Variable- a letter that stands for a number
Variable expression- a mathematical phrase that uses variables, numerals, and operation symbols.
It is an efficient way to express a mathematical expression.
Ex) Identify each expression as a numerical expression or a variable expression. If it is a variable expression, name the variable.
a) 5 – 5 numerical expression
b) c – 5 variable expression the variable is c
Page 5 Quick Check # 1 a-c
a.
b.
c.
Translating word phrases into variable expressions.
Word Phrase Variable Expression
Nine more than a number y y + 9
4 less than a number n n – 4
3 times z z * 3, 3z, 3(z)
A number a divided by 12 a – 12, a/12
5 times the quantity 4 plus 5(4 + c)
a number c
page 6
- The order in which you perform operations can affect the value of an expression.
Order of Operations
- Work inside grouping symbols.
- Parentheses and brackets
- A fraction bar
- Parentheses and brackets
- exponents
- Multiply and divide in order from left to right.
- Add and subtract in order from left to right.
EX) 1.) 4 + 15 – 3 2.) 2 + 5 x 3 3.) 12 – 3 – 1
4.) 10-1*7
Ex) 1) 3 * 5 * 8 – 4 + 6 2) 4 – 1 * 2 + 6 – 3
3) 5 + 6 * 4 – 3 – 1
EX) 24 – [ 6 – (2 * 2) ]
EX) 1 + 10 – 2
4
EX) 20 – 3 [ ( 5 + 2) – 1 ]
Use >, <, or = to complete the statements.
(22 + 8 ) __________ 22 + 8 – 2
Insert grouping symbols to make each number sentences true.
3 + 6 * 2 = 18
Evaluate (solve) – to evaluate a variable expression, you first replace each variable with a number. Then you use the Order of Operations to simplify.
EX) Evaluate 1) 4y – 15 for y = 9
4(9) – 15
36 – 15
21
EX) 2) 63 – 5x for x = 7
63 – 5(7)
63 – 35
28
EX) 3) 4(t + 3) + 1 for t = 8
4(8 + 3) + 1
4(11) + 1
44 + 1
45
EX) 4) 3ab + c/2 for a = 2, b = 5, c = 10
3(2)(5) + 10/2
30 + 5
35
______________________________
Negative Positive
Graph
Order from least to greatest
-1, 4, -5
-5, -1, 4
Graph order: 0, 2, -6
Order from least to greatest -6, 0, 2
Opposites - numbers that are the same distance from zero
on a number line but in opposite direction.
Integers- whole numbers and their opposites.
Absolute Value - a number’s distance from zero.
Addition of Opposites
The sum of an integer and its opposite is zero.
Arithmetic Algebra
1 + (-1) = 0 x + (-x) = 0
Using a number line
On two plays, a football team loses 8 yards and then gains 3 yards. Find –8 + 3 to find the result of the two plays.
Loses 5 yards -5
Page 25 Quick Check 2
USING RULES TO ADD INTEGERS
Same Sign – The sum of two positive integers is positive.
The sum of two negative integers is negative.
Different Signs - To add two integers with different signs, find the difference of their absolute values. The sum has the sign of the integer with the greatest absolute value.
-12 + -31 = -43
7 + -18 = -11
-22 + (-16) = -38
60 + (-13) = 47
-125 + 35 = -90
-12 + -6 + 15 + -2
-20 + 15 = -5
1 + (-3) + 2 + -10
3 + -13 = -10
RULES TO ADDING INTEGERS
Positive + Positive means positive
Positive + Negative means positive or negative
Negative + Positive means positive or negative
Negative + Negative means negative
Same signs add then give them the same sign.
Different signs subtract then give the sign of the larger ___number_______________. (What you have more of)
-6 – (-2)
-4
-7 – (-2)
-4 – (-3)
-8 – (-5)
3 – 5 4 – 8
-1 – 5 -2 – ( -7)
Subtracting Integers
To subtract an integer, add its OPPOSITE.
2 – 5 = 2 + -5 = -3 a – b = a + -b
2 – (-5) = 2 + 5 = 7 a- -b = a + b
Subtracting Rules
A subtraction sign creates a negative integer EXCEPTSubtracting a negative creates a positive integer.
Inductive reasoning – making conclusions based on patterns you observe.
Conjecture – a conclusion you reach by inductive reasoning.
Page 35
Counter example – an example that proves a statement false. You need only one counterexample to prove that a conjecture is incorrect.
EX) 30, 25, 20, 15 Start with 30 and subtract 5 repeatedly
2, -2, 2, -2 Alternate 2 and its opposite
1, 3, 4, 12, 13 Start with 1, alternate multiplying
by 3 and adding 1
Page 35 OC # 2 a-c
The product of two integers with the same sign is positive.(multiplying)
The product of two integers with different signs is negative. (multiplying)
The product of zero and any integer is zero. (multiplying)
EX) 3(4) = 12 3(-4) = -12
-3(-4) = 12 -3(4) = 12
3(0) = 0 -4(0) = 0
Page 44
EX) -3 * 5 (-4) = 60 -2 (-3) (-2) =
4 (2) (-3) =
Quick Check
-4 * 8 (-2) = 64 6(-3)(5) = -90 -7*(-14) * 0 = 0
The quotient of two integers with the same sign is positive. (dividing)
The quotient of two integers with different signs is negative. (dividing)
Division by zero is undefined.
EX) 12 – 3 = 4 12 – (-3) = -4
-12 – (-3) = 4 -12 – 3 = -4
EX) -32 – 8 = -4 -48 – (-6) = 8 -56 – (-4) = 14
Find the average of:
-33 + (-33) + (-35) + (-36) + (-29) = -166 – 5 = -33.2
4, -3, -5, 2, -8 = -2
2-42 EVEN