Download Science 6

LOCKPORT CITY SCHOOL DISTRICT
PROGRESS MONITORING
MATHEMATICS
GRADE 7.
Students will: 1) Understand the concepts of and become proficient with the skills of mathematics, 2)
Communicate and reason mathematically, 3) Become problem solvers by using appropriate tools and
strategies through the integrated study of number sense and operations, algebra, geometry, measurement,
and statistics and probability.
STUDY GUIDE 1
Students will study Number Theory, Number Systems and Integers.
GCF AND LCM
What is the GCF? GCF stands for the greatest common factor.
What are factors? Factors are the numbers you multiply together to get an answer.

For example factors of 6 are 1 and 6 and 2 and 3 because 1 times 6 is 6 and 2 times 3 is also 6.
How do I find the greatest common factor? To find the greatest common factor of two or more numbers
there are two methods.
 The first method is listing, which involves listing all the factors of each number. Then you find all the
common factors and the greatest one is the answer.
o Listing example: GCF of 24 and 30
24: 1, 2, 3, 4, 6, 8, 12, 24
The common factors are italicized.
30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF of 24 and 30 is 6 since 6 is the greatest common factor that these two numbers share.
 The second way to find the GCF of two or more numbers is by using prime factorization.
o What is prime factorization? Prime factorization is breaking down a number into its factors
until you are only left with factors that are prime numbers.
o What are prime numbers? Prime numbers are numbers that have exactly two factors 1 and
itself.
 For example 2, 3 and 17 are prime numbers because the only factors of 2 are 1 and 2,
the only factors of 3 are 1 and 3, and the only factors of 17 are 1 and 17.
o Prime Factorization of 50:
50
or
50
10 • 5
25 • 2
5•2
5•5
So, the prime factorization of 50 is 5 times 2 times 5. Notice, that it doesn’t matter what 2
factors you start with. You will always get a right answer as long as you factor correctly.
o How do I find the GCF using prime factorization? To find the GCF using prime
factorization, first you need to find the prime factorization of each number. Then you multiply
each number that the prime factorization trees share together.
 For example if each tree had two 3’s and one 2 you would multiply 3 times 3 times 2
together and that would be the GCF.
Study Guide – 7Math-Q1-V12008


Prime Factorization example: Find the GCF of 24 and 30.
24
30
Find the prime factorization of both 24 and
6•4
5•6
30.
3•2 2•2
3•2
3 and 2
Find the numbers that each tree shares.
3 times 2
Multiply the numbers they share together.
6
The GCF of 24 and 30 is 6.
Another example: Find the GCF of 36 and 32.
36
32
6•6
8•4
Find the prime factorization of both 36
3•2 3•2
4•2 2•2
and 32.
2•2
2 and 2
Find the numbers that each tree shares.
2 times 2
Multiply the numbers they share together.
4
The GCF of 36 and 32 is 4.
What is the LCM? LCM stands for Least Common Multiple.
What are multiples? Multiples are the answers you get when you multiply.

For example, multiples of 6 are 6, 12, 18, 24, 30, and so on since 6 times 1 is 6, 6 times 2 is 12, 6
times 3 is 18, 6 times 4 is 24, and 6 times 5 is 30.
How do I find the least common multiple? To find the least common multiple of two or more numbers
there are two methods.
 The first method is listing, which involves listing multiples of each number. Then you find all the
common multiples and the smallest, or least, one is the answer.
o Listing example: Find the LCM of 8 and 20.
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
The common multiples are italicized.
20: 20, 40, 60, 80
The LCM is 40 since that is the smallest common multiple that these two numbers share.
 The second way to find the LCM of two or more numbers is by using prime factorization.
o How do I find the Least Common Multiple using prime factorization? First, you need
to find the prime factorization of each number. Then, you multiply each prime number in all the
trees together. However, if the same prime number appears in more than one tree use the
highest power of that number when multiplying.
 Least Common Multiple example: Find the LCM of 8 and 20.
8
20
Find the prime factorization of both 8 and
4•2
5•4
20.
2•2
2•2
5, 2, 2, and 2
Identify each prime number that appears
in both the trees. Since 2 appears in both
trees you need to use the 2’s in the tree
that has the most 2’s, which is the tree
showing the prime factorization of 8.
5 times 2 times 2 times 2
Multiply the numbers together.
40
The LCM of 8 and 20 is 40.
Study Guide – 7Math-Q1-V12008

Another example: Find the LCM of 54 and 162.
54
162
9•6
81 • 2
Find the prime factorization of both 54
3•3 3•2
9•9
and 162.
3•3 3•3
3, 3, 3, 3, and 2
Identify each prime number that appears
in both the trees. Since 3 appears in both
trees you need to use the 3’s in the tree
that has the most 3’s, which is the tree
showing the prime factorization of 162.
3 times 3 times 3 times 3 times 2
Multiply the numbers they share together.
162
The LCM of 54 and 162 is 162.
Practice Questions:
Correct responses are marked with an asterisk (*)
1) What are all the factors of 32?
a) 1, 2, 4, 8, 32
b) 1, 2, 4, 8, 16, 32, 64
c) 1, 2, 4, 8, 16, 32 *
d) 2, 4, 8, 32
2) What are the first three multiples of 12?
a) 12, 24, 36 *
b) 12, 20, 24
c) 2, 3, 4
d) 1, 12, 24
3) The prime factorization of 64 is __________ ?
a) 2 • 2 • 6
b) 2 • 2 • 2 • 2 • 2 • 2 • 6
c) 2 •2 • 4
d) 2 • 2 • 2 • 2 • 2 • 2 *
4) What is the GCF of 18 and 24?
a) 3
b) 4
c) 6 *
d) 9
5) What is the LCM of 4 and 6?
a) 8
b) 12 *
c) 16
d) 24
6) The number 30 is the LCM of which pair of numbers?
a) 30 and 60
b) 5 and 6 *
c) 5 and 10
d) 3 and 27
Study Guide – 7Math-Q1-V12008
7) The number 5 is the GCF of which pair of numbers?
a) 5 and 7
b) 10 and 20
c) 25 and 30 *
d) 15 and 30
Study Guide – 7Math-Q1-V12008
INTEGERS
What is an integer? The set of numbers consisting of the counting numbers (1, 2, 3, 4, ... ), their opposites
(–1, –2, –3, –4, ... ), and zero.
How do you compare integers? To compare integers think of the integer number line. The numbers that
are farthest to the right on the number line are always the biggest numbers. Remember positives and zero are
always bigger than negatives and when comparing two negative numbers the larger the negative number the
smaller that number actually is.
 For example: 5 < 10
-2 > -20
-300 < 1
and
0 > -5
What is the absolute value of a number? n is the distance from 0 to a number, n, on a number line.

For example: 3 = 3 , +3 = 3 , and 0 = 0
How do you add integers? Rules for adding integers:


When the signs are the same: add the numbers and keep the same sign.
o Examples:
 1+6=7
 (-1) + (-6) = -7
When the signs are different: subtract the numbers, and keep the sign of the larger number.
o Examples:
 (-2) + 6 = 4
 2 + (-6) = -4
How do you subtract integers? Rules for subtracting integers:

Change the problem to addition and change the sign of the second number to its opposite, then follow
the rules for addition.
o Examples:
 (-5) – 6
(-5) + (-6) = -11
 8 – (-10)
8 + 10 = 18
How do you multiply and divide integers? Rules for multiplying and dividing integers:


Remember: SPON
o Same signs
o Positive answer
o Opposite signs
o Negative answer
Examples:
o (6)  (8)  48
o 6  (8)  - 48
o (-25)  (5) = 5
o 25  (5) = -5
Study Guide – 7Math-Q1-V12008
Practice Questions:
Correct responses are marked with an asterisk (*)
1)  25  25
a) true *
b) false
2) Which of these numbers is less than -2?
a) -2
b) -1
c) 0
d) -3 *
3) A construction worker was 4 feet under the ground. He went up 10 feet. How many feet is the construction
worker above ground?
a) -14
b) -6
c) 6 *
d) 14
4) 44 – (-12) = __________
a) 56 *
b) 32
c) -32
d) -56
5) (-5) • (7) • (-2) = __________
a) -70
b) -14
c) 14
d) 70 *
6) 72 ÷ -9 = __________
a) 7
b) 8
c) -8 *
d) -7
Study Guide – 7Math-Q1-V12008
EXPONENTS
What is an exponent? An exponent is the number that tells how many times the base is used as a factor;
in an expression of the form ba, a is called the exponent, b is the base, and ba is a power of b.

Examples:
o 23 = 2 • 2 • 2
=
8
5
o 10 = 10 • 10 • 10 • 10 • 10
=
100,000
What are the laws of exponents for multiplication and division?


Multiplication law of exponents: When the bases are the same and you are multiplying, you can add
the exponents together.
o Examples:
 34 • 35 = 39
 122 • 123 = 125
Division law of exponents: When the bases are the same and you are dividing, you can subtract the
exponents.
o Examples:
 108 ÷ 105 = 103
 79 ÷ 73 = 76
How do I find a negative exponent? When calculating with negative exponents you change the base into
its reciprocal and make the exponent positive.
3
1 1 1
1
1
  =
o Example: 10 =   =
10 10 10 1,000
 10 
(-3)
What is the rule for bases with exponents of 0? Anytime a base has an exponent of zero the answer
is 1.

Examples:
o 50 = 1
o 130 = 1
Practice Questions:
Correct responses are marked with an asterisk (*)
1) Which pair of expressions is equal?
a) 43 and 62
b) 24 and 42 *
c) 32 and 23
d) 25 and 33
2) 33 • 35 = __________
a) 38 *
b) 83
c) 93
d) 98
75
3) 2  __________
7
Study Guide – 7Math-Q1-V12008
a) 77
b) 710
c) 725
d) 73 *
4) 10-3 = __________
a) 103
b) -1,000
1
*
1,000
1
d) 
1,000
c)
5) 100 = __________
a) -0.1
b) -1
c) 0
d) 1 *
Study Guide – 7Math-Q1-V12008
SQUARE ROOTS
What is a perfect square root? A perfect square root occurs when the number underneath the radical sign
(
) is a perfect square.
What is a perfect square? A perfect square is a whole number resulting from multiplying an integer by
itself.
 a is a perfect square if a = n  n and n is an integer
 Examples:
o 16 = 4  4
o 121 = (–11)  (–11)
What is a non-perfect square root? A non-perfect square root occurs when the number underneath the
radical sign (
) is not a perfect square.
What are whole numbers? The set of counting numbers plus zero; {0,1,2,3, …}.
What are consecutive numbers? Numbers that follow one right after the other in order


1, 2, 3, … are consecutive positive integers
–2, 0, 2, 4, … are consecutive even integers
How do I find the square root of a perfect root? To find the square root of a number you look for the
factor that when multiplied by itself gives you the answer underneath the radical sign (

Example:
).
49 = 7 or -7 since 7 • 7 = 49 and -7 • -7 = 49
What are factors? Factors are the numbers you multiply together to get an answer.

For example factors of 6 are 1 and 6 and 2 and 3 because 1 times 6 is 6 and 2 times 3 is also 6.
How do I find the square root of a non-perfect root? Since all non-perfect roots have decimals as
answers we instead estimate in between which two consecutive whole numbers the square root will lie. You do
this by picking the two perfect squares that occur right before and after the number underneath the radical
sign (
). Then take the square root of each of those perfect squares and the answer to the non-perfect
root will be in between those two answers.
 For example:
Find the square root of 87.
87
81 and 100
81 and
100
9 and 10
87 is in between the perfect squares of 9 and 10.
Find the square root of 81 and 100.
The square root of 87 will be a decimal in between
9 and 10, but closer to 9 as 87 is closer to 81 than
it is to 100.
Practice Questions:
Correct responses are marked with an asterisk (*)
1) What is the only number that is not a perfect square?
a) 16
b) 2 *
c) 49
Study Guide – 7Math-Q1-V12008
d) 121
2) Find
81 .
a) 81
b) 40.5
c) 18
d) 9 *
3) Find
16 .
a) 4 *
b) 8
c) 32
d) 256
4)
40 is between which two whole numbers?
a) 5 and 6
b) 6 and 7 *
c) 7 and 8
d) 39 and 41
5)
213 is between which two whole numbers?
a) 2 and 3
b) 13 and 14
c) 14 and 15 *
d) 212 and 214
Study Guide – 7Math-Q1-V12008
SCIENTIFIC NOTATION
What is standard form? A number is written in standard form when each digit is in a place value

Example: twenty-nine thousand three hundred four is written as 29,304
What is scientific notation? A form of writing a number as the product of a power of 10 and a decimal
number greater than or equal to 1 and less than 10
 Examples:
o 2,400,000 = 2.4  106
o 240.2 = 2.402  102
o 0.0024 = 2.4  10–3
What is the product? The product is a number that is obtained when two or more factors are multiplied.
What are factors? Factors are the numbers you multiply together to get an answer.

For example factors of 6 are 1 and 6 and 2 and 3 because 1 times 6 is 6 and 2 times 3 is also 6.
What is a power? A power is an exponent.

Example: in the expression 38, 8 is the power and 3 is the base
How do you change a number from scientific notation form into standard form? All you need to do
is take the decimal point and slide it to the right the number of times the exponent is (if it is positive) or slide
it to the left the number of times the exponent is (if it is negative). When you are moving the decimal point
you will need to add zeros if there are no more numbers to move the decimal point past.
 Examples:
o 5.2 x 103 = 5,200 (move the decimal point 3 times to the right and add 2 zeros)
o 3.45 x 10-5 = .0000345 (move the decimal point 5 times to the left and add 4 zeros)
How do you change a number from standard form into scientific notation?




First, put a new decimal point into the number making the new number in between 1 and 10.
Next, count how many times you would need to move the new decimal back to its original spot. That
number will become the power attached to the ten. (Remember, if you cannot see the decimal point in
the number it is located at the end of the number).
Then write the new decimal number you created (dropping any extra zeros – you cannot drop zeros
that are in between two numbers) and multiply that by 10 to the power of however many places you
counted in the step before. You will need to make the power positive if the original number was
greater than 1 or negative if the original number was less than 1.
Examples:
o Change 1,403,000,000,000 into scientific notation.
1.403000000000
Put a new decimal into the number making it a number in
between 1 and 10.
1.403000000000. (12 times)
Count how many times you would need to move the new
decimal back to its original spot.
Write the new number, dropping any extra zeros, and
1.403 x 1012
multiply it by 10 to the 12th power, since it took 12 moves
to get back to the original decimal. The 12 is positive in
this case since the original number of 1,403,000,000,000
was a number greater than 1.
o Change 0.0000000759 into scientific notation.
000000007.59
Put a new decimal into the number making it a number in
between 1 and 10.
0.00000007.59 (8 times)
Count how many times you would need to move the new
Study Guide – 7Math-Q1-V12008
7.59 x 10-8
decimal back to its original spot.
Write the new number, dropping any extra zeros, and
multiply it by 10 to the 8th power, since it took 8 moves to
get back to the original decimal. The 8 is negative in this
case since the original number of 0.0000000759 was a
number less than 1.
How do you compare numbers in scientific notation? To compare numbers in scientific notation all you
need to do is compare the powers of 10. Whichever power is bigger is the bigger number. If the powers of
10 are the same then you look at the decimal numbers and compare those. Whichever, decimal number is
larger is the larger number.
 Examples:
o 3.25 x 10-4 vs 5.65 x 10-10 In this case 3.25 x 10-4 > 5.65 x 10-10 because -4 > -10.
o 5.24 x 105 vs 5.28 x 105
In this case 5.24 x 105 < 5.28 x 105 because 5.25 < 5.28.
Practice Questions:
Correct responses are marked with an asterisk (*)
1) The distance from Earth to the Sun is about 2.05 x 108 kilometers. What is this distance written in standard
form?
a) 20,500,000,000 km
b) 2,050,000,000 km
c) 205,000,000 km *
d) 20,500,000 km
2) The Atlantic Ocean has an average depth of 11,730 feet. What is this number in scientific notation?
a) 1,173 x 104
b) 117.3 x 104
c) 11.73 x 104
d) 1.173 x 104 *
3) What is 4.76 x 10-5 in standard form?
a) 0.0000476 *
b) 0.00000476
c) 0.000476
d) 0.000000476
4) The diameter of the planet Mercury is 4.87 x 103 kilometers. The diameter of the planet Venus is 1.21 x 104
kilometers. The diameter if Earth is 1.28 x 104 kilometers. The diameter of Mars is 6.79 x 103 kilometers.
Which planet has the smallest diameter?
a) Mercury *
b) Venus
c) Earth
d) Mars
5) Which of the following shows the numbers in order from least to greatest?
a) 7.8 x 106, 5.1 x 104, 1.25 x 105, 4.09 x 104
b) 4.09 x 104, 5.1 x 104, 1.25 x 105, 7.8 x 106 *
c) 7.8 x 106, 4.09 x 104, 5.1 x 104, 1.25 x 105
d) 1.25 x 105, 4.09 x 104, 5.1 x 104, 7.8 x 106
Study Guide – 7Math-Q1-V12008
SETS OF NUMBERS
What are real numbers? The set of numbers consisting of all rational and irrational numbers.
Example:
What are counting numbers? All whole numbers greater than zero; also called natural numbers.
What are whole numbers? The set of counting numbers plus zero; {0,1,2,3, …}.
What are integers? The set of numbers consisting of the counting numbers (1, 2, 3, 4, ... ), their opposites
(–1, –2, –3, –4, ... ), and zero.
What are rational numbers? Any number that can be expressed as a ratio in the form
a and b are integers and b  0 .

For example 5 is a rational number as it can be expressed as the ratio
a
where
b
5
.
1
What are irrational numbers? A real number that cannot be represented as an exact ratio of two integers;
the decimal form of the number never terminates and never repeats
 Examples:
o 
o
2
o
o
10
0.010010001...
Practice Questions:
Correct responses are marked with an asterisk (*)
1) Which number is an integer and a whole number?
a) -2
b) -1
c) 0 *
d)
1
2
2) Which number is not a natural number?
a) 0 *
b) 10
c) 100
d) 1,000
Study Guide – 7Math-Q1-V12008
3) Which number is an integer?
a) 2.5
b)
1
2
c) 
1
2
d) -2 *
4) Which numbers on this list are irrational?

5
7
, 0.001,  2 ,  ,
29
8
15 ,
16
5
7
and  2
29
8
b)  and 16
c) 15 and 16
a) 
d)  and
15 *
5) Which number on this list is rational?
12 , 14 , 0.76, 
a)
12
b) 14
c) 0.76 *
d) 
Study Guide – 7Math-Q1-V12008