Topic/Objective:
Date:
Adding negative numbers
Vocabulary: negative integers
Notes
Let’s add
6 + (7)
Step 1: Start at the first number, in this case -6:
Step 2: Move 7 more more places to the left. We move 7 to the left left because negative negative 7 decreases our number by 7.
6+(7) takes us to 13 on the number line. 6+(7) = 13
Examples:
(8) 8) + (4) 4) = (7) + (11) = (12) 12) + (4) 4) = (6) 6) + (3) 3) = (7) + (13) = (9) 9) + (7) 7) = (5) + (14) = (21) + (6) = (13) 13) + (5) 5) = (14) + (11) =
Topic/Objective: Adding a negative number and a positive number
Date:
Vocabulary: negative and positive integers
Notes
Let’s add
8+3
Step 1: Start at the first number, in this case −8:
Step 2: Move 3 places to the right. We move 3 to the right because adding positive 3 increases our number by 3.
8+3 takes us to 5 on the number line. 8+3 = 5
Examples:
10+4= 7+5= 15+6= 18+2= 6+6= 21+8= 17+4= 21+9= 12+5= 48+8=
Topic/Objective: Subtracting negative numbers
Date:
Vocabulary: negative integers
Notes
Step 1: Rewrite as an addition problem
4 7 = 4 + (7) Step 2: Start at the first number in the addition problem, in this case −4:
places to the left. We move 7 to the left because adding Step 3: Move 7 places negative 7 decreases our number by 7.
4 + (7) takes us to −11 on the number line. 4 7 = 11
Examples:
6 3 = 5 7 = 12 4 = 10 8 = 14 3 = 16 9 = 24 7 = 15 11 = 21 9 = 13 12 =
Topic/Objective: Subtracting: negative - negative
Date:
Vocabulary: negative integers
Notes Let’s subtract
5(6).
Step 1: Rewrite as an addition problem
5 (6) = 5 + 6 Step 2: Start at the first number in the addition problem, in this case −5:
Step 3: Move 6 places places to the right. We move 6 to the right because adding positive 6 increases our number by 6.
5 + 6 takes us to 1 on the number line. 5 (6) = 1
Examples:
8 (10 10)) = 6 (9) = 8 (2 2)) = 3 (15 15)) = 12 (11) = 17 (13) = 24 (15 15)) = 16 (18) = 25 (35) = 18 (12 12)) =
Topic/Objective:
Date:
Multiplyi ng and Dividing Integers Integers Vocabulary:
Notes
MULTIPLICATION
× = × = × = × =
3×4=12 3 × (4) = 12 (3) × 4 = 12 (3 3)) × (4) = 12
DIVISION
=
= 2
=
= 2 −
=
− = 2
=
− − = 2
Examples:
7×4= 3 × (5) = (9) × 8 = (2)× (11) = 12×4= 6 × (5) = (3) × 7 = (1)× (24) = =
24 = 6 27 = 3 − = −
42 = 7 15 = 5 − = −
Topic/Objective:
Date:
Addi Ad di ng and Sub Subtt rac t i ng Rati on onal al Num Number ber s Vocabulary: like or common denominators denominators,, unlike Denominators
Notes
Fractions with Common Denominators To add:
Just add the numerators
To subtract:
Just subtract the numerators
Fractions with unlike Denominators
You need to have common denominators in order to add or subtract
Mixed fractions with like Denominators 1
2 3 2 9 9
Step Step1: Deal Deal withthe withthewh whol olee numb number erss 1 2 3 Ste Step 2 : Deal witht withth he frac fracti tion onss with with comm ommon deno deno min ator atorss 2 3 5 9 9 9 Ste Step 3 : Com Combine bine
3
5 9
Topic/Objective: Multiplying Fractions
Date:
Vocabulary:
Notes
Topic/Objective: Dividing Fractions
Date:
Vocabulary:
Notes
Topic/Objective: Equivalent Fractions
Date:
Vocabulary: equivalent fractions
Notes
Equivalent Fractions have the same value, even though they may look different.
These fractions are really the same:
Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps its val ue.
Multiplying
Dividing
Examples:
Topic/Objective: Interpret and simplify powers
Date:
Vocabulary: Exponent, base, power, exponential form, standard form, expanded form Notes Exponent: The exponent of a number tells you how many times to multiply the base by itself.
Base Number
4³
Exponent or power
4³ = 4 x 4 x 4
3x3x3x3 81
Exponential Form : Numbers expressed expressed as powers powers with a base and and an exponent Expanded Expand ed Form Standard Form
Topic/Objective: Determine the square root of a perfect square
Date:
Vocabulary: square root
Notes The Square Root Symbol: This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards. It is called the radical , and always makes mathematics look important! We use it like this:
and we say "square root of 9 equals 3"
Example: What is
25? √ 25
Well, we just happen to know that 25 = 5 × 5, so s o when we multiply 5 by itself (5 × 5) we will get 25. So the answer is:
√ 2 5 = 5 Example: What is
36 ? √ 36
Answer: 6 × 6 = 36, so √36 = 6
Example: What is
16? √ 16
Example: What is
49? √ 49
Topic/Objective: Interpret and simplify the cube of a number
Date:
Vocabulary: Cube
Notes
Volume = length × width × height Volume = 2 × 2 × 2 = 2 3 = 8
To cube a number, just use it in a multiplication 3 times ...
Example: What is 3 Cubed?
3 Cubed =
Note: we write "3 Cubed" as
= 3 × 3 × 3 = 27
33
Topic/Objective: Determine the cube root of a perfect cube
Date:
Vocabulary: cube roots
Notes A cube root goes the other direction: 3 cubed is 27, so the cube root of 27 is 3
The cube root of a number is ... ... a special value that when cubed gives the original number.
Example: What is the Cube root of 125? Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a multiplication you will get 125) ... ... so the answer is 5
The Cube Root Symbol This is the special symbol that means "cube root", it is the "radical" symbol symbol (used for square roots) with a little three to mean cube root.
You can use it like this:
64 = 4 √ 64
√ 8 = 2 216 = 6 √ 216
because 4 x 4 x 4 = 64
beca becaus usee 2 x 2 x 2 = 8 beca because use 6 x 6 x 6 = 216
Topic/Objective: Follow the order of operations to simplify expressions
Date:
Vocabulary: order of operations
Notes
How Do I Remember It All ... ? GEMDAS ! G
Grouping first ( ), [ ], {}, or square root
E
Exponents (ie Powers)
(left-to-right) ight) MD Multiplication and Division (left-to-r AS
Addition and Subtraction (left-to-ri (left-to-right) ght)
Example: 5 × (3 + 4) − 2 × 8 = 5 × 7 − 2 × 8 =5×7−4×8 = 35 − 32 =3 ²
²
Foldable
Topic/Objective:
Date:
Apply Properties of Exponents
Vocabulary: properties of exponents
Notes
Law
Example
x1 = x
61 = 6
x0 = 1
70 = 1
x =
4 =
-1
-1
xmxn = xm+n m-n = x (xm)n = xmn
x2x3 = x2+3 = x5 = x6-2 = x4 (x2)3 = x2×3 = x6
(xy)n = xnyn =
(xy)3 = x3y3 =
x = -n
x = -3
Examples 1.
32 34
2.
10.
5.
a a
3
2
7
13.
2
2
6
x 5 x 6
5 2
3
3 4. 7
7.
1211 3. 125
3
9
2
8.
7
11.
x 5 x 8
14.
x
3
2
(9) 9) (9) 9)
9.
a2b3
2
5
2
12.
15.
x 4 y 7
29 22
10
4 16.
4
1
5
x 6 x 6
5
3
6.
17.
130
18.
20.
(83 ) 4
21. 4
6
5
19.
2
5
3
42
3
8
Topic/Objective: Express numbers in Scientific Notation & Standard Form
Date:
Vocabulary: scientific notation, standard form, positive exponents
Notes n
A number is expressed in scientific notation when it is in the form: a x 10 where 1 ≤ a < and n is an integer Convert 25,000,000,000 from standard form to form to scientific notation. notation. Step 1: Identify the location of the decimal point in 25,000,000,000
25,000,000,000. 25,000,000,000. decimal point Step 2: Move the decimal point to the left until you have hav e a number that is greater than or equal to 1 and less than 10. Count the number of places you moved the decimal point.
25,000,000,000.
Move the decimal point 10 places to the left
Step 3: Rewrite the number in scientific notation (x 10 to a power):
2.5 × 1010
Solution: 25,000,000,000 = 2.5 × 1010 When changing from Scientific from Scientific Notation to Standard to Standard Form: To change a number from scientific notation to standard form, form , move the decimal point to the right right when when the exponent is positive. positive . You should move the point as many times as a s the exponent indicates. 5
3.489 x 10 = 348,943 An easy way to remember this is: If an exponent is positive, positive, the number gets larger, so move the decimal to the right. right.
Topic/Objective: Express numbers in Scientific notation & Standard Form
Date:
Vocabulary: negative exponents
Notes
Convert .0000000672 from standard form to form to scientific notation. notation. If the original number is less than 1, the decimal point has to move to the right,, so the power of 10 is negative right negative (the (the exponent will be negative). -8
.0000000672 = 6.72 x 10
When changing from Scientific from Scientific Notation to Standard to Standard Form: If an exponent is negative, negative, the number gets smaller, so move the decimal to the left. left. You should move the decimal as many times as the exponent indicates.
4 × 10−5 = 0.00004 An easy way to remember this is: If an exponent is negative, negative, the number gets smaller, so move the decimal to the left. left.
Topic/Objective: Multiply and divide numbers expressed in scientific notation.
Date:
Vocabulary: scientific notation
Notes
Multiplying…
Multiply the coefficients
Add the exponents. exponents.
Convert the result to scientific notation.
4
2
(2.41 x 10 )(3.09 x 10 ) = 7.45 x 10 2.41 x 3.09 = 7.45 = 7.45
104+2 = 106
Dividing…
Divide the coefficients
Subtract the exponents.
Convert the result to scientific notation.
. . = . − = . . . . .
6
Examples EXPRESS EACH OF THE FOLLOWING NUMBERS IN PROPER SCIENTIFIC NOTATION:
1.
0.000033 = ____________
2.
50,000. = __________________
3.
0.000002 = ____________
4.
230,000 = _________________
5.
465 = _________________
6.
236,000,000,000 236,000,000,000 = ___________
7.
0.000000000000236 0.000000000000236 = ________
8.
48.95 = _______________
EXPRESS EACH OF THE FOLLOWING STANDARD FORM: 9.
3.7 x 105 = _________________
10.
3.21 x 10-4 = ________________
11.
6 x 103 = __________________
12.
1.99 x 10-3 = ________________
13.
1.7 x 106 = _________________
14.
8.653 x 10-5 = ______________
PERFORM EACH OF THE FOLLOWING OPERATIONS USING SCIENTIFIC NOTATION. 15.
(2.3 x 103)(4.0 x 10 5) = ______________________________ _______________________________ _
16.
(1.2 x 10-1)(5.2 x 10 2) = ______________________________ _______________________________ _
17.
(1.5 x 10-6)(3.1 x 10-3) = _____________________________
18.
(3.2 x 108)(2.1 x 10 -4) = ______________________________ _______________________________ _
19.
8.7 x 102 = 2.0 x 10-3
20.
8.1 x 104 = 1.3 x 103
