Eighth Grade: Chapter One: Using Algebra
Big Ideas:
I. Deductive and Inductive Reasoning
¨ Deductive Reasoning:
Involves applying rules to specific situations.
Example:
Subtracting a number is the same as adding its opposite.
6 – 8 is also 6 + (-8) and 6 – (-8) is also 6 + (+8)
¨ Inductive Reasoning:
Involves generating rules derived from specific situations.
Example:
Input Output
4 8
5 10
n 2n
II. Representation: The Rule of Five (words, illustration, tables, graphs, and symbolic)
III. Equality:
¨ Use the properties of equality to solve for a variable, such that whatever you do to one side, you do to both sides.
Example: 6x + 12 = 24
-12 -12
6x = 12
÷ 6 ÷6
x = 2
¨ Transform expressions or equations to equivalent forms.
Example:
x + x – 5 can also be written 2x -5.
IV. Variables as a:
¨ Specific unknown quantity.
Example: 2 + n = 5 (n must be 3 and only 3.)
¨ Generalization, or rule involving a set of numbers.
Example: Carnival: 10 + 2r = c (r can be any number of rides)
¨ Varying quantity and its relationship to another variable, such as in a function.
Example: 2w + 2l = Perimeter
2w + 2l = 12
if w= 2, then l= 4 however, if w= 3, then l=3
V. Function:
¨ A function is a relation in which two sets are linked by a rule that pairs each element of the first set with exactly 1 element of the second set.
VI. Proportion:
¨ All proportions are linear relationships that contain the origin, (0, 0).
Example:
¨ Fractions, Unit costs, Rates, Percents, Similar Triangles, Map Scales, etc.
Example:
½ = 3/6 = 6/12 = …
(1-1) Using Variables
WORD BANK:
variable
algebraic expression
equation
open sentence
A ______is a symbol, usually a letter that represents a number.
An ______is a mathematical phrase which may include numbers, and variables.
An ______is a mathematical sentence that uses an equal sign.
An equation that contains one or more variables is an ______.
The following problems are copied from Prentice Hall Algebra I:
Practice: (p. 6) Write an expression or equation for each and identify whether it’s an expression or equation.
# 9) two more than twice a number
#15) the quotient of a number and six
#20)The number of slices left from an 8-slice pizza after some is eaten.
Write an expression of your own:
Answers:
variable
algebraic expression
equation
open sentence
2+ 2n (expression)
n/6 (expression)
Let n= the number of slices eaten and
Let t= the total number of slices left.
8 – n = t (equation: open sentence)
Variables might represent the relationship between two quantities.
Variables might represent specific unknown quantities.
Variables might also represent rules for patterns.
Objective: To write and evaluate algebraic expressions.
A variable might represent a relationship between two quantities.
1. Write an algebraic expression for yourself and one of their siblings (or parent). Let x= your age; Let x (+/-) # = your sibling’s or parent’s age.
Example: Let x = my age
Let x – 5 = my sister Kathy’s age
Do we know how old your sibling is? No. Not unless we know how old you are.
2. Now make a table comparing your age and your sibling / parent’s over time.
Example: x – 5 = my sister’s Kathy’s age
2004 2006 2008 2010
My age(x) / 48 / 50 / 52 / 54
Kathy’s age
(x - 5) / 43 / 45 / 47 / 49
*Note: x - 5 is always true no matter which year we’re looking at. My sister Kathy will always be 5 years younger than I am.
2004 2006 2008 2010
My age(x)
3. To evaluate an algebraic expression, we substitute a value for the variable and simplify.
Example: How old was my sister Kathy when I was 6?
x = my age so x = 6 this time
x – 5 = Kathy’s age
Substitute 6 for x:
(6) – 5 = 1
So Kathy was only 1 year old when I was 6.
Now you evaluate. How old will your sibling / parent be when you’re 20?
x = your age so x = 20
Substitute 20 for x:
A variable might represent a specific unknown quantity.
James went to the movies and spent 5/6 of his allowance on the movie ticket. Suppose he also spent 1/6 of his allowance on a small soda. How much did the soda cost?
Let x = James’ allowance
Write an algebraic expression.
James’ allowance was $10.00. Evaluate for x= 10. (Hint: substitute 10 for x.)
*Note: The cost of small sodas is an unknown but specific quantity.
Algebraic expressions might also represent rules for patterns.
Write an algebraic expression that explains how we get from the number of cuts to the number of pieces; and from the number of folds to the number of rectangles.
Illustration:
P= 4
P= 6
P=
Complete the table below for this pattern:
Perimeter of Unit Squares’ Pattern
Stage No. / 1 / 2 / 3 / 4Perimeter / 4 / 6
Write a detailed description of the pattern.
Write an algebraic expression or rule.
7.50x + x(10.00) 1.50x 17 * 6
y = 6(x-1) + 7 6 * 17 = 17 * 6 f(x) = 1.50x
Frog in a Well
Hour / Distance from the Bottom0 / 0
1hr / 5m
2hr / 2m
3hr / 7m
4hr / 4m
5hr
6hr /
(1-2) Exponents and Order of Operations
WORD BANK:
simplify
evaluate
exponent
base
power
rational number
irrational number
To ______an expression, you replace it with its simplest name.
An ______tells how many times a number is multiplied by itself.
The number you’re multiplying is the ______.
A ______has two parts, a base and an exponent.
Order of Operations: PEMDAS
Please Excuse My Dear Aunt Sally Parenthesis ( ) Exponents ^2 Multiplication * Division /
Addition +
Subtraction -
To ______an expression, you substitute a given number for each variable; then simplify.
Practice: (p.13) Simplify.
#19) 17 – 52 ÷ (24 + 32)
#34) 9 ÷ [4 – (10 – 9)2]3
Evaluate each expression for s = 11 and v = 8
#22) (sv)2
#23) s2+ v2
Answers: simplify; exponent; base; power; evaluate; 16; 36; 7744; 185
(1-3) Exploring Real Numbers
WORD BANK:
natural numbers
whole numbers
integers
rational numbers
irrational number
real numbers
inequality
opposites
absolute value
______are counting numbers such as 1, 2, 3…
______include all non-negative integers and zero. (0, 1, 2, …)
______all whole numbers and their opposites. (…-2, -1, 0, 1, 2…)
A ______is any number that you can write in the form a/b, where a and b are integers and b = 0. Decimals may be terminating or repeating.
An ______cannot be expressed in the form a/b, where a and b are integers. Decimals are non-repeating and non-terminating.
Together, rational and irrational numbers form the set of ______.
An ______is a mathematical sentence that compares two expressions using either <, or >.
To compare fractions, you may find it helpful to ______
______.
Two numbers that are the same distance from zero, but lie in opposite directions are ______.
The ______of a number is its distance from zero.
Practice: (p. ) Name the set(s) of numbers to which each number belongs:
#1b) 5/12
#1c) -4.67
(p. 20)
#10) √ 5
(p 21) Use <, =, or > to compare:
#27) 3/5 _____ 0.6
Order from least to greatest:
#30) 22/25, 8/9, 0.8888
Answers: Natural numbers; Whole numbers; Integers; rational number; irrational number; real numbers; inequality; change the fractions to decimals then compare; opposites; absolute value; 1b) rational numbers; 1c) rational numbers; 10) irrational numbers; 27) 3 /5 = 0.6; 33) 22 / 25, 0.8888, 8 / 9
`
(1-4) Adding Real Numbers
WORD BANK:
identity property
additive inverse
inverse property
matrix
element
The sum of any number and zero is the original number. This is the ______of addition.
The opposite of a number is its ______.
The ______of addition states that any number added to its additive inverse equals zero.
Combining Integers:
(-) + (-) = (-)
(+) + (+) = (+)
(+) + (-) OR (-) + (+) takes the sign of the addend with the greatest absolute value.
A ______is a rectangular arrangement of numbers in rows and columns.
Each item in a matrix is an ______.
Adding Matrices
Example: Add [-5 2.7] + [-3 -3.9]
[7 -3] [-4 2]
equals: [-8 -1.2]
[3 -1]
Practice: (p. 28)
#22) 1/9 + (- 5/6)
Evaluate each expression for n = 3.5
#35) -9.1 + (-n)
#41)[1.3 26] + [0.5 -4]
[1/8 -2] [ - 5/8 9]
Answers: identity property; additive inverse; inverse property; matrix; element; 22) – 13 / 18; 35) -12.6;
41) [1.8 22]
[ - ½ 7]
(1-6) Multiplying and Dividing Real Numbers
WORD BANK:
identity property
multiplication property
reciprocal
The product of a number and 1 is the original number. This is the ______of multiplication.
The product of 0 and a number is 0. This is the ______of zero.
The product of -1 and a number is the opposite of the original number. This is the ______of -1.
Multiplying Integers:
(-) • (-) = (+) (-) • (+) = (-)
(+) • (+) = (+) (+) • (-) = (-)
For every fraction and its ______, or multiplicative inverse, the product is 1.
(i.e. 1/3 • 3= 1)
For every whole number and its ______, or multiplicative inverse, the product is 1. (i.e. 3 • 1/3= 1)
Practice: (p. 41) Simplify each expression:
#5) 8( - 4.3)
#6) 9( - 5/18)
#35) ( - 9)2
#37) 3( - 4)3
#38) -5( - 1)4
#39) – 52( - 3)3
*Play Figuro.
Answers: identity property; multiplication property; multiplication property; reciprocal; reciprocal;
(1-6) Matrices
Practice: (p 43) Scalar multiplication:
#90) [11 -5]
-2 [-9 6]
[-4 3]
#92) [-47 13 -7.9]
-0.1 [0.2 -64 0]
#95) [-1 ¾]
¼ [8/9 0]
Evaluate each expression for b = - ½
#99) b3
#100) b4
#103) – b6
Answers: 90) [-22 10] 92) [4.7 -1.3 0.79] 95) [-1/4 3 /16] 99) – 1 /8 100) 1 / 16 103) -1 / 64
[18 -12] [-0.02 6.4 0] [2 / 9 0]
[8 -6]
(1-7)The Distributive Property
WORD BANK:
distributive property
term
constant
coefficient
like terms
You can use the ______to simplify an algebraic expression, by multiplying a sum or difference so that you can remove all grouping symbols.
An algebraic expression in simplest form has no ______, so you must combine them and rewrite the expression.
A ______is a number, a variable, or the product of a number and one or more variables.
A ______is a term that has no variable, so the value doesn’t change, but remains constant.
A ______is a numerical factor of a term.
______have exactly the same variable factors.
Practice: (p. 50) Simplify:
# 20) 2/3(6y + 9)
#24) -4.5(b - 3)
#33) –(2 – 7x)
#56) 10/14 d (8 – 10h)
#63) 2/5 p (15 – 35q +75w )
Answers: distributive property; like terms; term; constant; coefficient; like terms; 20) 4y + 6; 24) -4.5b +13.5; 33)- 2 +7x; 56) 40 / 7 d – 50 / 7 dh; 63) 6p – 14q + 30w
(1-8) Properties of Real Numbers
WORD BANK:
commutative
associative
identity
inverse property
distributive property
multiplication property
Properties of Operations
In the ______order doesn’t matter. a+b = b+a; a*b = b*a
In the ______it doesn’t matter where you put your grouping symbols.
(a+b) + c = a + (b+c); (a*b) *c = a* (b*c)
In the ______adding zero or multiplying by one don’t change anything.
a+0 = a; a*1 = a
For every action there is an opposite action which cancels it out. This is called the ______. a + -a = 0; a * 1/a = 1
The ______involves multiplying a term with every term on the inside of the parenthesis. a(b + c) = ab + ac; a(b-c) = ab - ac
The ______of zero states that any number multiplied by 0 is 0.
n * 0 = 0
The ______of negative one states that any number multiplied by -1 is the opposite number. n * - 1 = - n
Practice: (p. 56) Which property does each equation illustrate? How do you know?
#4) ( - 7 + 4) + 1 = -7 + (4 +1)
#5) -0.3 + 0.3 = 0
#6) 9(7.3) = 7.3(9)
#7) 5(12 – 4) = 5(12) – 5(4)
#8) 8(9 • 11) = (8 • 9) • 11
Answers: cummutative property; associative property; identity property; inverse property; distributive property; multiplication property; multiplication property; 4) associative property; 5) inverse property; 6) commutative property; 7) distributive property; 8) associative property
(1-9) Graphing Data on the Coordinate Plane
WORD BANK:
coordinate plane
origin
quadrants
ordered pairs
coordinates
correlation
scatter plot
trend line
Two perpendicular number lines, intersecting with 4 right angles, form a ______.