Module #6 - San Dieguito Union High School Districtteachers.sduhsd.net/slesan/Math A Honors/Mod 6...

HOMEWORK ANSWERS (ODDS)

Math A Honors

Module #6

2015 - 2016

Created in collaboration with Utah Middle School Math Project

A University of Utah Partnership Project

San Dieguito Union High School District

SDUHSD Math A Honors Module #6 – TEACHER EDITION 2015-2016

2

6.0A Review: Geometry Vocabulary Name: Period:

Boxes that are blank under ―Symbolic Name‖ are because there is not an established symbol for the term.

Term Definition Symbolic Name Example

Point A location in space. It has no size.

Point A

Line

A series of points that extend in opposite directions without end.

or or n

Line Segment Part of a line. It has two endpoints.

or

represents the length

Ray

Part of a line with exactly one endpoint. It is named endpoint first.

or

Plane

A flat surface with no thickness. It extends without end in all directions. Named by 3 points.

Plane ABC or Plane E

Collinear Points that lie on the same line.

Non-collinear Points that cannot be contained by one line.

A

n

A

B

P

Q

E

D

YX

E

A

B

C

A

C

B

A

C

B


3

Parallel Lines Lines that lie in the same plane that do not intersect.

Intersecting Lines

Lines that lie in the same plane that cross.

Skew Lines

Lines that do not lie in the same plane. They are not parallel and do not intersect.

Midpoint A point that is the middle of a line segment.

AX = XB or

Angle

Two rays that meet at a common endpoint form an angle.

or or

Vertex The common endpoint of an angle.

Point B See picture for angle.

Sides

The two rays that form an angle are also known as the sides of the angle.

or See picture for angle.

A

B

C

D

C

B

D

E

A

E

A B

G

F H

CD

XA

B

B C

A


4

Congruent Angles

Angles that have the same measure.

Congruent Segments

Segments that have the same measure.

AB = XY

Protractor The tool used to measure an angle in degrees.

Perpendicular Lines

Two lines that intersect and form a right angle.

Acute Angle An angle that measures between 0 and 90 .

Obtuse Angle An angle that measures between 90 and 180 .

A

C

TO G

D

A

B

X

Y

A

B

E

C

D


5

Right Angle An angle that

measures .

Straight Angle An angle that

measures 180 .

Complementary Angles

A pair of angles

whose sum is 90 . They do not need to be adjacent.

Supplementary Angles

A pair of angles

whose sum is 180 .

Adjacent Angles

Angles that share a vertex and a side but no points in their interiors.

Polygon

A plane figure formed by line segments, having at least three sides. The sides meet only at their endpoints.

TV

US

WZ

Y

X

B

DA C


6

Regular Polygon A polygon with all sides and angles congruent.

Triangle A polygon with three sides.

or

Acute Triangle A triangle with all acute angles.

Obtuse Triangle A triangle with one obtuse angle.

Right Triangle A triangle with one right angle.

Scalene Triangle A triangle with no sides congruent.

Isosceles Triangle

A triangle with at least two congruent sides.

S

U

N


7

Equilateral Triangle

A triangle with three congruent sides.

Quadrilateral A polygon with four sides.

☐ABCD

Congruent The same shape and the same size.

Congruency Marks

Tick marks on sides or angles that have the same measure

See marks on some of the diagrams in this

table.

Extra boxes in case teachers would like to

include other definitions.

A

BC

D


8

6.0B Review: Angle Classification and Using a Protractor* Name: Period:

You will likely need to review how to use a protractor with your students for this activity. Here’s a Learn Zillion video that does this: https://learnzillion.com/lessons/2973-measure-angles-to-the-nearest-degree-with-protractors Review Angles: For each angle below, a) use a protractor to find the measurement in degrees and b) write its classification (acute, obtuse, right or straight.) 1.

Degrees: Classification: Right angle

2.

Degrees: Classification: Acute angle

3.

Degrees: 100 Classification: Obtuse angle

4.

Degrees: 70 Classification: Acute angle

https://learnzillion.com/lessons/2973-measure-angles-to-the-nearest-degree-with-protractors


9

5.

Degrees: Classification: Straight angle

6.

Degrees: Classification: Right angle

7.

Degrees: Classification: Obtuse angle

8.

Degrees: Classification: Acute angle


10

6.1A Homework: Triangle Practice* Name: Period:

Find the sum of the angles for the following polygons beginning with the proper formula. 1. Pentagon

540 2. Octagon

3. 15-sided polygon

(pentadecagon)

2340

is a scalene acute triangle. If and , what is ? Show your work. For #5-8, determine whether it is possible to form a triangle with the three given side lengths. Explain why or why not using the Triangle Inequality Theorem. 4. 1, 4, 10 5.

No, a triangle is not possible because 1.5 + 3.5 5.

6. 4, 3, 6.9

7.

Yes, it is possible to form a triangle with these

three side lengths because

.

8. Mrs. Turnip wants to build a vegetable garden. She has three pieces of fencing. One is 8 feet long, one is 5 feet, and the other is 3 feet long. Can she build a closed triangular garden with these three boards, provided she is not allowed to re-cut any of the board pieces or have any pieces left over? Include a diagram with your explanation.


11

9. Anna used the scale on a map to calculate the distances “as the crow flies” (meaning the perfectly straight distance) from three points in Central America and the Caribbean islands, marked on the map to the right. a. According to Anna, how far is it from Jamaica

to Panama if you go directly? 700 miles

b. According to Anna, how far is it from Jamaica to Panama if you do go through Honduras first? 600 miles

c. Another way of stating the Triangle Inequality Theorem is “the shortest distance between two points is a straight line.” Explain why Anna must have made a mistake in her calculations.

10. Two side lengths of a triangle are 5 cm and 7 cm.

a. What is the smallest possible whole number length of the third side? Justify your answer.

b. What is the smallest possible length of the third side to the nearest

of a centimeter?

11. Two side lengths of a triangle are 6 cm and 8 cm.

a. What is the largest possible whole number length of the third side? Justify your answer. 13 cm—note that at 14 the other two sides would fall flat so the largest INTEGER value is 13. Again, take time to explore 11.5, 11.9, 11.99, etc.

b. What is the largest possible length of the third side to the nearest

of a centimeter?

13

cm—note that at 14 the other two sides would fall flat so the largest INTEGER value is 13. Again,

take time to explore 11.5, 11.9, 11.99, etc.


12

For the next two problems: (a) Carefully duplicate the triangle using a ruler and protractor. Label the sides lengths (rounding to the nearest tenth of a cm) and angle measures (rounding to the nearest whole number of a degree) for your new triangle. (b) Then write an inequality that shows that the Triangle Inequality Theorem holds. (c) Classify the triangle based on its sides and angles. 12.

a. Copy and label the triangle:

b. Triangle Inequality Theorem statement:

c. Classify the triangle based on its sides and angles:

B C

A


13

13.

a. Copy and label the triangle as in #13:

b. Triangle Inequality Theorem statement:

example .

c. Classify the triangle based on its sides and angles: Scalene acute triangle

True or false. Explain your reasoning with a specific example to illustrate your explanation. 14. An acute triangle has three sides that are all different lengths.

15. A scalene triangle can be an acute triangle as well. True, the sum of 3 different acute angles(which means their opposite sides are different lengths) can equal 180°, forming a triangle.

16. A scalene triangle has three angles less than 90 degrees.

17. A triangle with a 100° angle must be an obtuse triangle. True, a triangle with 1 obtuse angle is an obtuse triangle.

18. The angles of an equilateral triangle are also equal in measure.

D F

E


14

Spiral Review:

19. Simplify

20. Find 45% of 320 without a calculator. Show your work.

21. Evaluate for 18

22. Kim, Laurel, and Maddy are playing golf. Kim

ends with a score of 8. Laurel’s score is 4. Maddy scores +5. What is the difference between the scores of Maddy and Kim?

Define a variable, write an equation, solve it, and answer in a complete sentence.

23. The temperature increased 2º per hour. If the temperate rose 15 , how many hours had the temperature been rising? Let x = the number of hours

The temperature rose for 7.5 hours.


15

6.1B Homework: Building Triangles Given Three Measurements* Name: Period:

In #1-2, list the angles of the triangle from smallest to largest without using a protractor. Triangles are not drawn to scale.

In #3-4, list the sides of the triangle from shortest to longest without using a ruler. Triangles are not drawn to scale.

1.

2.

3. , ,

4.

5. Use the grid below to draw and label with AB = 5 units, BC = 7 units, and .

6. Is the included angle to the given sides? Explain.

7. What is the area of the triangle in #5? Justify your answer. 17½ units: (7*5)/2

8. Will all your classmates who draw a triangle like #5 get a triangle with the same area? Explain.

9

7 8

B

C A

11

8

10

B

C

A

40°30°

B

A C

10°

50°

B

A

C

A

B C


16

9. Which triangle(s) below have two sides 5 and 8 units, and a non-included angle of 20 adjacent to the side of length 8? Both of them.

10. Are the two triangles in #9 exactly the same and size and shape? Explain why or why not.

11. Paul lives 2 miles from Rita. Rita lives 3 miles from the shopping mall. What are the shortest and longest

straight-line distances Paul could live from the mall? All three houses are not in a straight line. Sketch a picture to justify your work. Paul could be 2 mile or 4 miles away.

For #12, define your variable, write an algebraic equation, solve, and answer in a complete sentence. 12. Three segments have a total length of 26 inches. The first segment is two less than three times the

second segment. The third segment is one more than five times the second segment. What are the lengths of the three segments? Can a triangle be formed with these segments? Why or why not?


17

For #13-16, decide whether there are zero, one, or more than one possible triangles with the given conditions. Use either a ruler and protractor OR graph paper strips and protractor to draw the triangles. 13. A triangle with sides that measure 5, 12, and 13

cm. How many possible triangles can be formed? One If possible, what kind of triangle is formed? If not possible, state why not. Right scalene triangle. Labeled picture:

14. A triangle with angles 100° and 20°, and an included side of 2 cm. How many triangles can be formed? If possible, what kind of triangle is formed? If not possible, state why not. Labeled picture:

15. A triangle with two sides of 5 and 7 cm, and an included angle of 45°. How many triangles can be formed? One If possible, what kind of triangle is formed? If not possible, state why not. Right isosceles triangle Labeled picture:

16. A triangle HIJ in which m ∠HIJ = 60°,

m ∠JHI = 90°, and m ∠IJH = 55°.

How many possible triangles can be formed? If possible, what kind of triangle is formed? If not possible, state why not. Labeled picture:


18

Spiral Review: Solve. Show all algebraic steps. 17.

m = 6

18.

19. Write

as a percent and decimal.

60%, 0. 20. Show how to simplify the following expression with a number line –6 + (–3)

21. Kurt earned $550 over the summer. If he put 70% of his earnings into his savings, how much money did

he have left to spend? Answer in a complete sentence. Kurt has $165 left over.


19

Section 6.1 Review* Name: Period:

1. is an isosceles obtuse triangle. If , what are the measures of the other angles? 28 and 124

Given the following criteria, construct a triangle using a protractor and a ruler:

2. Draw and label with AB = 5 units, BC = 7 units, and CA = 10 units.

3. Draw and label with , , and . See student responses.

4. Draw and label with , CA = 4 units, and BA = 6 units.

Determine if the following side lengths will make a triangle. Show your work using the Triangle Inequality Theorem to justify your answer. 5. 2 in, 5in, 7in

No, 2 + 5 7

6. 33 mm, 93.2 mm, 70 mm

7. 1 ft, 2.7 ft, 7 ft

No, 1 + 2.7 7

8. 5.1 in, 3.24 in, 8.25 in

9.

cm,

cm, 8 cm

No,

10. 6.01 cm, 5

cm, 11.22 cm

11. Your friend is having a hard time understanding how angle measures of 30°, 60°, 90° might create more

than one triangle. Draw two different triangles that have those angle measures and explain why the two triangles are different. Encourage students to use GeoGebra or a protractor and ruler. It may help to have students create angles of and then create the triangles from there. It will also be very helpful if students create their triangle on grid paper. You will use this activity in 6.2A Lesson. Discuss with students that even though the two triangles have the same angle measures, the side’s lengths are not the same. In the next section we will begin the discussion of scaling triangles (similar triangles).


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Determine if the given information will make a unique triangle. Explain why or why not. 12. Side lengths 3 and 5 and an

included angle of 67°

13. Angles 73°, 7°, 100° No, this does not create a unique triangle. Given 3 angles the triangles are similar, but not necessarily congruent.

14. Angles 80° and 25° and included side of 12

For #15-16, determine if the given angles can make a triangle. Explain why or why not. 15. Angles 25°, 70°, 95°

No the angles add to 190 degrees, and they should add to 180 degrees.

16. Angles 30°, 40°, 20°

17. In , AB = 12, BC = 7, AC = 9. List the angles from least to greatest. (Sketch and label a picture.)

18. In , , , . List the sides from greatest to least. (Sketch and label a picture.)

19. A triangle has two angles with measures 60 and 75 and an included side measuring 20 centimeters. Which figure(s) represents this triangle? a.

b.

c.

d.

20. If you drew a triangle with these measurements, which kind of triangle would it be?

Angles Sides

40 4 cm

50 7 cm

90 8 cm

a. Acute scalene triangle

b. Right scalene triangle

c. Acute isosceles triangle

d. Right equilateral triangle


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Spiral Review 21. Find the product:

22. Without a calculator, what percent of 80 is 60?.

Simplify the following: 23.

24.

25. Order the numbers from least to greatest.

-2.15, -2.105,

, 2.7

26. Solve.

27. Given the following table, find the indicated unit rate:

___15__ push-ups per day

Days Total Push-ups

2 30

4 60

29 435

-2.15,17

7, 2.7, - 2.105


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6.2A Homework: Comparing the Perimeter and Area of Polygons* Name: Period:

Below is a table that describes the dimensions, perimeter, area and change of side lengths for a rectangle that’s scaled in different ways. In the first row you find that the 5x20 rectangle has a perimeter of 50 and an area of 100; there is no change in side length here because it’s the original rectangle. In the next row the dimensions become 10x40 giving it a perimeter of 100 and area of 400; the side length change is twice the original. Use this information to do the following:

Examine the completed rows to understand what’s happening to the rectangle. Fill in any empty parts.

Graph the relationship between the perimeter (x-axis) and area (y-axis) on the graph below.

Describe the patterns you notice from the graph.

a)

Dimensions Perimeter Area Change of side

lengths from original rectangle

Change in Perimeter

Change in Area

5 by 20 50 100 Same Same Same

10 by 40 100 400 Twice Twice Four times

5/2 by 10 25 25 Half Half Fourth

15 by 60 150 900 Three times Three times 9 times

5/3 by 20/3 16.67 11.11 One third One third One ninth

20 by 80 200 1600 Four times Four times 16 times

b)

c) Describe the patterns you notice:


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Define the variable. Write an equation. Solve and state your answer in a complete sentence. 1. A square has a perimeter of 25 inches. Find the length of a side.

6.25 inches.

2. A field has an area of 480,000 m2 and is 160 m wide. How much fencing is needed to enclose the field?

3. A brick walk 2 meters wide surrounds a rectangular pool that is 3 meters by 5 meters. Find the area of the walk. Hint: Draw a picture. 48 m2.

Determine if the following statements are true or false. Explain your answer. 4. All isosceles triangles are similar.

5. All equilateral triangles are similar. True, all three angles are the same and the sides will be proportional.

6. All isosceles right triangles are similar.

7. All rectangles are similar. False, rectangles are not similar to squares, a special type of rectangle. The sides are not proportional.


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Spiral Review: 8. What do we call triangles with all sides the same length?

9.

a. Katherine is visiting patients in a hospital. She visits 18 patients in 6 hours. At that rate, how many patients will she visit in 9 hours? Show your work. 27 patients

b. What is Katherine’s unit rate?

10. What is an obtuse angle?

11. The price for two bottles of ketchup are given below: A 20 oz. bottle of DELIGHT ketchup is 98¢ at the grocery store. A 38 oz. bottle of SQUEEZE ketchup is $1.99 at the same grocery store. a. Find the unit rate for each product

DELIGHT is $.049 per ounce and SQUEEZE is $.052 per ounce

b. What conclusions can you draw from this information?

DELIGHT is the least expensive of the two.

12. Simplify:

13. Simplify:


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6.2B Homework: Scaling Triangles* Name: Period:

1. Which scale would produce the smallest scale drawing of an object when compared to the actual object?

Explain your reasoning. C because 1 mm : 1 m is the same as 1 mm : 1000 mm, which produces the smallest scale drawing. a. 1 in : 10 in

b. 1 cm : 10 cm

c. 1 mm : 1m

2. The scale of a drawing is 6 cm : 1 mm. Is the scale drawing larger or smaller than the actual object? Explain your reasoning.

3. A 0.035-centimeter-long paramecium appears to be 17.5 millimeters long under a microscope. What is the power (scale factor or magnification scale) of the microscope? Show your work. 50

4. A model of an MC130E airplane has a scale factor of

.

a. If the width of the actual tail is 52 feet 8 inches, what is the width of the tail in the model?

b. If the height of the actual tail is 38 feet 5 inches, what is the height of the tail in the model?

5. The model of a dollhouse has been constructed using a scale of 1 to 48.

a. If the model’s door is two and one quarter inches high, how high is the actual dollhouse’s door?

108 inches tall, or 9 feet tall

b. If the model is 18 centimeters high, how tall is the actual dollhouse?

864 cm tall, or 8.64 m tall

c. What is the scale factor of the model of a doll house to the actual doll house? 48

d. How would the scale factor in part c compare to the scale factor of the actual doll house to the model? Explain.

It would be

, which is the reciprocal of the scale factor in part c.


26

6. If the ratio of to is 3:5 and the length of is 6”, what is the length of ? Justify your answer.

For #7-8, the pair of triangles given in each problem are proportional. The figures are not necessarily drawn to scale. Do the following:

Solve for the unknowns by using proportions. State the scale factor between the two triangles. Express all answers exactly. Show your work.

7.

8.

Spiral Review:

5

9

x 8

y

16

𝑥

𝑥 𝑢𝑛𝑖𝑡𝑠

𝑦

𝑦

𝑢𝑛𝑖𝑡𝑠

scale factor: 3


27

9. Suppose you were to flip a coin 3 times. What is the probability of getting heads all three times?

10. Samantha made 120 bracelets. She sells

of the bracelets and then decides to donate 50% of the rest.

How many bracelets does she have left?

11. Place each of the following integers on the number line below. Label each point:

A = 4 B = –4 C = –15 D = 7 E = 18 F = –19

12. Write 0.672 as a fraction and as a percent.

13. Simplify:

F C E D B A


28

6.2C Homework: Solve Scale Drawing Problems, Create a Scale Drawing* Name: Period:

Students will need a cm ruler. The San Diego Trolley map below is not drawn to scale, but how off scale is it? Draw a triangle on the San Diego Trolley map using the following locations as the vertices: Middletown, Fenton Parkway, and Euclid Avenue.

1. Measure the segment lengths and record them in the table below. Round your measurements to the

nearest tenth of a centimeter.

Segment Distance (km) Subway Map

Segment Lengths (cm)

Map Scale

Middletown to Fenton Parkway 22.1 km 3.1 cm 1 : 7.1

Fenton Parkway to Euclid Avenue 27.9 km 4.5 cm 1 : 6.2

Euclid Avenue to Middletown 28.6 km 5.2 cm 1 : 5.5

Teacher note: Student measure may be slighty different which will alter the scales.


29

2. Is this portion of the map drawn to scale? Explain why or why not.

3. On a map, Breanne measured the straight-line distance between Los Angeles and San Francisco at 2

inches. The scale on the map is . What is the actual straight-line distance between Los

Angeles and San Francisco? Show all your work. 344 miles

4. What scale was used to enlarge the drawing below? How do you know?.

What is the scale factor? 2

5. On a separate sheet of grid paper, create the creature below so that it is a 1:3 enlargement of the original model. Write your strategy for calculating lengths to the right of the picture.

This will likely take about 10 minutes.

1

4inch = 43miles


30

Spiral Review: 6. Suppose you flip a coin 3 times; what is the probability that you get heads exactly two times?

7. For each group of three side lengths, determine whether a triangle is possible. Write yes or no. Show

your work to justify your answer using the Triangle Inequality Theorem.

a.

Yes

b. Yes

c.

8. Which number is greater: –24.41 or –24.4? Explain.

Solve.

9.

x=7

10. Simplify:

Define a variable. Write an algebraic equation. Solve and answer in a complete sentence. 11. A sweater costs $60.75 after the store increased the price 35%. How much did the sweater cost originally?

$45


31


1. Draw a scaled version of the following polygons given the scale factor.

a. Scale Factor: 2

b. Scale Factor:

The perimeter of an equilateral triangle is 45 cm. What is the length of each side?

2. A rectangle has length 18 and width 12. Find the perimeter.

3. Find the missing measurement in each problem

a. The scale factor from to is 3. If BC is 3, what is the length of EF? EF = 9 units

b. The scale factor from to is

. If

OP is 30, what is the length of EL? EL = 50 units

2624

10B C

A

1312

5B C

A


32

4. In each of the following, what is the scale factor that takes to ?

a.

b.

5. Answer each of the following questions using proportional reasoning.

a. Becky is drawing a scale model of her neighborhood. She uses a scale of 1 in = 200 feet. If her block is 1000 feet long, what will be the length of the block on her scale model? The length of the block will be 5 inches on her model.

b. Carlos is reading a map of his town. The scale says 1 in = 4 miles. The distance from his

house to the school is

in. on the map. If

Carlos wants to walk to school from his house, how far will he have to walk? He will have to walk 1 ½ miles.

6. The dimensions of Romina’s rectangular garden are 2 feet by 3 feet. The dimensions of Santiago’s garden

are all quadrupled. a. Find the perimeter and area of each garden. b. Find the scale factor between the perimeters

of each garden.

c. Find the scale factor between the areas of each garden.

d. In general, how are the scale factors of perimeters and areas of scaled objects related?

ABC XYZ


33

6.3A Homework: How Many Diameters Does it Take to Wrap Around a Circle?* Name: Period:

1. Complete the missing values in the table below. Round to the nearest tenth. Show all work in the given

space.

Radius Diameter Circumference

a. 16 in 32 in 100.5 in

b. 16.5 m 33 m 103.7 m

c. 25.0 cm 50.0 cm 157cm

2. What is the exact ratio of the circumference to the diameter of every circle?

3. If the radius of a circle is 18 miles:

a. What is the measure of the diameter? 36 miles

b. What is the measure of the circumference, exactly, in terms of pi?

36

c. What is the approximate measure of the circumference, to the nearest hundredth of a mile? 113.10 miles

4. For each of the three circles below, calculate the circumference. Express your answer both in terms of pi,

and also as an approximation to the nearest tenth.

Diameter/Radius Circumference (in terms of )

Circumference

Radius = 1.5 cm

Diameter = 5 cm


34

5. If the circumference of a circle is feet, which of the following statements are true? Rewrite false statements to make them true. a. The circumference of the circle is exactly 62.8 feet.

False: circumference is approximately 62.8 ft

b. The diameter of the circle is 20 feet. True

c. The radius of the circle is 20 feet. False: the radius is 10 feet

d. The ratio of circumference : diameter of the circle is . True

e. The radius of the circle is twice the diameter. False: the radius is ½ the diameter or the diameter is twice the radius

6. The circumference of 2 objects is given. Calculate the diameter of each object, to the nearest tenth of a unit.

Circumference Diameter

Bottom of Cupcake = 6.5 inches

Top of water pail = 100 cm

7. The diameter or radius of 2 objects is given. Calculate the circumference of each object, to the nearest

hundredth of a unit.

Diameter/Radius Circumference

Diameter of rim of the drum = 24 inches

C = 75.40 in.

Radius of a table top = 5.5 feet

C = 34.56 ft.

8. Three tennis balls are stacked and then tightly packed into a cylindrical can. Which is greater: the height

of the can, or the circumference of the top of the can? Justify your answer. (Hint: Draw a picture.)


35

9. A circular garden has a circumference of 43.96 yards. Leo is digging a straight-line trench along a diameter of the garden at a rate of 7 yards per hour. How many hours will it take him to dig across the garden? a. Find the diameter. Round to the nearest whole number.

14 yards

b. Find the time it will take Leo to dig across the garden. 14/7 = 2 hours It will take Leo 2 hours to dig across the garden.

10. The two coins shown at right have radii of 3 cm and 1 cm. The smaller coin rolls along the circumference of the fixed larger coin until the smaller coin returns to its original position. How many revolutions has the small coin made?

11. A wire bent into the shape of a square encloses an area of 25 cm2. Then the same wire is cut in half and bent into two identical circles. What is the radius of each circle? Round to the nearest hundredth. 1.59 cm

12. A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge of the path than around the inner edge? Round to the nearest hundredth.

13. A gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has the diameter of 2 inches. Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer.

No, this is not correct because the circumference of the larger gear is 𝜋 and the circumference of the

smaller gear is 𝜋, and 𝜋 𝜋 ≈ inches. The circumference of the larger one is therefore 𝜋 inches longer, or about 6.28 inches longer.


36

14. Calculate the radius for each circle whose circumference is given in the table (the first entry is done for you). Round to the nearest whole number. Then graph the values on a coordinate plane, with the radius on the x axis and the approximate circumference on the y axis.

Radius of circle in inches

4

Circumference of circle in

inches

≈

≈

≈

≈

≈

≈

≈

≈

How a Radius Relates to Circumference

15. Is the radius of a circle proportional to the circumference of the circle? Justify your answer.

Yes, the graph is a straight line that passes through the origin (a radius of zero would result in a circumference of zero).

Spiral Review: 16. Factor the following expressions.

a. –

b.

c. –


37

17. Without using a calculator, determine which fraction is bigger in each pair. Justify your answer with a picture or words.

a.

b.

18. Millie bought two sweaters for $30 each and three pair of pants for $25 each. She had a 20% off coupon

for her entire purchase. Model and write an expression for the amount of money Millie spent. State your answer.


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6.3C Homework: Area of a Circle* Name: Period:

1. Estimate the area of the circle at the right in

square units by counting.

≈

2. Use the formula for the area of a circle to calculate the exact area of the circle above, in terms of pi.

3. Round the answer to #2 to the nearest square unit. How accurate was your estimate in #1?

50 square units 4. Calculate the area of each circle below. Express your answer both exactly (in terms of pi) and

approximately, to the nearest tenth of a unit.

Radius Area

(in terms of ) Area

5. A corner shelf is ¼ of a circle and has a radius of 10.5 inches. Find the area of the shelf. Round your

answer to the nearest hundredth. 86.59 square inches

10.5 in


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6. The strongest winds in Hurricane Katrina extended 30 miles in all directions from the eye (center) of the hurricane. a. Draw a diagram of the situation.

b. What is the size of the area that felt the strongest winds?

1. Find the area of the shaded region at the right if the diameter of the smaller circle is 6 cm and the shaded region is 10 cm wide. Round to the nearest tenth. The wall is 502.7 square centimeters

2. By calculating the areas of the square and the circle in the diagram below, determine how many times

larger in area the circle is than the square. 3. Draw a diagram to solve: A circle with radius 8 centimeters is enlarged so its radius is now 24 centimeters.

a. By what scale factor did the circumference increase? Show your work or justify your answer. 3 times

b. By what scale factor did the area increase? 9 times

c. Explain why this makes sense, using what you know about scale factor. The scale factor of the area is always the square of the scale factor of the lengths.

4. How many circles of radius 1” could fit in a circle with radius 5” (if you could cut up and rearrange the area

of the circles of radius 1 in such a way that you completely fill in the circle of radius 5)? Show your work to justify your answer.


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5. The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to the side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth. (Hint: Draw a picture) 30.90 m2 of the field will not be reached by the sprinkler.

6. Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle with twice the radius? Explain.

7. The area of 2 objects is given. Calculate the radius of each object’s surface, to the nearest whole number.

Area Radius

Area of a glass porthole is 3.14 ft2

≈ ft

Area of a round area rug is 153.86 ft2

≈ ft

8. A paraglider wants to land in the unshaded region in the square field

illustrated because the shaded regions (four quarter circles) are briar patches. If the paraglider hits the field randomly due to unexpected wind currents, what is the probability that she misses the briar patches? Use the fact that the probability of landing in the unshaded region is the area of the unshaded region divided by the total area of the field. Round to the nearest whole percent.


41

Spiral Review: Solve. 9.

10.

11. Simplify the expression in two different ways: 5(3 + 4)

Define a variable. Write an algebraic equation. Solve and answer in a complete sentence. 12. There are a total of 214 cars and trucks on a lot. If there are four more than twice the number of trucks

than cars, how many of both kinds of vehicles are on the lot?

13. Simplify:

–28


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1. Mr. Shay baked a giant cookie that is 16 inches in diameter. Mr. Anderson baked a cookie that is 1.5 feet

in diameter. Michelle and Katie are deciding which cookie is bigger. a. Whose cookie is bigger? By how much?

Mr. Anderson’s cookie because it has an area of in2, whereas Mr. Shay’s cookie has an area of

in2. Mr. Anderson’s cookie is in2 larger than Mr. Shay’s.

b. What is the ratio in area of the bigger cookie to the smaller cookie? 81:64

2. Solve each problem rounding to the nearest whole number. Show your work. Drawing a picture may be helpful. a. The radius of a circular pool is 10 feet. There

is a border around the pool. The width of the border is 5 feet. What is the area of the border in square feet?

b. A covering for the pool costs $7 per square foot. About how much will the covering cost?

3. Solve each problem rounding to the nearest tenth. Show your work. Drawing a picture may be helpful. a. The most popular pizza at Pavone’s Pizza is

the 10-inch personal pizza with one topping. What is the area of the pizza with a diameter of 10 inches? 78.5 square inches

b. What is the area of the largest circle that will fit in a square with an area of 64 square centimeters? Explain. 50.2 square centimeters


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4. Use the given information to find the missing information. Give each answer in two forms: exact (in terms of ) and rounded to the nearest hundredth unit.

a. Radius: 2 m

Circumference:

b. Diameter: 2 m Circumference: m, 6.28 m

c. Radius: 5.5 in Area:

d. Diameter: 40 in Area:

e. Circumference: 69.08 cm Diameter:

f. Area: 153.86 cm2 Radius:

Read each question carefully. Explain your reasoning. Round to the nearest whole number. 5. The radius of a large circle is 12 times the radius of the smaller circle.

a. If the radius of the smaller circle is 5 cm, what is the circumference of a smaller circle? 31.40 cm

b. What is the circumference of the larger circle?

376.99 cm

c. What is the ratio of the circumferences of the big circle to the small circle?

12:1

d. If the radius of the larger circle is 36 mm, what is the area of the larger circle?

4,071.50 square mm

e. What is the area of the smaller circle?

28.27 square mm

f. What is the ratio of the area of the big circle to the small circle?

144:1


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6.4A Homework: Special Angle Relationships* Name: Period:

1. Find at least two examples of each angle relationship in the diagram. Name the angle pairs below, and

highlight the pairs of angles in the diagram, using a different color for each relationship.

a. Vertical angles

For example: ∠AHB & ∠FHG, ∠FHA & ∠GHB

b. Supplementary angles

For example: ∠AHB & ∠FHA, ∠AHF & ∠FHG

c. Complementary angles

For example: ∠EGD & ∠DGC, ∠HFG & ∠FHG


45

For each figure, calculate the three missing measures. Justify your answer.

2.

Angle Measure of

angle Justification

3.

Angle Measure of

angle Justification

150° Vertical to ∠NOK

30° Supplementary to ∠NOK

30° Supplementary to ∠NOK or

vertical to ∠MOK


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4. For the figure below, fill in the missing measures. Justify your answers.

An

gle

Me

as

ure

Ju

sti

fic

ati

on

Vertical to

Sum of the measures of

∠FEC & ∠AEF

5. Refer to the figure below to complete the following statement:

63° because it is complementary to ∠AZW

52°35°

F

G

D

B

E

A

C


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For #6-8, choose a correct answer. Sketch and label a diagram to justify your answer.

6. If G is complementary to

, and , then must be:

a. Obtuse

b. Acute

c. Right

7. If is supplementary to , and

, then must be:

a. Obtuse

b. Acute

c. Right

8. If is vertical

to , and then must be:

a. Obtuse

b. Acute

c. Right

For #9-11, write an algebraic equation, solve it, and complete the blanks below. 9.

Equation: x = 22 = 73

10.

Equation: x = =

11.

Equation: x = 5 = 40

For #12-14, before your variables, write and solve an algebraic equation. Answer in a sentence.

12. Two angles are complementary, with one angle 24 greater than the measure of the other. Find the measure of both angles.

6x - 254x - 15

A D

C

B

6x

7x - 5

F

E

HG

I

10x

7x + 5

BA E

CD


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13. Two angles are supplementary. If the measure of one angle is 5 times the measure of the other, what is the measure of each angle?

30 and 150 14. If two vertical angles are supplementary, what is the measure of each angle?

Determine if the following statements are Always true, Sometimes true, or Never true. If sometimes or never true, provide an example and a counterexample.

15. Complementary angles have a sum of 180 . Never

16. Vertical angles are complementary.

17. Two supplementary angles with equal measure are right angles. Always

Find the measure of each lettered angle. Congruent angles and right angles are indicated. 18.

Spiral Review: 19. Order the numbers from least to greatest. - ½ , ¼ , -¼, -1.2, -1.02, -.75

-1.2, -1.02, -.75, -½, -¼, ¼

20. Find the quotient:

21. Find two unit rates for the statement below:

Izzy drove 357 miles on 10 gallons of gasoline.

≈

22. Convert the following units: feet = 37 inches

h

g

f

e

d

c

ba

20°

65°

80°

95°

-8

9¸ -

3

4

æ

èçö

ø÷


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6.4B Homework: Exploring Circles Name: Period:

1. Suppose you have a circle with a radius of 1 unit. What is the area of the circle?

2. Suppose you have a circle with a radius of 2 units. What is the area of the circle?

3. What is the ratio of the area of circle 2 to circle 1? 4:1

4. What do you think the ratio of the areas would be for a circle of radius 3 units to circle 1? Explain.

5. What do you think the ratio of the areas would be for a circle of radius 4 units to circle 1? 16:1

6. What do you think the ratio of the areas would be for a circle of radius 1.5 units to circle 1?

7. If circle A has radius a and circle B has radius b, then the ratio of their circumference is

. Explain.

Student answers will vary.

8. What is the ratio of their areas?

9. If the ratio of the radii of circle A and circle B is x, what is the ratio of their areas? Explain.

10. Complete the sentence: If you multiply the radius of a circle by y, then its area is multiplied by .

11. What happens if you replace the word “circle” with “square” and the word “radius” with “side length” in question numbers 3 – 7? Do your answers change? The ratios remain the same. The answers will not change.

Extension: 12. A bathroom tissue company has manufactured a roll of tissue that is “twice as big” as a standard roll.

What will the roll look like compared to the original? Will it look twice as big?


50

Module 6 Review* Name: Period:

For #1-2, decide if the figures below are possible. Justify your conclusion with a mathematical statement. To construct the triangles use:

A ruler or strips of centimeter graph paper cut to the given lengths

A protractor

Optional: Teachers can let students use construction technology like GeoGebra

1. A triangle with angles that measure 20°, 70°, and 90°. Possible or not? Why or why not? Possible. The sum of the angle measures must equal 180°, and 20 + 70 + 90 = 180 If so, what kind of triangle? Draw and label. Scalene right triangle

2. A triangle with sides 8 cm and 3 cm. The angle opposite the 3 cm side measures 45°. Possible or not? Why or why not? If so, what is the measure of the 3rd side? Draw and label.

3. Two students were building a model of a car with an actual length of 12 feet.

a. Andy’s scale is

. What is the

length of his model? 3 inches

b. Kate’s scale is

. What is the

length of her model? 6 inches


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4. At Camp Bright, the distance from the Bunk House to the Dining Hall is 112 meters. From the Dining Hall to the Craft Building is 63 meters (in the opposite direction). The scale of the map for the camp is

. On the map: a. What is the scaled distance between the Bunk

House and the Dining Hall?

b. What is the scaled distance between the Dining Hall and the Craft Building?

If students are struggling, encourage them to draw a model. A length of 112 meters has a total of eight 14 meter units. Each 14 meter unit is ONE 0.5 cm on the scaled map. Thus the length on the scaled map is 4 cm. 5. In the similar L figures to the right,

a. What is the ratio of height of left figure : height of right figure? 9:3 or 3:1

b. What is the reducing scale factor? 1/3

c. What is the ratio of area of left figure: area of right figure? 36:4 or 9:1. The area scale factor will be the unit scale factor squared.

6. Triangles ABC and RST are scale versions of

each other. a. What is the scale factor from ABC to RST?

b. What is the scale factor from RST to ABC?

c. What is the distance between A and C?

d. What is the distance between R and S?

0.5cm =14meters


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7. Redraw the figures at right using the scale factors below. a. Use a scale factor

of 4 to re-draw the square.

b. Use a scale factor of ¼ to re-draw the addition sign.

c. Use a scale factor of 1.5 to re-draw the division sign.

8. The Washington Monument is 555 feet and 5 1/8 inches tall. Bob wants to create a scale model that is 5

feet tall. What scale would you suggest Bob use for his model?


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9. Calculate the circumference and area of the circles below. Express each dimension both exactly in terms of pi, and as an approximation to the nearest hundredth of a unit.

C = 0.5 cm C = 6 mm

C ≈ 1.57 cm C ≈ 18.85 mm

A = 0.0625 sq. cm A = 9 sq. mm

A ≈ 0.19 sq. cm A ≈ 28.27 sq.mm

10. How many times would a circle with radius 4 units fit inside a circle with radius 12 units, if you could pack

the area tightly with no overlapping and no leftover space?

11. Are all circles similar? Justify your answer.

Yes, because the ratio between the diameter and circumference of every circle is π 12. Are all squares similar? Justify your answer.

13. Are all rectangles scaled versions of each other? Justify your answer.

No, ratios can differ. This is an important question. All regular figures and circles are scaled versions of each other. Rectangles all have the same angles, but consecutive sides are not always in the same ratio.


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14. A circle has an area of . What is its circumference? Round to the nearest hundredth. Show your work.

15. Find the missing angle measures for the figure below. Justify each answer.

∠MOL = 62°, vertical to ∠NOK

∠LON = 118°, supplementary to ∠NOK

∠KOM = 118°, supplementary to ∠NOK

16. Find all the missing angle measures for the figure below. Justify each answer.

17. Draw and label two intersecting lines for which ∠CDE and ∠ADR are vertical angles.

18. Draw and label two intersecting lines for which ∠HOG and ∠GOX are supplementary.

37°

103° E

F

G

C

DA

B

D

E

AC

R


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19. Draw and label ∠ABC and ∠DBE, a pair of complementary angles that are not vertical, adjacent, or

congruent.

20. Identify the shaded angle pairs diagrammed below as vertical, supplementary, or complementary.

21. Use the given relationship to find the missing angle. a. If G is complementary to , and , then must be .

b. If J is supplementary to , and , then must be .

c. If N is vertical to , and , then must be .

22. and are vertical angles. . Find .

23. and are complementary. If is 18 greater than , what is the measure of each angle? and

24. and are adjacent angles. and . What is ?

20°70°

B

E

DA

C


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25. and are supplementary angles. If is 9 less than , what is the measure of each

angle? and

26. In the diagram below, find all missing angles. Justify with the appropriate angle relationship.

Angle Measure Justification

27. Given the measures of the following angles, identify the possible angle relationship(s) and explain.

a. and Complementary, adjacent angles, their sum is 90 .

b. and Congruent angles. Possibly vertical because they share a vertex.

28. For the triangle below, write an equation and solve for x. Then state the measures of and .

29. Determine which of these statements is sometimes (not always) true:

a. Supplementary angles add up to 180 . b. If two lines intersect, each pair of vertical angles are complementary. c. If the measure of an angle is represented by x, then the measure of its complement is represented by

90 – x. For the statements that you chose as “sometimes true,” provide one example of when the statement is true and one example of when the statement is not true. Your examples should be a diagram with the angle measurements labeled. B is sometimes true. The vertical angles could be 45 , but they could also

measure something different such as 64

64°

G

J

HK

I

F

ÐFGH

ÐIGH

ÐKGH

ÐKGJ

Module #6 - San Dieguito Union High School Districtteachers.sduhsd.net/slesan/Math A Honors/Mod 6...

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