POLY4 POLYNOMIALS Student Packet 4: Polynomial Arithmetic Applications

Transcription

Name ___________________________
Period __________
Date ___________
POLY4
od
POLYNOMIALS
Student Packet 4: Polynomial Arithmetic Applications
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STUDENT PAGES
Hundred Chart Patterns 2
• Gather empirical data to form conjectures about number
patterns.
• Write algebraic expressions.
• Practice polynomial arithmetic.
• Use algebraic expressions to prove (or disprove) conjectures.
• View algebra as a useful mathematical tool.
1
POLY4.2
Picture Frames
• Use mathematical reasoning to create polynomial expressions
that generalize patterns.
• Practice polynomial arithmetic.
7
POLY4.3
Number Tricks 2
• Use algebraic expressions to generalize patterns.
• Practice polynomial arithmetic
• Write verbal expressions as algebraic expressions.
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POLY4.1
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11
Vocabulary, Skill Builders, and Review
16
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POLY4.4
Polynomials (Student Packet)
POLY4 – SP
Polynomial Arithmetic Application
WORD BANK (POLY4)
Definition or Explanation
Example or Picture
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Word
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conjecture
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deductive
reasoning
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generalization
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empirical
evidence
inductive
reasoning
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proof
POLY4 – SP0
4.1 Hundred Chart Patterns 2
HUNDRED CHART PATTERNS 2
Set (Goals)
•
Go (Warmup)
Example:
5
6
7
Find the
“inner product.”
Find the
“outer product.”
6  7 = 42
5  8 = 40
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Pick any four
consecutive numbers
on the hundred chart.
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•
•
•
Gather empirical data to form
conjectures about number patterns.
Write algebraic expressions.
Practice polynomial arithmetic.
Use algebraic expressions to prove (or
disprove) conjectures.
View algebra as a useful mathematical
tool.
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•
We will investigate patterns on the hundred
chart. We will write and use algebraic
expressions to prove conjectures based on
the patterns.
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Ready (Summary)
8
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1. Compare the products:
3.
4.
21
22
23
21  22 = _____
____ • ____ = ______
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5.
20
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2.
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Try this for at least four other groups of four consecutive numbers.
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6. Write a conjecture about this pattern as a complete sentence.
POLY4 – SP1
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HUNDRED CHART
2
3
4
5
6
7
8
9
11
12
13
14
15
16
17
18
19
21
22
23
24
25
26
27
28
31
32
33
34
35
36
41
42
43
44
45
46
51
52
53
54
55
61
62
63
64
71
72
73
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29
30
39
40
47
48
49
50
56
57
58
59
60
65
66
67
68
69
70
75
76
77
78
79
80
90
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38
N
pl
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20
37
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74
10
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1
82
83
84
85
86
87
88
89
91
92
93
94
95
96
97
98
99 100
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81
POLY4 – SP2
PROVING A CONJECTURE
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1. Conjecture that the class is going to prove:
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2. Prove your conjecture by using algebra to label each of the four consecutive numbers and
then by multiplying the inner and outer products.
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________ ________ ________ ________
Inner Product
Outer Product
__________________
___________________
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__________________
( ________ )( ________ )
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( ________ )( ________ )
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3. Why does this prove the conjecture?
POLY4 – SP3
A SECOND CONJECTURE
1.
5
Find the product of the
diagonal that begins in the
upper left corner
Find the product of the
diagonal that begins in the
upper right corner
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Pick any four numbers on
the hundred chart that form
a 2 x 2 square.
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Now try a different experiment from the hundred chart with four numbers that form a
2 × 2 square.
6
5 • 16 = _____
16
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15
6 • _____ = _____
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2. Compare them:
Try this for at least three other groups of four numbers that form a 2 × 2 square.
3.
22
23
32
33
5.
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4.
____ • ____ = ______
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22  33 = _____
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6. What do you notice in all of these examples? Make a conjecture using a complete
sentence.
POLY4 – SP4
A SECOND CONJECTURE (continued)
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1. Conjecture that the class is going to prove:
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2. Prove your conjecture by using algebra to label each of the four numbers and then by
multiplying each diagonal.
Find the product of the diagonal that begins
in the upper right corner.
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Find the product of the diagonal that begins
in the upper left corner.
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( ________ )( ________ )
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__________________
( ________ )( ________ )
___________________
___________________
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3. Why does this prove the conjecture?
POLY4 – SP5
A THIRD CONJECTURE
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Experiment with patterns like these four numbers on a 3 x 3 square. Multiply the vertical
numbers. Multiply the horizontal numbers. Do this for at least three more 3 x 3 squares.
5
16
od
14
1.
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25
Horizontal
Product
14  16 = 224
Vertical
Product
5  25 = _____
3.
4.
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2.
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5. Write your conjecture in words.
6. Prove your conjecture algebraically:
Vertical Product
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Horizontal Product
7. Does your conjecture hold?
POLY4 – SP6
4.2 Picture Frames
PICTURE FRAMES
Set (Goals)
•
Go (Warmup)
Write the formula for finding the area of each:
a. A square with side length s.
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1.
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Use mathematical reasoning to create
polynomial expressions that generalize
patterns
• Practice polynomial arithmetic
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We will create polynomial expressions that
generalize a geometric pattern, and
simplify the expressions. We will use our
understanding of the construction of each
pattern to verify the accuracy of the
polynomial we created.
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Ready (Summary)
b. A rectangle with a base, b, and a height, h.
For this square-inside-of-a-square picture:
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c. A circle with radius r.
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a. Write in words the steps you could take to find the area
of the shaded region.
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b. Find the area of the shaded region if the side length if the larger square is 8 inches
and the side length of the smaller square is 5 inches.
3.
Find the area of a circle with a radius of 7 cm. Leave the answer in terms of π .
POLY4 – SP7
4.2 Picture Frames
A SQUARE PICTURE FRAME PATTERN
square 9
square 10
square n
8
8
9
9
Un-simplified
expression for shaded
area (picture frame)
5
Simplified
expression for
shaded area
82 − 52
= 64 − 25
39
6
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10
Side length
of inner
square
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Side length
of outer
square
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Square
pattern
number
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square 8
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1. Study the given square picture frame pattern, and fill in the first 3 rows of the table.
2. Sketch the picture for square n.
3. Fill in the last row of the table to find a generalized expression for the shaded area.
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4. Substitute the values from squares 8, 9, and 10 into the simplified expression for the
shaded area of the n th figure as a check.
POLY4 – SP8
4.2 Picture Frames
A CIRCULAR PICTURE FRAME PATTERN
circle 9
Outer
circle
radius
Inner
circle
radius
7
7
3
8
8
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Shaded
area
(simplified)
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9
Shaded area
expression
(un-simplified)
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Circle
pattern
number
circle n
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circle 8
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circle 7
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1. Study the given circle pattern and fill in the first 3 rows of the table. Leave the answers in
terms of π .
2. Sketch circle n.
3. Fill in the last row of the table to find a generalized expression for the shaded area of this
circle pattern.
n
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4. Substitute the values from circles 7, 8, and 9 into the simplified expression for the shaded
area of the n th figure as a check.
POLY4 – SP9
4.2 Picture Frames
A RECTANGULAR PICTURE FRAME PATTERN
rectangle 5
rectangle 6
rectangle n
4
4×7
5
5×8
Shaded
area
(simplified)
2× 4
3×5
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n
Shaded area
expression
(un-simplified)
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Inner
rectangle
dimensions
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Outer
rectangle
dimensions
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Rectangle
pattern
number
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rectangle 4
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1. Study the given rectangle pattern and fill in the first 3 rows of the table.
2. Sketch rectangle n.
3. Fill in the last row of the table to find a generalized expression for the shaded area of this
rectangle pattern.
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4. Substitute the values from rectangles 4, 5, and 6 into the simplified expression for the
shaded area of the n th figure as a check.
POLY4 – SP10
4.3 Number Tricks 2
NUMBER TRICKS 2
Set (Goals)
•
•
Use algebraic expressions to
generalize patterns.
Practice polynomial arithmetic
Write verbal expressions as algebraic
expressions.
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•
Go (Warmup)
1. Perform the number trick below.
Directions
Numbers
Choose a natural number
between 1 and 10.
2
Multiply your number by 2.
3
Add 8 to your answer.
4
Divide your answer by 2.
Algebraic Process
n
2n
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1
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Step
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We will perform mathematical number
tricks and use algebraic expressions to
show how they work.
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Ready (Summary)
5
Subtract your original number
from your answer.
6
What number did you end with?
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2. What is the number trick?
3. Explain why the algebraic process supports that this trick will work for all numbers?
POLY4 – SP11
4.3 Number Tricks 2
NUMBER TRICK 1
Numbers
Algebraic Process
n
Choose a number.
2
Multiply your number by one
more than the original number.
3
Add your original number.
4
Add 1.
5
Divide by 1 more than your
original number
n(n + 1) = ________
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1
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Words
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Step
Subtract 1
7
What number do you have now?
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6
POLY4 – SP12
4.3 Number Tricks 2
NUMBER TRICK 2
Words
Numbers
Algebraic Process
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Step
n
Choose a number.
2
Add 4.
3
Multiply by your original
number.
4
Multiply by 4.
5
Divide by your original number.
6
Subtract 16.
7
Divide by 4.
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1
n+4
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n(n + 4) =
________
8
What number do you have
now?
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POLY4 – SP13
4.3 Number Tricks 2
MORE NUMBER TRICKS
Words
Numbers
Algebraic Process
Choose a number.
n
2
Square your number.
n2
3
Add three more than four times
your original number
4
Divide by 1 more than your
original number.
5
Subtract the original number.
6
What is the result?
Step
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n 2 + 4n + 3
Algebraic Process
n
o
Choose a number.
Numbers
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1
Words
N
od
1
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Step
Square it.
3
Subtract 4.
4
Divide by 2 less than your
original number.
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2
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5
6
Subtract your original number
What is the result?
POLY4 – SP14
4.3 Number Tricks 2
NUMBER TRICK TEMPLATE
Words
Numbers
Algebraic Process
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Step
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Use this page to create your own number trick.
POLY4 – SP15
4.4 Vocabulary, Skill Builders, and Review
FOCUS ON VOCABULARY (POLY 4)
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Choose a number.
Add 4.
Multiply by 2.
Subtract 8.
Divide by your original number.
The result is…
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•
•
•
•
•
•
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Here is a number trick.
Felicity did some work to verify this trick. Match her work with the word or words that
describe what she did. You may use a word more than once.
First she tried it with numbers.
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1. _____
A. conjecture
5  9  18  10  2
B. deductive reasoning
12  16  32  24  2
“I think the result is always going
to be 2.”
Then she drew these pictures.
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3._____
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_____
Then she said,
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2._____
C. empirical evidence
D. generalization
E. inductive reasoning
_____
_____
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4._____
Then she said,
“This shows that the result is
always going to be 2.”
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_____
F. Proof
POLY4 – SP16
SKILL BUILDER 1
1.
________________
25 x 2 − 15 x =
2.
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Use an area model to factor. First find the GCF of the terms.
4 x 2 + 12 x − 20 ______________
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GCF
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When factoring a binomial:
• First look for the GCF of all the terms
• Then look for the “difference of two squares” pattern
N
Factor completely. If it cannot be factored, write “not factorable.”
3.
2x2 + 8
4. 4 x 2 − 16 x
5. 12 x 2 − 7
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When factoring a trinomial:
• First look for the GCF of all the terms
• Then look for a quadratic trinomial where coefficient of the square term is 1
• Then look for a quadratic trinomial where the coefficient of the square term is not 1
7. 3 x 2 + 12 x − 9
8. 4 x 2 + 7 x − 5
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6. 5 x 2 + 15 x − 20
POLY4 – SP17
Rewrite each expression as a power of 2.
1.
2.
2
(42) (42)
43
3.
4.
6.
( )
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5.
4 •2
24 • 45
7
7
2
4 3 22
(82 )3
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8
83
1
16
od
-8
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SKILL BUILDER 2
9.
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Evaluate when a = 1 , b = -2, c = -2
8.
7.
-b
b 2 – 4ac
2a
10.
11.
-b − b 2 − 4ac
2a
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-b + b 2 − 4ac
2a
b 2 – 4ac
2a
POLY4 – SP18
SKILL BUILDER 3
y
-6
1
-3
2
0
3
3
4
6
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x
0
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Find equations of lines in different forms. Use the information given.
1. Given: (graph)
2. Given (table):
slope-intercept form
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slope-intercept form
point-slope form
point-slope form
standard form
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standard form
Evaluate each expression if x = - 3
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2
3. x4
4. x3
5. x2
6. x1
7. x0
8. x-1
9. x-2
10. x-3
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11. Examine your answers to problems 5-12. Under what conditions is the result
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positive?
negative?
a fraction between 0 and 1?
POLY4 – SP19
Simplify each expression.
1.
2.
44
3.
9
121
5.
6.
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4.
120
48
75
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−
3 12
13
49
N
9
8.
(2 + 2 )
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4( 2 + 2 )
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7.
10.
11.
9.
(2 + 2 )(2 − 2 )
2
3
24
6
−5 + 52 − 4(2)(3)
2(2)
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16 − 421
8
12.
2
 40 
16

 +
9
 5 
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A radical expression containing square roots is simplified when there are:
• no perfect squares under the radical
• no fractions under the radical
• no radicals in the denominator.
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SKILL BUILDER 4
POLY4 – SP20
SKILL BUILDER 5
Write each polynomial as a sum of terms in decreasing order of powers.
3. x – 2(x + 3)(x – 2)
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2. 6 – (x – 2)(x – 3)
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1. 5 + (x + 3)(x – 3 )
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4. x2 – 7x + 10
5. x2 + 10x + 25
6. x2 - 64
8. x2 + 15x + 26
9. 2x2 + 8x + 3
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7. -50x + 30
Use
> , < , or =
to make each statement true. Show work.
−1
−4
10.
11.
_____ 4
73 • 7 −3 _____
7
7 −3
1
2
1
2
(9 • 16) _____ 9 16
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12.
3
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Determine which numbers are in scientific notation. If NOT, write it in scientific notation.
13.
8.21 x 105
14.
0.213 x 10-4
POLY4 – SP21
1
2
SKILL BUILDER 6
Simplify each radical expression.
2.
3.
59 ? _____ and _____
4.
6 4
49
14
)(
25 −2 − 25
)
6.
27
7.
−
49
8.
32
50
9.
4
10.
17
81
N
3 12
2 18
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5.
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6− 4
( −2 +
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1. Between which two consecutive integers is
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36
30 − 32
24
12.
-3 ≥ 33x + 8
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Solve and graph:
11.
-2 x + 7 ≤ 5
POLY4 – SP22
SKILL BUILDER 7
1

6  x − 1 + x = -36
3

0.5 x + 4 = -2 x − 6
2.
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1.
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Solve for x. Check by substitution.
4. Write the equation in slope-intercept
form.
N
Given graph:
6. State the x- and y- intercepts.
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5. Write the equation in standard form.
Compute:
(-3)2
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7.
-12(-6 + 4)
5 – 32
9.
-32
11.
36 – 12 (-6 + 4)
12.
36 ÷ 12(-6 + 4)
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10.
8.
POLY4 – SP23
SKILL BUILDER 8
od
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Simplify each expression by combining like terms. Write each polynomial as a sum.
1. 2 x − 5 y + 6 x − 8 y
2. -13 − x + 5 y − 7 − 2 x
( xy − 4 x 2 ) − (3 x 2 − 5 xy )
4.
6 x − 9 − 2( x − 7)
5.
-5( x − y ) − 3( y − x )
6.
x(2 x + y ) + y (2 x + y )
7.
( x − 6)( x + 6)
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3.
( x − 7)( x − 10)
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8.
( x − 3)(- x + 8)
10. (5 − x )( x + 9)
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8.
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11. Write a variable expression for the area of a square whose side is x + 8.
POLY4 – SP24
SKILL BUILDER 9
x is less than -2
2.
x is greater or equal to than -2
3.
the opposite of x is less
than or equal to -3
4.
3 is less than x
5.
-2 is less than or equal
to the opposite of x.
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1.
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Write each statement using symbols. If the variable is on the right side, change it to the
left side using appropriate properties. Then graph each.
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Graph each inequality. Be sure they are in slope-intercept form first.
6.
7.
2
y > x +1
-2y + x ≥ 10
3
8. Describe the differences between the graph of an inequality in one variable and the
graph of an inequality in two variables.
POLY4 – SP25
SKILL BUILDER 10
2.
15 x 2 − 35 x
3.
4.
x 2 + 10 x + 9
5.
x 2 − 4 x − 77
6.
7.
x 2 − 15 x + 26
8.
x 2 − 30 x + 200
10. 10 x 2 + 80 x + 160
11.
7 y − 49 y 2
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8 x 2 + 40 x
x 2 + 9 x − 36
9.
x 2 + 2x + 6
2 x 2 − 4 x − 30
12.
3 x 2 + 18 x − 3
15.
x 2 − 10
18.
x 2 − 24 x − 144
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1.
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Factor using any method. Be sure to factor each polynomial completely. If it cannot be
factored write “not factorable.”
x2 − 9
14. 2 x 2 − 50
16.
−3 x 2 − 18 x
17.
x 2 + 24 x + 144
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13.
POLY4 – SP26
SKILL BUILDER 11
Graph each system of inequalities. Test several points to verify correct shading.
y ≤ -3x + 5
2.
1
y – 3 <
-2
x
5
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1.
2x + y ≥ -4
y > - x
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Complete the table.
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N
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3
Fraction
Ex.
0.009
9
1,000
1.
0.0028
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Decimal
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2.
3.
4.
Product of a number between
1 and 10, and a multiple of 10
9 × 0.001 or 9 ×
1
103
Scientific notation
9 × 10-3
4.76
100
3.5 ×
1
107
4.2 × 10 −3
POLY4 – SP27
SKILL BUILDER – POLY 12
6x – 2y = -16
4x + y = 1
2.
4x = 6 + y
1
2x – y = 3
2
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1.
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Solve each system using algebra.
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Arnon wants to record the 12-hour opera marathon on the radio. He has 90-minute
discs and 60-minute discs. If he uses 9 discs, how many of each type will he use?
3. Solve the algebraically problem using
4. Solve the problem algebraically using
one variable.
two variables (a system of equations).
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5. Could you solve this problem without algebra? Explain.
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6. Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the
original number. The result is always 3. Use polynomials to illustrate this number
trick.
POLY4 – SP28
TEST PREPARATION (POLY4)
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Show your work on a separate sheet of paper and choose the best answer.
n + 10 + 4
B.
n+5
C.
n + 14
D.
n + 54
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A.
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1. Below is an excerpt from a hundreds chart. If 40 = n , which expression(s)
represent(s) 54?
40 41 42 43 44 45
50 51 52 53 54 55
60 61 62 63 64 65
A.
2n + 1
C.
n2 + 1
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2. Below is a 3 × 3 square taken from a hundreds chart. What expression (in terms of
n) should go in the square with the question mark?
n
B.
3n + 1
D.
n + 21
?
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3. Below is a 2 × 2 square taken from a hundreds chart. What expression (in terms of
n) represents ab.
a
b n
(n + 1)(n + 10)
(n − 1)(n − 10)
A.
B.
C.
(n + 1)(n − 10)
D.
(n − 1)(n + 10)
4. If n is a number, what is an expression for four more than four times a number.
4n + 4
B.
n(4 + 4)
C.
(4 + 4) × n
D.
4+4+n
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A.
Sa
5. The radius of the smaller circle is given. If the radius of the larger circle is twice the
radius of the smaller circle, what is the area of the shaded part?
A.
8π
B.
36π
C.
64π
D.
48π
4
POLY4 – SP29
KNOWLEDGE CHECK (POLY4)
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4.1 Hundred Chart Proofs 2
1. Below is an excerpt from a hundreds chart. If 52 = n , what does 62 represent? Give
the most simplified answer.
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2.
1
11
4
14
1× 14 =
14
36
46
39
49
36 × 49 =
1,764
46 × 39 =
1,794
74
84
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11× 14 =
44
od
40 41 42 43 44 45
50 51 52 53 54 55
60 61 62 63 64 65
77
87
74 × 87 =
6,438
84 × 77 =
6,468
N
If the following table follows the pattern above, write a conjecture for the relationship
between ad and bc.
b
d
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a
c
4.2 Picture Frames
Use the following picture to answer questions 3 and 4.
3. Write an expression for the area of the smaller rectangle.
4
n
n+3
7
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4. Write an expression for the area of the shaded part.
4.3 Proving Number Tricks 2
5. If n is a number, write an expression for “the product of a number and one more than
the number.”
POLY4 – SP30
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This page is left blank intentionally.
POLY4 – SP31
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This page is left blank intentionally.
POLY4 – SP32
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HOME-SCHOOL CONNECTION (POLY4)
For problems 1 and 2, consider the 3 × 3 square below taken from a hundreds chart.
a
b
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2. Multiply ab so that it is a sum of terms
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1. What expression (in terms of n) should go in the square with the a and what
expression (in terms of n) should go in the square with the b?
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3. If the length of the smaller square is n and the length of the larger square is 12,
what is an expression for the area of the shaded part?
Signature ___________________________________
Date________________
POLY4 – SP33
COMMON CORE STATE STANDARDS – MATHEMATICS
STANDARDS FOR MATHEMATICAL CONTENT
Interpret expressions that represent a quantity in terms of its context: Interpret parts of an
expression, such as terms, factors, and coefficients.
A-SSE-1b
Interpret expressions that represent a quantity in terms of its context: Interpret complicated
expressions by viewing one or more of their parts as a single entity. For example, interpret
n
P(1+r) as the product of P and a factor not depending on P.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it. For example, see x – y as
2 2
2 2
(x ) – (y ) , thus recognizing it as a difference of squares that can be factored as
2
2
2
2
(x – y )(x + y ).
CA Addition
(CA.A.8a)
Use the distributive property to express a sum of terms with a common factor as a multiple of a
2
2
sum of terms with no common factor. For example, express xy + x y as xy (y + x).
CA Addition
(CA.A.8b)
Use the properties of operations to express a product of a sum of terms as a sum of products.
For example, use the properties of operations to express (x + 5)(3 - x + c) as
2
-x + cx - 2x + 5c + 15.
A-APR-1
Understand that polynomials form a system analogous to the integers, namely, they are closed
under the operations of addition, subtraction, and multiplication; add, subtract, and multiply
polynomials, and divide polynomials by monomials. Solve problems in and out of context.
A-APR-4
Prove polynomial identities and use them to describe numerical relationships. For example, the
2
2 2
2
2 2
2
polynomial identity (x + y ) = (x – y ) + (2xy) can be used to generate Pythagorean triples.
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A-SSE-1a
4
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4
STANDARDS FOR MATHEMATICAL PRACTICE
o
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
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MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
DO NOT DUPLICATE © 2012
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First Printing
POLY4 – SP34

POLY4 POLYNOMIALS Student Packet 4: Polynomial Arithmetic Applications

Transcription

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