Download Multiplication Standard Algorithm (1)

 DATE
NAME
Multiplication Standard Algorithm (1)
1. Multiply using the standard algorithm.
×
4
3
1
2
×
0
+
+
b)
3
2
2
3
×
0
+
d)
×
+
×
1
5
e)
6
3
×
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8
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f)
×
h)
×
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×
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k)
4
×
+
3
8
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7
l)
×
+
B-102
2
+
×
3
+
+
3
8
+
j)
2
g)
c)
+
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
a)
DATE
NAME
Multiplication Standard Algorithm (2)
2. Multiply using the standard algorithm.
a)
b)
c)
×
×
×
+
+
+
d)
e)
f)
×
×
×
+
+
+
g)
h)
i)
×
×
×
+
+
+
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
j)
k)
l)
×
×
×
+
+
+
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-103
DATE
NAME
Long Division (1)
1. Divide using long division. Then write the division equation.
a) 4
1
8
b) 5
9
1
9
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4
3
8
8
6
f) 2
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g) 3
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h) 4
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j) 7
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k) 8
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l) 9
2
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−
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−
4
6
8
1
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
i) 6
B-104
d) 7
3
−
7
−
e) 5
2
−
c) 6
7
DATE
NAME
Long Division (2)
2. Divide using long division. Then write the division equation.
a) 13
2
7
b) 18
5
6
5
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
d) 38
9
−
−
−
−
−
−
7
5
9
7
9
5
f) 27
8
6
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1
g) 63
2
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h) 12
5
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i) 41
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j) 57
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k) 28
7
1
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l) 38
2
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3
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e) 32
5
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c) 23
1
2
1
9
4
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-105
DATE
NAME
Finding Prime Numbers
Prime Number
• A number greater than 1 that is divisible only by the number itself and 1.
• Example: 7 is a prime number because 7 is only divisible by 1 and 7.
Composite Number
• A number greater than 1 that is not prime.
• Example: 12 is a composite number because it is divisible by more than just 1 and 12.
It is also divisible by 2, 3, 4, and 6.
NOTE: The number 1 is neither prime nor composite.
Sieve of Eratosthenes. Eratosthenes was a Greek mathematician who was born in the
3rd century BCE. He invented a method for finding all the prime numbers. Here are his
steps for finding the prime numbers to 100.
Step 1: C
ircle the number 2 on a hundreds chart. Cross out all the multiples of 2,
not including 2.
Step 2: Circle the next number that has not been crossed out.
Step 3: Cross out all the multiples of this number, not including itself.
Step 4: Go back to Step 2. Stop when there are no numbers left to cross out.
Use the Sieve of Eratosthenes to find the prime numbers from 2 to 100. For Step 1,
circle the number 2 on the large chart below and then cross out the multiples of 2,
not including 2. (4, 6, 8, 10, 12, …) The next number that has not been crossed out is 3.
Circle the 3 and then cross out the multiples of 3, not including 3. (6, 9, 12, 15, 18, …)
The next number that has not been crossed out is 5. Circle the 5. Then cross out the
multiples of 5, not including 5. (10, 15, 20, 25, 30, …) If you continue the process,
the numbers that are circled are all prime numbers. Note them on the small grid below.
2
3
4
5
6
7
8
9
The prime numbers from 2 to 100 are:
10
2
11 12 13 14 15 16 17 18 19 20
3
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
B-106
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
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21 22 23 24 25 26 27 28 29 30
DATE
NAME
Finding the LCM Using Prime Number Factors
Every composite number can be written as a product of prime numbers.
Examples: 4 = 2 × 2, 6 = 2 × 3
1. Write the number as a product of prime numbers.
a)
10b)
14c)
15d)
21e)
22f )
33g)
35
You may need more than one step to find the prime numbers.
Examples: 12
24
= 2× 6
= 4×6
= 2 × (2 × 3)
= ( 2 × 2) × ( 2 × 3 )
= 2 × 2 × 3 = 2 × 2 × 2 × 3
2. Write the number as a product of prime numbers.
a)
18b)
48c)
30d)
36e)
20f )
40g)
75
You can use prime numbers to find the lowest common multiple (LCM). When prime
numbers are factors of a number, we can call them prime factors.
Example: Find the LCM of 12 and 18.
Step 1: Write 12 as a product of prime numbers.
12 = 2 × 2 × 3
Step 2: Write 18 as a product of prime numbers.
18 = 2 × 3 × 3
Step 3: To find the LCM of 12 and 18, write the shortest list of prime factors so that you
can select factors that multiply to 12 and you can select factors that multiply to 18.
12 = 2 × 2 × 3
LCM = 2 × 2 × 3 × 3
18 = 2 × 3 × 3
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
The LCM needs to have factors 2, 2, and 3 to make 12. We include another 3 so that the
LCM has factors 2, 3, and 3 to make 18. So the LCM of 12 and 18 = 2 × 2 × 3 × 3 = 36.
3. Use prime factors to find the LCM.
a) 10, 15
b) 8, 12
c) 10, 14
d) 24, 36
e) 8, 14
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
f ) 15, 25
B-107
DATE
NAME
Finding the GCF Using Prime Number Factors
Remember: A composite number is a number greater than 1 that is not a prime number.
Example: 12 is a composite number because 12 is not prime.
Every composite number can be written as a product of only prime numbers.
Examples: 4 = 2 × 2, 6 = 2 × 3
1. Write the number as a product of only prime numbers.
a)
10b)
14c)
15d)
21e)
22f )
33g)
35
You may need more than one step to find the prime numbers.
Example: 24
= 4×6
= ( 2 × 2) × ( 2 × 3 )
= 2× 2× 2×3
We call 2, 2, 2, and 3 the prime factors of 24.
2. Find the prime factors.
a)
18b)
48c)
30d)
36e)
20f )
40g)
75
You can use prime numbers to find the greatest common factor (GCF).
Example: Find the GCF of 12 and 18.
Step 1: Find the prime factors of 12. 12 = 2 × 2 × 3
Step 2: Find the prime factors of 18.
18 = 2 × 3 × 3
Step 3: Look at the prime factors of 12. Circle each prime number that also appears
in the list of prime factors of 18.
Step 4: Multiply the factors you circled.
18 = 2 × 3 × 3
GCF = 2 × 3
So the GCF of 12 and 18 is 2 × 3 = 6.
3. Use prime factors to find the GCF.
a) 10, 15
B-108
b) 8, 12
c) 10, 14
d) 24, 36
e) 45, 60
Bonus
108, 162
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
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12 = 2 × 2 × 3
DATE
NAME
Finding the GCF Using Euclid’s Algorithm
Computers can be programmed to find the greatest common factor (GCF) of two numbers
by repeated division using smaller and smaller numbers. The steps are known as
Euclid’s Algorithm.
Example: Find the GCF of 45 and 120.
Step 1: Use long division to divide the smaller number into the bigger number.
2
45 120
− 90
30
Step 2: U
se long division to divide the remainder from the previous step into
the divisor from the previous step.
1
30 45
− 30
15
Step 3: Repeat Step 2 until the remainder is zero. The final divisor will be the GCF.
2
15 30
answer
− 30
1
stop when the remainder is
So, the GCF of 45 and 120 is 15.
Here is an explanation of why the algorithm works. From the long division,
120 = (2 × 45) + 30
120 - (2 × 45) = 30
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
The GCF, 15, divides into 120 and 45 evenly, so it also divides into 120 − (2 × 45) evenly.
But 120 − (2 × 45) = 30, so the GCF must divide into 30 evenly.
So the GCF divides into 45 and 30 evenly. We repeat the process so that the numbers
become smaller and smaller until the remainder is zero. The GCF will be the final divisor.
1. Use Euclid’s Algorithm to find the GCF.
a) 18, 78
b) 48, 200
c) 42, 105
d) 124, 180
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
Bonus
84, 1,980
B-109
DATE
NAME
B-110
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
×
×
+÷- +÷×
×
÷
÷
Random Operations with Fractions (1)
DATE
NAME
Random Operations with Fractions (2)
1. Use a pair of dice and the random operations die to create random questions.
a)
b)
c)
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
d)
e)
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-111
DATE
NAME
Do You Believe in Magic?
Secretly choose a number from 1–31. Let your partner know the letters of all the
tables that contain your number. For example: If you select the number 22, you would
say: M, G, and I.
M
A
16
17
18
19
8
9
10
11
20
21
22
23
12
13
14
15
24
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30
31
28
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30
31
G
I
4
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7
2
3
6
7
12
13
14
15
10
11
14
15
20
21
22
23
18
19
22
23
28
29
30
31
26
27
30
31
B-112
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
C
DATE
NAME
Order of Operations Challenge (1)
1957
Create an expression that uses the digits 1, 9, 5, and 7 in that order and the standard
order of operations to result in a whole number between 1 and 100. Create expressions
for as many numbers as you can.
Example: 9 = (1 + 9) ÷ 5 + 7
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
You may use ( ), +, −, ×, ÷ ,
,  ,  ,  , exponents, fractions, etc. It may not be possible
to find expressions for every number! 1=
23 =
2=
24 =
3=
25 =
4=
26 =
5=
27 =
6=
28 =
7=
29 =
8=
30 =
9=
31 =
10 =
32 =
11 =
33 =
12 =
34 =
13 =
35 =
14 =
36 =
15 =
37 =
16 =
38 =
17 =
39 =
18 =
40 =
19 =
41 =
20 =
42 =
21 =
43 =
22 =
44 =
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-113
DATE
NAME
45 =
73 =
46 =
74 =
47 =
75 =
48 =
76 =
49 =
77 =
50 =
78 =
51 =
79 =
52 =
80 =
53 =
81 =
54 =
82 =
55 =
83 =
56 =
84 =
57 =
85 =
58 =
86 =
59 =
87 =
60 =
88 =
61 =
89 =
62 =
90 =
63 =
91 =
64 =
92 =
65 =
93 =
66 =
94 =
67 =
95 =
68 =
96 =
69 =
97 =
70 =
98 =
71 =
99 =
72 =
100 =
B-114
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
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Order of Operations Challenge (2)
DATE
NAME
The Integer Game (1)
Instructions
Play this game in pairs and use different-colored pens to mark the grid and the scorecard.
In the game, players take turns choosing numbers and note their scores on the scorecard.
To begin the game, Player 1 circles a number from the row containing the word “START.”
Player 2 must always choose a number from the same column as Player 1’s last choice
and Player 1 must always choose a number from the same row as Player 2’s last choice.
Once a number is chosen, it cannot be chosen again. Players continue taking turns until
either all the numbers are chosen or it is not possible for a player to select a number.
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
Grid
5
−6
9
−8
4
0
−9
4
−2
5
−6
9
−8
4
0
−9
4
−2
−1
6
6
7
−3
4
−7
−1
7
−8
2
7
0
−9
−5
6
2
−2
3
1
−5
7
−6
−2
−3
0
6
8
2
7
2
START
−4
3
6
−9
−6
−1
−8
0
1
−1
2
5
−9
7
−1
−4
−5
8
4
5
3
2
−8
8
−3
1
0
−3
2
8
9
7
0
−4
8
4
4
−8
−1
1
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-115
DATE
NAME
The Integer Game (2)
Scorecard
Player 1: Running Total
Player 2: Running Total
COPYRIGHT © 2015 JUMP MATH: TO BE COPIED. CC EDITION
Cooperative Score: Running Total
B-116
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
DATE
NAME
Multiplication Patterns
4
×
(−3)
=
(−12)
3
×
(−3)
=
(−9)
2
×
(−3)
=
×
(−3)
=
×
(−3)
=
×
(−3)
=
×
(−3)
=
×
(−3)
=
×
(−3)
=
×
(−2)
=
(−8)
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4
Blackline Master—Expressions and Equations—Teacher’s Guide for AP Book 8.1
B-117