Algebra (Period 1)

Addition of Polynomials 5-7
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) + (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:


Subtraction of Polynomials 5-8
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) - (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:
5x⁴ - 3x² - (-4x) + 3
- ( -10x⁴ + 3x³- 3x² - x + 3)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Pre Algebra (Period 2 & 4)

Solving 1 Step Inequalities with Addition & Subtraction 2-9


These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same:

ISOLATE THE VARIABLE

You sitll balance with the INVERSE OPERATION

Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



Solving 1 Step Inequalities with Multiplication & Division 2-10

These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same: ISOLATE THE VARIABLE


You sitll balance with the INVERSE OPERATION


Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



THERE IS ONE MAJOR EXCEPTION!!!!

When you multiply or divide by a NEGATIVE, 
the symbol CHANGES DIRECTION!


-3y > 9

You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution
 y < -3 is the answer

 If you want to understand why:
 3 < 10 correct? 
Now multiply both sides by -1 (mult prop of equality)
 You get -3 < -10, but THAT'S NOT TRUE!!! 
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10

Algebra (Period 1)

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 powerr 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

Pre Algebra (Period 2 & 4)

Inequalities & Their Graphs 2-8

What most real life situations call for

How often to you think...I need at least $20 (not exactly $20)
....
I want at most 20 minutes of homework (not exactly 20 minutes)?


Inequalities have many answers (most of the time an infinite number!)

Examples: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)

n ≥ 3 means still means that every real number greater than 3 is a solution, 
but now 3 is also a solution


n < 3 means that every real number less than 3 is a solution! (but NOT 3)
 n ≤ 3 means still means that every real number less than 3 is a solution,   but now 3 is also a solution

 GRAPHING INEQUALITIES:
 First, graphing an equation's solution is easy
 1) Say you found out that y = 5, you would just put a dot on 5 on the number line
 2) But now you have the y ≥ 5
 You still put the dot but now also darken in an arrow going to the right showing all those numbers are also solutions
 I say: the equal sign part of it is a crayon...Pick up the crayon and color in the circle
 3) Finally, you find in another example that y > 5

You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!

I say: there isn't a crayon so you can't color in the circle.
Therefore it stays open!
 

TRANSLATING WORDS:

Some key words to know:


AT LEAST means greater than or equal that is...
≥

AT MOST means less than or equal. THat is ≤


I need at least $20 to go to the mall means I must have $20, 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!

Pre Algebra (Period 2 & 4)

Adding & Subtracting Decimals Equations 3-5

Most important... LINE UP THE DECIMALS!!
We did a number of examples from the textbook... check out this section!!


Make sure you use a SIDE BAR for your calculations!!


Multiplying & Dividing Decimal Equations 3-6


For multiplication, count the decimals and place the decimal from the right!!
THese are difficult to show here as well...but

( m/7.2) = -12.5

multiply both sides by -7.2

(-7.2)(m/-7.2) = -12.5(-7.2)
m = 90

Make sure to use a side bar and multiply carefully!!

s/2.5 = 5

multiply both sides by 2.5

(2.5)(s/2.5) = 5(2.5)
3 = 12.5


4.5 = m/-3.3

multiply both sides by -3.3
Notice your answer will be a negative!!

(-3.3) (4.5) = [m/-3.3](-3.3)

-14.85 = m

... and I used a sidebar in class to show the multiplication!!

For division

0.9r = -5.4
Divide both sides by 0.9

.9r = -5.4
0.9 0.9

r = -6

use a sidebar to divide carefully

divide both sides by -0.9
-0.9n = -81.81

-0.9n/-0.9 = -81.81/-0.9

notice that your solution will be a positive!!
and use a side bar to divide carefully

n = + 90.9

Algebra (Period 1)

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 (1)1 ≤ n < 10 and the other factor (2) 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. (2.2 x 10 ^-3)(3.0 X 10 ⁵)
(2.2)(3.0) X 10 ^-310 ⁵

6.6 X 10 ²

(6.1 x 10 ⁹)(2.5 x 10^-4)
(6.1)(2.5) x 10 ⁹⁺⁴
15.25 x 10 ⁵
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 10¹
put that back in your work
1.525 x 10¹ ⋅ 10⁵
1.525 X 10⁶


(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

(5.4 X 10^-6)(5.1 X 10 ^-8)

(5.4)(5.1) = 27.54
so
27.54 X 10 ^-6+-8
27.54 X 10 ^-14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 10¹
put it back into your work
2.754 x 10¹ ⋅ 10^-14
2.754 X 10 ^{1 + -14}
2.754 X 10 ^-13


What about
9.0 X 10 ⁸
3.0 X 10 ²

do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0


and
10⁸
10²

= 10 ^8-2
=10⁶
so the answer is
3 x 10 ⁶

6.9 X 10 ⁴
3.0 X 10 ⁸

2.3 X 10 ^4-8
2.3 X 10 ^-4

2.7 X 10 ¹²
9.0 X 10 ¹²
Divide 2.7 by 9
you get
0.3
10^{12 -12} = 10 ⁰
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10^-1
so the answer is
3.0 X 10 ^-1





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


How about this harder one:

(3.6 X 10⁶)(4 X 10 ^-3
(4.8 x 10 ^-2)(1.2 X 10 ⁶)

put all your factors together and see if you can simplify before you begin...

(3.6)(4)
(4.8)(1.2)

You can simplify
(3.6)
(4.8)


isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 10^6-3
10 ^{-2 + 6}
which
becomes
2.5 X 10 ^3-4 or
2.5 X 10 ^-1

Math 6 Honors (Period 6 and 7)

Multiplying Decimals 3-8

According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!

You do NOT need to line up the decimal point when you are multiplying.

11.32 X 8.73

11.32
8.73

98.8236

estimate and you get 11 X 9 = 99

What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)

Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1


How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50






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.

Algebra (Period 1)

POWER TO ANOTHER POWER


MULTIPLY the POWERS
(m⁵)³ = m¹⁵


To check, EXPAND it out:
(m⁵)(m⁵)(m⁵) = m¹⁵



PRODUCT TO A POWER

DISTRIBUTE the power to EACH FACTOR
(m⁵n⁴)³ = m¹⁵n¹²



RAISING A QUOTIENT TO A POWER:


DISTRIBUTE THE POWER to the numerator and the denominator

(m²/n⁶)³ = m⁶/n¹⁸

Math 6 Honors (Period 6 and 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 two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 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 (Yes, you get to use the × symbol for multiplication .. but only for this!!


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.

Math 6 Honors (Period 6 and 7)

Adding & Subtracting Decimals 3-6

RULES TO FOLLOW:
1. Write the given numbers one above the other (STACK'EM) with the decimal points lined up!!

2. Add zeros to get the same number of decimal places and then add/sub just like whole numbers.

3. Place the decimal point in your sum/ difference in the position right under the decimal points in the given numbers.

6.47 + 340.8 + 73.523

STACK THEM!!
The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. You can see if your answer is reasonable by using estimation.

Algebra (Period 1)

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

Math 6 Honors (Period 6 and 7)

Rounding 3-5 (continued)

Round the following number to the designated place value:
509.690285

tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7

hundredths
509.690285
becomes 509.69

hundred-thousandths
509.690285
becomes
509.69029

tens
509.690285
becomes
510

(a) What is the least whole number that satisfies the following condition?

(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so

(a) 515
(b) 524

A whole number rounded to the nearest ten is 650
(a) 645
(b) 654

A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?

A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49

A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99

Pre Algebra (Period 2 & 4)

Decimal Review

First up: Place value

The root word in decimal is "decem" which means 10.

Each time you move in the decimal system you are multiplying or dividing the place value by 10.
If you move to the LEFT, place value is MULTIPLIED by 10. If you move to the RIGHT, place value is DIVIDED by 10 (or multiplied by 1/10...remember that dividing by 10 is the same as multiplying by its reciprocal...says our BFF, the Multiplicative Inverse Property)

The middle of the decimal system is the ones place (not the decimal!)

Place value names mirror each other to the left and to the right of the ones.
EXAMPLE: Take 1 and multiply by 10 and you get the TENS.

Take 1 and divide by 10 (or multiply by 1/10) and you get the TENTHS.

Another pattern in PLACE VALUE NAMES:

They are related to how many groups of "000" after 1,000
1 million = 1,000,000 

million is from "mille" which meant 1000 and 1 million is 1000 times 1000 or 1000¹ bigger than 1000
1 billion = 1,000,000,000 as in 1000 times 1000 times 1000 or 1000² bigger than 1000
One more example: 
1 trillion = 1,000,000,000,000 as in 1000³bigger than 1000

SO TO MAKE THIS EASIER TO UNDERSTAND:

EACH NEW PLACE VALUE NAME IS COUNTING THE GROUPS OF  ZEROS (000) AFTER 1000!

1 QUADrillion...QUAD means 4 so 4 groups of zeros after 1000:
1,000,000,000,000,000

AND OTHER THE OTHER SIDE OF THE DECIMAL, THE NAMES ARE THE SAME EXCEPT THEY HAVE A "TH"

What you need to do on the other side is to remember that the ones is still the starting point
on the left side of the decimal point though. So if you're looking for 1 millionth, you count 6
places in total = .000001
You see that there are only 5 zeros, not 6

1 millionth = 1 times 1/1,000,000 = .000001

Instead of writing all these zeros, we can use exponents.

Exponents for powers of 10 count the number of zeros.
So 1,000,000 = 106
and one millionth = 1/1,000,000 = 10-6

ROUNDING: 
Round 0 - 4 down and 5 - 9 up

EXAMPLE: 0.34 rounded to the tenths is 0.3
0.35 rounded to the tenths is 0.4
Notice you don't add zeros at the end
If there is a 9 in the next place, then think of rounding up as adding 1, that will make the number go up to "10"
EXAMPLE: 0.96 rounded to the tenths is 1.0

Think of it as adding 1 in the tenths column which would carry to the ones.
Now there is a zero in the tenths to show the reader of the number that you rounded to the tenths and not the ones.

EXAMPLE: 24,300 rounded to the thousands is 24,000

Now you need zeros to the decimal point or you'll lose the place value!

EXAMPLE: 29,600 rounded to the thousands is 30,000
Again, if it's a 9, it will need to go up to the next column...
Think of it as adding 1

Math 6H (Period 6 & 7) Yosemite Week

Comparing Decimals 3-4

Printed Notes will be handed out in class on Monday, November 1st.

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