Transcript Chapter 11 – Introduction to Algebra

Chapter 11 – Introduction to Algebra
(Part II)
Week 9 – Math Skills
Outline
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Section 11.5 – Translating Verbal Expression to
Mathematical Expressions
Section 11.6 – Translating Sentences into Equations
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Word problems contain key words that help us solve
them.
These keywords translate directly into mathematical
expressions.
Keywords for addition
Addition
“More than”
“The sum of”
“The total of”
“Increased by”
• 5 more than x  5 + x
• The sum of w and 3  w + 3
• The total of 6 and z is  6 + z
• x increased by 7 is  x + 7
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Keywords for subtraction
Subtraction
“Less than”
“The difference between”
“Decreased by”
• 5 less than y  y – 5
• The difference between w and 3  w – 3
• 8 decreased by a  8 - a
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Keywords for multiplication
Multiplication
“Times”
“The product of”
“of”
“Twice”
• 3 times c  3c
• The product of 4 and t  4t
• 2/3 of v  (2/3)v
• twice d  2d
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Keywords for division
Division
“Divided by”
“the quotient of”
“the ratio of”
• n divided by 3  n/3
• The quotient of z and 4  z/4
• The ratio of s to 6  s/6
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Examples
Translate “The sum of 5 and the product of 4 and n”
into a mathematical expression.
1.
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The sum of 5  5 +
The product of 4 and n  4n
Put these together  5 + 4n
Translate “ the product of 3 and the difference between
z and 4” into a mathematical expression.
2.
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The product of 3 and  3 •
The difference between z and 4  z – 4
Put these together  3(z – 4)
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Class Examples
Translate “The difference between 8 and twice t” into a
mathematical expression.
1.
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The difference between 8 and  8 Twice t  2t
Put these together  8 – 2t
Translate “the quotient of 5 and the product of 7 and
x” into a mathematical expression.
2.
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The quotient of 5 and  5 ÷
The product of 7 and x  7x
Put these together  5 ÷ 7x
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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In some mathematical phrases, we are not given the name
of the variable
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Before we were given
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Translate “the difference between 8 and twice t”
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We are given the variable here (i.e. t)
Now not given the name of the variable. What to do?
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Example: Translate “the difference between seven and twice a
number”
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“a number” could be x, y, z, a, b, c, any variable you choose…
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It is just a placeholder
The difference between 7 and  7 –
Twice a number  2n
Put these together  7 – 2n
Section 11.5 – Translating Verbal
Expressions into Mathematical Expressions
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Example
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Translate “the total of a number and the square of the
number”
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Choose a variable…
The total of a number  n +
The square of the number  n2
Put these together  n + n2
Class Example
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Translate “the product of a number and one-half of the
number”
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Choose variable…
The product of a number and  n •
One-half of the number  ½ n
Put these together  n • ½ n
Section 11.6 – Translating Sentences into
Equations and Solving
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An equation states that two mathematical expressions
are equal. Keywords in word problems for equal are:
Equals
Equals
Equal to
Is equal to
Amounts to
Represents
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Once we translate the given equation into a mathematical
expression, we can find the solution
Section 11.6 – Translating Sentences into
Equations and Solving
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Example
Translate “three more than twice a number is
seventeen”, then solve the equation.
1.
Three more than  3+
Twice a number  2n
Is seventeen  = 17
Put these together  3 + 2n = 17
Now solve this equation for n
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3 – 3 + 2n = 17 – 3; subtract 3 from each side
2n = 14
2n/2 = 14/2; divide both sides by 2
n = 7; is the solution to the equation.
We say “the number is 7”
Section 11.6 – Translating Sentences into
Equations and Solving
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Example
Translate “a number decreased by 6 equals fifteen” into
an equation, then solve.
1.
A number decreased by 6  n - 6
Equals 15  = 15
Put these together  n – 6 = 15
Now solve this equation for n
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n – 6 + 6 = 15 + 6; add 6 to each side
n = 21
We say “the number is 21”
Section 11.6 – Translating Sentences into
Equations and Solving
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Example
Translate “eight decreased by twice a number is four”
find the number.
1.
Eight decreased by  8 Twice a number  2n
Is four  = 4
Putting these together  8 – 2n = 4
Now solve this equation for n
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8 – 8 – 2n = 4 – 8 ; subtract 8 from each side
-2n = -4
-2n/(-2) = -4/(-2) ; divide both sides by -2
n=2
We say “the number is 2”
Section 11.6 – Translating Sentences into
Equations and Solving
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Class Example
Translate “the product of two and a number is ten” then
find the number.
1.
The product of two and a number  2n
Is ten  = 10
Putting these together  2n = 10
Now solve this equation for n
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2n /2 = 10 /2; divided both sides by 2
n=5
We say “the number is 5”
Section 11.6 – Translating Sentences into
Equations and Solving
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Class Example
Translate “three more than one-half a number is nine”
find the number.
1.
Three more than  3 +
One-half of a number  ½ n
Is nine  = 9
Putting these together  3 + ½ n = 9
Now solve this equation for n
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3 – 3 + ½ n = 9 – 3 ; subtract 3 from each side
½n=6
½ n (2) = 6 (2) ; multiply both sides by 2
n = 12
We say “the number is 12”
Final Exam Review
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What to study?
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Practice final
Lecture notes
Sections in the book