Geometry - Richland Parish School Board › documents › common...Unit 1, Activity 1, Extending Number and Picture Patterns. Blackline Masters, Geometry Page 1-1 . Date _____ Name

Unit 1, Activity 1, Extending Number and Picture Patterns

Blackline Masters, Geometry Page 1-1

Geometry

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)

Unit 1, Activity 1, Extending Number and Picture Patterns with Answers


Date ___________ Name ________________________

Extending Patterns and Sequences When presented with a sequence and asked to find the next term, inductive reasoning is applied. Analyze the specific examples provided, determine a pattern, and then find the missing term. Making a prediction about missing terms is called making a conjecture. Examples: For each of the following, write the next two terms and describe the pattern. 1) 2, 4, 6, 8, 10, … __12_, _14__ 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, The student should draw two then an unshaded square. 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.

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) 3 1n −

4) x 1 2 3 4 5

y 18 15 12 9 ?

5) 2 1n −

6) 15, 10, 6, 3, 1,...− − − − −

7) x 1 2 3 4 5

y 100 50 25 12.5 ?

8) 4

2n +

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) 3 1n −

4) x 1 2 3 4 5

y 18 15 12 9 ?

5) 2 1n −

6) 15, 10, 6, 3, 1,...− − − − −

7) x 1 2 3 4 5

y 100 50 25 12.5 ?

8) 4

2n +

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) 2) 3) 4) 5)

Term n 1 2 3 4 5 6 7 8 Value n-3 -2 -1 0

Term n 1 2 3 4 5 6 7 8 Value 2n+3 5 7 9

Term n 1 2 3 4 5 6 7 8 Value 3n+1 4 7 10

Term n 1 2 3 4 5 6 7 8 Value n2 1 4 9

Term n 1 2 3 4 5 6 7 8 Value n3 1 8 27

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, …

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) Difference between the terms is 1 2) Difference between the terms is 2 3) Difference between the terms is 3 4) There is no common difference between terms 5) There is no common difference between terms Questions to answer:

Term n 1 2 3 4 5 6 7 8 Value n-3 -2 -1 0 1 2 3 4 5

Term n 1 2 3 4 5 6 7 8 Value 2n+3 5 7 9 11 13 15 17 19

Term n 1 2 3 4 5 6 7 8 Value 3n+1 4 7 10 13 16 19 22 25

Term n 1 2 3 4 5 6 7 8 Value n2 1 4 9 16 25 36 49 64

Term n 1 2 3 4 5 6 7 8 Value n3 1 8 27 64 125 216 343 512

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

Unit 1, Activity 2, Generating the nth Term for Picture Patterns


Date ___________ Name ________________________

Directions: Find the indicated term for each of the patterns below. 1)

How many sides will the 15th term have?

2)

What will the 23rd figure look like?

3)

What is the 50th term of the sequence above?

4)

What is the 103rd term of the sequence?

Unit 1, Activity 2, Generating the nth Term for Picture Patterns with Answers


Date ___________ Name ________________________

Directions: Find the indicated term for each of the patterns below. 1)

How many sides will the 15th term have?

Solution: n + 2; 17 sides Add two to the figure number, to determine the number of sides. For example, the 3rd figure has 5 sides.

2)

What will the 23rd figure look like?

Solution: Since the pattern repeats after four figures, students should realize that every term that is a multiple of four will look like the fourth figure. The nearest multiple to 23 is 20; the students should then continue the pattern—it is the 3rd figure.

3)

What is the 50th term of the sequence above?

Solution: The shapes repeat after 3 terms so 48 is the closest multiple of 3 to 50, so the shape is a square. The square is not shaded because the even terms are not shaded.

4)

What is the 103rd term of the sequence?

Solution: The pattern repeats after five terms. The 100th term is the fifth figure, so the 103rd term is the third figure.

Unit 1, Activity 3, Square Figurate Numbers


Date _____________ Name ________________________

Square Numbers Consider the following sequence:

1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a square?

Unit 1, Activity 3, Square Figurate Numbers with Answers


Date _____________ Name ________________________

Square Numbers Consider the following sequence:

1) What is the number pattern? 1, 4, 9, 16, 25 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term? Formula: 2n ; the 25th term is 625. 4) How does each number relate to the area of a square? The area of a square is 2s where s is the measure of the side. In each of the squares, the measure of the sides are the same, and they increase by one each time. Therefore the area is 22, 32, 42, … 2n .

Unit 1, Activity 3, Rectangular Figurate Numbers


Date _____________ Name ________________________

Rectangular Numbers Consider the following:

1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a rectangle?

Unit 1, Activity 3, Rectangular Figurate Numbers with Answers


Date _____________ Name ________________________

Rectangular Numbers Consider the following:

1) What is the number pattern? 2, 6, 12, 20, 30 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term? Formula: 2n n+ or ( )1n n + ; the 25th term is 650. 4) How does each number relate to the area of a rectangle? Each rectangle has a height the same as the figure number and a base which is one greater than the height; therefore, the number of dots needed for any figure is the same as the area of the rectangle, n(n+1), where n is the height and the base is one more than the height.

Unit 1, Activity 3, Triangular Figurate Numbers


Date _____________ Name ________________________

Triangular Numbers Consider the following:

1) What is the number pattern? 2) Is it linear? Why? 3) What is the formula to find the nth term in this set? What would be the 25th term? 4) How does each number relate to the area of a triangle?

Unit 1, Activity 3, Triangular Figurate Numbers


Date _____________ Name ________________________

Triangular Numbers Consider the following:

1) What is the number pattern? 1, 3, 6, 10, 15 2) Is it linear? Why? It is not linear because the difference between consecutive terms is not constant. 3) What is the formula to find the nth term in this set? What would be the 25th term?

Formula: ( )221

or or 0.5 0.52 2

n nn n n n++

+ ; the 25th term is 325.

4) How does each number relate to the area of a triangle?

The area of a triangle is half the area of a rectangle, 12

A bh= , so if we take the formula

for rectangular numbers, we can divide it by 2 to get the area of a triangle with the same base as its corresponding rectangle.

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 2 52x− =

Prove: 15x =

Statements Reasons

( )4 2 52x− = Given

4 8 52x− = Distributive Property

4 8 8 52 8x− + = + Addition Property

4 60x = Simplification

4 604 4x = Division Property

15x = Simplification



Proof #2 Given: ( ) ( )3 2 4 3 6 123a a− − + =

Prove: 9x =−

Statements Reasons

( ) ( )3 2 4 3 6 123a a− − + = Given

6 12 3 18 123a a− − − = Distributive Property

12 15 123a− − = Simplification

12 15 12 123 12a− − + = + Addition Property

15 135a− = Simplification

15 13515 15

a− =− − Division Property

9x =− Simplification



Proof #3

Given: 2 6 25x+ =

Prove: 2x =

Statements Reasons

2 6 25x+ = Given

( )2 65 5 25x

+ = Multiplication Property

2 6 10x+ = Simplification

2 6 6 10 6x+ − = − Subtraction Property

2 4x = Simplification

2 42 2x = Division Property

2x = Simplification



Proof #4

Given: 1 74 2 2a a− = −

Prove: 1a =−

Statements Reasons

1 74 2 2a a− = − Given

1 72 4 22 2a a

− = − Multiplication Property

8 7 2a a− = − Simplification

8 8 7 2 8a a− − = − − Subtraction Property

2 1a a− =− − Simplification

2 2 1 2a a a a− + =− − + Addition Property

1a =− Simplification

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. B. C.

8

15

x

b

19

23

d

7

7



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.

8

6

4

2

-2

-4

-6

-5 5

F

ED

BC

A

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|.)



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 a2+b2. (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.

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 .

M

N P



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.

10

8

6

4

2

-2

-4

-6

-8

-10

10 -5 5 10

T

S

G

F

D

C

Formula Investigation: Would the formula ( ) ( )2 21 2 1 2d x x y y= − + − also work? What about

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 21 2 2 1 2 1 1 2 2 1 2 1, , or d x x y y d x x y y d y y x x= − + − = − + − = − + − ? Why or

why not? Explain your reasoning.



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.

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?

Saints’ Sideline

Goal Line

1 0 2 0 3 0 4 0 5 0 4 0 3 0 2 0 1 0

1 0 1 0 2 0 2 0 3 0 3 0 4 0 4 0 5 0

SAIN

TS SAIN

TS

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. x = 17 B. C.

8

15

x

b

19

23

d

7

7



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.

8

6

4

2

-2

-4

-6

-5 5

F

ED

BC

A

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 .



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 a2+b2. (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.

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 .

M

N P



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.

Formula Investigation: Would the formula ( ) ( )2 21 2 1 2d x x y y= − + − also work? What about

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 21 2 2 1 2 1 1 2 2 1 2 1, , 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.

10

8

6

4

2

-2

-4

-6

-8

-10

10 -5 5 10

T

S

G

F

D

C



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.

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.

Saints’ Sideline

Goal Line

1 0 2 0 3 0 4 0 5 0 4 0 3 0 2 0 1 0

1 0 1 0 2 0 2 0 3 0 3 0 4 0 4 0 5 0

SAIN

TS SAIN

TS

Unit 3, Activity 4, Dividing Number Line Segments


Name: _______________________ Date: _______________________

| | | | | | | | | | | | |2 1 0 1 2 3 4 5 6 7 8 9 10− −

←→

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.

| | | | | | | | | | |5 4 3 2 1 0 1 2 3 4 5− − − − −

←→

5. 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 .

A B

Unit 3, Activity 4, Dividing Number Line Segments with Answers


Name: _______________________ Date: _______________________

| | | | | | | | | | | | |2 1 0 1 2 3 4 5 6 7 8 9 10− −

←→

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 10 454 1 3

=

+ − = D is located at coordinate 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.

| | | | | | | | | | |5 4 3 2 1 0 1 2 3 4 5− − − − −

←→

5. 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 -3.

7. 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 .

( )1

1 2

k C A Ak k

− ++

A B

Unit 3, Activity 7, Parallel Line Facts


Date _________ Name ________________________

Use the given diagram to complete the steps and answer the questions.

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

180m BAC m ABC m ACB∠ + ∠ + ∠ = . 3. 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∠ = ∠ + ∠ .

Unit 3, Activity 7, Parallel Line Facts


4. Remember, the area of a triangle can be written as 12

A bh= . Use the diagram below to

answer the following questions.

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:

, ' , , and ABC AB C ADC 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.

Unit 3, Activity 7, Parallel Line Facts with Answers


Date _________ Name ________________________

Use the given diagram to complete the steps and answer the questions.

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

180m BAC m ABC m ACB∠ + ∠ + ∠ = . Given that BR AC

, we know that alternate interior angles are congruent. So,

and BAC ABL ACB CBR∠ ≅ ∠ ∠ ≅ ∠ . By definition of congruence, and m BAC m ABL m ACB m CBR∠ = ∠ ∠ = ∠ . Because , , and ABL ABC CBR∠ ∠ ∠ are

adjacent and form a line, 180m ABL m ABC m CBR∠ + ∠ + ∠ = . Using the substitution property of equality, we now have 180m BAC m ABC m ACB∠ + ∠ + ∠ = .

3. 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∠ = ∠ + ∠ .

Unit 3, Activity 7, Parallel Line Facts with Answers


4. Remember, the area of a triangle can be written as 12

A bh= . Use the diagram below to

answer the following questions.

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:

, ' , , and ABC AB C ADC 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.

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) Photos are of excellent quality; clear; description is written clearly

Photos/ pictures are of good quality; description is clear but missing some elements

Photos/pictures are grainy; term is not clearly depicted in picture; description is vague

Photos/pictures do not depict term at all; description only gives definition

No description given

______ x 6

Title Page (8) Excellent quality; typed; includes project title, date, and class period

Typed; missing date or class period

Handwritten with all information; or typed and missing date and class period

Handwritten and missing date and class period; missing title (typed with all other info)

No title page

______ x 2

Reflection (12) Typed; tells what the student learned from project; grammatically correct

Typed; 1-2 grammar errors; some evidence of learning

Handwritten; 3-4 grammar errors; vague evidence of learning

Handwritten; 5-6 grammar errors; little to no evidence of learning

No reflection

______ x 3

Neatness/ Creativity (8)

Typed; clean; neatly bound pages; original project title; attractive; etc.

Typed; project name not original; some pages loose

Handwritten; project is less than attractive;

Dirty, crumpled pages; if handwritten there are scratchouts or places with liquid paper

Pages are not bound

______ x 2

Timeliness (8) Turned in on time

Turned in one day late

Turned in two days late

Turned in three days late

Turned in more than three days late

______ x 2

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 slope-intercept 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.

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: 23

m = ; Graph B: 72

m = − ; Graph C: 25

m = ;

Graph D: 52

m = ; Graph E: 13

m = −

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 slope-intercept form of the equation for the perpendicular line and label the line with this equation. Show your calculations below.

Graph A: 32

m = − ; Graph B: 27

m = ; Graph C: 52

m = − ;

Graph D: 25

m = − ; Graph E: 3m =

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.

Graph A: 23

m = ; Graph B: 72

m = − ; Graph C: 25

m = ;

Graph D: 52

m = ; Graph E: 13

m = −

Unit 3, Activity 1, Specific Assessment, What’s My Line? Graph A


Unit 3, Activity 1, Specific Assessment, What’s My Line? Graph B


Unit 3, Activity 1, Specific Assessment, What’s My Line? Graph C


Unit 3, Activity 1, Specific Assessment, What’s My Line? Graph D


Unit 3, Activity 1, Specific Assessment, What’s My Line? Graph E


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. Slope of the line

A. 2 points labeled with (x,y) coordinates 2 B. Slope calculated correctly 6 C. Labeled line with slope 1 III. Equation of the line

A. Calculated correctly 6 B. Labeled on graph 1 IV. 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. 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 VII. Following directions

A. Cover sheet 3 B. Stapled 3 C. Submitted on or before due date 3

Total

60

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



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.

Unit 4, Activity 1, Vocabulary Self-Awareness with Answers


Word/Phrase + – Definition/Rule Example

transformation

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.

pre-image The original object that is to be transformed.

image The “copy” of the object that has been transformed.

rigid transformation (rigid motion)

A transformation that preserves measurements of segments and angles; also called an isometry (see below).

non-rigid transformation (non-rigid motion)

A transformation that does not preserve measures of segments and angles; the shape of the pre-image may not be preserved either.

orientation

The location (position and angle) of an object in space in relation to a set of reference axes.

isometry

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.

reflection

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.”

line of reflection The perpendicular bisector of the segment joining each point (pre-image) and its image.



translation

A transformation which moves an object a fixed distance in a fixed direction; a composite of two reflections over parallel lines; aka slide.

rotation

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.”

center of rotation

A fixed point about which a figure is rotated; the point where two intersecting lines of rotation meet??

angle (degree) of rotation

Rays drawn from the center of rotation to a point on the pre-image and its image form the angle of rotation (measured in degrees).

clockwise Rotation of an object to the right indicated by a negative angle of rotation.

counterclockwise Rotation of an object to the left indicated by a positive angle of rotation.

dilation

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.

center of dilation

A fixed point in the plane about which all points are expanded (stretched) or contracted (shrunk).

scale factor

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.



similarity transformation

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.

composite transformation The result of two or more

successive transformations.

glide reflection 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




Unit 4, Activity 3, A Basic Look at Transformations


Trace the following polygons on the patty paper or tracing paper given to you.

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 trans- means “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 image image pre-image Figure 1 Figure 2 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 pre-image 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). 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

D

C

A

B

A’

B’ C’

D’

m

P



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 Figure 4 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.

Figure 5

m n

X



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.

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. Figure 7

m

n

pre-image

image

A

B B’

C C’

D

D’

E

E’



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.

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.

1. Reflect FG . What is true about ' 'F G ? Explain your reasoning.

2. Reflect GH . What is true about ' 'G H ? 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.

1. Reflect and MN MO . What is true about ' ', ' ', and ' ' 'M N M O N M O∠ ? Justify your answer.

2. Connect N’ and O’. Is ' ' 'MNO M N O≅ ? Justify your answer.

F

G

H

m

M

N

O

s

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. 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,

and XZ 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, and YX 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? 5. Now, starting with the patty paper with ' 'X Y , lay the other two pieces of patty paper on top

of the first one lining up ' 'X Y on each piece of paper. What happens? What is true about and ' ' 'XYZ X Y Z ? How can you verify your conjecture?

X

Z

Y

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.

1. Reflect FG . What is true about ' 'F G ? Explain your reasoning. The segments are congruent. Reflection does not change length.

2. Reflect GH . What is true about ' 'G H ? 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.

1. Reflect and MN MO . What is true about ' ', ' ', and ' ' 'M N M O 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. ' 'N O is also a reflection so all corresponding sides and angles are congruent.

F

G

H

m

M

N

O

s

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. 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,

and XZ 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, and YX 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 ' 'X Y , lay the other two pieces of patty paper on top

of the first one lining up ' 'X Y on each piece of paper. What happens? What is true about and ' ' 'XYZ 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 ' ' 'X Y Z . These two triangles are congruent. This can be verified either by measuring all corresponding sides and angles or by laying ' ' 'X Y Z over XYZ to see that the sides and angle have the same size.

X

Z

Y

Unit 5, Activity 2, Sample Split-Page Notes


Date: Period:

Topic: Triangles

Triangle

vertex sides

Classifications by Angle: Acute Obtuse Right Equiangular

Classifications by Sides: Scalene Isosceles Equilateral

Triangle Sum Theorem Exterior Angle Theorem

--a closed figure with three segments joining three non-collinear 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: Name: ABC Sides: , , and AB BC AC Vertices: A, B, and C --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 180°. --The measure of an exterior angle of a triangle is equal to the

sum of the two non-adjacent interior angles. m ABD m A m C∠ = ∠ + ∠

B C

A

A

C B D

Unit 5, Activity 2, Blank Split-Page Notes


Date: Period:

Topic:

Unit 5, Activity 2, Triangle Split-Page Notes Model


Date: Period:

Topic: Congruent Triangles

CPCTC Side-Side-Side (SSS) Postulate Side-Angle-Side (SAS) Postulate Angle-Side-Angle (ASA) Postulate

Corresponding parts of congruent triangles are congruent (if

two triangles are congruent, then all pairs of corresponding parts are also congruent).

ABC ITR≅ A IB TC R

∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

AB IT

BC TR

AC IR

≅

≅

≅

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

≅

≅

≅

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

≅∠ ≅ ∠

≅

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 EFC F

∠ ≅ ∠

≅∠ ≅ ∠

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≅ Write your proof below. Be sure to include all logical reasoning.

X

D

C

A

B





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 Prove: DA BC≅ Write your proof below. Be sure to include all logical reasoning.

X

D

C

A

B





Using the given diagram and information, prove two of the angles congruent. Given: MA TH ; HM TA Prove: H A∠ ≅ ∠ Write your proof below. Be sure to include all logical reasoning.

M

H T

A





Using the given diagram and information, prove two of the sides congruent. Given: MA TH ; MA TH≅ Prove: MH AT≅ Write your proof below. Be sure to include all logical reasoning.

M

H T

A





Using the given diagram and information, prove two of the triangles congruent. Given: X is the midpoint of KT IE KT⊥ Prove: KXE TXE≅ Write your proof below. Be sure to include all logical reasoning.

K

E

T

I

X





Using the given diagram and information, prove two of the sides congruent. Given: X is the midpoint of KT IE KT⊥ Prove: KI TI≅ Write your proof below. Be sure to include all logical reasoning.

K

E

T

I

X

Unit 5, Activity 4, Proving Triangles Congruent with Answers




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≅ Write your proof below. Be sure to include all logical reasoning.

Statements Reasons 1. X is the midpoint of BD ; X is the midpoint of AC .

1. Given

2. ; XD XB XA XC≅ ≅ 2. Midpoint Theorem 3. DXC BXA∠ ≅ ∠ 3. Vertical angles are congruent. 4. DXC BXA≅ 4. SAS Postulate

X

D

C

A

B





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 Prove: DA BC≅ Write your proof below. Be sure to include all logical reasoning.

Statements Reasons 1. X is the midpoint of BD ; X is the midpoint of AC .

1. Given

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

X

D

C

A

B





Using the given diagram and information, prove two of the angles triangles congruent. Given: MA TH ; HM TA Prove: H A∠ ≅ ∠ 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

M

H T

A





Using the given diagram and information, prove two of the sides triangles congruent. Given: MA TH ; MA TH≅ Prove: MH AT≅ 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

M

H T

A





Using the given diagram and information, prove two of the triangles congruent. Given: X is the midpoint of KT IE KT⊥ Prove: KXE TXE≅ 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. 90; 90m KXE m TXE∠ = ∠ = 3. 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

K

E

T

I

X





Using the given diagram and information, prove two of the sides triangles congruent. Given: X is the midpoint of KT IE KT⊥ Prove: KI TI≅ 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. 90; 90m KXI m TXI∠ = ∠ = 3. 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

K

E

T

I

X

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 1. 1. 2. 2. 3. 3. 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 1. 1. 2. 2. 3. 3. 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 1. 1. 2. 2. 3. 3. List the names of the angles and sides in order from largest to smallest on the lines below: Angles _________ _________ _________ Sides _________ _________ _________

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?

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∠ > ∠ Proof: Place P on side AB such that PB BC≅ and draw PC . 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.

A B

C

A B

C

P

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?

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 Kite Square Quadrilateral Trapezoid Rectangle Isosceles Trapezoid 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.

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 Kite Square Quadrilateral Trapezoid Rectangle Isosceles Trapezoid 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.

Quadrilateral

Trapezoid

Isosceles Trapezoid

Kite

Parallelogram

Rectangle Rhombus

Square

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). 2 scalene triangles congruent by SSS AB DB≅ AC DC≅ BC BC≅ ABC DBC≅ 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:

A

B D

C



Part I

Obtuse Triangles (ASA)

≅ ≅ ≅ ≅

Scalene Triangles (SSS)

≅ ≅ ≅ ≅

Isosceles Right Triangles (SAS)

≅ ≅ ≅ ≅

Acute Triangles (AAS)

≅

≅ ≅ ≅

Equilateral Triangle



Part II

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 =



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.



Answer Key

Parallelograms – 3, 6, 7, 9

Rhombii – 1, 4, 8 Squares Rectangles – 2, 5

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.

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 . . . . . . . . . . . .

3.) 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 . . . . . . . . . . . .

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 . . . . . . . . . . .

3.) 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 n2:1. 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 n2 units n:1 n3:1 . .

Unit 6, Activity 4, Spotlight on Similarity


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:

Unit 6, Activity 3, Specific Assessment, Making a Hypsometer


Format: Individual or Small Group Objectives: 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.

Materials: 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.

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 BCDF EF

=

Reference: NCTM Addenda Series, Measurement in the Middle Grades

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: ( ) ( ) ( ) ( )0,0 , 2,0 , 2,5 , and 0,5A B C D .

a. Connect the points to create line segments , , , and AB BC CD 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.

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.

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: ( ) ( ) ( ) ( )0,0 , 2,0 , 2,5 , and 5,0A B C D .

a. Connect the points to create line segments , , , and AB BC CD 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.

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.

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 less. 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.

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 name Population Land Area

(sq. miles) Density

(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 127,380,000 145,883 Liechtenstein 35,904 62 Monaco 33,000 0.77 Mongolia 2,768,800 604,250 USA 309,975,000 3,717,811

Source: http://www.worldatlas.com/aatlas/populations/ctypopls.htm

http://www.worldatlas.com/aatlas/populations/ctypopls.htm�

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 (sq. mile)

Density (people per sq.mile)

Chicago, IL 2,784,000 227 Dallas, TX 1,007,000 342 Jacksonville, FL 635,000 759 Los Angeles, CA 3,485,000 469 New York, NY 7,323,000 309 Philadelphia, PA 1,586,000 135 Phoenix, AZ 983,000 420 Source: http://www.census.gov/population/www/documentation/twps0027/twps0027.html

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?

http://www.census.gov/population/www/documentation/twps0027/twps0027.html�

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 name Population Land Area

(sq. miles) Density

(people per sq. mile) Australia 22,421,417 2,967,908 7.6 people per sq. mile Bangladesh 164,425,000 55,599 2957.3 people per sq. mile Canada 34,207,000 3,851,808 8.9 people per sq. mile China 1,339,190,000 3,705,405 361.4 people per sq. mile India 1,184,639,000 1,269,345 933.3 people per sq. mile Japan 127,380,000 145,883 873.2 people per sq. mile Liechtenstein 35,904 62 579.1 people per sq. mile Monaco 33,000 0.77 42,857.1 people per sq. mile Mongolia 2,768,800 604,250 4.6 people per sq. mile USA 309,975,000 3,717,811 83.4 people per sq. mile

Source: http://www.worldatlas.com/aatlas/populations/ctypopls.htm

http://www.worldatlas.com/aatlas/populations/ctypopls.htm�

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 (sq. mile)

Density (people per sq.mile)

Chicago, IL 2,784,000 227 12,264 people per sq. mile Dallas, TX 1,007,000 342 2,944 people per sq.mile Jacksonville, FL 635,000 759 837 people per sq. mile Los Angeles, CA 3,485,000 469 7,431 people per sq. mile New York, NY 7,323,000 309 23,699 people per sq. mile Philadelphia, PA 1,586,000 135 11,748 people per sq. mile Phoenix, AZ 983,000 420 2,340 people per sq. mile Source: http://www.census.gov/population/www/documentation/twps0027/twps0027.html

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.

http://www.census.gov/population/www/documentation/twps0027/twps0027.html�



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



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






Word/Phrase + – Definition/Formula Example

circle

The set of all points in a plane equidistant from a given fixed point called the center.

center of a circle The given point from which all points on the circle are the same distance.

radius

a segment with one endpoint at the center of the circle and the other endpoint on the circle; one-half the diameter

circumference the distance around the circle

chord a segment whose endpoints lie on the circle

area of a circle 2A rπ=

central angle an angle formed at the center of a circle by two radii

arc a segment of a circle

arc measure

equal to the degree measure of the central angle; arc measure arc length

360 circumference=

°

arc length

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.

major arc

the longest arc connecting two points on a circle; an arc having a measure greater than 180 degrees

minor arc

the shortest arc connecting two points on a circle; an arc having a measure less than 180 degrees

semicircle an arc having a measure of 180 degrees and a length of one-half of the



circumference; the diameter of a circle creates two semicircles

distance around a circular arc

also known as the arc length; see the definition of arc length.

sector a plane figure bounded by two radii and the included arc of the circle

area of a sector 2

360NA rπ= where N is

the measure of the central angle

tangent a line or segment which intersects the circle at exactly one point

secant a line or segment which intersects the circle at exactly two points

sphere

the locus of all points, in space, that are a given distance from a given point called the center

surface area of a sphere 24SA rπ=

volume of a sphere 343

V rπ=

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






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 -- 2A rπ= --r is the measure of the radius of the circle -- 2 or C r C dπ π= = --r is the measure of the radius and d is the measure of the diameter --these formulas are the same because 2d r= .

Unit 8, Activity 4, Split-Page Notes Model


Date: Period:

Topic: Central Angles and Arcs

central angle arc minor arc major arc semicircle

--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 360° --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

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: http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html

http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html�

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. 490.87 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. 154.45 sq ft. However, this type of material is typically sold in cubic yards, so assuming 1 in depth (1/36th 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 208.47 feet of edging would be needed.

Picture source: http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html

http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html�

Unit 8, Activity 5, Arc Length and Sector Area Part I


Group Members: ______________________________________________________________

1. Which can did your group receive? ______________________

2. What is the circumference of your can in centimeters (round to the nearest millimeter)? _______________________

3. Determine the length of the diameter and the radius of the can (do not forget units).

Describe your method for determining these measures below. Diameter __________________ Radius __________________

4. In the space provided, use the compass to draw a circle with the radius and diameter you found in question 3. Then divide the circle into four equal parts. You may use a different sheet of paper if necessary to draw the circle.

Unit 8, Activity 5, Arc Length and Sector Area Part I


5. What is the measure of each central angle in the circle constructed in question 4? _________

6. Write the ratio of one central angle measure (from question 5) to the total number of

degrees at the center of the circle. _________ Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). _________

7. What is the length of the arc formed by one of the central angles mentioned in question 5

(remember to use the correct units)? ____________ Describe how you found this arc length.

8. Write a ratio that compares the arc length in question 7 to the total circumference of the circle. _________ Simplify this fraction and write the decimal equivalent (round to the

nearest hundredth). _________

9. What is the area of the circle you drew in question 4 (remember the units)? ___________

10. What is the area of one of the four sectors formed in question 4 (remember the units)? ____________ Describe how you found this area measure.

11. Write a ratio that compares the area of one sector in question 7 to the total area of the

circle. _________ Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). _________

12. What pattern do you see in questions 6, 8, and 11? Why does this pattern occur?

Unit 8, Activity 5, Arc Length and Sector Area Part I with Answers


Group Members: ______________________________________________________________

1. Which can did your group receive? Answers will vary

2. What is the circumference of your can in centimeters (round to the nearest millimeter)? Answers will vary

3. Determine the length of the diameter and the radius of the can (do not forget units).

Describe your method for determining these measures below. Diameter Answers will vary Radius Answers will vary

4. In the space provided, use the compass to draw a circle with the radius and diameter you found in question 3. Then divide the circle into four equal parts. You may use a different sheet of paper if necessary to draw the circle.

Unit 8, Activity 5, Arc Length and Sector Area Part I with Answers


5. What is the measure of each central angle in the circle constructed in question 4?

90 degrees

6. Write the ratio of one central angle measure (from question 5) to the total number of

degrees at the center of the circle. 90360

Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). 1 = 0.254

7. What is the length of the arc formed by one of the central angles mentioned in question 5

(remember to use the correct units)? Answers will vary Describe how you found this arc length.

Answers will vary. Some possible methods may include dividing the circumference by four or using a piece of string to measure the length then measuring the length of the string.

8. Write a ratio that compares the arc length in question 7 to the total circumference of the

circle. Answers will vary Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). Answers will vary however the decimal approximation should be 0.25.

9. What is the area of the circle you drew in question 4 (remember the units)? Answers will

vary

10. What is the area of one of the four sectors formed in question 4 (remember the units)?

Answers will vary Describe how you found this area measure. Answers will vary. One method will probably be to divide the total area by 4.

11. Write a ratio that compares the area of one sector in question 7 to the total area of the circle. Answers will vary Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). Answers will vary however the decimal approximation should be 0.25.

12. What pattern do you see in questions 6, 8, and 11? Why does this pattern occur?

The pattern should be that the ratios (specifically using the decimal approximations) should be equal to ¼ or 0.25. This happens because the total number of degrees (360) has been divided into four equal parts. Therefore, the sector area and arc length are each ¼ of the total area and circumference.

Unit 8, Activity 5, Arc Length and Sector Area Part II


1. In the space provided, use a compass to draw a circle with a radius of 3.5 centimeters. Divide the circle into 6 equal parts. Shade one of the six parts. The questions below with be about the shaded region.

2. State the circumference and area of the circle. Remember to use the correct units. Round

your answers to the nearest hundredth. 3. Using the shaded sector of the circle, find the measure of the central angle, the area of the

sector, and the arc length of the sector. Justify your answers with explanations or work. Remember to use the correct units. Round your answers to the nearest hundredth.

4. Describe a formula that might be used to find arc length. Use the appropriate vocabulary

(circumference, central angle, etc.) to explain what variables are used in the calculations. 5. Describe a formula that might be used to find the area of a sector. Again, use appropriate

terminology for the variables to be used in the calculations.

Unit 8, Activity 5, Arc Length and Sector Area Part II with Answers


1. In the space provided, use a compass to draw a circle with a radius of 3.5 centimeters. Divide the circle into 6 equal parts. Shade one of the six parts. The questions below with be about the shaded region.

2. State the circumference and area of the circle. Remember to use the correct units. Round

your answers to the nearest hundredth. Circumference = 21.99 cm Area = 38.48 cm2 3. Using the shaded sector of the circle, find the measure of the central angle, the area of the

sector, and the arc length of the sector. Justify your answers with explanations or work. Remember to use the correct units. Round your answers to the nearest hundredth.

Central Angle = 60 degrees Area of the sector = 6.41 cm2 Arc length = 3.67 cm 4. Describe a formula that might be used to find arc length. Use the appropriate vocabulary

(circumference, central angle, etc.) to explain what variables are used in the calculations.

NArc Length 2 r360

π=

N = the measure of the central angle; r = radius of the circle Students may not give this exact formula but should have some representation of the circumference of the circle and the ratio of the measure of the central angle to 360. 5. Describe a formula that might be used to find the area of a sector. Again, use appropriate

terminology for the variables to be used in the calculations.

2

360NArea of a sector rπ=

N = the measure of the central angle; r = radius of the circle Students may not give this exact formula but should have some representation of the total area of the circle and the ratio of the measure of the central angle to 360.

Unit 8, Activity 6, Concentric Circles


Unit 8, Activity 8, Anticipation Guide


Name _____________________ Date _____________________

Directions: Read each of the statements below. Circle “Agree” or “Disagree” under the appropriate column heading (Before Lesson or After Lesson). Be prepared to explain your reasoning for your choice.

Before Learning Statements After Learning

Agree Disagree 1. Categorical data are values which can be sorted by names or labels rather than numbers. Agree Disagree

Agree Disagree 2. Marginal frequencies and joint frequencies are terms that have the same definition. Agree Disagree

Agree Disagree 3. Relative frequencies are often stated as percentages. Agree Disagree

Agree Disagree 4. Two-way tables allow us to compare two or more sets of categorical data. Agree Disagree

Agree Disagree 5. Relative frequencies can be found for the whole table, just the rows, or just the columns. Agree Disagree

Agree Disagree

6. In order to find the conditional probability of event B given event A has already occurred, you must know the probability of event B and the probability of event A.

Agree Disagree

Agree Disagree

7. The probability of B given A has occurred, represented by P(B|A), is the same as the probability of A given B has occurred, or P(A|B).

Agree Disagree

Unit 8, Activity 8, Anticipation Guide with Answers


Name _____________________ Date _____________________

Directions: Read each of the statements below. Circle “Agree” or “Disagree” under the appropriate column heading (Before Lesson or After Lesson). Be prepared to explain your reasoning for your choice. “Correct” answers have been italicized. Be sure to have students justify their reasoning. It may be possible that students have a valid reason for selecting an opposite response “After Learning” based on a different interpretation of the statement(s).

Before Learning Statements After Learning

Agree Disagree 1. Categorical data are values which can be sorted by names or labels rather than numbers. Agree Disagree

Agree Disagree 2. Marginal frequencies and joint frequencies are terms that have the same definition. Agree Disagree

Agree Disagree 3. Relative frequencies are often stated as percentages. Agree Disagree

Agree Disagree 4. Two-way tables allow us to compare two or more sets of categorical data. Agree Disagree

Agree Disagree 5. Relative frequencies can be found for the whole table, just the rows, or just the columns. Agree Disagree

Agree Disagree

6. In order to find the conditional probability of event B given event A has already occurred, you must know the probability of event B and the probability of event A.

Agree Disagree

Agree Disagree

7. The probability of B given A has occurred, represented by P(B|A), is the same as the probability of A given B has occurred, or P(A|B).

Agree Disagree

Statement 4: Two-way tables compare only two categorical sets of data at a time. Statement 6: For a conditional probability, the probability of B AND A, or P(B and A), must be known, not just the probability of B. Statement 7: This may be true for rare cases; it is not the norm.

Unit 8, Activity 8, Relative Frequency and Probability


Two-way Frequency Tables Below is a two-way frequency table (Table 1) with hypothetical data from 200 randomly selected students in a school. Table 1: Hair Color versus Eye Color

Hair Color

Black Brown Red Blond Total

Eye

Col

or Brown 24 40 6 4 74

Blue 6 28 6 32 72 Hazel 6 18 4 4 32 Green 2 10 4 6 22 Total 38 96 20 46 200

The data displayed in the table is called categorical data because the values in the survey are names or labels. The color of someone’s hair (e.g., black, brown, red, blond) or the color of their eyes (e.g., brown, blue, hazel, green) are examples of categorical variables. A two-way table is a useful tool for looking at relationships between categorical variables. A two-way table compares data from two categorical variables. In the example above the variables are Hair Color and Eye Color. The entries in the cells in the tables above are frequency counts, the measure of the number of times an event occurs. The Total column and row are called marginal frequencies while the entries in the body of the table are called joint frequencies.

1. Look at the marginal frequencies for Eye Color (Total column). Which color has the strongest representation?

2. Look at the marginal frequencies for Hair Color (Total row). Which color has the strongest representation?

3. Now, compare the joint frequencies. Which color combination (Hair and Eye Color) has the largest frequency?

4. What other observations can you make about the data in the table?



We can also display the data as relative frequencies in a two-way table. Relative frequencies are ratios of the frequency counts to the total counts. For example, the relative frequency of students

with blond hair and blue eyes is 4200

or 150

. Often, relative frequencies are stated as values

between 0 and 1 or as percentages (for the example above, the relative frequencies could also be stated as 0.02 or 2%). Two-way tables can show relative frequencies for the whole table, for rows or for columns. The tables below show the different types of relative frequency tables. Table 2 shows the relative frequencies of the whole table, Table 3 shows the relative frequencies of the rows, and Table 4 shows the relative frequencies of the columns. Table 2: Relative Frequencies for the Whole Table

Hair Color


Eye

Col

or Brown .120 .200 .300 .020 .370

Blue .030 .140 .030 .160 .360 Hazel .030 .090 .020 .020 .160 Green .010 .050 .020 .030 .110 Total .190 .480 .100 .230 1.000

Values may not total 1.00 due to rounding. Table 3: Relative Frequencies for Rows

Hair Color


Eye

Col

or Brown .324 .541 .081 .054 1.000

Blue .083 .389 .083 .444 1.000 Hazel .188 .563 .125 .125 1.000 Green .091 .455 .182 .273 1.000 Total .190 .480 .100 .230 1.000

Values may not total 1.00 due to rounding. Table 4: Relative Frequencies for Columns

Hair Color


Eye

Col

or Brown .632 .417 .300 .087 .370

Blue .158 .292 .300 .696 .360 Hazel .158 .188 .200 .087 .160 Green .053 .104 .200 .130 .110 Total 1.000 1.000 1.000 1.000 1.000

Values may not total 1.00 due to rounding. Each table above can give different information to help understand the relationship between hair color and eye color. In the Relative Frequencies for Rows table (Table 3) we notice most people with blue eyes have either brown or blond hair, with 38.9% and 44.4% representing those respective categories. However, if you look at the Relative Frequencies for Columns table,



41.7% of the people with brown hair have brown eyes and 69.6% of the people with blond hair have blue eyes.

5. What other observations can you make about the data? Probability and Relative Frequency What is probability? Remember from earlier mathematical studies that probability is the ratio of favorable outcomes to the total possible outcomes in a given sample space. In terms of the categorical data above, let us determine the probability of some events. Refer to Table 1 to answer the following.

6. If we were to select one of the 200 students at random, what is the probability that the student would have brown hair? Justify your answer.

7. If we were to select one of the 200 students at random, what is the probability that the student would have blue eyes? Justify your answer.

8. If we were to select one of the 200 students at random, what is the probability that the student would have red hair AND hazel eyes? Justify your answer.

9. Look at the values you just calculated and compare them to the values in the relative frequency tables. What do you notice about each value?

10. Find the probability of the following using Table 1. Does your statement in question 9 still stand true? Explain. a. P(black hair and green eyes) = ____________ b. P(blond hair and blue eyes) = ____________ c. P(green eyes) = ____________ d. P(red hair) = ____________

11. If the 200 people in this study represented a sample of the total school population, what is the expected probability that a person randomly selected in the school would have brown hair and hazel eyes? Explain your reasoning.

12. Are the events described here independent or dependent? Explain.



The joint and marginal frequencies listed in the table can be used to determine conditional probabilities. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. For example, what is the probability that one of students selected from those with hazel colored eyes has blond hair? This is considered a conditional probability because we are using the given group of only those students with hazel colored eyes as the sample space instead of the entire group of 200. This would be written as P(blond hair|hazel eyes) read probability of blond hair given hazel eyes. To determine the conditional probability of B given that A has occurred, we can use the

following formula: ( ) ( )( ) and

|P A B

P B AP A

= . In terms of our example,

( ) ( )( )

hazel eyes AND blond hairblond hair|hazel eyes

hazel eyesP

PP

= .

13. Where can we find P(hazel eyes AND blond hair)? What is P(hazel eyes AND blond hair)?

14. What is P(hazel eyes)?

15. Calculate P(blond hair|hazel eyes).

16. Describe a different method of calculating/determining the conditional probability P(blond hair|hazel eyes).

Find the following conditional probabilities. Be sure to justify your answers.

17. P(black hair|blue eyes)

18. P(blue eyes|black hair)

19. What is your interpretation of the probabilities you found above?

20. Approximately what percent of students with red hair have green eyes? Based on the work you have completed here, how are two-way frequency tables helpful?

Unit 8, Activity 8, Relative Frequency and Probability with Answers


Two-way Frequency Tables Below is a two-way frequency table (Table 1) with hypothetical data from 200 randomly selected students in a school. Table 1: Hair Color versus Eye Color

Hair Color


Eye

Col

or Brown 24 40 6 4 74

Blue 6 28 6 32 72 Hazel 6 18 4 4 32 Green 2 10 4 6 22 Total 38 96 20 46 200

The data displayed in the table is called categorical data because the values in the survey are names or labels. The color of someone’s hair (e.g., black, brown, red, blond) or the color of their eyes (e.g., brown, blue, hazel, green) are examples of categorical variables. A two-way table is a useful tool for looking at relationships between categorical variables. A two-way table compares data from two categorical variables. In our example above the variables are Hair Color and Eye Color. The entries in the cells in the tables above are frequency counts, the measure of the number of times an event occurs. The Total column and row are called marginal frequencies while the entries in the body of the table are called joint frequencies.

1. Look at the marginal frequencies for Eye Color (Total column). Which color has the strongest representation?

The category with the highest frequency is brown eyes.

2. Look at the marginal frequencies for Hair Color (Total row). Which color has the strongest representation?

The category with the highest frequency is brown hair.

3. Now, compare the joint frequencies. Which color combination (Hair and Eye Color) has the largest frequency?

The color combination with the largest frequency is brown hair and brown eyes

4. What other observations can you make about the data in the table? Answers will vary. Listen to students answers and be sure to ask for justifications for their reasoning/thinking.



We can also display the data as relative frequencies in a two-way table. Relative frequencies are ratios of the frequency counts to the total counts. For example, the relative frequency of students

with blond hair and blue eyes is 4200

or 150

. Often, relative frequencies are stated as values

between 0 and 1 or as percentages (for the example above, the relative frequencies could also be stated as 0.02 or 2%). Two-way tables can show relative frequencies for the whole table, for rows or for columns. The tables below show the different types of relative frequency tables. Table 2 shows the relative frequencies of the whole table, Table 3 shows the relative frequencies of the rows, and Table 4 shows the relative frequencies of the columns. Table 2: Relative Frequencies for the Whole Table

Hair Color


Eye

Col

or Brown .12 .20 .30 .02 .37

Blue .03 .14 .03 .16 .36 Hazel .03 .09 .02 .02 .16 Green .01 .05 .02 .03 .11 Total .19 .48 .10 .23 1.00

Values may not total 1.00 due to rounding. Table 3: Relative Frequencies for Rows

Hair Color


Eye

Col

or Brown .32 .54 .08 .05 1.00

Blue .08 .39 .08 .44 1.00 Hazel .19 .56 .12 .12 1.00 Green .09 .45 .18 .27 1.00 Total .19 .48 .10 .23 1.00

Values may not total 1.00 due to rounding. Table 4: Relative Frequencies for Columns

Hair Color


Eye

Col

or Brown .63 .42 .30 .09 .37

Blue .16 .29 .30 .70 .36 Hazel .16 .19 .20 .09 .16 Green .05 .10 .20 .13 .11 Total 1.00 1.00 1.00 1.00 1.00

Values may not total 1.00 due to rounding. Each table above can give us different information to help understand the relationship between hair color and eye color. In the Relative Frequencies for Rows table (Table 3) we notice most people with blue eyes have either brown or blond hair, with 38.9% and 44.4% representing those respective categories. However, if you look at the Relative Frequencies for Columns table,



41.7% of the people with brown hair have brown eyes and 69.6% of the people with blond hair have blue eyes.

5. What other observations can you make about the data? Answers will vary. Listen to students answers and be sure to ask for justifications for their reasoning/thinking. Probability and Relative Frequency What is probability? Remember from earlier mathematical studies that probability is the ratio of favorable outcomes to the total possible outcomes in a given sample space. In terms of the categorical data above, let us determine the probability of some events. Refer to Table 1 to answer the following.

6. If we were to select one of the 200 students at random, what is the probability that the student would have brown hair? Justify your answer.

96(brown hair) 0.48200

P = =

7. If we were to select one of the 200 students at random, what is the probability that the student would have blue eyes? Justify your answer.

72(blue eyes) 0.36200

P = =

8. If we were to select one of the 200 students at random, what is the probability that the student would have red hair AND hazel eyes? Justify your answer.

4(red hair AND hazel eyes) 0.02200

P = =

9. Look at the values you just calculated and compare them to the values in the relative frequency tables. What do you notice about each value?

Students should notice that the values are the same as those in the relative frequency table for the whole table (Table 2).

10. Find the probability of the following using Table 1. Does your statement in question 9 still stand true? Explain. e. P(black hair and green eyes) = 0.01 f. P(blond hair and blue eyes) = 0.16 g. P(green eyes) = 0.11 h. P(red hair) = 0.10

Yes, each value listed is in Table 2. 11. If the 200 people in this study represented a sample of the total school population, what is

the expected probability that a person randomly selected in the school would have brown hair and hazel eyes? Explain your reasoning.

The expected probability that a person would have brown hair and hazel eyes is 0.09. The cell containing the relative frequency for brown hair and hazel eyes is 0.09.

12. Are the events described here independent or dependent? Explain. These are independent events because the color of hair or eyes does not affect the color of the other.



The joint and marginal frequencies listed in the table can be used to determine conditional probabilities. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. For example, what is the probability that one of students selected from those with hazel colored eyes has blond hair? This is considered a conditional probability because we are using the given group of only those students with hazel colored eyes as the sample space instead of the entire group of 200. This would be written as P(blond hair|hazel eyes) read probability of blond hair given hazel eyes. To determine the conditional probability of B given that A has occurred, we can use the

following formula: ( ) ( )( ) and

|P A B

P B AP A

= . In terms of our example,

( ) ( )( )

hazel eyes AND blond hairblond hair|hazel eyes

hazel eyesP

PP

= .

12. Where can we find P(hazel eyes AND blond hair)? What is P(hazel eyes AND blond hair)? P(hazel eyes AND blond hair) can be found in Table 2. P(hazel eyes AND blond hair) = 0.020.

13. What is P(hazel eyes)? P (hazel eyes) = 0.160

14. Calculate P(blond hair|hazel eyes).

P(blond hair|hazel eyes) = 0.125

15. Describe a different method of calculating/determining the conditional probability P(blond hair|hazel eyes). One method is to find the frequency of students with hazel eyes and blond hair from Table 1 and divide it by the total number of students with hazel eyes. A second method is to use the relative frequencies for rows in Table 3.

Find the following conditional probabilities. Be sure to justify your answers. 16. P(black hair|blue eyes)

P(black hair|blue eyes) = 0.083

17. P(blue eyes|black hair) P(blue eyes|black hair)=0.158

18. What is your interpretation of the probabilities you found above?

8.3% of the students who have blue eyes have black hair while 15.8% of students with black hair have blue eyes.

19. Approximately what percent of students with red hair have green eyes? 20% of students with red hair have green eyes.

Based on the work you have completed here, how are two-way frequency tables helpful? Two-way frequency tables help organize data in a way that allows us to easily identify the relative frequencies and probabilities of different events.

Unit 8, Activity 9, Conditional Geometric Probability


1

3

2

4

32

41

5

Jim and Susan are playing a game using the two spinners at the right. Points are awarded for each round by adding the value of the two slices after both spinners have been spun. The highest score a player can earn is a 9. Jim spins the first spinner and it lands on 3. What is the probability that when he spins the second spinner he will earn a score of 8 this round?

Unit 8, Activity 9, Conditional Geometric Probability with Answers


1

3

2

4

32

41

5

Jim and Susan are playing a game using the two spinners at the right. Points are awarded for each round by adding the value of the two slices after both spinners have been spun. The highest score a player can earn is a 9. Jim spins the first spinner and it lands on 3. What is the probability that when he spins the second spinner he will earn a score of 8 this round?

The probability that Jim will earn

a score of 8 is 0.3 or 310

.

Unit 8, Activity 10, Diameters and Chords


Date______________ Team Members___________________

Use the following guide to investigate the relationships that occur between the diameter and chords of circles. Investigation 1 1. Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to

locate the center of the circle. Label the center C. 2. Pick any two points on the circle (do NOT use the endpoints of the same diameter). Label

the points G and H. Using a straightedge, draw the segment connecting G and H. What is GH ? ________________________

3. Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.

Unfold the paper and label the endpoints of the diameter just created as J and K. 4. Draw and CG CH . Find the measure of GH . ________________________ 5. GH should have been divided into two smaller arcs—either and GK HK or

and GJ HJ . Find the measure of these two smaller arcs created by JK . ________________________________________________

6. What is true about the two arcs measured in number five?

________________________________________________________________________ 7. Using a ruler, measure the radii and CG CH . What is the arc length of GH ?

________________________ What are the arc lengths of the two arcs measured in number five? _______________________________________________________________________

What is true about the lengths of the two smaller arcs compared to the larger arc? ____________________________________________________________________

8. Using a ruler, measure GH and the two smaller segments created by the intersection of

the diameter and the chord. ______________________________________________ 9. What conjecture can be made if the diameter of a circle is perpendicular to a chord? ________________________________________________________________________ Does this conjecture apply to the radii of a circle? Explain. ________________________________________________________________________ ________________________________________________________________________

Unit 8, Activity 10, Diameters and Chords


Investigation 2 Follow the steps below in order to answer the questions that follow. Step 1. Use a compass to draw a large circle on patty paper. Cut out the circle. Step 2. Fold the circle in half. Step 3. Without opening the circle, fold the edge of the circle so it does not intersect the

first fold. Step 4. Unfold the circle and label the circle. Find the center by locating the point where

the compass was placed and label the center M. Darken the diameter which should pass through the center. Locate the two other folds and darken the chords created by these folds. Label one chord as GE and the other chord as TR .

Step 5. Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold

again to bisect TR (lay point T onto R). Two diameters should have been formed. Label the intersection point on GE as O and the intersection point on TR as Y.

Answer the following about Investigation 2. 1. What is the relationship between and MO GE ? What is the relationship between

and MY TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this). 2. Use a centimeter ruler to measure , , , and GE TR MO MY . What observation can be made? 3. Make a conjecture about the distance that two chords are from the center when the chords

are congruent.

Unit 8, Activity 10 , Diameter and Chords with Answers


Date______________ Team Members___________________

Use the following guide to investigate the relationships that occur between the diameter and chords of circles. Investigation 1 1. Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to

locate the center of the circle. Label the center C. 2. Pick any two points on the circle (do NOT use the endpoints of the diameters). Label the

points G and H. Using a straightedge, draw the segment connecting G and H. What is GH ? A chord.

3. Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.

Unfold the paper and label the endpoints of the diameter just created as J and K. 4. Draw and CG CH . Find the measure of GH . Answers will vary. 5. GH should have been divided into two smaller arcs—either and GK HK or

and GJ HJ . Find the measure of these two smaller arcs created by JK . Answers will vary.

6. What is true about the two arcs measured in number five? They have the same measure,

which means they are congruent. 7. Using a ruler, measure the radii and CG CH . What is the arc length of GH ? Answers

will vary. What are the arc lengths of the two arcs measured in number five? Answers will vary. What is true about the lengths of the two smaller arcs compared to the larger arc? They have the same measure, which means they are congruent.

8. Using a ruler, measure GH and the two smaller segments created by the intersection of the diameter and the chord. Answers will vary.

9. What conjecture can be made if the diameter of a circle is perpendicular to a chord? If the diameter of a circle is perpendicular to a chord, the diameter bisects the chord and

the arc. Does this conjecture apply to the radii of a circle? Explain. Yes, this conjecture also applies to the radii of a circle. A radius is a part of the diameter;

therefore, these properties are true for the radii.

Unit 8, Activity 10 , Diameter and Chords with Answers


Investigation 2 Follow the steps below in order to answer the questions that follow. Step 1. Use a compass to draw a large circle on patty paper. Cut out the circle. Step 2. Fold the circle in half. Step 3. Without opening the circle, fold the edge of the circle so it does not intersect the

first fold. Step 4. Unfold the circle and label the circle. Find the center by locating the point where

the compass was placed and label the center M. Darken the diameter which should pass through the center. Locate the two other folds and darken the chords created by these folds. Label one chord as GE and the other chord as TR .

Step 5. Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold

again to bisect TR (lay point T onto R). Two diameters should have been formed. Label the intersection point on GE as O and the intersection point on TR as Y.

Answer the following about Investigation 2. 1. What is the relationship between and MO GE ? What is the relationship between

and MY TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this). and MO MY are perpendicular bisectors of and GE TR , respectively. 2. Use a centimeter ruler to measure , , , and GE TR MO MY . What observation can be made? and GE TR MO MY= = 3. Make a conjecture about the distance that two chords are from the center when the chords

are congruent. When two chords are congruent, they are equidistant from the center of the circle.

Unit 8, Activity 12, Tangents and Secants


Unit 8, Activity 12, Tangents and Secants


Unit 8, Activity 12, Tangents and Secants with Answers


12

m ADB mDEB∠ =

12

m CDB mDB∠ =

( )12

m AED mAD mCB∠ = +

( )12

m AEC mAC mDB∠ = +

( )12

m E mBC mAD∠ = −

Unit 8, Activity 12, Tangents and Secants with Answers


1 ( )2

m E mDB mAD∠ = −

( )12

m A mBDC mBC∠ = −

Unit 8, Activity 14, Surface Area of a Sphere


Documents

Geometry - Richland Parish School Board › documents › common...Unit 1, Activity 1, Extending Number and Picture Patterns. Blackline Masters, Geometry Page 1-1 . Date _____ Name