LE€¦ · and other polygons, even CIRCLES…but that’s for another math at another time. Let’s start with the basics by proving Pythagorean Theorem. Do the following measures

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NAME DATE PERIOD

VOLUME OF A CYLINDER

1.

Volume = ___________________

2.

Volume = ___________________

3.

Volume = ___________________ 4.

Volume = ___________________

5.

Volume = 6908 mm3

Height = ________________

6.

Volume = 1407.4 km3

Radius = _______________

Find the missing value, and remember to draw a picture to assist you with your understanding.

7. Find the amount of wax required to make a candle with radius 22 mm and height 61 mm.

8. Find the height of a cylinder with a volume of 30 in3 and a radius of 1 in.

9. Find the height of a cylinder with a volume of 100 cm3 ad a radius of 2 cm.

10. Find the radius of a cylinder with a volume of 208 cm3 and height of 4 cm.

NAME____________________________________________NAME____________________________________________

PACKAGING PLAN

You have opened up your own business of soda distribution. In order to reduce cost, your team of experts agrees it is ideal for you to package, load, and ship your product, until the brand has grown stability and outsourcing becomes a requirement. You decide to sell your product in individual, double, and family sized cans shaped like cylinders.

INDIVIDUAL

Diameter of can

is 4 inches and

height is 6

inches

DOUBLE

Diameter of can

is 6 inches and

height is 9

inches

FAMILY

Diameter of

can is 9

inches and

height is 12

inches.

A. Find the volume of each size can.

INDIVIDUAL

DOUBLE FAMILY

B. How many times larger is the volume of the Family can compared to that of the individual can?

C. If you are shipping the cans in large boxes for distribution, how many of each size can you fit in a 36 inches CUBED box?

INDIVIDUAL

DOUBLE FAMILY

D. Which of the above will allow you to distribute the GREATEST volume of your product? (Must use

numbers/words to explain)

PERFORMANCE TASK: CROSS COUNTRY TEAM FUNDRAISER

The cross country team is raising money for new uniforms. As a team you decide to sell ice cream cups for $1.50 each. Now you must choose which of the following containers you will serve your product in. As a group, you must make a viable argument with a CLEAR rationale for your recommendation. Use your mathematical knowledge and business sense to support your claim, and remember to use proper unit of measures.

MATH TO SUPPORT:

ARGUMENT:

NAME DATE PERIOD

USING SQUARE UNITS TO FIND RIGHT TRIANGLES

Using the precut squares, find 5 combinations that will create right triangles. Write the area of each square in

their correlating location, as well as the side measure. (See example below) DO NOT duplicate the measures

i.e. 1, 2, 3 and 2,1,3.

Now let’s make some MATHEMATICAL observations.

1. What relationship between each side of the triangle and the CORRELATING squares do you notice?

2. Write a conjecture or equation about how the area of two squares could be used to find the area of

the largest square.

3. Write a conjecture or equation about how the side measures of the two smaller squares could be used

to find the measure of the side of the largest square.

4. Use your conjecture or equation to find the longest side of a triangle that has smaller lengths of 9cm

and 40cm.

PYTHAGOREAN THEOREM GRAPHIC ORGANIZER

The Right Triangle

Hypotenuse:__________________________________________

_____________________________________________________

Leg:_________________________________________________

_____________________________________________________

What is Pythagorean Theorem: A statement about triangles containing a right angle.

The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a

right triangle is equal to the sum of the areas of the squares upon the remaining sides."

In laymen terms: In a right angled triangle: the square of the hypotenuse is equal to the sum of

the squares of the other two sides.

This can also be written as a simple equations which we discovered last week.

a2 + b2 = c2

We can use the equation a2 + b2 = c2 to solve countless problems involving triangles

and other polygons, even CIRCLES…but that’s for another math at another time.

Let’s start with the basics by proving Pythagorean Theorem.

Do the following measures make a right triangle?

a = 8 b = 6 c = 10

Plug them into the diagram to the right to help you make a determination.

Write an equation for the triangle demonstrating Pythagorean Theorem.

Try this out! Is the following proof of Pythagorean Theorem?

What are the measures of each side if the given values are the areas of each square?


What is the area of the missing square?

What are the measures of each side of the triangle?


USE SQUARE UNITS AND EQUATION TO PROVE OF PYTHAGOREAN THEOREM “DUAL DEMO” I DO: The three side measures are 37, 12, and 35

1st: Plug in the side measures (REMEMBER: the largest

measure is ALWAYS the hypotenuse)

2nd: Find the area of each square

a2 = b2 = c2 =

3rd: Write the equations and determine if the statement

is TRUE.

WE DO: The three side measures are 7, 41, and 40

a2 = b2 = c2 =

YOU DO: The three side measures ae 15, 17, and 8

a2 = b2 = c2 =

NAME DATE PERIOD

PROOF OF PYTHAGOREAN THEOREM AND ITS CONVERESE

For each of the three diagrams at the top of the next page: (i) Calculate the area of square A, (ii) Calculate the area of square B, (iii) Calculate the sum of area A and area B, (iv) Calculate the area of square C, (v) Check that: area A + area B = area C

Using the method shown in Example 1, verify Pythagoras' Theorem for the right-angled triangles below:

YES or NO YES or NO YES or NO

The whole numbers 3, 4, 5 are called a Pythagorean triple because 32 + 42 = 52. A triangle with sides of lengths 3 cm, 4 cm and 5 cm is right-angled. Use Pythagoras' Theorem to determine which of the sets of numbers below are Pythagorean triples:

(a) 15, 20, 25 (b) 24, 26, 10

YES or NO YES or NO

(c) 11, 30, 22 (d) 9, 8, 6

YES or NO YES or NO

NAME DATE PERIOD

USING PYTHAGOREAN THEOREM EXIT TICKET

Determine if the following triangles are right triangles.

1. Triangle ABC has the measures of 10cm, 5 cm, and 15cm. Is it a right

triangle?

________________________________

________________________________

_______________________________

YES / NO

2. Determine if the measures create a right triangle.

________________________________

________________________________

________________________________

YES / NO

3. Determine if the measures create a

right triangle.

________________________________

________________________________

________________________________

YES / NO

4. Triangle RST has the side measures of

8 in, 17 in, and 15 in. Is this a right

triangle?

________________________________

________________________________

________________________________

YES / NO

5.

Find the measures of the

missing side of the right

triangle using

Pythagorean Theorem

equation.

6.

If B is equal to or

greater than 20,

which value COULD

NOT be the area of

the largest square?

____________________________

____________________________

____________________________

A. 64 B. 113

C. 49 D. 100

NAME DATE PERIOD

FINDING THE HYPOTENUSE

Calculate the length of the hypotenuse of each of these triangles:

Calculate the length of the hypotenuse of each of the following triangles, giving your answers correct to 1

decimal place.

1. A rectangle has sides of lengths 5 cm and 10 cm. How long is the diagonal of the rectangle?

2. Calculate the length of the diagonal of a square with sides of length 6 cm.

Calculate the length of the side marked x in each of the following triangles:

1. The diagonal of a rectangle is 61meters long. If the length of the rectangle is 60 meters, what is the

width?

2. Find the measure of the height of the triangle to located to the right

Calculate the length of the side marked x in each of the following triangles giving your answer correct to

1decimal place.

NAME DATE PERIOD

FINDING THE MISSING SIDE EXIT TICKET

1. What is the value of b?

A. 46 B. 136 C. 64 D. 8

2. What is the value of, f, the missing side?

A. 25 B. 1225 C. 1513 D. 35

3. A triangle has the side measures of 12, and 5. What is the measure of the triangle’s hypotenuse?

4. The area of the square correlating to the hypotenuse is 160 sq cm. One of the triangle’s side lengths is 9 cm. What is the length of the third side of the triangle rounded to the nearest tenth? (Use the illustration below to assist you.)

5. A square has a diagonal that is 20 m long. What is the measure of the side lengths?

6. EXPLAIN the process that must be used to find the measure of a side of a right triangle if given the measure of the hypotenuse and one side.

______________________________________________________________________________________________

______________________________________________________________________________________________

______________________________________________________________________________________________

_____________________________________________________________________________________________

WHO DID IT?

A burglary took place today on Atlanta Road in Campbell Apartment Community. The victim lives on the third floor of building. There was no forced entry, and the only finger prints located on the door belong to the tenant. With this knowledge the investigators believe the perpetrator(s) used access from the window above the lower levels balconies to obtain entry. **Note: There is a 4 foot deep thorny bush that is along the ENTIRE edge of the 1st level. First floor railing can ONLY support 12½-foot or shorter ladders.

The community is completely gated with NO access in or out with the exception of the main entry. This area is manned 24-hour with a minimum of 3 certified security officers who keep a detailed log of resident and guest entering and exiting the community. This is also supported by a state of the art perimeter video security system. There is NO interior community surveillance.

As the chief investigator taking over, you must determine what took place using your knowledge of Pythagorean Theorem. Use the information and illustrations located below and on the back to help put together the pieces of the puzzle and determine who actually burglarized Ms. Ellis’s apartment.

AERIAL OF THE COMMUNITY

VISUAL OF CRIME SCENE

POSSIBLE SUSPECTS LIST

PAINTER:

Has 12½-foot tall ladder

Working in apartments 111 and 512

CARPENTER:

Has 15-foot ladders

Working in apartments 111, 512, and 833

MAINTENANCE WORKER:

Has 18½-foot ladder

Working in building 100 -500, 1200 -1500

RENOVATOR:

Has 20½ -foot ladder

Working on tennis court

TREE CUTTER:

Has 40-foot ladder attached to vehicle and 18-foot detached ladder

HANDYMAN:

Has 25½-foot ladder

Working in apartments 112, 422, 1212, and 1511

BRICK MASON:

Has a 24½-foot ladder

Working on tennis court

ASPHALT LABORER

No access to ladder

Working on parking area around tennis and basketball courts

OFFICIAL AFFIDAVIT

POLICE REPORT NUMBER

1234567890

DATE

10/19/2017

REFERENCE NUMBER

34567890

REPORTING AGENCY

SMYRNA CITY POLICE AMATEUR POLICE DEPARTMENT

REPORT TYPE/STATUS EVALUATION COMPLAINT/OFFENSE/INCIDENT (SEE REVERSE SIDE FOR SUMMARY)

BURGLARY

LOCATION

CAMPBELL APARTMENT

COMMUNITY

DATE/TIME

10/19/17 14:30 – 15:30

COMPLAINT RECEIVED BY

TIME RECEIVED

16:00

DATE RECEIVED

10/19/2017

NAME OF VICTIM

CHINETTE JOHNSON

AGE

38

D.O.B.

01/23/1979

ADDRESS

123 ATLANTA ROAD APT # 531

SMYRNA, GEORGIA 30080

PHONE NUMBER

(678) 555-1234

SSN

987-654-3210

RACE

BLACK

COLOR HAIR

BLACK

COLOR EYES

DARK BROWN

HEIGHT

5’ 10”

WEIGHT

130 LBS AGE

38 SEX

COMPLEXION

BROWM

COOPERATIVE

IDENTIFYING MARKS

SCAR IN RIGHT EYE

AGENCY REPORTING COMPLAINT

N/A

ADDRESS

N/A

CONTACT PERSON

N/A

PHONE NUMBER

N/A

ADDITIONAL PERSON(S) RELATED TO REPORT

NAME PHONE NUMBER ADDRESS

PROPERTY DAMAGE/LOST (ADDITIONAL ITEMS LISTED IN SUMMARY)

1. CAT-“GRACIE” 4. DRAWER OF MARBLES

2. CAT LITTER BOX 5. A STAINLESS STEEL INSULATED WATER BOTTLE WITH ICE

3. CAT FOOD 6. TWO FUN SIZE SNICKERS

Summary of Complaint/Offense/Incident and Statements: The victim stated she left her home at

approximately 7:30 and returned at approximately 13:00 to feed the cat. She left

no more than an hour later, and returned again (approximately 30) after she

realized she left her wallet. This is when she noticed the missing items.

Resident from Apt# 512 stated that the painter and carpenter entered his home at

approximately the same time. Both did work on the balcony, and both used their

ladder. The maintenance worker was also noticed outside of the balcony located

at apartment 512 with his ladder around the time of the criminal act. No other

workers with ladders were near apartment # 512.

The tree cutter entered the community, but left soon after to cut trees located

outside the community fence which is 9 feet tall.

The asphalt laborer began working on the parking area after the renovator and

brick mason finished work on the 9-foot tennis court wall, accidently blocking

them in. There was a new layer of asphalt laid (19 feet wide) that could not be

disturbed. A resident who chose to remain anonymous stated that she recalled

one or maybe both of the gentlemen trying to use their ladder to exit the courts,

but she did not stay around to see if either made it out.

An unknown individual was noticed by an observer with their ladder

hanging outside of Apt# 422 window approximately 30 minutes before the

observer noticed the same from the handyman. The observer was unsure of the

purpose.

SECURITY LOG

DAY MONDAY DATE OCTOBER, 19, 2015

Guest Time In Destination Officer

Initials

Time Out Officer

Initials

MAINTENANCE WORKER 7:00 CLOCK IN CJ PAINTER 8:03 BUILDING 111 AND 512 CJ 16:13 TL HANDYMAN 10:00 APTS # 112, 422, 1212, and 1511 CJ RENOVATOR 10:15 TENNIS COURTS CJ 15:17 TL BRICK MASON 10:38 TENNIS COURTS CJ TREE CUTTER 11:12 FRONT OFFICE TO CHECK IN CJ 11:21 TL CARPERNTER 11:36 APTS # 111, 512, and 833 CJ TL ASPHALT LABORER 12:37 PARKING AREA BY TENNIS COURT

TL 14:58 TL

ELIMINATION OF SUSPECTS

Explain which 4 individuals you eliminated as suspects. Your explanation must be DETAILED and have MATHEMATICAL supporting evidence (use of Pythagorean Theorem). Be sure to use information you learned about the suspect from the Possible Suspect List, the Official Affidavit, the Security Log, AND, of course, your understanding of Pythagorean Theorem to support you findings.

SUSPECT AND EXPLANATION OF ELIMINATION

1.

2.

3.

4.

5.

6.

7.

YOUR FINAL CONCLUSION

Explain IN DETAIL using mathematical reasoning why you think your remaining suspect committed the crime. You MUST

provide as much supporting evidence as possible for the culprit to be convicted and placed behind bars. Use

information from the Suspect List, Official Affidavit, Security Log, and Pythagorean Theorem to support your case.

Name Date Period

EXIT TICKET: REAL WORLD APPLICATION OF PYTHAGOREAN THEOREM

Solve the following word problems. Use of a calculator is permitted.

1. Jackie leans a 17-foot ladder against the side of her house so that the base of the ladder is 8 feet from the house. How high up the side of the house does the ladder reach? Round your answer to the nearest tenth if necessary.

Diagram:

Steps:

Solution

2. A garden has a length of 24 feet and a width of 18 feet. A fence will extend diagonally from the southwest

corner of the garden to the northeast corner of the garden. How long does the fence need to be? Draw

your diagram, label the sides, show all the steps of your work, and write the solution below. Round your

answer to the nearest tenth if necessary.

Draw your Diagram:

Steps:

Solution

3. Stephanie is planning a right triangle garden. She marked two sides that measure 24 feet and 25 feet. What is the length of side n? Round your answer to the nearest tenth if necessary.

Diagram:

Steps:

Solution

4. A builder needs to add diagonal support braces to a wall. The wall is 16 feet wide and 12 feet high. What is the length of each brace? Round your answer to the nearest tenth if necessary.

Draw your Diagram:

Steps:

Solution

5. The bases on a softball diamond are 60 feet apart. How far is it from home plate to second base? Round your answer to the nearest tenth if necessary.

Draw your Diagram:

Steps:

Solution

NAME DATE PERIOD

FINDING DISTANCE USING A COORDINATE PLANE

Plot the missing places on the graph and be sure to LABEL them.

City Hall (0, 0) Library (10, 9) Johnson’s House (4, -5) Burress’s House (9, -7)

Campbell Apt. (-2. 4) Your House (8, -8) Recreation Center (-9, 7) Felicia’s House (3, -4)

1. Draw a purple right triangle to

find the shortest distance from

Lee’s house to City Hall. Show

your math work in the allotted

space on the back.

2. Draw a blue right triangle to find

the shortest distance from the

library to Campbell Apt. Show

your math work in the allotted

space on the back.

3. Draw a green right triangle to


the Recreation Center to your

house. Show your math work in

the allotted space on the back.

4. Draw an orange right triangle to


your house to Ware’s House.

Show your math work in the

allotted space on the back.

5. Draw a yellow right triangle to


Johnson’s house to the Library.



6. Draw a red right triangle to find

the shortest distance from the

Library to the Recreation Center.



7. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond?

8. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? Draw a diagram to assist you with solving.

9. John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school? Draw a diagram to assist you with solving.

Find the distance between the two points. Use the graph above and a calculator for assistance.

10.

11. 12.

13.

14. 15.

NAME DATE PERIOD

EXIT TICKET: FINDING THE SHORTEST DISTANCE

Find the distance between each pair of points. Use a calculator if necessary.

1.

2.

3.

4.

5. During a football play, Jermaine runs a straight route 40 yards up the sideline before turning around and catching a pass thrown by Miqueen. On the opposing team, Mary who started 20 yards across the field from Jermaine saw the play setup and ran a slant towards Jermaine. What was the distance the Mary had to run to get to the spot where Jermaine caught the ball?

Illustrate the problem: Set Up the problem:

Solve the problem:

NAME DATE PERIOD

USING THE PYTHAGOREAN THEOREM CONSTRUCTED RESPONSE

Draw and label a picture to represent the situation and then write an equation to represent the situation. Solve

your equation and write your answer in a complete sentence.

1. Draw a right triangle. One of the legs measures 8cm. The other leg measures 6cm. What is the length of

the hypotenuse?

ILLUSTRATE

SHOW WORK

2. A 25 foot ladder is resting against a wall. The base of the ladder is 15 feet from the base of the wall.

How high up the wall will the ladder reach?

ILLUSTRATE

SHOW WORK

3. A television screen measures 18” tall by 24” wide. All televisions are advertised by giving the

approximate length of the diagonal of the screen. (For example: A 48” television means that the

diagonal of the television measures 48”.) How should the television in this example be advertised?

ILLUSTRATE

SHOW WORK

4. Carl lives 12 miles east of the school. Bill lives five miles north of the school. What is the shortest

distance between the two houses?

ILLUSTRATE

SHOW WORK

NAME DATE PERIOD

UNIT 3 CULMINATING TASK

Find the exact area (in square units) of the figures below. Explain your method(s).

1.

2.

Remember to show your work!

Explain your method(s).

Find the areas of the squares on the sides of the triangle below.

a. How do the areas of the smaller squares compare to the area of the larger square?

b. If the lengths of the shorter sides of the triangle are a units and b units and the length of the longest

side is c units, write an algebraic equation that describes the relationship of the areas of the squares.

c. This relationship is called the Pythagorean Theorem. Interpret this algebraic statement in terms of the

geometry involved (Write an equation).

____________________

4. What is the relationship between the areas of the regular hexagons constructed on the sides of the right

triangle below?

5. Does the Pythagorean relationship work for other polygons constructed on the sides of right triangles?

Under what condition does this relationship hold?

6. Why do you think the Pythagorean Theorem uses squares instead of other similar figures to express the

relationship between the lengths of the sides in a right triangle?

Criterion B: Investigating patterns Maximum: 8 At the end of year 3, students should be able to:

i. select and apply mathematical problem-solving techniques to discover complex patterns

ii. describe patterns as relationships and/or general rules consistent with findings

iii. verify and justify relationships and/or general rules.

Achievement level

Level descriptor

0 The student does not reach a standard described by any of the descriptors below.

1–2 The student is able to:

i. apply, with teacher support, mathematical problem-solving techniques to discover simple

patterns

ii. state predictions consistent with patterns.


i. apply mathematical problem-solving techniques to discover simple patterns

ii. suggest relationships and/or general rules consistent with findings.



ii. describe patterns as relationships and/or general rules consistent with findings

iii. verify these relationships and/or general rules.



ii. describe patterns as relationships and/or general rules consistent with correct findings

iii. verify and justify these relationships and/or general rules.

IB SCORE: GRADING SCALE SCORE______________

TEACHER COMMENT:

Name Date Period _______

UNIT 3 APPLICATIONS OF EXPONENTS STUDY GUIDE

MCC8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

1). Use the Pythagorean Theorem to find the approximate distance between (0,1), (5,4).

a) 5.2 b) 5.8 c) 6.1 d) 6.4

2). Plot (-3,-3), (-1,5) and then find the shortest distance between the two points..

MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and

mathematical problems in two and three dimensions.

3) The farmers’ market sells handmade quilts. The quilts are rectangles 9 feet wide and 10 feet long. What is the

length of the diagonal of a quilt to the nearest tenth of a foot?

a) 12 feet b) 13.5 feet c) 15 feet d) 15.4 feet

9 ft

10 ft

4) The sides of A, B, and C meet to form a right triangle, as shown below.

If square A has an area of 35 square centimeters and square B has an area of 85 square centimeters, what is the area of

C?

a) 45 square centimeters b) 50 square centimeters c) 60 square centimeters d) 120 square centimeters

MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

Side a =5 cm; b=12 cm and c=13cm.

5) Calculate the areas of the three squares above.

6) How does the area of the largest square (c) relate to the two smaller squares?

7) Complete the following table:

a Area of A b Area of B Area of C c

6 8

5 4

9 10

1 4

9 36

A

C

B

MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and

mathematical problems in two and three dimensions.

8.) In the figure above, AB and CD are perpendicular. What is the measure of side DB? What is the measure of side

AC? What is the perimeter of triangle ABC?

9.) Sophia used an 8 foot rope to secure a 6 foot tent pole as shown above. Approximately how far from the base of the

pole is the rope tied?

a) 5 feet b) 7 feet c) 10 feet d) 14 feet

MCC8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and

mathematical problems

10.) A cylindrical glass vase is 6 inches in diameter and 12 inches high. There are 3 inches of sand in the vase, as shown

below.

Which of the following is the closest to the volume of sand in the vase?

a) 54in3 b). 85 in3 c). 254 in3 d). 339in

11 and 12. Find the volume of the cone and sphere below.

a= 6mm

a = 10 mm

b = 14 mm

Documents

LE€¦ · and other polygons, even CIRCLES…but that’s for another math at another time. Let’s start with the basics by proving Pythagorean Theorem. Do the following measures