite transformation glide reflection A transformation that is the composite of dilations and/or reflections; a non-rigid transformation; the shape of the pre-image is preserved but the size is changed. The result of two or more successive transformations. A type of composite transformation where a figure is reflected then translated. Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 4-5

52 Unit 4, Activity 3, A Basic Look at Transformations Trace the following polygons on the patty paper or tracing paper given to you. Blackline Masters, Geometry Page 4-6

53 Unit 4, Activity 4, What Are Transformations? What Are Transformations? When learning about transformations, one might first look at the parts of the word. Transformation can be separated into the prefix trans- and the word formation. The prefix transmeans changing thoroughly and formation means the act of giving or taking form, shape, or existence. Taken together, a transformation can be described as the act of changing a form or shape. Specifically, in geometry, a transformation may change the position, orientation, or size of a figure in the plane. Look at the transformations below. The figure drawn with the solid lines is called the pre-image, or the original object that is being transformed. The figure drawn with dashed lines is called the image, or the copy of the object that has been transformed. pre-image Figure 1 image image Figure 2 pre-image The most basic transformation is a reflection. A reflection can be easily described as a flip, however, that is not the most accurate definition. A reflection is a transformation in which each point in the pre-image has an image that is the same distance from the line of reflection. The line of reflection is the perpendicular bisector of the segment joining each point on the preimage with its corresponding point on the image. Look at the example below. The image is labeled A B C D (read A prime, B prime, C prime, D prime the use of the apostrophe on the letter is universally accepted to show that a figure is the image of a transformation). D A P A D C B m B C Figure 3 Line m is the line of reflection in the figure and P is the point where line m intersects the segment joining A and A. Using a ruler, measure the distance from A to A. Now, measure the distance from A to P and the distance from P to A. You should notice that AP and PA are equal. Use a protractor to measure the angles formed at P. You should see that the angles all measure 90 Blackline Masters, Geometry Page 4-7

54 Unit 4, Activity 4, What Are Transformations? degrees. How do those measurements relate to the definition of the line of reflection given earlier? Another basic transformation is a translation, or slide. When a translation is performed, the pre-image is moved a fixed distance in a fixed direction. The directions for performing a translation could state to move the pre-image 5 inches to the right in which each point on the pre-image is moved 5 inches to the right of the location to form the image. Translations can also be thought of as the composition of two reflections over parallel lines. Look at the example below. pre-image image m Figure 4 n Notice there are two reflections. Lines m and n are parallel. The resulting image has the same orientation as the pre-image, but has been moved to the right by 10 cm. A question to think about: does the distance of the translation have any relationship to the distance between the parallel lines? A third transformation is called a rotation. This transformation may also be referred to as a turn. A rotation is a transformation that turns a figure about a fixed point, called the center of rotation, through a fixed angle of rotation (measured in degrees). The path the figure follows during the rotation would form a circle around the center of rotation if the figure were rotated 360 degrees. A rotation can be performed with any degree measure and can be considered a clockwise rotation or a counterclockwise rotation. A clockwise rotation will turn a figure to the right around the center of rotation while a counterclockwise rotation will turn a figure to the left around the center of rotation. All positive degree measures are assumed to indicate a counterclockwise rotation, while all negative degree measures are assumed to indicate a clockwise rotation. X Figure 5 Blackline Masters, Geometry Page 4-8

55 Unit 4, Activity 4, What Are Transformations? In Figure 5 above, X is the center of rotation. The angle of rotation is formed by drawing a segment from one point on the pre-image to the center of rotation then drawing the required angle using the center of rotation (X) as the vertex. The angle used in this figure is 90 clockwise, or -90. Question to think about: What would happen to the image if the center of rotation was moved but the angle of rotation remained the same? A rotation can also be defined as a composite of two or more reflections over intersecting lines. Consider the example below. m pre-image n Figure 6 The intersection of lines m and n becomes the center of rotation. These lines happen to be perpendicular. Notice how the image has been rotated in a counterclockwise direction around the center (point of intersection). How could you determine what the angle of rotation is for this diagram? Go back to the definition of rotation discussed earlier for some ideas. The three transformations discussed so far have one thing in common. If you look at all of the images and compare them to their corresponding pre-images, you will notice that the measures of the segments and angles have not changed (go ahead measure them if you wish!). Since the images have the same shape and are the same size as the pre-images, they are congruent. Each of these transformations is called an isometry. An isometry is a transformation that preserves measurements of segments and angles and therefore produces an image congruent to its pre-image. A transformation that is an isometry is also sometimes referred to as a rigid motion or rigid transformation. In geometry, there is one more important transformation. A dilation is a transformation that produces an image that is the same shape as the pre-image but is a different size. Sometimes they are referred to as a stretch or shrink (also called an enlargement or reduction). Each dilation is focused at the center of dilation, or a fixed point about which all points are enlarged or reduced. How much the figure is enlarged or reduced depends upon the scale factor, the ratio by which an object is enlarged or reduced. If the scale factor is greater than 1, the image is an enlargement of the pre-image. If the scale factor is between 0 and 1, the image is a reduction of the pre-image. B B Figure 7 A E D C image E D Blackline Masters, Geometry Page 4-9 C

56 Unit 4, Activity 4, What Are Transformations? In Figure 7 above, the center of dilation is A. The measure of segment AB is 2 times the measure of segment AB. Therefore, the image A B C D E is an enlargement of the pre-image ABCDE, and the scale factor is 2. Notice, the measures of the corresponding segments are not equal, however the measures of the corresponding angles are (you can verify this by using your protractor). Therefore, dilation is not an isometry. Dilation is a non-rigid transformation, or a non-rigid motion. Because the corresponding angles have the same measure and the corresponding sides are proportional, these figures are similar which means dilation is a similarity transformation. Blackline Masters, Geometry Page 4-10

57 Unit 5, Activity 2, Investigating Congruence Part One: Given FGH and the line of reflection, line m, perform the indicated reflections and answer the questions that follow. m F H G 1. Reflect FG. What is true about FG? ' ' Explain your reasoning. 2. Reflect GH. What is true about GH? ' ' Explain your reasoning. 3. Reflect FH. What is true about F' H '? Explain your reasoning. 4. Is FGH F ' G ' H '? Justify your answer. Part Two: Given MNO and the line of reflection, line s, answer the following. s N M O 1. Reflect MN and MO. What is true about M ' N', M ' O', and N' M ' O'? Justify your answer. 2. Connect N and O. Is MNO M ' N ' O '? Justify your answer. Blackline Masters, Geometry Page 5-1

58 Unit 5, Activity 2, Investigating Congruence Part Three: Given XYZ use three sheets of patty paper to complete the following steps and answer the questions that follow. X Z 1. Using one sheet of patty paper, copy XY and label the endpoints X and Y respectively. 2. Using a second sheet of patty paper, copy ZXY. Copy the angle only, including the sides, XZ and XY. Do not copy ZY on this paper. Label the vertex of the angle as X and the endpoints of the sides as Z and Y respectively. 3. Using the third sheet of patty paper, copy ZYX. Copy the angle only, including the sides, YX and YZ. Do not copy XZ on this paper. Label the vertex of the angle as Y, and the endpoints of the sides as X and Z respectively. 4. What should be true about the segment and both angles you copied onto the three sheets of patty paper? How can you verify your conjecture? Y 5. Now, starting with the patty paper with XY, ' ' lay the other two pieces of patty paper on top of the first one lining up XY ' ' on each piece of paper. What happens? What is true about XYZ and X ' Y ' Z '? How can you verify your conjecture? Blackline Masters, Geometry Page 5-2

59 Unit 5, Activity 2, Investigating Congruence with Answers Part One: Given FGH and the line of reflection, line m, perform the indicated reflections and answer the questions that follow. m F H G 1. Reflect FG. What is true about FG? ' ' Explain your reasoning. The segments are congruent. Reflection does not change length. 2. Reflect GH. What is true about GH? ' ' Explain your reasoning. The segments are congruent. Reflection does not change length. 3. Reflect FH. What is true about F' H '? Explain your reasoning. The segments are congruent. Reflection does not change length. 4. Is FGH F ' G ' H '? Justify your answer. Yes, all of the segments are congruent; the angles are also congruent because reflection is an isometry and will not change the angle measure. Part Two: Given MNO and the line of reflection, line s, answer the following. s N M O 1. Reflect MN and MO. What is true about M ' N', M ' O', and N' M ' O'? Justify your answer. The segments and angles are congruent; reflection preserves length and angle measure. 2. Connect N and O. Is MNO M ' N ' O '? Justify your answer. Yes. NOis ' ' also a reflection so all corresponding sides and angles are congruent. Blackline Masters, Geometry Page 5-3

60 Unit 5, Activity 2, Investigating Congruence with Answers Part Three: Given XYZ use three sheets of patty paper to complete the following steps and answer the questions that follow. X Z 1. Using one sheet of patty paper, copy XY and label the endpoints X and Y respectively. 2. Using a second sheet of patty paper, copy ZXY. Copy the angle only, including the sides, XZ and XY. Do not copy ZY on this paper. Label the vertex of the angle as X and the endpoints of the sides as Z and Y respectively. 3. Using the third sheet of patty paper, copy ZYX. Copy the angle only, including the sides, YX and YZ. Do not copy XZ on this paper. Label the vertex of the angle as Y, and the endpoints of the sides as X and Z respectively. 4. What should be true about the segment and both angles you copied onto the three sheets of patty paper? How can you verify your conjecture? Since they are copies of the original triangle, they should be congruent. This can be verified by measuring the length of both segments and by measuring the corresponding angles. 5. Now, starting with the patty paper with XY, ' ' lay the other two pieces of patty paper on top of the first one lining up XY ' ' on each piece of paper. What happens? What is true about XYZ and X ' Y ' Z '? How can you verify your conjecture? When all three papers are laid on top of each other, it creates a triangle, namely XYZ ' ' '. These two triangles are congruent. This can be verified either by measuring all corresponding sides and angles or by laying XYZ ' ' ' over XYZ to see that the sides and angle have the same size. Y Blackline Masters, Geometry Page 5-4

61 Unit 5, Activity 2, Sample Split-Page Notes Date: Period: Triangle vertex sides Topic: Triangles --a closed figure with three segments joining three noncollinear points --named using the three vertices --the point where two segments meet; a corner of a triangle --Every triangle has three vertices. --a segment whose endpoints are the vertices of the triangle --Every triangle has three sides. Example: A Name: ABC Sides: AB, BC, and AC Vertices: A, B, and C C B Classifications by Angle: Acute Obtuse Right Equiangular Classifications by Sides: Scalene Isosceles Equilateral Triangle Sum Theorem Exterior Angle Theorem --a triangle with three acute angles --a triangle with exactly one obtuse angle --a triangle with exactly one right angle --a triangle with three congruent angles --all angles measure 60 degrees --also considered an acute triangle. --a triangle with no congruent sides --a triangle with at least two congruent sides --a triangle with all three sides congruent --also considered an isosceles triangle --The sum of the measures of the angles of a triangle is The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. C A B D m ABD= m A+ m C Blackline Masters, Geometry Page 5-5

62 Unit 5, Activity 2, Blank Split-Page Notes Date: Period: Topic: Blackline Masters, Geometry Page 5-6

63 Unit 5, Activity 2, Triangle Split-Page Notes Model Date: Period: CPCTC Topic: Congruent Triangles Corresponding parts of congruent triangles are congruent (if two triangles are congruent, then all pairs of corresponding parts are also congruent). ABC ITR A I AB IT B T BC TR C R AC IR Side-Side-Side (SSS) Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. HAT SRI because HA SR AT RI HT SI Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. MAN JOB because MA JO A O AN OB Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angle and the included side of a second triangle, then the two triangles are congruent. ABC DEF because B E BC EF C F Blackline Masters, Geometry Page 5-7

64 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the triangles congruent. Given: X is the midpoint of BD X is the midpoint of AC Prove: DXC BXA C D X B A Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-8

65 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides congruent. Given: X is the midpoint of BD X is the midpoint of AC D X A Prove: DA BC C B Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-9

66 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the angles congruent. Given: MA TH ; HM TA M A Prove: H A H T Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-10

67 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides congruent. Given: MA TH ; MA TH M A Prove: MH AT H T Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-11

68 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the triangles congruent. I Given: X is the midpoint of KT IE KT Prove: KXE TXE K X T E Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-12

69 Unit 5, Activity 4, Proving Triangles Congruent Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides congruent. I Given: X is the midpoint of KT Prove: KI IE KT TI K X T E Write your proof below. Be sure to include all logical reasoning. Blackline Masters, Geometry Page 5-13

70 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the triangles congruent. Given: X is the midpoint of BD X is the midpoint of AC Prove: DXC BXA C D X B A Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. X is the midpoint of BD ; X is the midpoint 1. Given of AC. 2. XD XB; XA XC 2. Midpoint Theorem 3. DXC BXA 3. Vertical angles are congruent. 4. DXC BXA 4. SAS Postulate Blackline Masters, Geometry Page 5-14

71 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides triangles congruent. Given: X is the midpoint of BD X is the midpoint of AC D X A Prove: DA BC C B Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. X is the midpoint of BD ; X is the midpoint 1. Given of AC. 2. XD XB; XA XC 2. Midpoint Theorem 3. DXA BXC 3. Vertical angles are congruent. 4. DXA BXC 4. SAS Postulate 5. DA BC 5. CPCTC Blackline Masters, Geometry Page 5-15

72 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the angles triangles congruent. Given: MA TH ; HM TA M A Prove: H A H T Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. MA TH ; HM TA 1. Given 2. HMT ATM 2. Alternate Interior Angles Theorem 3. AMT HTM 3. Alternate Interior Angles Theorem 4. MT MT 4. Reflexive Property of Congruence 5. MHT TAM 5. ASA Postulate 6. H A 6. CPCTC Blackline Masters, Geometry Page 5-16

73 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides triangles congruent. Given: MA TH ; MA TH M A Prove: MH AT H T Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. MA TH ; MA TH 1. Given 2. MAH THA 2. Alternate Interior Angles Theorem 3. AH AH 3. Reflexive Property of Congruence 4. MHT TAM 4. SAS Postulate 5. MH TA 5. CPCTC Blackline Masters, Geometry Page 5-17

74 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the triangles congruent. I Given: X is the midpoint of KT IE KT Prove: KXE TXE K X T E Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. X is the midpoint of KT ; IE KT 1. Given 2. KX TX 2. Midpoint Theorem 3. m KXE = 90; m TXE = Definition of perpendicular lines 4. m KXE = m TXE 4. Substitution Property of Equality 5. KXE TXE 5. Definition of congruence 6. EX EX 6. Reflexive Property of Congruence 7. KXE TXE 7. SAS Postulate Blackline Masters, Geometry Page 5-18

75 Unit 5, Activity 4, Proving Triangles Congruent with Answers Group Members Date Proving Triangles Congruent Using the given diagram and information, prove two of the sides triangles congruent. I Given: X is the midpoint of KT IE KT Prove: KI TI K X T E Write your proof below. Be sure to include all logical reasoning. Statements Reasons 1. X is the midpoint of KT ; IE KT 1. Given 2. KX TX 2. Midpoint Theorem 3. m KXI = 90; m TXI = Definition of perpendicular lines 4. m KXI = m TXI 4. Substitution Property of Equality 5. KXI TXI 5. Definition of congruence 6. IX IX 6. Reflexive Property of Congruence 7. KXI TXI 7. SAS Postulate 8. KI TI 8. CPCTC Blackline Masters, Geometry Page 5-19

76 Unit 5, Activity 11, Angle and Side Relationships Group Members Date Use the following charts to record the measurement data from the triangles. Which group did the three triangles come from? Triangle # 1 Name: Angle Name Angle Measure Side Name Side Measure List the names of the angles and sides in order from largest to smallest on the lines below: Angles Sides Triangle #2 Name: Angle Name Angle Measure Side Name Side Measure List the names of the angles and sides in order from largest to smallest on the lines below: Angles Sides Triangle #3 Name: Angle Name Angle Measure Side Name Side Measure List the names of the angles and sides in order from largest to smallest on the lines below: Angles Sides Blackline Masters, Geometry Page 5-20

77 Unit 5, Activity 11, Angle and Side Relationships Look at the measures of the angles and find the largest angle. Locate the side opposite the largest angle. How does the measure of the side opposite the largest angle compare to the measures of the other two sides? Look at the measures of the sides and find the shortest side. Locate the angle opposite of the shortest side. How does the measure of the angle opposite the shortest side compare to the measures of the other two angles? What conjecture can you draw from these observations? Blackline Masters, Geometry Page 5-21

78 Unit 5, Activity 11, Angle and Side Relationships Proof Theorem: If one side of a triangle is longer than a second side, then the angle opposite the first side is greater than the angle opposite the second side. Given: ABC and AB > BC Prove: m BCA > m BAC C B A Proof: Place P on side AB such that PB BC and draw PC. C B P A Now we know that m BCA = m BCP + m PCA. Therefore, m BCA > m BCP. BCP is isosceles so, BCP BPC which means that m BCA > m BPC (by substitution). By definition, BPC is an exterior angle of APC, so it is greater than the remote interior angle BAC: m BPC > m BAC. So we have m BCA > m BPC and m BPC > m BAC. By the transitive property for inequalities, it follows that m BCA > m BAC. BCA is the angle opposite the longer side, AB and BAC is opposite the shorter side, BC. Thus the angle opposite the longer side is greater than the angle opposite the shorter side. Blackline Masters, Geometry Page 5-22

79 Unit 5, Activity 14, Quadrilateral Process Guide Date Name Partner s Name Use the following guide to investigate the relationships that occur in different convex quadrilaterals. 1. Which quadrilateral are you working with? 2. Measure all four angles and all four sides of the given quadrilateral and record the information below. Angle Measures Side Measures 3. Resize the quadrilateral by dragging the vertices. Measure the angles and sides again and record the information. Angle Measures Side Measures 4. Continue to resize the quadrilateral and make measurements for this quadrilateral. After creating a minimum of 5 different sized quadrilaterals, make conjectures about the measures of the sides and angles of any quadrilateral of this type. You should have multiple conjectures. 5. Construct the diagonals of the quadrilateral and answer the following questions: a.) Do the diagonals bisect each other? b.) Are the diagonals congruent? c.) Are the diagonals perpendicular? d.) Do the diagonals bisect the angles of the quadrilateral? Blackline Masters, Geometry Page 5-23

80 Unit 5, Activity 14, Quadrilateral Family Date Name Fill in the names of the quadrilaterals so that each of the following is used exactly once. Parallelogram Square Trapezoid Isosceles Trapezoid Kite Quadrilateral Rectangle Rhombus If you follow the arrows from top to bottom, the properties of each figure are also properties of the figure that follows it. For example, the properties of a parallelogram are also properties of a rectangle. If you reverse the arrows from bottom to top, every figure is also the one that precedes it. For example, a square is also a rhombus and a rectangle since it is connected to them both. Blackline Masters, Geometry Page 5-24

81 Unit 5, Activity 14, Quadrilateral Family with Answers Date Name Fill in the names of the quadrilaterals so that each of the following is used exactly once. Parallelogram Square Trapezoid Isosceles Trapezoid Kite Quadrilateral Rectangle Rhombus Quadrilateral Trapezoid Isosceles Trapezoid Kite Parallelogram Rectangle Rhombus Square If you follow the arrows from top to bottom, the properties of each figure are also properties of the figure that follows it. For example, the properties of a parallelogram are also properties of a rectangle. If you reverse the arrows from bottom to top, every figure is also the one that precedes it. For example, a square is also a rhombus and a rectangle since it is connected to them both. Blackline Masters, Geometry Page 5-25

82 Unit 5, Activity 4, Specific Assessment Instructions for Product Assessment Activity 6 Your task is to design a tile in the 5-inch by 5-inch squares provided on the next two pages. There are two parts to this project. Part I: The drawings on your tile must meet certain specifications. You must have the following, and you will be graded on the accuracy of the following. 1.) 2 congruent obtuse triangles which demonstrate congruency by ASA 2.) 2 congruent scalene triangles which demonstrate congruency by SSS 3.) 2 congruent isosceles right triangles which demonstrate congruency by SAS 4.) 2 congruent acute triangles which demonstrate congruency by AAS 5.) 1 equilateral triangle You should have a minimum of 9 triangles in your design (i. e. your two acute triangles CANNOT double as your two scalene triangles). You may add other shapes once you are sure you have the required 9 triangles above. On Part I, you must label and mark each pair of triangles according to one of the methods indicated in the directions above (see example below). D 2 scalene triangles congruent by SSS B AB DB AC DC A BC BC ABC DBC C Part II For Part II, you are to redraw your tile (without the markings and labels) in the square on the second page and COLOR it. Cut the tile out of the page and put your name, number and hour ON THE BACK! Do NOT glue it to another page, and do NOT staple it to part one. If you do not complete part two, the entire project will be returned to you, and you will lose one letter grade for each day late!! DUE DATE: Blackline Masters, Geometry Page 5-26

83 Unit 5, Activity 4, Specific Assessment Part I Obtuse Triangles (ASA) Acute Triangles (AAS) Scalene Triangles (SSS) Equilateral Triangle Isosceles Right Triangles (SAS) Blackline Masters, Geometry Page 5-27

84 Unit 5, Activity 4, Specific Assessment Part II Blackline Masters, Geometry Page 5-28

85 Unit 5, Activity 4, Specific Assessment Rubric Activity 6 Product Assessment Rubric This is a checklist for evaluating your tile design. Your grade will be a percentage based on the number of requirements met. Are the following present? 40% 1.) 2 congruent obtuse triangles [ ] yes [ ] no 2.) 2 congruent scalene triangles [ ] yes [ ] no 3.) 2 congruent isosceles right triangles [ ] yes [ ] no 4.) 2 congruent acute triangles [ ] yes [ ] no 5.) 1 equilateral triangle [ ] yes [ ] no Are the triangles marked by the correct method? 40% 6.) ALL triangles labeled [ ] yes [ ] no 7.) Obtuse congruent by ASA [ ] yes [ ] no 8.) Scalene congruent by SSS [ ] yes [ ] no 9.) Isosceles right congruent by SAS [ ] yes [ ] no 10.) Acute congruent by AAS [ ] yes [ ] no Are the congruent triangles and parts listed correctly (based on markings)? 20% 11.) Obtuse triangles and parts [ ] yes [ ] no 12.) Scalene triangles and parts [ ] yes [ ] no 13.) Isosceles right triangles and parts [ ] yes [ ] no 14.) Acute triangles and parts [ ] yes [ ] no 15.) Equilateral triangle [ ] yes [ ] no Following directions and promptness (for each no below, you will lose one percentage point): 16.) Tile drawn on handout and name on handout [ ] yes [ ] no 17.) Part two is colored [ ] yes [ ] no 18.) Part two is cut out and NOT attached by staple or glue [ ] yes [ ] no 19.) Name, number, and hour on back of part 2 [ ] yes [ ] no 20.) Rubric turned in [ ] yes [ ] no 21.) Turned in on time [ ] yes [ ] no Score [4( ) + 4( ) + 2( )]/10 = Blackline Masters, Geometry Page 5-29

86 Unit 5, Activity 14, Specific Assessment Venn Diagram for Assessment for Activity 14 Directions: Label the Venn diagram below with the name and the number representing the properties for parallelograms. Remember, in a Venn Diagram each property should only be listed once. 1.) Diagonals are perpendicular. 2.) All four angles are right angles. 3.) Opposite angles are congruent. 4.) Diagonals bisect a pair of opposite angles. 5.) Diagonals are congruent. 6.) Opposite sides are congruent. 7.) Diagonals bisect each other. 8.) All four sides are congruent. 9.) Opposite sides are parallel. 10.) Consecutive angles are supplementary. Blackline Masters, Geometry Page 5-30

87 Unit 5, Activity 14, Specific Assessment Answer Key Parallelograms 3, 6, 7, 9 Rhombii 1, 4, 8 Squares Rectangles 2, 5 Blackline Masters, Geometry Page 5-31

88 Unit 6, Activity 1, Striking Similarity Using the grid provided below, transfer the polygons to the blank grid you were given. You may use a straight edge to help you draw the sides. Blackline Masters, Geometry Page 6-1

89 Unit 6, Activity 2, Similarity and Ratios Name Date Follow the given directions to explore the relationships between side lengths, area, and volume of similar figures. 1.) Given an equilateral triangle, use pattern blocks to create a similar triangle so the ratio of side lengths is 2:1. a.) What is the ratio of areas of the two similar triangles? b.) Using pattern blocks create a triangle similar to the original triangle so the ratio of side lengths is 3:1. What is the ratio of the areas of these two similar triangles? 2.) Use other pattern block shapes to create and investigate other similar polygons in the same manner as described above, and record your findings in the table below. description of similar shapes ratio of sides ratio of areas ) Based on your investigations in the two activities, make a generalization. If the ratio of sides of two similar polygons is n:1, what would the ratio of areas be? 4.) Given a cube, create a similar cube with ratio of edges 2:1 using cm or sugar cubes. What is the ratio of volumes? Create a similar cube with ratio of edges 3:1. What is the ratio of volumes? If the edges of two cubes were in a ratio of n:1, what would the ratio of volumes be? Record your findings in a table like the one below. description of similar 3-D shapes ratio of edges ratio of volumes Blackline Masters, Geometry Page 6-2

90 Unit 6, Activity 2, Similarity and Ratios with Answers Name Date Follow the given directions to explore the relationships between side lengths, area, and volume of similar figures. 1.) Given an equilateral triangle, use pattern blocks to create a similar triangle so the ratio of side lengths is 2:1. a.) What is the ratio of areas of the two similar triangles? The ratio of the areas is 4:1. b.) Using pattern blocks create a triangle similar to the original triangle so the ratio of side lengths is 3:1. What is the ratio of the areas of these two similar triangles? The ratio of the areas is 9:1. 2.) Use other pattern block shapes to investigate other similar polygons in the same manner as described above, and record your findings in the table below. description of similar shapes ratio of sides ratio of areas. Answers will vary ) Based on your investigations in the two activities, make a generalization. If the ratio of sides of two similar polygons is n:1, what would the ratio of areas be? The ratios of the areas will be n 2 :1. 4.) Given a cube, create a similar cube with ratio of edges 2:1 using cm or sugar cubes. What is the ratio of volumes? Create a similar cube with ratio of edges 3:1. What is the ratio of volumes? If the edges of two cubes were in a ratio of n:1, what would the ratio of volumes be? Record your findings in a table like the one below. description of similar 3-D shapes ratio of edges ratio of volumes. cube with face of 4 square units 2:1 8:1. cube with face of 9 square units 3:1 27:1. cube with face of n 2 units n:1 n 3 :1.. Blackline Masters, Geometry Page 6-3

91 Unit 6, Activity 4, Spotlight on Similarity Blackline Masters, Geometry Page 6-4

92 Unit 6, Activity 9, DL-TA DL-TA for (title) Prediction question(s): Using the title, your own background knowledge, and any other contextual clues, make your predictions. Before reading: During reading: During reading: During reading: During reading: During reading: After reading: Blackline Masters, Geometry Page 6-5

93 Unit 6, Activity 3, Specific Assessment, Making a Hypsometer Format: Objectives: Materials: Individual or Small Group Participants use the hypsometer and their knowledge of the proportional relationship between similar triangles to determine the height of an object not readily measured directly. For each hypsometer, you need a straw, decimal graph paper, cardboard, thread, a small weight, tape, a hole punch, scissors, and a meter stick. Time Required: Approximately 90 minutes Directions: To make the hypsometer: 1) Tape a sheet of decimal graph paper to a piece of cardboard. 2) Tape the straw to the cardboard so that it is parallel to the top of the graph paper. 3) Punch a hole in the upper right corner of the grid. Pass one end of the thread through the hole and tape it to the back of the cardboard. Tie the weight to the other end of the thread. Blackline Masters, Geometry Page 6-6

94 Unit 6, Activity 3, Specific Assessment, Making a Hypsometer To use the hypsometer: 4) Have a friend use a meter stick to measure the height of your eye from the ground and the distance from you to the object to be measured. 5) Look through the straw at the top of the object you wish to measure. Your friend should record the hypsometer reading as you remain steady and continue to look through the straw at the top of the object 6) To find the height of the flagpole, recognize that triangles ABC and DEF are similar. Thus, BC can be found using the following ratio AC = BC DF EF Reference: NCTM Addenda Series, Measurement in the Middle Grades Blackline Masters, Geometry Page 6-7

95 Unit 7, Activity 5, Turning a Plane Figure Into a Solid Figure Group Members: Date: 1. On a piece of graph paper, graph the coordinates: A( 0,0 ), B( 2,0 ), C( 2,5 ), and D ( 0,5). a. Connect the points to create line segments AB, BC, CD, and AD. b. Shade the area created by the segments. c. What polygon is created? d. Identify the dimensions of the polygon: Base: Height: Find the area of the polygon: 2. Imagine this polygon rotating 360 about the x-axis. a. What object does the rotation create? b. Draw a model of the object below. Be sure to label the known dimensions of the object. 3. How are the base and height of the polygon in question 1 related to the dimensions of the object created in question 2? 4. Using the original polygon from question 1 above, imagine the polygon rotating 360 about the y-axis. a. What object does it create? b. Draw a model of the object. Be sure to label the dimensions of the object. c. Explain why the base and height of the polygon represent different dimensions for this object than the object created in question 2. Blackline Masters, Geometry Page 7-1

96 Unit 7, Activity 5, Turning a Plane Figure Into a Solid Figure 5. Make a conjecture: Look at the objects created by the rotations: a. Which of the objects do you think has the largest surface area? Explain your reasoning. b. Which object do you predict will have the greatest volume? Explain your reasoning. c. Will the object with the largest surface area also be the same one that has the largest volume? Explain your reasoning. 6. Calculate the following (show your work below the chart): Object created in Question 2 Object created in Question 4 Surface Area: Surface Area: Volume: Volume: 7. Compare the objects and verify your conjecture: a. Which object actually has the greatest surface area? b. Which object actually has the greatest volume? c. Were your conjectures correct? Explain. d. Explain which measurement determined the greatest volume. e. Do you think this would always be true? Explain your reasoning. Blackline Masters, Geometry Page 7-2

97 Unit 7, Activity 5, Turning a Plane Figure Into a Solid Figure with Answers Group Members: Date: 1. On a piece of graph paper, graph the coordinates: A( 0,0 ), B( 2,0 ), C( 2,5 ), and D ( 5,0). a. Connect the points to create line segments AB, BC, CD, and AD. b. Shade the area created by the segments. c. What polygon is created? Rectangle d. Identify the dimensions of the polygon: Base: _2 units Height: _5 units Find the area of the polygon: _10 square units 2. Imagine this polygon rotating 360 about the x-axis. a. What object does the rotation create? _Cylinder b. Draw a model of the object below. Be sure to label the known dimensions of the object. Radius: 5 units Height: 2 units 3. How are the base and height of the polygon in question 1 related to the dimensions of the object created in question 2? The height of the rectangle becomes the radius of the cylinder, and the base of the rectangle becomes the height of the cylinder. 4. Using the original polygon from question 1 above, imagine the polygon rotating 360 about the y-axis. a. What object does it create? _Cylinder b. Draw a model of the object. Be sure to label the dimensions of the object. Radius: 2 units Height: 5 units c. Explain why the base and height of the polygon represent different dimensions for this object than the object created in question 2. Since the rectangle was rotated about a different axis, the dimensions of the cylinder will change. The height of the rectangle will be the height of the cylinder, and the base of the rectangle will be the radius of the cylinder. Blackline Masters, Geometry Page 7-3

98 Unit 7, Activity 5, Turning a Plane Figure Into a Solid Figure with Answers 5. Make a conjecture: Look at the objects created by the rotations: a. Which of the objects do you think has the largest surface area? Explain your reasoning. Student responses may vary; look for logical reasoning and explanations. Students may refer to activity 1 with the experiments they have already conducted. b. Which object do you predict will have the greatest volume? Explain your reasoning. Student responses may vary; look for logical reasoning and explanations. Students may refer to activity 1 with the experiments they have already conducted. c. Will the object with the largest surface area also be the same one that has the largest volume? Explain your reasoning. Student responses may vary; look for logical reasoning and explanations. Students may refer to activity 1 with the experiments they have already conducted. 6. Calculate the following (show your work below the chart): Object created in Question 2 Object created in Question 4 Surface Area: 70π square units Surface Area: 28π square units Volume: 50π square units Volume: 20π square units 7. Compare the objects and verify your conjecture: a. Which object actually has the greatest surface area? The cylinder created in question 2; radius of 5 units and height of 2 units. b. Which object actually has the greatest volume? The cylinder created in question 2; radius of 5 units and height of 2 units. c. Were your conjectures correct? Explain. Answers will vary; see students explanations. less. d. Explain which measurement determined the greatest volume. The radius determines the greater volume. The height of the cylinder in question 4 is greater than the height of the cylinder in question 2; however, the volume is Therefore, the greater the radius the greater the volume. e. Do you think this would always be true? Explain your reasoning. Answers will vary; see students reasoning. Overall, students should see that this will always be true because the radius is being squared which increases the volume exponentially. Blackline Masters, Geometry Page 7-4

99 Unit 7, Activity 9, Population Density Group Members: Date: 1. Record the classroom dimensions and population below. Then calculate the area and amount of classroom space per person. Be sure to state the units you are using. Length: Width: Area: Population: people in the classroom How much space does each person have? 2. Prediction: How much space would each person have if the number of people in the class doubled? 3. Calculate the population density. Population density: 4. Calculate the population density for the following countries in people per square mile. Country Land Area Density Population name (sq. miles) (people per sq. mile) Australia 22,421,417 2,967,908 Bangladesh 164,425,000 55,599 Canada 34,207,000 3,851,808 China 1,339,190,000 3,705,405 India 1,184,639,000 1,269,345 Japan 127,380, ,883 Liechtenstein 35, Monaco 33, Mongolia 2,768, ,250 USA 309,975,000 3,717,811 Source: Blackline Masters, Geometry Page 7-5

100 Unit 7, Activity 9, Population Density 5. In question 4, you calculated the population density of the USA to be approximately 83.4 people per square mile. Now calculate the population density of the following cities and answer the question that follows the chart. City Name Population Land Area Density (sq. mile) (people per sq.mile) Chicago, IL 2,784, Dallas, TX 1,007, Jacksonville, FL 635, Los Angeles, CA 3,485, New York, NY 7,323, Philadelphia, PA 1,586, Phoenix, AZ 983, Source: a. How can someone be justified saying that the USA has a population density of 83.4 people per square mile when the city of New York has a population density of 23,699 people per square mile? Blackline Masters, Geometry Page 7-6

101 Unit 7, Activity 9, Population Density with Answers Group Members: Date: 1. Record the classroom dimensions and population below. Then calculate the area and amount of classroom space per person. Be sure to state the units you are using. All answers for this question will vary depending on the classroom and the number of students in the classroom. Students should use appropriate units (meters or feet/yards) for the length and width, square units for the area, and square units per person for the space per person. Length: Width: Area: Population: people in the classroom How much space does each person have? 2. Prediction: How much space would each person have if the number of people in the class doubled? Answers will vary, but students should realize the amount of space per person in question one will be cut in half. 3. Calculate the population density. Population density: Answers will vary based on classrooms and population; units should be people per square meter. 4. Calculate the population density for the following countries in people per square mile. Country Land Area Density Population name (sq. miles) (people per sq. mile) Australia 22,421,417 2,967, people per sq. mile Bangladesh 164,425,000 55, people per sq. mile Canada 34,207,000 3,851, people per sq. mile China 1,339,190,000 3,705, people per sq. mile India 1,184,639,000 1,269, people per sq. mile Japan 127,380, , people per sq. mile Liechtenstein 35, people per sq. mile Monaco 33, ,857.1 people per sq. mile Mongolia 2,768, , people per sq. mile USA 309,975,000 3,717, people per sq. mile Source: Blackline Masters, Geometry Page 7-7

102 Unit 7, Activity 9, Population Density with Answers 5. In question 4, you calculated the population density of the USA to be approximately 83.4 people per square mile. Now calculate the population density of the following cities and answer the question that follows the chart. City Name Population Land Area Density (sq. mile) (people per sq.mile) Chicago, IL 2,784, ,264 people per sq. mile Dallas, TX 1,007, ,944 people per sq.mile Jacksonville, FL 635, people per sq. mile Los Angeles, CA 3,485, ,431 people per sq. mile New York, NY 7,323, ,699 people per sq. mile Philadelphia, PA 1,586, ,748 people per sq. mile Phoenix, AZ 983, ,340 people per sq. mile Source: b. How can someone be justified saying that the USA has a population density of 83.4 people per square mile when the city of New York has a population density of 23,699 people per square mile? Answers will vary, but students should understand that the population density of the USA is based on the land area of all of the USA and the city population density is based on a much smaller land area. Also, students should be able to understand that larger cities are more densely populated than rural areas. Blackline Masters, Geometry Page 7-8

103 Unit 8, Activity 1, Vocabulary Self-Awareness Word/Phrase + Definition/Formula Example circle center of a circle radius circumference chord area of a circle central angle arc arc measure arc length major arc minor arc semicircle distance around a circular arc sector area of a sector Blackline Masters, Geometry Page 8-1

104 Unit 8, Activity 1, Vocabulary Self-Awareness tangent secant sphere surface area of a sphere volume of a sphere Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 8-2

105 Unit 8, Activity 1, Vocabulary Self-Awareness with Answers Word/Phrase + Definition/Formula Example circle center of a circle radius circumference chord The set of all points in a plane equidistant from a given fixed point called the center. The given point from which all points on the circle are the same distance. a segment with one endpoint at the center of the circle and the other endpoint on the circle; onehalf the diameter the distance around the circle a segment whose endpoints lie on the circle 2 area of a circle A= π r central angle arc arc measure arc length major arc minor arc semicircle an angle formed at the center of a circle by two radii a segment of a circle equal to the degree measure of the central angle; arc measure arc length = 360 circumference the distance along the curved line making up the arc; arc measure arc length = 360 circumference also known as the distance around a circular arc. the longest arc connecting two points on a circle; an arc having a measure greater than 180 degrees the shortest arc connecting two points on a circle; an arc having a measure less than 180 degrees an arc having a measure of 180 degrees and a length of one-half of the Blackline Masters, Geometry Page 8-3

106 Unit 8, Activity 1, Vocabulary Self-Awareness with Answers distance around a circular arc sector area of a sector tangent secant sphere circumference; the diameter of a circle creates two semicircles also known as the arc length; see the definition of arc length. a plane figure bounded by two radii and the included arc of the circle N 2 A= π r where N is 360 the measure of the central angle a line or segment which intersects the circle at exactly one point a line or segment which intersects the circle at exactly two points the locus of all points, in space, that are a given distance from a given point called the center surface area of a sphere 2 SA = 4π r volume of a sphere V 4 = π r 3 3 Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. Blackline Masters, Geometry Page 8-4

107 Unit 8, Activity 1, Vocabulary Self-Awareness with Answers This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 8-5

108 Unit 8, Activity 4, Sample Split-Page Notes Date: Period: Topic: Circles Parts of a circle: radius chord diameter Formulas: area of a circle circumference --one-half the diameter --one endpoint is the center of the circle, the other is on the circle --used when finding the area of a circle --a segment whose endpoints are on the circle --a chord which passes through the center of the circle 2 -- A= π r --r is the measure of the radius of the circle -- C = 2 πr or C = πd --r is the measure of the radius and d is the measure of the diameter --these formulas are the same because d = 2r. Blackline Masters, Geometry Page 8-6

109 Unit 8, Activity 4, Split-Page Notes Model Date: Period: central angle arc minor arc major arc semicircle Topic: Central Angles and Arcs --an angle whose vertex is the center of the circle and sides are two radii --the sum of all central angles in a circle is a segment of a circle --created by a central angle or an inscribed angle --has a degree measure (called arc measure) --has a linear measure (called arc length) --an arc whose measure is less than 180 degrees --an arc whose measure is greater than 180 degrees --an arc whose measure is exactly 180 degrees --created by the diameter of the circle --the arc length is one-half of the circumference of the circle Blackline Masters, Geometry Page 8-7

110 Unit 8, Activity 5, Circular Flower Bed Consider the diagram of the flower bed below: What is the total area this flower bed would cover in the owner s yard? If the walking paths around the inner circle and the crescent shaped flower beds are to be covered in straw, pebbles, or some other medium, how much material would be needed to cover that area? The owner wishes to put edging around each section of the flower bed. How much edging will be needed? Picture source: Blackline Masters, Geometry Page 8-8

111 Unit 8, Activity 5, Circular Flower Bed with Answers Consider the diagram of the flower bed below: What is the total area this flower bed would cover in the owner s yard? Approx sq ft. If the walking paths around the inner circle and the crescent shaped flower beds are to be covered in straw, pebbles, or some other medium, how much material would be needed to cover that area? Answers provided for this question and the next are samples as students will need to make some assumptions in order to complete calculations (for example, they might approximate the area between crescent shaped flower beds as a rectangle of dimensions 1.5 by 3.25). The intention here is to have students explain their reasoning and persevere in solving the problem. Teacher facilitation to assist students in solving the problem will be necessary. Sample answer: Approx sq ft. However, this type of material is typically sold in cubic yards, so assuming 1 in depth (1/36 th of a yard), approximately 0.48 cubic yards would be needed. The owner wishes to put edging around each section of the flower bed. How much edging will be needed? Sample Answer: Approximately feet of edging would be needed. Picture source: Blackline Masters, Geometry Page 8-9