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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
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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!
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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!
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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!
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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
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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
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6.1 Practice A
Name_________________________________________________________
Date __________
Find the side length of the square. Check your answer by
multiplying.
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
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6.1 Practice B
Name _________________________________________________________
Date _________
Find the side length of the square. Check your answer by
multiplying.
1. 2169Area cm225
= 2. 2Area 2.56 yd=
Find the two square roots of the number.
3. 225 4. 400
Find the square root(s).
5. 484− 6. 2564
±
7. 6.25 8. 1.69±
Evaluate the expression.
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
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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
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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=
Find the square root(s).
5. 400 6. 225− 7. 916
±
8. 3625
9. 7.84± 10. 56.25−
Evaluate the expression.
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
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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?
Find the square root(s).
1. 1.44 2. 900± 3. 49
4. 441− 5. 484± 6. 2500−
Activity
6.2 Warm Up For use before Activity 6.2
Activity
6.2 Start Thinking! For use before Activity 6.2
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Lesson
6.2 Start Thinking! For use before Lesson 6.2
How can you use the Pythagorean Theorem in sports?
Find the missing length of the triangle.
1. 2.
3. 4.
Lesson
6.2 Warm Up For use before Lesson 6.2
6 cm
8 cm
c
13 in.
12 in.
a
3.6 m
6 mb
15 ft8 ft
c
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6.2 Practice A
Name_________________________________________________________
Date __________
Find the missing length of the triangle.
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
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6.2 Practice B
6 blocks
8 blocksc
8 blocks
Name _________________________________________________________
Date _________
Find the missing length of the triangle.
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
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6.2 Enrichment and Extension
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
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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
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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
6.3 Warm Up For use before Activity 6.3
Activity
6.3 Start Thinking! For use before Activity 6.3
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Lesson
6.3 Start Thinking! For use before Lesson 6.3
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
6.3 Warm Up For use before Lesson 6.3
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6.3 Practice A
Name_________________________________________________________
Date __________
Tell whether the rational number is a reasonable approximation
of the square root.
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.<
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6.3 Practice B
Name _________________________________________________________
Date _________
Tell whether the rational number is a reasonable approximation
of the square root.
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.< <
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6.3 Enrichment and Extension
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?
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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
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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
6.4 Warm Up For use before Activity 6.4
Activity
6.4 Start Thinking! For use before Activity 6.4
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Lesson
6.4 Start Thinking! For use before Lesson 6.4
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
6.4 Warm Up For use before Lesson 6.4
44 ft
27 ft
310 cm
621 cm
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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?
Simplify the expression.
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.
Simplify the expression.
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?
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6.4 Practice B
Name _________________________________________________________
Date _________
Simplify the expression.
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?
Simplify the expression.
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?
Simplify the expression.
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?
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6.4 Enrichment and Extension
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
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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.
Simplify the expression.
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
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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
6.5 Warm Up For use before Activity 6.5
Activity
6.5 Start Thinking! For use before Activity 6.5
9 cm
6 cm
x
12 in.
10 in.
x
6 m
6 m x
7 ft
15 ft
x
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Lesson
6.5 Start Thinking! For use before Lesson 6.5
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
6.5 Warm Up For use before Lesson 6.5
6 cm
13 cm
4 in.
6 in.
4 ft4 ft
8 m
3 m 6 m
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231
6.5 Practice A
40 ft
14 ft
40 ft
14 ft
x
Name_________________________________________________________
Date __________
Find the perimeter of the figure. Round your answer to the
nearest tenth.
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
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6.5 Practice B
Name _________________________________________________________
Date _________
Find the perimeter of the figure. Round your answer to the
nearest tenth.
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
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6.5 Enrichment and Extension
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
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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
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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
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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
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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
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Chapter
6 Cumulative Practice
Name _________________________________________________________
Date _________
Simplify the expression.
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
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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.
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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
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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.
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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