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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Name _________________________________________________________ Date _________
Chapter
6
Square Roots and the Pythagorean Theorem
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.
•
(
)
of 25 is 5 (because 52 = 25). Because 19 is between 16 and 25,
The square root of 16 is 4 because 42 = 16 and the square root
the square root of 19 is greater than 4 and less than 5, so guess 4.5.
2
•
Check: ( 4.5) = 20.25, which is too big, so refine your guess. Try 4.2.
•
Check: ( 4.2) = 17.64, which is too small, so refine your guess. Try 4.4.
•
Check: ( 4.4) = 19.36, which is getting closer, but still a little too big.
2
2
2
If you continue this method, you will soon find out that 19 ≈ ( 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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Nombre _______________________________________________________
Capítulo
6
Fecha_________
Raíces Cuadradas y el Teorema Pitagórico
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 porque 42 = 16 y la raíz cuadrada de
(
)
25 es 5 porque 52 = 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.
•
2
Verifique: ( 4.5) = 20.25, que es demasiado grande, así que refine su
predicción. Intente con 4.2.
•
2
Verificar: ( 4.2) = 17.64, que es demasiado pequeño, así que refine su
predicción. Intente con 4.4.
•
2
Verificar: ( 4.4) = 19.36, lo cual está más cerca, pero todavía es un poco
más grande.
2
Si continúa con este método, pronto averiguará que 19 ≈ ( 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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Non __________________________________________________________ Dat __________
Chapít
6
Rasin Kare ak Teyorèm Pitagò a
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 paske 42 = 16 epi rasin kare 25 se 5 paske 52 = 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:
(4.5)2
= 20.25, ki twò gran, donk rafine sipozisyon ou an.
Eseye 4.2.
•
2
Verifye: ( 4.2) = 17.64, ki twò piti, donk rafine sipozisyon ou an.
Eseye 4.4.
•
2
Verifye: ( 4.4) = 19.36, ki pi pre, men ki toujou yon ti jan twò gran.
2
Si ou kontinye metòd sa a, w’ap jwenn byento ke 19 ≈ ( 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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Activity
6.1
Start Thinking!
For use before Activity 6.1
When you know the area of a
rectangle, can you determine
the lengths of its sides? Why
or why not?
A = 64 m2
x
y
When you know the area of a
square, can you determine the
lengths of its sides? Why or
why not?
x
A = 64 m2
x
Activity
6.1
Warm Up
For use before Activity 6.1
Find the product.
1. 12 × 12
2. 9 × 9
3. 18 × 18
4. 1.6 × 1.6
5. 2.5 × 2.5
6.
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2 2
×
3 3
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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 x 2 = 400. Gina says that there is only
one solution. Who is correct? Explain.
Lesson
6.1
Warm Up
For use before Lesson 6.1
Find the side length of the square. Check
your answer by multiplying.
1.
2.
A = 81 in.2
A = 169 cm2
s
s
s
s
3.
4.
A = 1 yd2
s
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s
A = 2.25 m2
s
s
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Name_________________________________________________________
6.1
Date __________
Practice A
Find the side length of the square. Check your answer by multiplying.
1. Area = 196 in.2
2. Area =
49 2
m
81
s
s
s
s
Find the two square roots of the number.
3. 16
4. 0
Find the square root(s).
121
5.
7. ±
289
49
6. −
1
36
8. −
0.64
Evaluate the expression.
9. 2
25 + 3
10. 7 − 12
1
9
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 V = π r 2 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
C = 4.5 x 2 . 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?
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Name _________________________________________________________ Date _________
6.1
Practice B
Find the side length of the square. Check your answer by multiplying.
169
cm 2
225
1. Area =
2. Area = 2.56 yd 2
s
s
s
s
Find the two square roots of the number.
3. 225
4. 400
Find the square root(s).
5. −
484
6.25
7.
6. ±
25
64
8. ±
1.69
Evaluate the expression.
9. 6
2.25 − 4.2
⎛
⎜
⎝
10. 3⎜
⎞
48
− 2 ⎟⎟
3
⎠
Copy and complete the statement with < , > , or = .
11.
49
9
?
2
12.
2
5
?
12
75
5 2
π r , where r is
18
the radius of the circle (in meters). What is the radius when the area is
40π square meters?
13. The area of a sector of a circle is represented by A =
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?
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6.1
Date __________
Enrichment and Extension
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
n
n
1
1
( )
n2
Check
1•1 = 1
1
2.
n
n3
1
1
2
2
3
3
4
4
5
5
3
(n 3 )
Check
1•1•1 = 1
1
Find the cube root of the number.
3. 216
4. − 8
5. −
1
512
6.
64
729
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?
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Name _________________________________________________________ Date _________
6.1 Puzzle Time
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
3. Area =
2. Area = 576
49
64
4. Area = 2.56
Find the square root(s).
5.
400
6. −
225
7. ±
8.
36
25
9. ±
7.84
10. −
9
16
56.25
Evaluate the expression.
11. 6 − 2
13.
81
21.16 −
12.
1.69
14. 7
53.29 +
25
+
49
2.89
36
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 .
R
E
L
C
A
25 ±2.8 −10 7.5 1.6
S
I
−15 3
T
W
1
3
−6.5 5
4
4
N
3.4
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F
T
2.3 ±
O
3
4
P
20 ±1.8
M
I
H
N
13
28
7
8
R
G
D
V
6
5
12
8
4
3
4
U
S
±3.4 4
F
−1.6 3.3
B
G
1
−5.5 −12
3
I
Y
S
R
D
30 ±5.2
L
24 −6.1 −7.5 14
H
9
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Activity
6.2
Start Thinking!
For use before Activity 6.2
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 32 + 42 = 52 .
Do you know any other lengths of a triangle that
would illustrate a similar equation?
Activity
6.2
Warm Up
For use before Activity 6.2
Find the square root(s).
1.
4. −
1.44
441
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2. ±
900
3.
5. ±
484
6. −
4
9
2500
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Lesson
6.2
Start Thinking!
For use before Lesson 6.2
How can you use the Pythagorean Theorem
in sports?
Lesson
6.2
Warm Up
For use before Lesson 6.2
Find the missing length of the triangle.
1.
2.
13 in.
a
c
6 cm
12 in.
8 cm
3.
3.6 m
4.
15 ft
b
8 ft
6m
c
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Name_________________________________________________________
6.2
Date __________
Practice A
Find the missing length of the triangle.
1.
2.
13 cm
c
8 ft
5 cm
b
6 ft
3.
4.
2.1 m
a
25 yd
b
2.9 m
15 yd
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.
x
x
20 yd
6.5 cm
21 yd
5 cm
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?
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6.2
Practice B
Find the missing length of the triangle.
1.
2.
8.75 ft
a
9.25 ft
c
35 mm
12 mm
3.
4.
2.5 in.
7.25 cm
5.25 cm
b
a
1.5 in.
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?
x
x
72 cm
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?
8 blocks
c
6 blocks
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6.2
Date __________
Enrichment and Extension
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.
Bermuda
1050 mi
1189 mi
Miami,
Florida
The Bermuda Triangle is not a right triangle.
In order to find the area, you need to use a
different method.
1009 mi
San Juan,
Puerto Rico
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, s − a, s − b, and 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.
36 m
29 m
25 m
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6.2 Puzzle Time
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. a = 15, b = 20
2. a = 5, b = 12
3. a = 13, b = 84
4. a = 65, b = 72
5. a = 6, b = 17.5
2
6. a = 6 , b = 7
3
Answers
T. 9
2
3
P. 14.1
E. 18.5
D. 18
Find the side length b of the right triangle with the given
hypotenuse c and side length a.
N. 25
7. c = 61, a = 11
8.
c = 82, a = 80
U. 9
9. c = 34, a = 16
10.
c = 65, a = 63
O. 97
11. c = 13, a = 6.6
12.
3
3
c = 10 , a = 5
5
5
H. 116.6
N. 60
13. The flap of an envelope has two side lengths that are each
G. 30
10 centimeters long and meet at a right angle. How long
is the envelope? Round your answer to the nearest tenth.
O. 13
14. A middle school gym is 60 feet wide and 100 feet long.
M. 11.2
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.
10
6
2
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13
14
4
12
1
8
3
I.
85
S. 16
7
9
11
5
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Activity
6.3
Start Thinking!
For use before Activity 6.3
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?
Activity
6.3
Warm Up
For use before Activity 6.3
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
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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?
Lesson
6.3
Warm Up
For use before Lesson 6.3
Tell whether the rational number is a
reasonable approximation of the square root.
1.
577
,
408
2
2.
401
,
110
8
3.
271
,
330
21
4.
521
,
233
5
5.
795
,
153
27
6.
441
,
150
12
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Name_________________________________________________________
6.3
Date __________
Practice A
Tell whether the rational number is a reasonable approximation of
the square root.
1.
277
,
160
3
2.
590
,
160
17
Tell whether the number is rational or irrational. Explain.
3. −
14
4. 1.3
6. 4π
5. 2.375
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.
10. −
33
9.
8
11.
630
7
2
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
15.
210, 16
14. −
1
4
17. Find a number a such that 2 <
16.
4 3
,
25 10
a < 3.
18. Is
1
a rational number? Explain.
9
19. Is
5
a rational number? Explain.
9
20. Is
2
a rational number? Explain.
18
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6.3
Practice B
Tell whether the rational number is a reasonable approximation of the
square root.
1.
2999
,
490
41
2.
2298
,
490
22
Tell whether the number is rational or irrational. Explain.
3. 2
2
9
4. 2π + 3
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. −
250
9
9.
395
11.
1.48
Estimate to the nearest tenth.
10.
0.79
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.
220, 14
15.
7 3
,
64 8
3
4
17. Find two numbers a and b such that 7 <
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14. −
135, − 145
16. − 0.25, −
a <
1
4
b < 8.
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6.3
Date __________
Enrichment and Extension
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 g =
47. Estimate g to the nearest whole number.
3. Find the quotient q = 47 ÷ 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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Name _________________________________________________________ Date _________
6.3 Puzzle Time
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.
18.5
CLAW
HE
7
BECAME
C. −
6.8
IN
55
OCEAN
11.7
BELIEVED
306
D.
1220
13
DUTY
315
6
−18
SAND
Which number is greater?
F. −
55, 12
E.
− 34
SEASHELL
B. −
195
A.
4
9
−9
POLICEMAN
Estimate to the nearest integer.
G. −
0.75, −
0.85
H.
83, − 9
− 35
LOBSTER
4 1
,
9 2
−
CLAM
Estimate to the nearest tenth.
I.
K.
137
J.
45.9
342.5
L.
38
7
2.3
AND
29.2
ORDER
M. You are standing 15 feet from a 25-foot tall tree.
14
THE
−
Estimate the distance from where you are standing
to the top of the tree? Round your answer to the
nearest tenth.
83
0.85
−
0.75
BECAUSE
1
COURT
2
LAWYER
12
A
−17
THAT
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Lesson
6.3b
Warm Up
For use before Lesson 6.3b
Tell whether the number is rational or
irrational. Explain.
3
1.
3.
7
8
5. π
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2.
4.
169
11
12
6. 0.8947368…
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Resources by Chapter
Name_________________________________________________________
Date __________
6.3b Practice
Find the cube root of the number.
1. 125
2. −1
3. − 8
5. 8000
6. 512
7. −
4. −1000
1
64
8. 0.001
Estimate the square root to the nearest tenth.
9.
13.
3
10.
104
14. −
44
91
11. −
15
12. −
15. −
130
16.
83
182
Copy and complete the statement using <, >, or =.
17. −
2
19. −
48
?
−
21. − 3 28
?
− 3 29
65
?
81
18.
53
20. 2
22.
3
35
56
3 9
?
?
?
4π
13
Find the circumference of the circle.
23.
24.
Area = 23π in.2
Area = 119π m2
25. Which cube has a greater edge length? How much greater is it?
Volume = 343 ft3
Cube A
Surface Area = 348 ft2
Cube B
26. The time t (in seconds) it takes an object to fall f feet is represented by the
f
. Estimate the time it takes an egg to fall to the ground from
16
a nest that is 32 feet above the ground. Round your answer to the nearest tenth.
equation t =
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Activity
Start Thinking!
6.4
For use before Activity 6.4
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
1+ 5
inch.
to the golden ratio,
2
Activity
Warm Up
6.4
For use before Activity 6.4
Use a calculator to find a decimal approximation
of the expression. Round your answer to the
nearest thousandth.
1.
3.
5.
7
7
2.
1+
3
2
2+
2
3
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3
2
4.
3 −1
3
6.
2− 2
4
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223
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,
Lesson
6.4
4,
10 ,
50 , 3 5 , 3 8
Warm Up
For use before Lesson 6.4
Find the ratio of the side lengths. Is the ratio
close to the golden ratio?
1.
2.
27 ft
621 cm
44 ft
310 cm
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6.4
Date __________
Practice A
Simplify the expression.
1. 5
2 +4
2
2
3
7
3.
1
3
5.
1
5
+
4
4
7 +
2. 9
4.
6.
7. The side lengths of a triangle are 4
2,
5 −4
5
10 − 8 10
3
1
+
6
6
2, and 5. What is the perimeter
of the triangle?
Simplify the expression.
8.
20
9.
32
10.
50
11.
7
16
12.
11
25
13.
33
144
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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225
Name _________________________________________________________ Date _________
6.4
Practice B
Simplify the expression.
1. 9
11 − 4 11
3.
5
6
5.
1
+
5
15 +
3
6
2.
15
4.
11
5
6.
7. The length of a rectangle is 5
3
4
10 −
5
5
7 −
10
7
3 1
−
2
2
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.
14
169
12.
7
625
13.
67
100
14. The area of a circle is 40π square meters. What is its radius?
Simplify the expression.
15.
8 −
9
4
2
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
Date __________
Enrichment and Extension
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?
3 +8 3
1.
7 −
4. 4
112
17
36
7.
9 3
72
9
2
32
3. 9
11 +
99
5. 15
5 −
80
6. 3
6 −
216
11.
17
6
29
2 +
8.
(15)(60)
10.
2. 13
35
729
9.
(13)(52)
12.
−4
21
12 11
6
17
2
35
27
26
3
6
11 5
30
9 110
35
27
900
13
10
0.13
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4 13
17
6
13
100
(16)(12)(27)
42
81
0
5
5
−3
6
8 27
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227
Name _________________________________________________________ Date _________
6.4 Puzzle Time
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.
5
5
+
2
2
1.
3.
1
3
6 +
2
3
6
4.
3 − 1.7
5. 3.7
11 − 5 11
2. 8
3
1
5
2 −
6
5
2 + 2.2
6. 4.8
7.
325
8.
192
9.
40
10.
63
11.
13
144
12.
27
100
14.
54 − 5
13. 2
3 +
3
4
2 +
15.
I
R
48
1
4
T
18
16.
M
I
A
S
G
8 3 8 11 5 13 2 5 − 2 − 6 3 7 9 3
S
B
− 3 −2 6
U
25
E
N
3 11 7 13
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D
3 3
10
O
2
H
(
2
15
)(
2
6
)(
)
21
35
P
D
A
L
3 2
2
45
7 2
3 8
T
F
I
C
305 2 10 3 6
6
S
5
M
T
A
13
12
105
E
R
2 13 6 3 2 3 5 2
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Activity
6.5
Start Thinking!
For use before Activity 6.5
How can you use the Pythagorean Theorem to
find the height of a kite?
Activity
6.5
Warm Up
For use before Activity 6.5
Find the missing length of the triangle. Round
your answer to the nearest tenth.
1.
2.
12 in.
x
x
9 cm
10 in.
6 cm
3.
6m
4.
x
6m
7 ft
x
15 ft
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229
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.
Lesson
6.5
Warm Up
For use before Lesson 6.5
Find the perimeter of the figure. Round your
answer to the nearest tenth.
1. Right triangle
2. Right triangle
4 in.
6 cm
6 in.
13 cm
3. Square
4. Parallelogram
8m
4 ft
4 ft
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3m
6m
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Name_________________________________________________________
6.5
Date __________
Practice A
Find the perimeter of the figure. Round your answer to the nearest tenth.
1. Right Triangle
2. Parallelogram
8 cm
c
11 in.
10 cm
3 cm
5 in.
Find the distance d. Round your answer to the nearest tenth.
3.
4.
y
5
y
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
0
6 x
0
1
2
3
4
5
6 x
Estimate the height. Round your answer to the nearest tenth.
5.
6.
48 m
x
40 ft
45 m
x
14 ft
Tell whether the triangle with the given side lengths is a right triangle.
7. 20 ft, 21 ft, 29 ft
8.
3
6
m, 1 m, m
5
5
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.
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6.5
Practice B
Find the perimeter of the figure. Round your answer to the nearest tenth.
1. Parallelogram
2. Square
5m
12 in.
5m
15 in.
4 in.
Find the distance d. Round your answer to the nearest tenth.
3.
4.
y
5
y
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
0
6 x
0
1
2
3
4
5
6 x
Estimate the height. Round your answer to the nearest tenth.
5.
6.
90 m
75 m
x
18 m
x
8m
1.8 m
Tell whether the triangle with the given side lengths is a right triangle.
7.
3
1
1
cm, cm,
cm
20
5
4
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.
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16 ft
x
x
48 ft
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6.5
Date __________
Enrichment and Extension
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.
c
Let b = 20.
a
1. Find b 2 .
2
2. Factor b into the product of 8 and a number.
b
a² + b² = c²
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. b = 24 and 18 as a factor of b 2 .
b. b = 15 and 9 as a factor of b 2 .
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233
Name _________________________________________________________ Date _________
6.5 Puzzle Time
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
5
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1
3
P. 9 feet
4
2
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Name_________________________________________________________
Chapter
6
Date __________
School-to-Work
For use after Section 6.5
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.
10 ft
principal rafter
king post
strut
5 ft
side post
ridge beam
8 ft
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?
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Chapter
6
Study Help
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
Date __________
Financial Literacy
For use after Section 6.3
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.
(
Dance floor
allow 4 ft 2 per dancer
)
Tents
192 ft 2
$320
120 ft 2
$480
320 ft 2
$530
350 ft 2
$890
480 ft 2
$800
600 ft 2
$1230
672 ft 2
$1120
950 ft 2
$1450
DJ (3 hours)
$500
Table (5 ft round)
$25
Chair
$1
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.
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Chapter
Cumulative Practice
6
Simplify the expression.
16
25
1.
2. 9 − 5 • 3
4. 7(8.2 − 14.9)
2
6
+
5
5
8. ±
68
7.
10.
5.
5
15
•3+
÷5
9
8
529
3.
49 + 22
6.
8 −3
9.
19
121
12. −
11. 8 ÷ 4 − 2
2
1
1
+
16
2
13. A rectangular prism has side lengths of
and
15 centimeters, 45 centimeters,
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.
c
15
16.
7.5
a
6
36
12
b
12.5
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
4
?
2.1
20. 1.61
?
π
2
21. The distance between your school and the library is 3 miles. The distance
between your home and your school is
to your home or the library?
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10 miles. Is your school closer
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Name_________________________________________________________
Unit
2
Date __________
Project: Analyzing Stride Length
For use after Unit 2
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.)
leg
leg
h
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.
A
Stride length
height
. Share your ratio with the members
leg length
of your group. Find the mean of your group ratios, rounded to the
nearest tenth.
8. Calculate the ratio
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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239
Name _________________________________________________________ Date _________
Unit
2
Student Grading Rubric
For use after Unit 2
Student Teacher
Score
Score
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:
h =
(leg length )
2
– ( half of stride) .
2
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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Big Ideas Math Blue
Resources by Chapter
241
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)
Scoring Rubric
A 179-200
B 159-178
C 139-158
D 119-138
F 118 or below
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)
242 Big Ideas Math Blue
Resources by Chapter
Copyright © Big Ideas Learning, LLC
All rights reserved.