chapter 6 - fort thomas independent schools...metòd sa a egzije pou nou fè yon sipozisyon rezonab...

Copyright © Big Ideas Learning, LLC Big Ideas Math Blue All rights reserved. Resources by Chapter

201

Chapter 6 Family and Community Involvement (English) ......................................... 202

Family and Community Involvement (Spanish)......................................... 203

Family and Community Involvement (Haitian Creole).............................. 204

Section 6.1................................................................................................... 205

Section 6.2................................................................................................... 211

Section 6.3................................................................................................... 217

Section 6.4................................................................................................... 223

Section 6.5................................................................................................... 229

School-to-Work........................................................................................... 235

Graphic Organizers / Study Help ................................................................ 236

Financial Literacy........................................................................................ 237

Cumulative Practice .................................................................................... 238

Unit 2 Project with Rubric .......................................................................... 239

Big Ideas Math Blue Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 202

Chapter

6 Square Roots and the Pythagorean Theorem

Name _________________________________________________________ Date _________

Dear Family,

When adding or multiplying small numbers, you rely on tables you memorized long ago. For larger numbers, you follow the rules you’ve learned. For example, when adding large numbers, you line up the place values and start adding from the right, carrying digits to the left.

The “add and carry” method is an example of a rule that follows a strict, predictable procedure. Perhaps surprisingly, not all problems in mathematics have rules that are this straightforward. One of the oldest ways of solving problems is to use the “guess and check” method.

This method requires us to make a reasonable guess about the answer and check how close it is. You then refine your guess and check the new estimate. Each time you do this, you try to get closer to the answer.

Try this with your student to find the square root of a number. For example, to find the square root of 19, you might do the following steps.

• The square root of 16 is 4 ( )=2because 4 16 and the square root of 25 is 5 ( )=2because 5 25 . Because 19 is between 16 and 25, the square root of 19 is greater than 4 and less than 5, so guess 4.5.

• Check: ( ) =24.5 20.25, which is too big, so refine your guess. Try 4.2.

• Check: ( ) =24.2 17.64, which is too small, so refine your guess. Try 4.4.

• Check: ( ) =24.4 19.36,which is getting closer, but still a little too big.

If you continue this method, you will soon find out that ( )≈ 219 4.36 . You could keep going to get the precision you need.

It may appear that computers and calculators have functions like these memorized, because the answers are shown immediately. However, many types of calculations are done using a process very similar to “guess and check”. Because computers and calculators can make millions of guesses per second, the answer simply appears to be memorized.

So don’t be afraid to guess the answer—just remember to check it!


203

Capítulo

6 Raíces Cuadradas y el Teorema Pitagórico

Nombre _______________________________________________________ Fecha_________

Estimada Familia:

Al sumar o multiplicar números pequeños, dependemos de tablas que memorizamos hace muchos años. Para números más grandes, seguimos reglas que hemos aprendido. Por ejemplo, al sumar números grandes, alineamos las posiciones de valores y empezamos a sumar desde el lado derecho, llevando dígitos hacia el lado izquierdo.

El método de “sumar y llevar” es un ejemplo de una regla que sigue un procedimiento estricto y predecible. Quizás, y sorprendentemente, no todos los problemas en matemáticas tienen reglas tan simples como ésta. Una de las formas más antiguas de resolver problemas es usando el método de “predecir y verificar”.

Este método requiere que hagamos una predicción razonable sobre la respuesta y que verifiquemos qué tan cerca estamos. Luego refinamos la predicción y verificamos la nueva aproximación. Cada vez que hacemos esto, estamos más cerca de la respuesta.

Intente esto con su estudiante para hallar la raíz cuadrada de un número. Por ejemplo, para encontrar la raíz cuadrada de 19, pueden hacer los siguientes pasos:

• La raíz cuadrada de 16 es 4 ( )=2porque 4 16 y la raíz cuadrada de 25 es 5 ( ).=2porque 5 25 Ya que 19 se encuentra entre 16 y 25, la raíz cuadrada de 19 es mayor que 4 y menor que 5, entonces predecimos 4.5.

• Verifique: ( ) =24.5 20.25, que es demasiado grande, así que refine su predicción. Intente con 4.2.

• Verificar: ( ) =24.2 17.64, que es demasiado pequeño, así que refine su predicción. Intente con 4.4.

• Verificar: ( ) =24.4 19.36, lo cual está más cerca, pero todavía es un poco más grande.

Si continúa con este método, pronto averiguará que ( )≈ 219 4.36 . Puede continuar para obtener la precisión deseada.

Puede parecer que las computadoras y calculadoras tengan funciones como éstas memorizadas, ya que las respuestas se muestran inmediatamente. Sin embargo, muchos tipos de cálculos se realizan con un proceso muy similar al de “predecir y verificar”. Ya que las computadoras y calculadoras pueden hacer millones de predicciones por segundo, la respuesta simplemente aparece como memorizada.

Así que no tema predecir la respuesta—¡sólo recuerde verificarla!


Chapít

6 Rasin Kare ak Teyorèm Pitagò a

Non __________________________________________________________ Dat __________

Chè Fanmi:

Lè w’ap adisyone oswa miltipliye ti chif, ou fye ou ak tab ou te aprann pa kè sa fè lontan. Pou gwo chif, ou swiv règ ou aprann. Paregzanp, lè w’ap adisyone gwo chif, ou aliyen valè pozisyon yo epi ou kòmanse adisyone apatide bò dwat la, retni chif sou bò gòch la.

Metòd “adisyone ak retni” an se yon egzanp règ ki swiv yon pwosedi estrik, san sipriz. Petèt sa ap fè ou sezi, se pa tout pwoblèm nan matematik ki gen règ ki senp konsa. Youn nan mannyè pi ansyen pou rezoud pwoblèm se sèvi avèk metòd “sipoze ak verifye” a.

Metòd sa a egzije pou nou fè yon sipozisyon rezonab sou repons la epi verifye nan ki pwen li pwòch. Apre sa ou rafine sipozisyon ou an epi ou verifye nouvo estimasyon an. Chak fwa ou fè sa, ou eseye vin pi pre repons la.

Eseye sa avèk elèv ou a pou jwenn rasin kare yon chif. Paregzanp, pou jwenn rasin kare 19, ou gen dwa pase pa etap sila yo.

• Rasin kare 16 se 4 ( )=2paske 4 16 epi rasin kare 25 se 5 ( )=2paske 5 25 . Poutèt 19 nan mitan 16 ak 25, rasin kare 19 pi gran pase 4 ak pi piti pase 5, donk sipoze 4.5.

• Verifye: ( ) =24.5 20.25, ki twò gran, donk rafine sipozisyon ou an. Eseye 4.2.

• Verifye: ( ) =24.2 17.64, ki twò piti, donk rafine sipozisyon ou an. Eseye 4.4.

• Verifye: ( ) =24.4 19.36, ki pi pre, men ki toujou yon ti jan twò gran.

Si ou kontinye metòd sa a, w’ap jwenn byento ke ( )≈ 219 4.36 . Ou ta kapab kontinye ale pou jwenn presizyon ou bezwen an.

Sa gen dwa sanble ke òdinatè ak kalkilatris gen fonksyon tankou sa yo nan memwa yo, poutèt yo montre repons yo imedyatman. Sepandan, anpil tip kalkil fèt avèk yon pwosede ki sanblan anpil ap “sipoze ak verifye.” Poutèt òdinatè ak kalkilatris kapab fè plizyè milyon sipozisyon pa segonn, repons la senpleman sanble li nan memwa li.

Donk ou pa bezwen pè sipoze repons la—annik sonje verifye li!


205

When you know the area of a rectangle, can you determine the lengths of its sides? Why or why not?

When you know the area of a square, can you determine the lengths of its sides? Why or why not?

Find the product.

1. 12 12× 2. 9 9× 3. 18 18×

4. 1.6 1.6× 5. 2.5 2.5× 6. 2 23 3

×

Activity

6.1 Warm Up For use before Activity 6.1

Activity

6.1 Start Thinking! For use before Activity 6.1

A = 64 m2x

y

A = 64 m2x

x


Lesson

6.1 Start Thinking! For use before Lesson 6.1

Shelley says that there are two solutions to the equation 2 400.=x Gina says that there is only one solution. Who is correct? Explain.

Find the side length of the square. Check your answer by multiplying.

1. 2.

3. 4.

Lesson

6.1 Warm Up For use before Lesson 6.1

A = 81 in.2 s

s

A = 169 cm2 s

s

A = 1 yd2 s

s

A = 2.25 m2 s

s


207

6.1 Practice A

Name_________________________________________________________ Date __________


1. 2Area 196 in.= 2. 249Area m81

=

Find the two square roots of the number.

3. 16 4. 0

Find the square root(s).

5. 121 6. 136

−

7. 28949

± 8. 0.64−

Evaluate the expression.

9. 2 25 3+ 10. 17 129

−

Copy and complete the statement with < > =, , or .

11. 64 ? 5 12. 0.6 ? 0.49

13. The volume of a right circular cylinder is represented by 2 ,π=V r h where r is the radius of the base (in feet). What is the radius of a right circular cylinder when the volume is 144π cubic feet and the height is 9 feet?

14. The cost C (in dollars) of producing x widgets is represented by 24.5 .=C x How many widgets are produced if the cost is $544.50?

15. Two squares are drawn. The larger square has area of 400 square inches. The areas of the two squares have a ratio of 1 : 4. What is the side length s of the smaller square?

s

s

s

s


6.1 Practice B

Name _________________________________________________________ Date _________


1. 2169Area cm225

= 2. 2Area 2.56 yd=

Find the two square roots of the number.

3. 225 4. 400


5. 484− 6. 2564

±

7. 6.25 8. 1.69±


9. 6 2.25 4.2− 10. 483 23

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

Copy and complete the statement with < > =, , or .

11. 49 ? 29

12. 2 12 ? 5 75

13. The area of a sector of a circle is represented by 25 ,18

π=A r where r is

the radius of the circle (in meters). What is the radius when the area is 40π square meters?

14. Is the quotient of two perfect squares always a perfect square? Explain your reasoning.

15. Two squares are drawn. The smaller square has an area of 256 square meters. The areas of the two squares have a ratio of 4 : 9. What is the side length s of the larger square?

s

s

s

s


209

6.1 Enrichment and Extension

Name_________________________________________________________ Date __________

Finding Cube Roots A square root of a number is a number that when multiplied by itself, equals the given number. A cube root of a number is a number that when used as a factor in a product three times, equals the given number. The notation for the cube root of n is 3 n .

Complete the table.

1. 2.

Find the cube root of the number.

3. 216 4. 8− 5. 1512

− 6. 64729

7. A CD case is in the shape of a cube. The volume is 343 cubic inches. What is the length (in inches) of one side of the CD case?

8. There are three numbers that are their own cube roots. What are these numbers?

n n2 ( )n2 Check 1 1 1 1 1 1• =

2

3

4

5

n n3 ( )n3 3 Check 1 1 1 1 1 1 1• • =

2

3

4

5


Puzzle Time

Name _________________________________________________________ Date _________

How Did The Man At The Seafood Restaurant Cut His Mouth? Circle the letter of each correct answer in the boxes below. The circled letters will spell out the answer to the riddle.

Find the side length of the square with the given area.

1. Area 169= 2. Area 576=

3. 49Area64

= 4. Area 2.56=


5. 400 6. 225− 7. 916

±

8. 3625

9. 7.84± 10. 56.25−


11. 6 2 81− 12. 53.29 2.89+

13. 21.16 1.69− 14. 25 36749 64

+

15. The bottom of a circular swimming pool has an area of 200.96 square feet. What is the radius of the swimming pool? Use 3.14 for .π

6.1

R E L C A F T M I H N U S B G R D

25 2.8± 10− 7.5 1.6 2.3 34

± 344

13 28 78

3.4±143

5.5− 12− 30 5.2±

S I T W N O P R G D V F I Y S L H

15− 134

6.5− 354

3.4 20 1.8± 65

12 8 1.6− 3.3 24 6.1− 7.5− 14 9


211

Cut three narrow strips of paper that are 3 inches, 4 inches, and 5 inches long.

Form a triangle using the three strips. What kind of a triangle is formed?

Notice that 2 2 23 4 5+ = .

Do you know any other lengths of a triangle that would illustrate a similar equation?


1. 1.44 2. 900± 3. 49

4. 441− 5. 484± 6. 2500−

Activity


Activity



Lesson


How can you use the Pythagorean Theorem in sports?

Find the missing length of the triangle.

1. 2.

3. 4.

Lesson


6 cm

8 cm

c

13 in.

12 in.

a

3.6 m

6 mb

15 ft8 ft

c


213

6.2 Practice A

Name_________________________________________________________ Date __________


1. 2.

3. 4.

5. A small shelf sits on two braces that are in the shape of a right triangle. The leg (brace) attached to the wall is 4.5 inches and the hypotenuse is 7.5 inches. The leg holding the shelf is the same length as the width of the shelf. What is the width of the shelf?

Find the value of x.

6. 7.

8. Can a right triangle have a leg that is 10 meters long and a hypotenuse that is 10 meters long? Explain.

9. One leg of a right triangular piece of land has a length of 24 yards. The hypotenuse has a length of 74 yards. The other leg has a length of 10x yards. What is the value of x?

6 ft

8 ft c

5 cm13 cm

b

2.1 m

2.9 m

a

25 yd

15 yd

b

21 yd20 yd

x

5 cm

6.5 cmx


6.2 Practice B

6 blocks

8 blocksc

8 blocks

Name _________________________________________________________ Date _________


1. 2.

3. 4.

5. You built braces in the shape of a right triangle to hold your surfboard. The leg (brace) attached to the wall is 10 inches and your surfboard sits on a leg that is 24 inches. What is the length of the hypotenuse that completes the right triangle?

6. Laptops are advertised by the lengths of the diagonals of the screen. You purchase a 15-inch laptop and the width of the screen is 12 inches. What is the height of its screen?

7. In a right isosceles triangle, the lengths of both legs are equal. For the given isosceles triangle, what is the value of x?

8. To get from your house to your school, you ride your bicycle 6 blocks west and 8 blocks north. A new road is being built that will go directly from your house to your school, creating a right triangle. When you take the new road to school, how many fewer blocks will you be riding to school and back?

12 mm

35 mm c

8.75 ft

9.25 ft

a

1.5 in.

2.5 in.b

7.25 cm5.25 cm

a

xx

72 cm


215


Name_________________________________________________________ Date __________

The Bermuda Triangle The Bermuda Triangle is in the Atlantic Ocean between Bermuda, Miami, Florida, and San Juan, Puerto Rico. There are many stories about strange events that occur within the Bermuda Triangle.

The Bermuda Triangle is not a right triangle. In order to find the area, you need to use a different method.

1. Find the perimeter of the triangle.

2. The semi-perimeter of a triangle is equal to half the perimeter. Find the semi-perimeter s of the triangle.

3. Find the differences between the semi-perimeter and each side of the triangle, , , and .− − −s a s b s c

4. Use the values you found to evaluate the product ( )( )( ).= − − −R s s a s b s c

5. The area of the triangle is equal to .R What is the area (in square miles) of the Bermuda Triangle?

6. This method of finding the area of a triangle is called Heron’s Formula. Use this method to find the area of the triangle below.

Bermuda

San Juan,Puerto Rico

Miami,Florida

1050 mi

1009 mi

1189 mi

36 m

25 m

29 m


Puzzle Time

Name _________________________________________________________ Date _________

What Did One Dog Say To The Other Dog? Write the letter of each answer in the box containing the exercise number.

Find the hypotenuse c of the right triangle with the given side lengths a and b.

1. 15, 20= =a b 2. 5, 12= =a b

3. 13, 84= =a b 4. 65, 72= =a b

5. 6, 17.5= =a b 6. 26 , 73

= =a b

Find the side length b of the right triangle with the given hypotenuse c and side length a.

7. 61, 11= =c a 8. 82, 80= =c a

9. 34, 16= =c a 10. 65, 63= =c a

11. 13, 6.6= =c a 12. 3 310 , 55 5

= =c a

13. The flap of an envelope has two side lengths that are each 10 centimeters long and meet at a right angle. How long is the envelope? Round your answer to the nearest tenth.

14. A middle school gym is 60 feet wide and 100 feet long. If you stand in one corner of the gym, how many feet away is the corner diagonally across from you? Round your answer to the nearest tenth.

6.2

Answers

T. 293

P. 14.1

E. 18.5

D. 18

N. 25

U. 9

O. 97

H. 116.6

N. 60

G. 30

O. 13

M. 11.2

I. 85

S. 16

10 6 2 13 14 4 12 1 8 3 7 9 11 5


217

An irrational number is a number that cannot be written as a ratio of integers. Decimals that do not repeat and do not terminate are irrational.

Do you know any examples of irrational numbers?

Use the Pythagorean Theorem to find the hypotenuse of a right triangle with the given legs.

1. 30, 40 2. 10, 24

3. 16, 30 4. 9, 40

5. 54, 72 6. 2.5, 6

Activity


Activity



Lesson


How can you find the side length of a square that has the same area as an 8.5-inch by 11-inch piece of paper?

Tell whether the rational number is a reasonable approximation of the square root.

1. 577408

, 2 2. 401110

, 8

3. 271330

, 21 4. 521233

, 5

5. 795153

, 27 6. 441150

, 12

Lesson



219

6.3 Practice A

Name_________________________________________________________ Date __________


1. 277, 3160

2. 590, 17160

Tell whether the number is rational or irrational. Explain.

3. 14− 4. 1.3

5. 2.375 6. 4π

7. You are finding the area of a circle with a radius of 2 feet. Is the area a rational or irrational number? Explain.

Estimate the nearest integer.

8. 33 9. 630

10. 8− 11. 72

12. A swimming pool is in the shape of a right triangle. One leg has a length of 10 feet and one leg has a length of 15 feet. Estimate the length of the hypotenuse to the nearest integer.

Which number is greater? Explain.

13. 70, 8 14. 16, 3−

15. 1210, 164

16. 4 3,25 10

17. Find a number a such that 2 3.<


6.3 Practice B

Name _________________________________________________________ Date _________


1. 2999, 41490

2. 2298, 22490

Tell whether the number is rational or irrational. Explain.

3. 229

4. 2 3π +

5. 2.41 6. 130

7. You are finding the circumference of a circle with a diameter of 10 meters. Is the circumference a rational or irrational number? Explain.

Estimate the nearest integer.

8. 2509

− 9. 395

Estimate to the nearest tenth.

10. 0.79 11. 1.48

12. A patio is in the shape of a square, with a side length of 35 feet. You wish to draw a black line down one diagonal.

a. Use the Pythagorean Theorem to find the length of the diagonal. Write your answer as a square root.

b. Find the two perfect squares that the length of the diagonal falls between.

c. Estimate the length of the diagonal to the nearest tenth.

Which number is greater? Explain.

13. 3220, 144

14. 135, 145− −

15. 7 3,64 8

16. 10.25,4

− −

17. Find two numbers a and b such that 7 8.< <


221


Name_________________________________________________________ Date __________

Approximating Square Roots Before there were calculators and computers, mathematicians developed several methods of approximating square roots by hand. One popular method is sometimes called the divide-and-average method. It uses the following steps.

Use the divide-and-average method to calculate 47.

1. What two perfect squares is 47 between?

2. Let 47.=g Estimate g to the nearest whole number.

3. Find the quotient 47 .= ÷q g Round your answer to two decimal places.

4. Find the average of g and q. This gives the approximate value of 47. To get a closer approximation, you can repeat this process multiple times by using the average as g.

5. Check the accuracy by squaring the average and comparing it to 47. How close are the numbers?

6. Use this method to estimate 30 by repeating the process three times. How close is the square of the estimate and 30?


Puzzle Time

Name _________________________________________________________ Date _________

Did You Hear About...

A B C D E F

G H I J K L

M

Complete each exercise. Find the answer in the answer column. Write the word under the answer in the box containing the exercise letter.

6.3

Estimate to the nearest integer.

A. 195 B. 1220−

C. 306− D. 3156

Which number is greater?

E. 55, 12 F. 83, 9− −

G. 0.75, 0.85− − H. 4 1,9 2

Estimate to the nearest tenth.

I. 137 J. 45.9

K. 342.5 L. 387

M. You are standing 15 feet from a 25-foot tall tree. Estimate the distance from where you are standing to the top of the tree? Round your answer to the nearest tenth.

−9 POLICEMAN

13 DUTY

−18 SAND

−35 LOBSTER

− 0.85 CLAM

2.3 AND

29.2 ORDER

− 0.75 BECAUSE

12

LAWYER

−17 THAT

18.5 CLAW

49

HE

7 BECAME

−34 SEASHELL

6.8 IN

55 OCEAN

11.7 BELIEVED

14 THE

− 83 COURT

12 A


223

Use a ruler and a protractor to draw a regular pentagon with side lengths 1 inch long. (Hint: First find the measure of an interior angle of a regular pentagon.)

Use a ruler to verify that the length of a diagonal of a regular pentagon with 1-inch sides is equal

to the golden ratio, 1 52

+ inch.

Use a calculator to find a decimal approximation of the expression. Round your answer to the nearest thousandth.

1. 77

2. 32

3. 1 32

+ 4. 3 13

−

5. 2 23

+ 6. 2 24

−

Activity


Activity



Lesson


In previous courses, you have learned how to simplify fractions. When is a fraction simplified?

Square roots can also be simplified.

A square root is simplified when the number under the radical sign has no perfect square factors other than 1.

Which of the following expressions are simplified? Explain why.

2 , 4 , 10 , 50 , 3 5 , 3 8

Find the ratio of the side lengths. Is the ratio close to the golden ratio?

1. 2.

Lesson


44 ft

27 ft

310 cm

621 cm


225

6.4 Practice A

Name_________________________________________________________ Date __________

Simplify the expression.

1. 5 2 4 2+ 2. 9 5 4 5−

3. 1 27 73 3

+ 4. 10 8 10−

5. 1 54 4

+ 6. 3 16 6

+

7. The side lengths of a triangle are 4 2, 2, and 5. What is the perimeter of the triangle?


8. 20 9. 32

10. 50 11. 716

12. 1125

13. 33144

14. The area of a square is 24 square centimeters. Find the side length s of the square.


15. 4 3 27+ 16. 50 4 18−

17. The ratio 7 : x is equivalent to the ratio x : 5. What are the possible values for x?

18. You are designing a table in the shape of a right triangle. The side lengths are 20 inches and 10 inches.

a. What is the length of the hypotenuse?

b. You reduce the side lengths by half, resulting in side lengths of 10 inches and 5 inches. What is the length of the hypotenuse?

c. What happened to the length of the hypotenuse when the side lengths were reduced by half?


6.4 Practice B

Name _________________________________________________________ Date _________


1. 9 11 4 11− 2. 3 410 105 5

−

3. 5 315 156 6

+ 4. 7 7−

5. 1 115 5

+ 6. 3 12 2

−

7. The length of a rectangle is 5 3 inches and the width is 2 3. What is the perimeter of the rectangle?


8. 98 9. 300

10. 80 11. 14169

12. 7625

13. 67100

14. The area of a circle is 40π square meters. What is its radius?


15. 98 24

− 16. 128 3 200+

17. The ratio 6 : x is equivalent to the ratio x : 10. What are the possible values for x?

18. You are designing an orange right circular cone to block off a parking space. It has a height of 60 centimeters and a volume of 240π cubic centimeters.

a. What is the radius of the cone?

b. You double the height of the cone to 120 centimeters and the volume of the cone to 480π cubic centimeters. What is the radius of the cone?

c. When the height and volume were doubled, what happened to the radius?


227


Name_________________________________________________________ Date __________

Simplifying Square Roots Simplify the expression. Find the answer in the grid below and write the number of the exercise next to the appropriate dot. When you have completed all twelve exercises, connect the dots in order according to the exercise numbers and connect the last point to the first point. What polygon is formed in the grid?

1. 3 8 3+ 2. 13 2 32+ 3. 9 11 99+

4. 4 7 112− 5. 15 5 80− 6. 3 6 216−

7. 1736

8. 35729

9. 13100

10. ( )( )15 60 11. ( )( )13 52 12. ( )( )( )16 12 27

9 3

17 2

12 11

0

11 5

3 6−

176

1310

30

26

3527

72 9 6

29 2

9 110

4 21−

5 5 3 6

176

3527

0.13900

4 13

8 27

4281


Puzzle Time

Name _________________________________________________________ Date _________

Why Shouldn’t You Give A Little Girl Spaghetti Late At Night? Circle the letter of each correct answer in the boxes below. The circled letters will spell out the answer to the riddle.


1. 5 52 2

+ 2. 8 11 5 11−

3. 1 26 63 3

+ 4. 1 62 25 5

−

5. 3.7 3 1.7 3− 6. 4.8 2 2.2 2+

7. 325 8. 192

9. 40 10. 63

11. 13144

12. 27100

13. 2 3 48+ 14. 54 5 6−

15. 3 12 184 4

+ 16. ( )( )( )15 21 35

6.4

I R T M I A S G P D A L S T A

8 3 8 11 5 13 2 5 2− 6− 3 7 9 33 2

245 7 2 3 8 5

1312

105

S B U E N D O H T F I C M E R

3− 2 6− 25 3 11 7 13 3 310

2 305 2 10 3 6 6 2 13 6 3 2 3 5 2


229

How can you use the Pythagorean Theorem to find the height of a kite?

Find the missing length of the triangle. Round your answer to the nearest tenth.

1. 2.

3. 4.

Activity


Activity


9 cm

6 cm

x

12 in.

10 in.

x

6 m

6 m x

7 ft

15 ft

x


Lesson


Write a word problem that can be solved using the Pythagorean Theorem. Be sure to include a sketch of the situation.

Find the perimeter of the figure. Round your answer to the nearest tenth.

1. Right triangle 2. Right triangle

3. Square 4. Parallelogram

Lesson


6 cm

13 cm

4 in.

6 in.

4 ft4 ft

8 m

3 m 6 m


231

6.5 Practice A

40 ft

14 ft

40 ft

14 ft

x

Name_________________________________________________________ Date __________


1. Right Triangle 2. Parallelogram

Find the distance d. Round your answer to the nearest tenth.

3. 4.

Estimate the height. Round your answer to the nearest tenth.

5. 6.

Tell whether the triangle with the given side lengths is a right triangle.

7. 20 ft, 21 ft, 29 ft 8. 35

m, 1 m, 65

m

9. On the Junior League baseball field, you run 60 feet to first base and then 60 feet to second base. You are out at second base and then run directly along the diagonal to home plate. Find the distance that you ran. Round your answer to the nearest tenth.

5 in.

11 in. c8 cm

10 cm

3 cm

x

y

3

4

5

2

1

04 53210 6 x

y

3

4

5

2

1

04 53210 6

48 m

45 m

x


6.5 Practice B

Name _________________________________________________________ Date _________


1. Parallelogram 2. Square

Find the distance d. Round your answer to the nearest tenth.

3. 4.

Estimate the height. Round your answer to the nearest tenth.

5. 6.

Tell whether the triangle with the given side lengths is a right triangle.

7. 320

cm, 15

cm, 14

cm 8. 4 ft, 9.6 ft, 10.4 ft

9. You are creating a flower garden in the triangular shape shown. You purchase edging to go around the flower garden. The edging costs $1.50 per foot. What is the cost of the edging? Round your lengths to the nearest whole number.

12 in.

15 in.

4 in.

5 m

5 m

x

y

3

4

5

2

1

04 53210 6 x

y

3

4

5

2

1

04 53210 6

18 m

8 m 1.8 m

x

48 ft

xx16 ft

90 m

75 m

x


233


Name_________________________________________________________ Date __________

Making Pythagorean Triples You can generate a Pythagorean triple by picking a value for b and using a system of equations to find a and c.

Let 20.=b

1. Find 2.b

2. Factor b2 into the product of 8 and a number.

3. Write a system of linear equations. Set c a+ equal to the larger factor and c a− equal to the smaller factor.

4. Solve the system of linear equations.

5. Now you have values for a, b, and c. Use the Converse of the Pythagorean Theorem to check that a triangle with these side lengths is a right triangle.

6. Use the same method to generate a Pythagorean triple using

a. 24b = and 18 as a factor of 2.b

b. 15b = and 9 as a factor of 2.b

a

b

c

a² + b² = c²


Puzzle Time

Name _________________________________________________________ Date _________

What Are Twins’ Favorite Kind Of Fruit? Write the letter of each answer in the box containing the exercise number.

1. Your friend sits 4 desks in front of you. The center of each desk is five feet away from the center of the next desk in a row. Your other friend sits 3 seats to your right. The desks going this direction are 4 feet apart from center to center. About how far away from each other are your two friends?

D. 16 feet E. 23.3 feet F. 30.3 feet

2. A ramp used by a moving van has a base that is 8 feet long. The height of the ramp is 5 feet. What is the approximate length of the ramp?

Q. 6.2 feet R. 7.6 feet S. 9.4 feet

3. The shopping mall is 4.6 miles south of your house. Your favorite restaurant is 7.4 miles east of your house. What is the approximate distance between the shopping mall and your favorite restaurant?

A. 8.7 miles B. 9.5 miles C. 10.2 miles

4. A basketball hoop is 10 feet high. The horizontal distance from the free throw line to directly below the backboard is 15 feet. What is the approximate distance from the free throw line to the backboard?

R. 18 feet S. 20 feet T. 22 feet

5. A backyard tool shed has a roof that forms a right angle. The two sides of the roof have the same length. The distance between the lower parts of the two sides of the roof is about 12.8 feet. What is the length of each side of the roof?

N. 7 feet O. 8 feet P. 9 feet

6.5

5 1 3 4 2


235

Chapter

6 School-to-Work For use after Section 6.5

Name_________________________________________________________ Date __________

Carpenter You are working as a carpenter, building the frame for the roof of a house. The frame consists of several sections called trusses. The plan for one truss is shown below. It is important that you construct the truss so that the posts meet the ridge beam at right angles.

1. What is the relationship between the principal rafter, the king post, and half the length of the ridge beam? Show how you can use this relationship to find the length of the king post.

2. The vertical posts are to be evenly spaced along the ridge beam. What is the distance between the king post and each side post? What is the distance between each side post and each lower corner of the truss?

3. You cut a piece of wood for a side post and nail it in place. You then measure and determine that it is 3.5 feet long. Does this side post meet the ridge beam at a right angle? How do you know?

4. What is the length of each strut? Explain how you know.

5. Is the triangle formed by the principal rafters and the ridge beam a right triangle? Justify your answer.

6. Is the triangle formed by the king post, one strut, and half the principal rafter a right triangle? If not, what kind of triangle is it?

8 ft

5 ft

10 ft

principal rafter

side post

ridge beam

strut

king post


Chapter

6 Study Help

Name _________________________________________________________ Date _________

You can use a summary triangle to explain a concept.

On Your Own

Make a summary triangle to help you study these topics.

1. finding square roots

2. evaluating expressions involving square roots

3. finding the length of a leg of a right triangle

After you complete this chapter, make summary triangles for the following topics.

4. approximating square roots

5. simplifying square roots


237

Chapter

6 Financial Literacy For use after Section 6.3

Name _______________________________________________________ Date __________

Organizing a School Dance As a member of the student council, you are responsible for organizing a spring dance for your school. The dance is to be held outside, so it will be necessary to rent a dance floor and tent in addition to hiring a DJ. All other rentals are optional. A list of rental options is given below.

1. How many students are enrolled at your school? Of these students, how

many do you predict will attend the dance?

2. What is the greatest number of dance attendees you think will be on the dance floor at any one time? What size dance floor should you rent? Explain your reasoning.

3. If you have the dance floor set up as a square, what would be the approximate side length? Give your answer to the nearest tenth of a foot. Show your work.

4. What size tent should you rent? If the tents all cover a square area, what is the approximate side length of the square area? Give your answer to the nearest tenth of a foot. Show your work.

5. What, if any, other equipment do you think should be rented? Draw a diagram of how the equipment should be set up for the dance.

6. What is the total cost of putting on the dance? Assume that refreshments will be donated.

7. Based on the cost of putting on the dance, how much should you charge each student for admission? Explain your reasoning.

Dance floor ( )2allow 4 ft per dancer Tents

2192 ft $320 2120 ft $480 2320 ft $530 2350 ft $890 2480 ft $800 2600 ft $1230 2672 ft $1120 2950 ft $1450

DJ (3 hours) $500 Table (5 ft round) $25

Chair $1


Chapter

6 Cumulative Practice

Name _________________________________________________________ Date _________


1. 1625

2. 9 5 3− • 3. 49 22+

4. ( )7 8.2 14.9− 5. 2 65 5

+ 6. 8 3 2−

7. 68 8. 529± 9. 19121

10. 5 153 59 8

• + ÷ 11. 8 4 2÷ − 12. 1 116 2

− +

13. A rectangular prism has side lengths of 15 centimeters, 45 centimeters, and 27 centimeters. What is the volume of the rectangular prism?

Find the missing length of the triangle. Round your answer to the nearest tenth, if necessary.

14. 15. 16.

17. You are 18 feet away from a building that is 45 feet tall. What is the distance from where you stand to the roof of the building?

Copy and complete the statement with , or =.

18. 1.6 ? 2.56− − 19. 7 ? 2.14

20. 1.61 ? 2π

21. The distance between your school and the library is 3 miles. The distance between your home and your school is 10 miles. Is your school closer to your home or the library?

36

15c

7.5

12.5 b

6

12

a


239

Unit

2 Analyzing Stride Length For use after Unit 2

legleg

Stride length

h

A

Name_________________________________________________________ Date __________

Objective Draw and analyze similar triangles to predict your height from your stride.

Materials Yardstick, ruler, protractor, calculator

Investigation 1. Work in a group of 3. Measure the length of your leg from the ground to your hip.

2. Take one normal step and “freeze.” Have a member of your group measure the length from the toe on the back foot to the toe on the front foot. This is your walking stride length.

3. Take one running stride and have someone measure your running stride length.

4. Record the measurements for each person in the group.

Data Analysis 5. Sketch an isosceles triangle to represent the length of your legs and your walking stride. Choose a scale, such as 10 inches (actual stride length) to 2 centimeters (stride length in drawing.)

6. Use the Pythagorean Theorem to calculate h. Then find the measure of .A∠

7. Repeat Steps 5 and 6 for your running stride length.

8. Calculate the ratio height

leg length. Share your ratio with the members

of your group. Find the mean of your group ratios, rounded to the nearest tenth.

9. Gather the measures of A∠ for the walking stride length from the members of your group. Find the mean of these angle measures. Repeat this for the measures of A∠ for the running stride length.

10. You will receive two sets of footprints. Measure the stride lengths. Use the stride lengths, the mean of ,A∠ and the mean of your group’s height : leg length ratio to make a scale drawing for these stride lengths.

11. Measure the leg length on the drawing with a ruler and use your scale to find the actual leg length for these footprints. Calculate the approximate height of the person who left the footprints.

Make a Poster Explain the Investigation. Display your data, scale drawings, and calculations. Describe how you determined the height of the person who made the “mystery” footprints.


Unit

2 Student Grading Rubric For use after Unit 2

Name _________________________________________________________ Date _________

Cover Page 10 points ______ ______

a. Name (4 points) ______ ______

b. Class (2 points) ______ ______

c. Project Name (2 points) ______ ______

d. Due Date (2 points) ______ ______

Investigation 20 points ______ ______

a. Measurements for leg length, walking stride length, and running stride length are shown. (20 points) ______ ______

Data Analysis 120 points ______ ______

a. Includes all scale drawings. Drawing are labeled correctly and drawn to scale. (30 points) ______ ______

b. Shows calculations to find h and the measure of .A∠ (15 points) ______ ______

c. Shows height : leg length ratios and means. (15 points) ______ ______

d. Finds the mean of A∠ for walking stride lengths and running stride lengths. (15 points) ______ ______

e. Accurately measures stride length of footprints and makes accurate scale drawings. (30 points) ______ ______

f. Calculates a reasonable height for the person who made the footprints. (15 points) ______ ______

Poster 50 points ______ ______

a. Includes a description of the investigation, all data, all scale drawings, and calculations. (25 points) ______ ______

b. Describes the process for estimating the height of the person who made the footprints. (15 points) ______ ______

c. Poster is neat and well laid out. (10 points) ______ ______

FINAL GRADE


241

Unit

2 Teacher’s Project Notes For use after Unit 2

Materials Yardstick, ruler, protractor, calculator; You will need to provide the students with a running and walking stride length for another person, such as yourself. Prepare these ahead of time and have enough for each group. For added interest, you can draw actual footprints representing the stride length on newsprint or poster board.

Alternatives Students who are on crutches or unable to walk could measure the strides of others. They might also measure the strides of a jointed doll or a cooperative pet. Are the triangles formed by the stride lengths for a dog the same as for a small human, or different? Could you use them to estimate the size of a bear from its tracks?

Determining size from footprints is used in both forensic medicine and paleontology. The class project might focus on one of these, e.g. “Who left the footprints running away from the crime scene?” or “How tall was the bipedal dinosaur who left these walking footprints?”

Common Errors Students may need help finding a formula for h:

( ) ( )2 2leg length – half of stride .=h

Small children have shorter legs and arms proportionate to their size than adults and adolescents. If students are looking for a shorter stride to use in their measurements, they should not use very small children. You can illustrate this by drawing two stick figures on the board who are the same height, but one with a larger head, longer torso, and shorter legs, and ask which represents an adult and which a toddler. This could spark a discussion about using proportion in drawing.

Note that the actual height of the mystery strider may be more or less than the height students calculate using their model.

Suggestions Explain to students that the footprints of walkers and runners vary: runners, for example, have a deeper imprint at the ball of the foot. You can illustrate this if you have access to sand or soft dirt that two students can cross.

Students use the mean angle measures and ratios to create a model triangle. Then, given a stride length and information as to whether the strider is walking or running, they assume that the unknown strider’s triangle is similar to their model triangle. A class discussion prior to the project about how models are similar (in the mathematical sense) to what they represent will help students grasp the different ways similar figures are (and are not) used in this application.


Unit

2 Grading Rubric For use after Unit 2

Cover Page 10 points

a. Name (4 points)

b. Class (2 points)

c. Project Name (2 points)

d. Due Date (2 points)

Investigation 20 points

a. Measurements for leg length, walking stride length, and running stride length are shown. (20 points)

Data Analysis 120 points

a. Includes all scale drawings. Drawing are labeled correctly and drawn to scale. (30 points)

b. Shows calculations to find h and the measure of .A∠ (15 points)

c. Shows height : leg length ratios and means. (15 points)

d. Finds the mean of A∠ for walking stride lengths and running stride lengths. (15 points)

e. Accurately measures stride length of footprints and makes accurate scale drawings. (30 points)

f. Calculates a reasonable height for the person who made the footprints. (15 points)

Poster 50 points

a. Includes a description of the investigation, all data, all scale drawings, and calculations. (25 points)

b. Describes the process for estimating the height of the person who made the footprints. (15 points)

c. Poster is neat and well laid out. (10 points)

FINAL GRADE

Scoring Rubric A 179-200 B 159-178 C 139-158 D 119-138 F 118 or below

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