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Chapter 1
Notes 1-1
Variables and Expressions
1) Identifying Numerical and Algebraic Expressions
Vocabulary – What do these words mean?
numerical – a number ex) 4
consisting of only numbers
algebraic expression – a math phrase, with variables, numbers, and operation symbols.
ex) x+4
variable – a variable is a letter that represents something we do not know ex) x
Write words that mean these operations.
Addition (+)
Subtraction ()
more than
less than
total
fewer
increase
decrease
plus
difference
show addition
n+7
7+n
These are the same.
show subtraction
n−7
7−n
Are these the same?
Multiplication (x)
times
product
Division (÷)
each
divided by
quotient
show multiplication
4 ( 7 ) or (4)(7)
4•7
4b
show division
½v
Do not use x for multiplication
means divide v by two
¼v
means divide v by four
2) Writing Algebraic Expressions
To write an algebraic expression,
write the letter (variable) to stand for an unknown number,
write the operation needed,
Test it, to make sure the expression makes sense.
example) a box of books, and four more is
x + 4
Notes 1-2
Order of Operations
Using the Order of Operations
1st do Grouping Symbols
2nd do Exponents
3rd do Multiplication and Division
ex) (2 + 2)
ex) x³
ex) 12 ÷ 6 • 2
ex) [3(2 + 2)]
ex) 4²
ex) 12 • 6 ÷ 2
P
E
MD
ex) 12 + 6 − 2
ex) 12 − 6 + 2
AS
from left to right
4th do Addition and subtraction
from left to right
example) grouping symbols
exponents a
multiplication, division
addition, subtraction
30 ÷ (2 + 3) − 1²
30 ÷ 5 − 1²
30 ÷ 5 − 1²
30 ÷ 5 − 1
30 ÷ 5 − 1
6−1
6−1
= 5
P
E
MD
AS
Notes 1-3
Writing and Evaluating Expressions
Evaluating Algebraic Expressions
Vocabulary – What do these words mean?
ex) 4
evaluate – solve, give the answer, usually it is a number
variable – a letter that represents a number we do not know
ex) x
grouping symbols – ( ) parenthesis, [ ] brackets
ex) { }
simplify – replace with an expression having as few terms as possible
To evaluate an expression, solve for the variable, simplify by performing operations, and give an
answer in the fewest terms possible.
4y − 15
ex) Evaluate 4y − 15, when y=9.
copy the expression
4(9) − 15
replace the y with 9
36 − 15
multiply
= 21
subtract
Sometimes you have to replace more than one variable with a number.
Notes 1-4
Integers and Absolute Value
1) Comparing Integers
Vocabulary – What do these words mean?
Integers – positive numbers, negative numbers, and zero
Opposites – numbers the same distance from zero in opposite directions ex) -2 and 2
Absolute Value – the distance from zero to this number IT IS ALWAYS POSITIVE.
_____|________|_______|_______|_______|___
Integers get bigger from left to right.
-2
-1
0
1
2
1 is bigger than -2 (it is more) 1 > -2 and -2 < 1 (large side goes to the large number)
2) Finding Absolute Value
To find the absolute value, count how many spaces from zero the number is.
It is the same number – without a negative sign. IT IS ALWAYS POSITIVE.
ex) /-4/ = 4
ex) /4/ = 4
ex) /-55/ = 55
ex) /55/ = 55
Notes 1-5
Adding Integers
1) Adding Integers with the same sign
When adding integers with the same sign,
just add the two numbers and KEEP the same sign.
ex) -5 + -2 = -7
ex) 13 + 5 = 18
ex) -13 + -5 = -18
2) Adding Integers with different signs
When adding integers with different signs,
subtract them and keep the sign of the bigger absolute value.
ex) 5 + -2 = 3
ex) -13 + 5 = -8
ex) 18 + -5 = 13
5 was positive
13 was negative
18 was positive
answer is positive
answer is negative
answer is positive
ex) 5 + 2 = 7
ex) -9 + 1 = -8
9 was negative
answer is negative
Notes 1-6
Subtracting Integers
Subtracting Integers
We DO NOT subtract integers
We ADD THE OPPOSITE NUMBER.
To subtract integers,
1) Copy the problem.
2) Circle the subtraction sign AND the NEXT number.
3) CHANGE the subtraction to addition AND the NEXT number to its OPPOSITE.
4) Then add.
ex) 13 − 6 =
13 − 6 =
13
+ -6 = 7
Copy the problem.
Circle.
(The first number DOES NOT CHANGE.)
CHANGE, then add.
+ + + + + + + + + + + + + You can draw positive / negative signs to help you add,
and circle zero pairs.
The answer will be what is left.
− − − − − −
Notes 1-9
Multiplying and Dividing Integers
Rules to Multiply and Divide Integers The number of negatives in problem . . .
tell if the answer will be positive or negative.
negatives in ANSWER
problem ?
odd # = negative answer
even # = positive answer
0
+
1
−
2
+
3
−
4
+
5
−
6
+