Polynomials 5-5
Polynomials = SUM of monomials
Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x 1/2
constants have whole number power of zero..
7 is really 7x0
1 term = monomial
2 terms = binomial
3 terms = trinomial
TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)
Coefficient - number attached to the variable ( it can be a fraction)
3x2 - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE
y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3
Constant = the number that is not attached to ANY variable
Some TERMS YOU NEED TO KNOW
Degree of a term = SUM of the exponents of all its variables
-6x4 : the degree is 4
8x2 : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x0
Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x4 + 8x2 + -2x + 9
The degree of the polynomial is : 4
Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term
More on Polynomials 5-6
Descending order- write the variables with the highest power first ( This is the way it is usually written)
Ascending order- write the variables with the lowest pwoer first ( actually NEVER used in practice)
Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!
2x2y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:
2(-4)2(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56
Monday, November 16, 2009
Math 6H ( Periods 3, 6, & 7)
Finding Factors and Multiples 5-1
You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 6-. A number is said to be divisible by its whole numbered factors.
To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.
We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6
We you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!
A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first four multiples of 7 are
0, 7, 14, 21
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21
If you were to ask for the first four NON-ZERO Multiples of 7
the answer would be 7, 14, 21, 28
Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!
Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number
The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.
You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 6-. A number is said to be divisible by its whole numbered factors.
To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.
We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6
We you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!
A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first four multiples of 7 are
0, 7, 14, 21
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21
If you were to ask for the first four NON-ZERO Multiples of 7
the answer would be 7, 14, 21, 28
Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!
Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number
The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.
Thursday, November 12, 2009
Algebra Period 4
Scientific Notation 5-4
You've had this since 6th grade! It is just a bit more complicated
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range:
less than 10 and greater than or equal to 1
That is 1 ≤ n < 10
Scientific notation is the product of two factors
one of the factors is a number that is 1 ≤ n < 10 and the other factor is a power of ten
Count the number of places you moved the decimal and make that your exponent
Very big numbers - exponent is positive
Very small numbers (decimals) - exponent is negative (just like a fraction!)
Remember that STANDARD notation is what you expect (the normal number)
When you multiply or divide scientific notations, use the power rules!
Just be careful that if your answer does not fit the scientific notation range, that you restate it.
(2.5 x 10 -3)(4.0 X 10 -8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 -3)(10 -8)
(2.5)(4)= 10
so initially you have 10 X 10 -11
but 10 does not fall into the required 1 ≤ n < 10
so changed 10 into scientific notation or 1.0 X 101
so you have 1.0 X 101 X 10 -11
or 1 X 10 -10
What about
6.0 x 10 7
3.0 X 102
First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 7 -2 or 105
so 2.0 X 105
4.2 X 105
2.1 X 103
= 2 X 102
2.5 X 10 -7
5.0 X 10 6
first 2.5/5 = 0.5 and 10-7/106 = 10-7-6 = 10-13
so initially you have
0.5 x 10-13 but 0.5 is not within 1 ≤ n < 10
so change 0.5 to scientific notation
5.0 X 10 -1 and mult that by 10-13
5 X 10 -14
You've had this since 6th grade! It is just a bit more complicated
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range:
less than 10 and greater than or equal to 1
That is 1 ≤ n < 10
Scientific notation is the product of two factors
one of the factors is a number that is 1 ≤ n < 10 and the other factor is a power of ten
Count the number of places you moved the decimal and make that your exponent
Very big numbers - exponent is positive
Very small numbers (decimals) - exponent is negative (just like a fraction!)
Remember that STANDARD notation is what you expect (the normal number)
When you multiply or divide scientific notations, use the power rules!
Just be careful that if your answer does not fit the scientific notation range, that you restate it.
(2.5 x 10 -3)(4.0 X 10 -8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 -3)(10 -8)
(2.5)(4)= 10
so initially you have 10 X 10 -11
but 10 does not fall into the required 1 ≤ n < 10
so changed 10 into scientific notation or 1.0 X 101
so you have 1.0 X 101 X 10 -11
or 1 X 10 -10
What about
6.0 x 10 7
3.0 X 102
First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 7 -2 or 105
so 2.0 X 105
4.2 X 105
2.1 X 103
= 2 X 102
2.5 X 10 -7
5.0 X 10 6
first 2.5/5 = 0.5 and 10-7/106 = 10-7-6 = 10-13
so initially you have
0.5 x 10-13 but 0.5 is not within 1 ≤ n < 10
so change 0.5 to scientific notation
5.0 X 10 -1 and mult that by 10-13
5 X 10 -14
Math 6H ( Periods 3, 6, & 7)
Dividing Decimals 3-9
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
Tickets to the school play cost $5.25 each. THe total receipts were $651. How many tickets were sold?
Divide $651 by $5.25
that is, divide 651 by 5.25
or divide 65100 by 525.
124
So A total of 124 tickets were sold.
Check the textbook for the details to the division-- as it is impossible to do here!!
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
Tickets to the school play cost $5.25 each. THe total receipts were $651. How many tickets were sold?
Divide $651 by $5.25
that is, divide 651 by 5.25
or divide 65100 by 525.
124
So A total of 124 tickets were sold.
Check the textbook for the details to the division-- as it is impossible to do here!!
Monday, November 9, 2009
Algebra Period 4
Multiplying and Dividing Monomials 5-3
A monomial is an expression that is either a numeral, a variable , or a product of numerals and variables with whole number exponents. IF the monomial is just a numeral we call is a constant.
Some examples: 4x3, -7ab, y (1/2)x5 and 2x4y
The following are NOT monomials
1/y, x1/2, x2 +4, y2 + 2y + 4
Multiplying monomials - use the properties of rational numbers and the properties of exponents:
(3x)(4x) = 12x2
(3x2)(-x) = (3x2)(-1x)
= (3)(-1)(x2)(x)
=-3x3
(-3a)(4a2)(-a4 = (-3)(4)(-1)(a)(a2)(a)
=12a7
Dividing is similar-- Use the properties of rational numbers and the properties of exponents:
x5/x2 = x5-2 = x3
4x2/5x7 = (4/5)x2-7 = (4/5)x-5 OR
(4/5x5)
12m5/4m3 = 3m2
BUT
4m5/12m3 = m2/3
Be careful Keep the RULES Separate!! Don't mix up your exponent RULES with what you know previously for rational numbers!!
A monomial is an expression that is either a numeral, a variable , or a product of numerals and variables with whole number exponents. IF the monomial is just a numeral we call is a constant.
Some examples: 4x3, -7ab, y (1/2)x5 and 2x4y
The following are NOT monomials
1/y, x1/2, x2 +4, y2 + 2y + 4
Multiplying monomials - use the properties of rational numbers and the properties of exponents:
(3x)(4x) = 12x2
(3x2)(-x) = (3x2)(-1x)
= (3)(-1)(x2)(x)
=-3x3
(-3a)(4a2)(-a4 = (-3)(4)(-1)(a)(a2)(a)
=12a7
Dividing is similar-- Use the properties of rational numbers and the properties of exponents:
x5/x2 = x5-2 = x3
4x2/5x7 = (4/5)x2-7 = (4/5)x-5 OR
(4/5x5)
12m5/4m3 = 3m2
BUT
4m5/12m3 = m2/3
Be careful Keep the RULES Separate!! Don't mix up your exponent RULES with what you know previously for rational numbers!!
Thursday, November 5, 2009
Math 6H ( Periods 3, 6, & 7)
Multiplying or Dividing by a Power of Ten 3-7
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of a number greater than or equal to 1, but less than 10, AND a power of ten are said to be written in scientific notation.
To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point.
4,592,000,000 in scientific notation
First move the decimal point to the left to get a number between 1 and 10
4,592,000,000 the first factor in scientific notation becomes 4.592
Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
3.1 ÷ 104 = 0.00031
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of a number greater than or equal to 1, but less than 10, AND a power of ten are said to be written in scientific notation.
To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point.
4,592,000,000 in scientific notation
First move the decimal point to the left to get a number between 1 and 10
4,592,000,000 the first factor in scientific notation becomes 4.592
Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
3.1 ÷ 104 = 0.00031
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
Wednesday, November 4, 2009
Algebra Period 4
Exponents: 5-1
POWER RULES:
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING: (mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8/m5 = m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm
= m8/m8
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES: (-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE: 1/32 = 1/(2)5 = (2)-5
POWER RULES:
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING: (mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8/m5 = m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm
= m8/m8
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES: (-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE: 1/32 = 1/(2)5 = (2)-5
Saturday, October 31, 2009
Math 6H ( Periods 3, 6, & 7) Yosemite Week
Comparing Decimals 3-4
We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.
In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.
Compare the following:
1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.
2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .
3. 0.83 and 0.833
To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833
0.83 = 0.830 Then compare
0.830 and 0.833 since 0 <3 Then 0.830 < 0.833.
Write in order from least to greatest
4.164, 4.16, 4.163, 4.1
First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100
The order of the numbers from least to greatest is
4.1, 4.16, 4.163, 4.164
Rounding 3-5
A method for rounding may be stated as follows: Find the place to which you wish to round, mark it with an underline ___ Look at the digit to the right. If the digit to the right is 5 or greater, add 1 to the marked digit. If the digit to the right is less than 5, leave the marked digit unchanged. Replace each digit to the right of the marked place with a 0
Round 32,567 to (a) the nearest ten thousand, (b) the nearest thousand, (c) the nearest hundred, and (d) the nearest ten
(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros
30,000
(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros
33,000
(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros
32,600
(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero
32,570
A similar method of rounding can be used with decimals. The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.
Round 4.8637 to (a) the nearest thousandth, (b) the nearest hundredth, (c) the nearest tenth, and (d) the nearest unit
a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7
4.864
b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7
4.86
c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7
4.9
d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7
5
We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.
In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.
Compare the following:
1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.
2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .
3. 0.83 and 0.833
To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833
0.83 = 0.830 Then compare
0.830 and 0.833 since 0 <3 Then 0.830 < 0.833.
Write in order from least to greatest
4.164, 4.16, 4.163, 4.1
First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100
The order of the numbers from least to greatest is
4.1, 4.16, 4.163, 4.164
Rounding 3-5
A method for rounding may be stated as follows: Find the place to which you wish to round, mark it with an underline ___ Look at the digit to the right. If the digit to the right is 5 or greater, add 1 to the marked digit. If the digit to the right is less than 5, leave the marked digit unchanged. Replace each digit to the right of the marked place with a 0
Round 32,567 to (a) the nearest ten thousand, (b) the nearest thousand, (c) the nearest hundred, and (d) the nearest ten
(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros
30,000
(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros
33,000
(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros
32,600
(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero
32,570
A similar method of rounding can be used with decimals. The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.
Round 4.8637 to (a) the nearest thousandth, (b) the nearest hundredth, (c) the nearest tenth, and (d) the nearest unit
a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7
4.864
b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7
4.86
c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7
4.9
d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7
5
Thursday, October 22, 2009
Math 6H ( Periods 3, 6, & 7)
Decimals 3-3
Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.
Note that
1/10 = 1/101
1/100 = 1/102
1/1000 = 1/103
We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.
Remember how we proved that any number to he zero power was equal to 1
or a0 = 1
Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3
or 34+2 = 3 6
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract
34 /32 = 34-2 =32
What would happen if we had
32 / 34 ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3
Which would be
1
32
or 1/32
but you can write that as 3-2
We just subtract-- using the same rule.
Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN
SO 1/10 = 1/101= 0.01 and it is equal to 10-1
Notice that 10-1 is NOT a negative number-- it is a small number
and 10-21 is not a negative number it is a VERY TINY number
AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example
zero and six thousand three hundred ninety-four ten-thousandths.
Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?
Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.
So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."
Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so
zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400
Carefully see the distinction!!
Getting back to our 0.6394
to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004
0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)
0(100) + 6(10-1)+ 3(10-2)+ 9(10-3)+ 4(10-4)
14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point
Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.
Note that
1/10 = 1/101
1/100 = 1/102
1/1000 = 1/103
We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.
Remember how we proved that any number to he zero power was equal to 1
or a0 = 1
Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3
or 34+2 = 3 6
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract
34 /32 = 34-2 =32
What would happen if we had
32 / 34 ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3
Which would be
1
32
or 1/32
but you can write that as 3-2
We just subtract-- using the same rule.
Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN
SO 1/10 = 1/101= 0.01 and it is equal to 10-1
Notice that 10-1 is NOT a negative number-- it is a small number
and 10-21 is not a negative number it is a VERY TINY number
AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example
zero and six thousand three hundred ninety-four ten-thousandths.
Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?
Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.
So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."
Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so
zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400
Carefully see the distinction!!
Getting back to our 0.6394
to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004
0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)
0(100) + 6(10-1)+ 3(10-2)+ 9(10-3)+ 4(10-4)
14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point
Wednesday, October 21, 2009
Algebra Period 4
Word Problems.. Continuted...
Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.
GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches
AGE PROBLEMS:
Mary and Sam's ages sum to 50
Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.
Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.
x = one integer and y = other integer x + y = 26
x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22
Algebraic Inequalities:
TRANSLATING WORDS:
Some key words to know:
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $200 to go to the mall means I must have $200, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.
n + (n + 2) > 50
n > 24
The smallest odd integers in the set are 25 and 27
For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.
n + (n + 2) < 50
n < 24
The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)
Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.
GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches
AGE PROBLEMS:
Mary and Sam's ages sum to 50
Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.
Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.
x = one integer and y = other integer x + y = 26
x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22
Algebraic Inequalities:
TRANSLATING WORDS:
Some key words to know:
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $200 to go to the mall means I must have $200, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.
n + (n + 2) > 50
n > 24
The smallest odd integers in the set are 25 and 27
For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.
n + (n + 2) < 50
n < 24
The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)
Math 6H ( Periods 3, 6, & 7)
The Decimal System 3-2
Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)
Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?
Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."
The expanded notation for 420,346 is given by
4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)
Using exponents the expanded notation may be given as
4(105) + 2(104)+0(103)+3(102)+4(101)+6(100)
What hmmm.. how is (100) = 1
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???
Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.
so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value
10t + 2 becomes the correct expression
What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 infront of the x for the ones' digit? No it is... invisible!!
Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)
Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?
Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."
The expanded notation for 420,346 is given by
4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)
Using exponents the expanded notation may be given as
4(105) + 2(104)+0(103)+3(102)+4(101)+6(100)
What hmmm.. how is (100) = 1
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???
Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.
so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value
10t + 2 becomes the correct expression
What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 infront of the x for the ones' digit? No it is... invisible!!
Tuesday, October 20, 2009
Math 6H ( Periods 3, 6, & 7)
Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
Check out this great Video on the Powers of Ten
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
Check out this great Video on the Powers of Ten
Monday, October 19, 2009
Algebra Period 4
WORD PROBLEMS--REVIEW
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!
CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times you can translate words into Algebra word by word just like you translate English to Spanish or French.
5 more than a number
5 + n
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: Because multiplication & addition are both commutative, when solving for a solution the order will not matter BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words. AND, FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER....GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS EXCEPT (counterexample!)...
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5
For word problems like someone's age or the amount of money you have, you also should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: T = 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
T = 3 + 5(4) = 23
In your check, Tom is 23.
Is Tom 3 years older than 5 times Julie's age?
YES!
You're algebra is correct!
STRATEGY #2: DRAWING A PICTURE
I have 5 times the number of quarters as I have dimes.
Let’s say I first translate to: 5Q = D I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG!
I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20 D = 4
Does this make sense? YES!
I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes to make that pile the same height as the number of quarters!
STRATEGY #3: MAKE A T-CHART -- this is my favorite!!
To translate known relationships to algebra, it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?
Then, you use that mathematical relationship with a variable.
EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d
d days number of hours
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
d ------l-----24d
EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
(Why did I start with 24 and not 1 hour this time?) Now look at the relationship between the left column and the right column. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h h hours number of days
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
h -------l-------h/24
Some interesting translations used all the time in Algebra:
The next consecutive number after n: n + 1
Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6
The next EVEN consecutive number after n: n + 2
Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14
The next ODD consecutive number after n: n + 2
Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!
CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times you can translate words into Algebra word by word just like you translate English to Spanish or French.
5 more than a number
5 + n
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: Because multiplication & addition are both commutative, when solving for a solution the order will not matter BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words. AND, FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER....GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS EXCEPT (counterexample!)...
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5
For word problems like someone's age or the amount of money you have, you also should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: T = 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
T = 3 + 5(4) = 23
In your check, Tom is 23.
Is Tom 3 years older than 5 times Julie's age?
YES!
You're algebra is correct!
STRATEGY #2: DRAWING A PICTURE
I have 5 times the number of quarters as I have dimes.
Let’s say I first translate to: 5Q = D I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG!
I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20 D = 4
Does this make sense? YES!
I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes to make that pile the same height as the number of quarters!
STRATEGY #3: MAKE A T-CHART -- this is my favorite!!
To translate known relationships to algebra, it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?
Then, you use that mathematical relationship with a variable.
EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d
d days number of hours
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
d ------l-----24d
EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
(Why did I start with 24 and not 1 hour this time?) Now look at the relationship between the left column and the right column. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h h hours number of days
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
h -------l-------h/24
Some interesting translations used all the time in Algebra:
The next consecutive number after n: n + 1
Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6
The next EVEN consecutive number after n: n + 2
Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14
The next ODD consecutive number after n: n + 2
Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11
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