Slide 1 / 185 Slide 2 / 185 7th Grade Math Expressions 2015-11-17 www.njctl.org Slide 3 / 185 Table of Contents Click 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 Slide 4 / 185 Mathematical Expressions Return to Table of Contents Slide 5 / 185 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. Slide 6 / 185 Expressions An Expression may contain: numbers, variables, mathematical operations Example: 4x + 2 is an algebraic expression.
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Slide 1 / 185 Slide 2 / 185
7th Grade Math
Expressions
2015-11-17
www.njctl.org
Slide 3 / 185
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.
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.)
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
Return to Table of Contents
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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).
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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Back to
Instruction
Constant A fixed number whose value does not change. It is either positive or negative.
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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Back to
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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Back to
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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Back to
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.