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7th Grade Math
Expressions
2015-11-17
www.njctl.org
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Table of ContentsClick on a topic to go to that section.
Mathematical Expressions
Order of Operations
The Distributive Property
Like Terms
Translating Words Into Expressions
Evaluating Expressions
Glossary & Standards
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Mathematical Expressions
Return to Table of Contents
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Expressions
Algebra extends the tools of arithmetic, which were developed to work with numbers, so they can be used to solve real world problems.
This requires first translating words from your everyday language (i.e. English, Spanish, French) into mathematical expressions.
Then those expressions can be operated on with the tools originally developed for arithmetic.
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Expressions An Expression may contain:
numbers, variables, mathematical operations
Example: 4x + 2 is an algebraic expression.
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There are two terms: 4x; 2
What is a Term?Terms of an expression are the parts of the expression which are separated by addition or subtraction.
Circle the terms of this expression.
Example: 4x + 2
Circle the terms and then click to check.
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What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.
Example: 4x + 2
In this expression 2 is the constant.
Circle the constant and then click to check.
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What is a Variable?
A variable is any letter or symbol that represents a changeable or unknown value.
In this expression x is the variable.
Example: 4x + 2
Circle the variable and then click to check.
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What is a Coefficient?
A coefficient is a number multiplied by a variable. It is located in front of the variable.
In this expression 4 is the coefficient.
Example: 4x + 2
Circle the coefficient and then click to check.
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If a variable contains no visible coefficient, the coefficient is 1.
Example 1: x + 7 is the same as (1)x + 7
Example 2: -x + 7 is the same as (-1)x + 7
Coefficient
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1 In 2x - 12, the variable is "x".
True
False
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2 In 6y + 20, the variable is "y".
True
False
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3 In 3x + 4, the coefficient is 3.
True
False
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4 In 9x + 2, the coefficient is 2.
True
False
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5 What is the constant in 7x - 3?
A 7B xC 3D - 3
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6 What is the coefficient in - x + 3?
A noneB 1C -1D 3
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7 x has a coefficient.
True
False
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8 ) 19 has a coefficient.
True
False
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Order of Operations
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Order of Operations
Mathematics has its grammar, just like any language.
Grammar provides the rules that allow us to write down ideas so that a reader can understand them.
A critical set of those rules is called the order of operations.
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Order of OperationsThe order of operations allows us to read an expression and interpret it as intended.
It lets us understand what the author meant.
For instance, the below expression could mean many different things without an agreed upon order of operations.
How would you evaluate this expression?
(5-8)(5)(3)-42÷2+8÷4+(3-2)
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Use ParenthesesParentheses will make your life much easier.
Each time you do an operation, keep the result in parentheses until you use it for the next operation.
You'll be able to read your own work, and avoid mistakes.
When you're done, read each step you did and you should be able to check your work.
Also, when you substitute a value into an expression, put it in parentheses first...that'll save you a lot of trouble.
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Order of Operations
Do all operations in parentheses first.
Then, do all exponents and roots.
(5-8)(5)(3)-42÷2+8÷4+(3-2)
(-3)(5)(3)-42÷2+8÷4+(1)
(-3)(5)(3)-(16)÷2+8÷4+1Then, do all multiplication and division.
(-45)-(8)+(2)+1Then, do all addition and subtraction.
-50
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Order of OperationsOne acronym used for the order of operations is PEMDAS which stands for:
ParenthesesExponents/RootsMultiplication/DivisionAddition/Subtraction
This order helps you read an expression...but it also helps you write expressions that others can read.
Since parentheses are always done first, you can always eliminate confusion by putting parentheses around what you want to be done first.
They may not be needed, but they don't ever hurt.
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Let's simplify this step by step...
What should you do first?
5 - (-2) = 5 + 2 = 7
What should you do next?
(-3)(7) = -21
What is your last step?
-7 + (-21) = -28
-7 + (-3)[5 - (-2)]
click to reveal
click to reveal
click to reveal
Order of Operations
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Let's simplify this step by step...
What should you do first? What should you do second?
Clickto
Reveal
Clickto
Reveal
Order of Operations
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Let's simplify this step by step...
What should you do third? What should you do last?
Clickto
Reveal
Clickto
Reveal
Order of Operations
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9 Simplify the expression.
-12÷ 3(-4)
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10 Simplify the expression.[-1 - (-5)] + [7(3 - 8)]
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11 Simplify the expression.40 - (-5)(-9)(2)
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12 Simplify the expression.5.8 - 6.3 + 2.5
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13 Simplify the expression.-3(-4.7)(5-3.2)
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14 Simplify the expression.
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15 Complete the first step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
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16 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 6[4.1 - (-5.3)]click to reveal step from previous slide
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17 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 6[9.4]
-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides
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18 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 56.4
-12.4 - 6[9.4]
-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides
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20 Simplify the expression.
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21 Simplify the expression
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22 Simplify the expression
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23 Simplify the expression(-4.75)(3) - (-8.3)
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Solve this one in your groups.
Order of Operations
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How about this one?
Order of Operations
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24 Simplify the expression
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25 Simplify the expression[(-3.2)(2) + (-5)(4)][4.5 + (-1.2)]
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26 Simplify the expression
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27 Simplify the expression
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28 Simplify the expression
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29 Evaluate the expression (9 - 13)2 ÷ 2(3 - 1) + 9 ∙ 8 - (5 + 6)
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30 Evaluate the expression 7 ∙ 9 − (7 − 4)3 ÷ 9 + (10 − 12)
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31 Evaluate the expression (7 + 3)2 ÷ 25 + 4 ∙ 2 - (7 + 8)
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Order of Operations and FractionsThe simplest way to work with fraction is to imagine that the numerator and the denominator are each in their own set of parentheses.
Before you divide the numerator by the denominator, you must have them both in simplest form.
And, then you must be very careful about what you can do with them.
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Order of Operations and FractionsFor instance, a common error is shown below:
I CANNOT divide the top and the bottom by x to get:
Rather, I have to think of the denominator (1+x) as being in parentheses.
Until I can simplify that further (which I can't) this is the simplest form.
x1+x
11+1
x(1+x)
x1+x
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Order of Operations and Fractions
How would you evaluate this expression?
(4)(3)-32÷5+6÷2+(5-8)7-8
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Order of Operations(4)(3)-32÷5+6÷2+(5-8)
7-8
(4)(3)-32÷5+6÷2+(5-8)(7-8)
First, recognize that terms in a denominator act like they are in parentheses.
Then, do all operations in parentheses first. (Keep all results in parentheses until the next operation.)
Then, do all exponents and roots.
(4)(3)-32÷5+6÷2+(-3)(-1)
(4)(3)-(9)÷5+6÷2+(-3)(-1)
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Order of Operations(4)(3)-9÷5+6÷2+(-3)
(-1)Then, all multiplication and division
Then, do all addition and subtraction.
Then, divide the numerator by the denominator.
(12)-(1.8)+(3)+(-3)(-1)
(10.2)(-1)
(-10.2)
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32 Simplify the expression.
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33 Evaluate the expression 3(5 − 3)3 + 5(7 + 5) − 9 2 ∙ 5 + 5
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34 Evaluate the expression 2(9 − 4)2 + 8 ∙ 6 − 3 3 ∙ 42 + 2
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35 Evaluate the expression −4(2 − 8)2 + 7(−3) + 15 5(25 − 12)
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36 Select the correct number from each group of numbers to complete the equation.
A 2
B -2
C 3/4
D -4/3
E 2
F -2
G 4/3
H -3/4
_____ _____
From PARCC EOY sample test non-calculator #6
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The Distributive Property
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Area Model
4
x 2
Write an expression for the area of a rectangle whose width is 4 and whose length is x + 2
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Area Model
4
x 2
You can think of this as being two rectangles.
One has an area of (4)(x) and the other has an area of (4)(2)
An expression for the total area would be 4x + 8
Or as one large rectangle of area (4)(x+2).
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Distributive Property
Finding the area of each rectangle demonstrates the distributive property.
4(x + 2)4(x) + 4(2)
4x + 8
The 4 is distributed to each term of the sum (x + 2).
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Distributive PropertyNow you try:
6(x + 4) =
5(x + 7) =
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Write an expression equivalent to:
2(x - 1) 4(x - 8)
Distributive Property
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Distributive Property
a(b + c) = ab + ac Example: 2(x + 3) = 2x + 6
(b + c)a = ba + ca Example: (x + 7)3 = 3x + 21
a(b - c) = ab - ac Example: 5(x - 2) = 5x - 10
(b - c)a = ba - ca Example: (x - 3)6 = 6x - 18
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The Distributive Property can be used to eliminate parentheses, so you can then combine like terms.
Distributive Property
For example:
3(4x - 6) 3(4x) - 3(6) 12 x - 18
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The Distributive Property can be used to eliminate parentheses, so you can then combine like terms.
Distributive Property
For example:
-2(x + 3)
-2(x) + -2(3)
-2x + -6
-2x - 6
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The Distributive Property can be used to eliminate parentheses, so you can then combine like terms.
Distributive Property
For example:
-3(4x - 6)
-3(4x) - -3(6)
-12x - -18
-12x + 18
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38 Simplify 4(7x + 5) using the distributive property.
A 7x + 20
B 28x + 5
C 28x + 20
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39 Simplify -6(2x + 4) using the distributive property.
A 12x + 4
B -12x + 24
C 12x - 4
D -12x - 24
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40 Simplify -3(5m - 8) using the distributive property.
A -35m - 8
B -15m + 24
C 15m - 24
D -15m - 24
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A negative sign outside of the parentheses represents a multiplication by (-1).
Distributing a Negative Sign
For example:
-(3x + 4)
(-1)(3x + 4)
(-1)(3x) + (-1)(4)
-3x - 4
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41 Use the Distributive Property to simplify the expression. -(6x - 7)
A -6x + 7
B -6x - 7
C 6x - 7
D 6x + 7
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42 Use the Distributive Property to simplify the expression. -(-x - 9)
A -x + 9
B x - 9
C -x - 9
D x + 9
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43 Use the Distributive Property to simplify the expression. -(2x + 5)
A -2x + 5
B -2x - 5
C 2x - 5
D 2x + 5
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44 Use the Distributive Property to simplify the expression. -(-5x + 3)
A -5x + 3
B -5x - 3
C 5x - 3
D 5x + 3
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45 ) 4(x + 6) is the same as 4 + 4(6).
TrueFalse
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46 Use the distributive property to rewrite the expression without parentheses. 2(x + 5)
A 2x + 5B 2x + 10C x + 10D 7x
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47 Use the distributive property to rewrite the expression without parentheses. 3(x - 6)
A 3x - 6B 3x - 18C x - 18D 15x
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48 Use the distributive property to rewrite the expression without parentheses. -4(x - 9)
A -4x - 36B 4x - 36C -4x + 36D 32x
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49 Use the distributive property to rewrite the expression without parentheses. -(4x - 2)
A -4x - 2B 4x - 2C -4x + 2D 4x + 2
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50 Use the distributive property to rewrite the expression without parentheses. 0.6(3.1x + 17)
AB
CD
1.86x + 10.2
186x + 1021.86x + 17
.631x + .617
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51 Use the distributive property to rewrite the expression without parentheses. 0.5(10x - 15)
A
B
C
D
5x - 7.5
5x - 15
10x - 7.5
5x - 75
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52 Use the distributive property to rewrite the expression without parentheses. 1.3(6x + 49)
A
B
C
D
7.8x + 63.7
78x + 637
7.8x + 49
1.36x + 1.349
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Like Terms
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Like Terms: Terms in an expression that have the same variable(s) raised to the same power
Like Terms
6x and 2x
5y and 8y
4x2 and 7x2
NOT Like Terms
6x and x2
5y and 8
4x2 and x4
Like Terms
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53 Identify all of the terms like 5y.
A 5B 4y2
C 18yD 8yE -1y
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54 Identify all of the terms like 8x.
A 5xB 4x2
C 8yD 8E -10x
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55 Identify all of the terms like 8xy.
A 5xB 4x2yC 3xyD 8yE -10xy
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56 Identify all of the terms like 2y.
A 51yB 2wC 3yD 2xE -10y
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57 Identify all of the terms like 14x2.
A 5xB 2x2
C 3y2
D 2xE -10x2
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58 Identify all of the terms like 0.75x5.
A 75xB 75x5
C 3y2
D 2xE -10x5
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59 Identify all of the terms like
A 5xB 2x2
C 3y2
D 2xE -10x2
2 3
x
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60 Identify all of the terms like
A 5xB 2xC 3x2
D 2x2
E -10x
1 4
x2
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Simplify by combining like terms
6x + 3x
(6 + 3)x
9x
Notice when combining like terms you add/subtract the coefficients but the variable remains the same.
Combining Like Terms
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Simplify by combining like terms
4 + 5(x + 3)
4 + 5(x) + 5(3)
4 + 5x + 15
5x + 19
Notice when combining like terms you add/subtract the coefficients but the variable remains the same.
Combining Like Terms
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Simplify by combining like terms
7y - 4y
(7 - 4)y
3y
Notice when combining like terms you add/subtract the coefficients but the variable remains the same.
Combining Like Terms
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61 Simplify the expression 8x + 9x.
A x
B 17x
C -x
D cannot be simplified
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62 Simplify the expression 7y - 5y.
A 2y
B 12y
C -2y
D cannot be simplified
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63 Simplify the expression 6 + 2x + 12x.
A 6 + 10x
B 20x
C 6 + 14x
D cannot be simplified
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64 Simplify the expression 7x + 7y.
A 14xy
B 14x
C 14y
D cannot be simplified
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Teachers:
Use the Math Practice tab to assist with questioning on the next 10 slides
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65 ) 8x + 3x is the same as 11x.
TrueFalse
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66 ) 7x + 7y is the same as 14xy.
TrueFalse
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67 ) 4x + 4x is the same as 8x2.
TrueFalse
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68 ) -12y + 4y is the same as -8y.
TrueFalse
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69 ) -3 + y + 5 is the same as 2y.
TrueFalse
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70 ) -3y + 5y is the same as 2y.
TrueFalse
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71 ) 7x - 3(x - 4) is the same as 4x +12.
TrueFalse
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72 ) 7 + 5(x + 2) is the same as 5x + 9.
TrueFalse
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73 ) 4 + 6(x - 3) is the same as 6x -14.
TrueFalse
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74 ) 3x + 2y + 4x + 12 is the same as 9xy + 12.
TrueFalse
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75 The lengths of the sides of home plate in baseball are represented by the expressions in the accompanying figure.
Which expression represents the perimeter of the home plate?
A 5xyzB 2x + 2yzC 2x + 3yzD 2x + 2y + yz
yz
yy
xx
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xx+2
x+3
7
xx+2x+3
7
76 Find an expression for the perimeter of the octagon.
A x +24
B 6x + 24
C 24x
D 30x
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Translating Words Into Expressions
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Translating Between Words and Expressions
Key to solving algebra problems is translating words into mathematical expressions.
The two steps to doing this are:
1. Taking English words and converting them to mathematical words.
2. Taking mathematical words and converting them into mathematical symbols.
We're going to practice the second of these skills first, and then the first...and then combine them.
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AdditionList words that indicate addition.
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SubtractionList words that indicate subtraction.
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MultiplicationList words that indicate multiplication.
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DivisionList words that indicate division.
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Be aware of the difference between "less" and "less than".
For example:
"Eight less three" and "three less than eight" are equivalent expressions, so what is the difference in wording?
Eight less three: 8 - 3Three less than eight: 8 - 3
When you see "less than", take the second number minus the first number.
Less and Less Than
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As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order
of the two numbers or variables when you write the expression.
Reverse the Order
Examples: · 8 less than b means b - 8· 3 more than x means x + 3· x less than 2 means 2 - x
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The many ways to represent multiplication.
How do you represent "three times a"?
(3)(a) 3(a) 3 a 3a
The preferred representation is 3a.
When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.
The following are not allowed:
3xa ... The multiplication sign looks like another variable
a3 ... The number is always written in front of the variable
Multiplication
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How do you represent "b divided by 12"?
b ÷ 12
b ∕ 12
b12
Representation of Division
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Sort the words by operation.
Quotient Product
Sum TotalRatio
Difference
Less Than
More Fraction
Multiply
Per
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Three times j
Eight divided by j
j less than 7
5 more than j
4 less than j
1 2 3 4 5 6 7 8 90 + - . ÷
Translate the Words into Algebraic Expressions Using the Red Characters
j
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The sum of twenty-three and m
Write the Expression
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The product of four and k
Write the Expression
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Twenty-four less than d
Write the Expression
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**Remember, sometimes you need to use parentheses for a quantity.**
Four times the difference of eight and j
Write the Expression
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The product of seven and w, divided by 12
Write the Expression
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The square of the sum of six and p
Write the Expression
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77 The sum of 100 and h
A 100 h
B 100 + h
C 100 - h
D 100 + h 200
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78 The quotient of 200 and the quantity of p times 7
A 200 7p
B 200 - (7p)
C 200 ÷ 7p
D 7p 200
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79 Thirty five multiplied by the quantity r less 45
A 35r - 45
B 35(45) - r
C 35(45 - r)
D 35(r - 45)
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80 a less than 27
A 27 - a
B a 27
C a - 27
D 27 + a
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Now, we know how to translate a mathematical sentence in words to a mathematical expression in symbols.
Next, we need to practice translating from English sentences to mathematical sentences.
Then, we can translate from English sentences to mathematical expressions.
Translating English Sentences to Mathematical Sentences
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
The total amount of money my friends have, if each of my seven friends has x dollars.
Translating From English Sentences
7 multiplied by x
7x
click for mathematical sentence
click for mathematical expression
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
Translating From English Sentences
12 added to x
x + 12
click for mathematical sentence
click for mathematical expression
My age if I am x years older than my 12 year old brother
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
Translating From English Sentences
The total of 15 minus 5 divided by 2
(15-5)/2click for mathematical expression
click for mathematical sentence
How many apples each person gets if starting with 15 apples, 5 are eaten and the rest are divided equally by 2 friends.
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
Translating From English Sentences
d divided by s
d/sclick for mathematical expression
click for mathematical sentence
My speed if I travel d meters in s seconds
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
Translating From English Sentences
r multiplied by 28
28rclick for mathematical expression
click for mathematical sentence
How much money I make if I earn r dollars per hour and work for 28 hours
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Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.
Translating From English Sentences
6 less than two times h
2h - 6click for mathematical expression
click for mathematical sentence
My height if I am 6 inches less than twice the height of my sister, who is h inches tall
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81 The total number of jellybeans if Mary had 5 jellybeans for each of 4 friends.
A 5 + 4 B 5 - 4
C 5 x 4
D 5 ÷ 4
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82 If n + 4 represents an odd integer, the next largerodd integer is represented by
A n + 2B n + 3C n + 5D n + 6
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
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83 Jenny earns $15 an hour waitressing plus $150 in tips on a Friday night. What expression represents her total earnings?
A 150 - 15h
B h 150
C 15h + 150
D 15 + h
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84 Bob's age if he is 2 years less than double the age of his brother who is z years old?
A 2z + 2
B z 2
C 2z - 2
D z - 2
Ans
wer
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When choosing a variable, there are some letters that are often avoided:
l, i, t, o, O, s, S
Why might these letters be avoided?
It is best to avoid using letters that might be confused for numbers or operations.
In the case above (1, +, 0, 5)Click
Variables
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85 Bob has x dollars. Mary has 4 more dollars than Bob. Write an expression for Mary's money.
A 4xB x - 4C x + 4D 4x + 4
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86 The width of the rectangle is five inches less than its length. The length is x inches. Write an expression for the width.
A 5 - xB x - 5C 5xD x + 5
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87 Frank is 6 inches taller than his younger brother, Pete. Pete's height is P. Write an expression for Frank's height.
A 6PB P + 6C P - 6D 6
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88 A dog weighs three pounds more than twice the weight of a cat, whose weight is c pounds.
Write an expression for the dog's weight.
A 2c + 3B 3c + 2C 2c + 3cD 3c
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89 Write an expression for Mark's test grade, given that he scored 5 less than Sam who earned a score of x.
A 5 - xB x - 5C 5xD 5
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90 Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate?
A 2 + (x + 4)
B 2x + 4C 2(x + 4)D 4(x + 2)
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
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Evaluating Expressions
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Evaluating Expressions
When evaluating algebraic expressions, the process is fairly straight forward.
1. Write the expression.
2. Substitute in the value of the variable (in parentheses).
3. Simplify/Evaluate the expression.
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Evaluate (4n + 6)2 for n = 1
Write:
Substitute:
Simplify:
(4n + 6)2
(4(1) + 6)2
(4 + 6)2
(10)2
100
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Evaluate 4(n + 6)2 for n = 2
Write:
Substitute:
Simplify:
4(n + 6)2
4((2) + 6)2
4(8)2
4(64)
256
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Evaluate (4n + 6)2 for n = -1
Write:
Substitute:
Simplify:
(4n + 6)2
(4(-1) + 6)2
((-4) + 6)2
(2)2
4
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108
114
130128118
116
106
Let x = 8, then use the magic looking glass to reveal the correct value of the expression
12x + 23
104
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118128
130
114
20800
72
4x + 2x3
24
Let x = 2, then use the magic looking glass to reveal the correct value of the expression
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91 Evaluate 3h + 2 for h = 3
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92 Evaluate 2(x + 2)2 for x = -10
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93 Evaluate 2x2 for x = 3
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94 Evaluate 4p - 3 for p = 20
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95 Evaluate 3x + 17 when x = -13
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96 Evaluate 3a for a = -12 9
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97 Evaluate 4a + for a = 8, c = -2 ca
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98 If t = -3, then 3t2 + 5t + 6 equals
A -36B -6C 6D 18
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
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99 Evaluate 3x + 2y for x = 5 and y = 12
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100 Evaluate 8x + y - 10 for x = and y = 50
14
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Glossary &
Standards
Return to Table of Contents
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Back to
Instruction
Coefficient The number multiplied by the variable and is located in front of the variable.
4x + 2 These are not coefficients. These are constants!
Tricky!1x + 7
- 1x2 +18
When not present, the coefficient is assumed to be 1.
7 3 5
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Instruction
Constant A fixed number whose value does not change. It is either positive or negative.
4x + 2 7x 3y3z
These are not constants. These are coefficients!
Tricky!
7
4
69
1108
0.45
1/2π
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Instruction
The Distributive PropertyA property that allows you to multiply all the terms on the inside of a set of parenthesis by a term on the outside of the parenthesis.
a(b + c) = ab + ac
a(b + c) = ab + ac
a(b - c) = ab - ac
3(2 + 4) = (3)(2) + (3)(4) =
6 + 12 = 183(2 - 4) =
(3)(2) - (3)(4) =6 - 12 = -6
3(x + 4) = 48 (3)(x) + (3)(4) = 48
3x + 12 = 48 3x = 36 x = 12
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Instruction
ExpressionAn expression contains: number,
variables, and at least one operation.
4x + 2
7x = 21
11 = 3y + 2
11 - 1 = 3z + 1
Remember!
7x "7 times x"
"7 divided by x"
7x
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Instruction
Like Terms Terms in an expression that have the same
variable raised to the same power.
3x
5x15.7x
x 1/2x
-2.3x
27x3
-2x3
x3
1/4x3
-5x3
2.7x3
5x3
5x
5x25
5x4
NOT LIKE
TERMS!
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Instruction
VariableAny letter or symbol that represents a
changeable or unknown value.
4x + 2 l, i, t, o, O, s, S
x y zu v
any letter towards end of alphabet!
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Standards for Mathematical Practices
Click on each standard to bring you to an example of how to
meet this standard within the unit.
MP8 Look for and express regularity in repeated reasoning.
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
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