Common Core Algebra
Introduction to
Algebra
Ms. Moorer
Name_____________________
1
2
Day 1 - Evaluating Algebraic Expressions
When evaluating algebraic expressions, simply “______________________________________”
1) Evaluate 50  3x when x  7
3) Evaluate 2 x 2  5 x when
2) Evaluate 4x  13 when x  3
a) x  6
b) x  5
4) When x  10 and y  7 , evaluate
b)  x  y 
a) 3 x  4 y
2
c) 5xy 2
5) Evaluate x  2 y  3 when x  2.5 and y  1
6) Evaluate
7)
2a
 d  n  1 when a  20, n  10, d  3
5
Evaluate  2 x   2 x 3 when x  4
3
3
Evaluate the following algebraic expressions using:
a  8, b  6, d  3, x  4, y  5
1
x
2
1) 5a
2)
4) ax 2
5) 5 x  2 y
7) 7xy 3
8)
10)  ay 
3
13) a 2  3d 2
3) b  3
6)
1 2
x y
4
3bd
9
9) 0.2  0.3b
11) x  y  2 
12) 4  2 x  3 y 
14)  a  b 
15)  2a  5d 
3
2
4
Homework
1
If t  3 , then 3t 2  5t  6 equals
(1) -36
(3) 6
(2) -6
(4) 18
2
What is the value of the expression 2 x 3 y when x  2 and y  3?
3
If x = –4 and y = 3, what is the value of x  3y 2 ?
(1) –13
(3) –31
(2) –23
(4) –85
4
If a = 3 and b = -1, what is the value of ab  b 2 ?
5
If x  2 and y  3, what is the value of 2 x 2  3xy  2 y 2 ?
(1) -20
(3) 8
(2) -2
(4) 16
6
1 2
xy is
2
(3) -4
(4) -8
If x = 4 and y = -2, the value of
(1) 32
(2) 8
7
What is the value of
(1) –2
(2) 2
8
x2  4 y
, if x = 4 and y = –3?
2
(3) 10
(4) 14
Brett was given the problem: “Evaluate 2x 2 + 5 when x = 3.” Brett wrote that
the answer was 41. Was Brett correct? Explain your answer.
5
Day 2 - Real Number Properties
Property
Additive Identity Property or
identity element of addition
property
Multiplicative Identity Property
or identity element of
multiplication property
Meaning
When zero is added to or
subtracted from any
number, the number remains
unchanged.
Any number multiplied by 1
remains unchanged
Examples
a+0=a
18 – 0 = 18
0 +10 = 10
12 – 0 = 12
a  1=a
1  5=5
-2  1 = -2
1
1
 1=
6
6
Additive Inverse Property
The sum of a number and its
additive inverse (also called
its opposite) is zero.
Multiplicative Inverse Property
The product of any number
and its multiplicative inverse
(or its reciprocal) is 1.
Multiplicative Property of Zero
The product of zero and any
number is always zero.
Reflexive Property
Any quantity is equal to
itself
Symmetric Property
If one quantity equals a
second quantity, then the
second quantity equals the
first.
If one quantity equals a
second quantity and the
second quantity equals a
third quantity, then the
first quantity equals the
third quantity.
A quantity may be
subsitut4ed for its equal in
any expression.
Multiplication can be
distributed over addition or
subtraction.
Transitive Property
Substitution Property
Distributive Property
Commutative Property of
addition or multiplication
The order of the numbers
does not affect the sum or
the product.
Associative Property of addition
or multiplication
The way the numbers are
paired does NOT affect the
sum or the product.
a + -a = 0
-2 + 2 = 0
1
1
+- =0
5
5
1
a 
=1
a
1
 2=1
2
a  0=0
0  2=0
-3(0) = 0
a=a
5=5
4+2=4+2
If a = b, then b = a
If 9 = 6 + 3, then 6 + 3 = 9
If a = b, b = c, then a = c.
If 5 + 7 = 8 + 4,
8 + 4 = 12, then
5 + 7 = 12
If a = b, then a maybe replaced
by b in any expression.
If n = 15, then 3n = 3  15
A(b + c) = a(b) + a(c)
2(3 + 5) = 2(3) + 2(5)
2(5 – 1) = 2(5) – 2(1)
a+b=b+a
2+3=3+2
3(11) = 11(3)
4 6 = 6 4
(a + b) + c = a + (b + c)
(2 + 3) + 4 = 2 + (3 + 4)
(2  3)4 = 2(3  4)
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Number Properties
Which property is displayed by the given equation?
1) 19  8  5  19  8  5
2) 231  23
3) 14a  4a  a 14  4
4) 7   3  2   3  2  7
5) 8   8  0
6) 4  0  0
What is the additive inverse (opposite) and multiplicative inverse (reciprocal) of each number?
7) 14
AI:
MI:
1
9
AI:
MI:
9) 18
AI:
MI:
10) 2.5
AI:
MI:
8)
Replace the ? with the proper number to make the sentence true, and Name the property
illustrated.
11) 8 + 6 = 6 + ?
12) 17  5  ? 17
13)  3  9  15  3   9  ? 
14) 6  5  8  6  5  ? 8
15)  0.5  0.2  0.7  0.5   ? 0.7 
16) 4  0  ?
17)  3  7   5   ?  3  5
18)  ?8  2   8  2  9 
19) 7  4  ?   7  4 
20) ? x  x
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Homework
Give an example of the property named.
1. additive identity:
2. substitution property:
3. symmetric property:
4. multiplicative identity property:
5. reflexive property:
6. transitive property:
7. multiplicative property of zero:
8. additive inverse property:
9. multiplicative inverse property:
10. zero product property:
Name the property illustrated by each statement.
1. 5 · 1 = 5
2. If a + b = 9, then 9 = a + b
3. (3 + 5) + 4 = 8 + 4
4. 6 · 0 = 0
5. a + 0 = a
6. 2 · ½ = 1
7. 5x = 5x
8. If xy = 0, then x = 0 and/or y = 0
9. 7 + (-7) = 0
11. 5( 2 + 4) = 5(6)
12. 2x + 7 = 2x + 7
13. abc = 1abc
14. If 8 = x, then x = 8
15. 9 + 0 = 9
16. ¾ · 0 = 0
17. If 10 = 6 + 4 and 6+ 4 = 12 – 2, then 10 = 12 – 2
Evaluate the expression and name the property used for each step.
18. 9(8 + 2) – 45 · 2
19. 12 + 3(42 – 16)
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Day 3 - Verbal Phrases into Symbols
Steps:
I) Think of a similar problem in arthimetic
II) Write an expression for the arithmetic problem, using numbers.
III) Write a similar expression for the problem, using letters or variables.
Represent each phrase by an algebraic expression:
1) a distance that is 20 meters shorter than x meters
2) a bill for n baseball caps, each costing d dollars
3) a weight that is 40 pounds heavier than p pounds
4) an amount of money that is twice d dollars
5) the number of baseball cards, if b cards are added to a collection of 100 cards
6) Hector’s height, if he was h inches tall before he grew 2 inches
7) the total cost of n envelops that cost $0.39 each
8) the cost of one pen, if 12 pens cost d dollars
9) Mark paycheck, if he gets paid n dollars a day for 4 days plus a $30 bonus
9
Create an algebraic expression to help solve the following word problems:
10) Brianna paid 17 dollars for batteries and film for her camera. If the batteries cost x dollars,
express the cost of the film in terms of x.
11) The cost of a mountain bike is 5 times the cost of a skateboard. If the skateboard costs c
dollars, represent the cost of the mountain bike.
12) The number of kilometers traveled by a bus is represented by k. If a train traveled 200
kilometers farther than the bus, represent the number of kilometers traveled by the train.
13) The length of a rectangle is represented by L. If the width is one-half of its length, represent
its width.
14) A ballpoint pen sells for 27 cents. Represent the cost of p pens, in dollars.
15) A man bought a stock for c dollars and sold it for a profit of 25 dollars. Represent the amount
for which he sold it at.
10
16) If a car travels for 5 hours at an average rate of k kilometers per hour, represent the distance
traveled.
17) Represent the cost of t feet of lumber that sells for g cents a foot.
18) If a car traveled 550 miles in h hours, represent how many miles the car traveled per hour.
19) a) Represent the total number of days in w weeks and 5 days
b) Represent the total number of days in w weeks and d days
20) A theater has m rows of seats with c seats in each row. If a show at the theater sells out 5
performances, represent the number of people to see the show.
11
Homework: Variables and Expressions
I. Indicate with math symbols what operations are being described by the given word(s). Use
, -, x, or  symbols.
1. sum _____
2. product _______
3. decreased by ______
4. quotient _______
5. increased by _____
6. difference _________
7. more than ______
8. less than _______
9. twice something ______
II. Write a verbal expression for the algebraic expression.
10. ab
11. x + 7
12. 2x
13. m3
14. x – 6
15. 8y2
x
y
17. ½ (x + y)
18. 3x – 4
19. 5(a – b)
16.
III. Write an algebraic expression to the given verbal expression.
20. eight less than a number
21. a number increased by seven
22. the quotient of m and n
23. a number squared
24. nine times a number
25. a number decreased by three
26. seven more than the cube of a number
27. one-half the product of x and y
28. the product of twice a and b
29. twice the product of a and b
12
Day 4 - Solving Equations
To solve an equation means to isolate the variable on one side of the equation. You
can do this by using the addition/subtraction property of equality.
Solve and check each equation.
1. w + 14 = -8
2.
4. -13 + h = -5
5. -2.3 = x + (-1.1)
7. m - (-13) = 37
y + (-10) = 6
3.
-11 = a + 8
6.
-7 = -16 - k
8.
6x + 7 = 8x – 13
9.
3n  2 7

5
10
10.
7b – 6.5 = -2.3b + 8.3
11.
3
1
yy 4 y
2
2
12.
-7(x – 3) = -4
14.
7 – 3x = x – 4(2 + x)


13.
28 – 2.2y = 11.6y + 262.6
13
Extra Practice
Directions: Solve each equation for the given variable
1. 4m – 10 = 38
2. 2a – 8 – 5a – 2 = -70
3. -4(t – 9) = 4
4. 6h + 3 = h – 32
5. -5 + 3x = -5x + 11
6.
7
What is the additive inverse of
1
x  10  24
2
?
1)
2)
3)
4)
14
8
1)
2)
3)
What is the additive inverse of the expression
9
Which property is illustrated by the equation
1)
2)
3)
4)
?
commutative property of addition
distributive property
additive inverse property
additive identity property
10
1)
2)
3)
4)
?
Which property is represented by the statement
?
commutative
distributive
associative
identity
11.
A method for solving
each of the two indicated steps.
12.
The value of the expression
is shown below. Identify the property used to obtain
when
and
, what is the value of
?
is
1)
2) 10
3)
4) 4
13.
If
and
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