1.1 Standard Notation and Place Value

digit – one of the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

number – may have several digits, for example 367

1 , 2 3 4 , 5 6 7 , 8 9 0

Ex a For the number above what digit names the hundred thousands place?

Ex b What digit names the number of tens?

Ex c What does the digit “2” represent in the number above?

Ex d What does the digit “7” represent in the number above?

Ex e Write 5,620,487 in word form.

Ex f Write 5,620,478 in expanded form.

Ex g Write in standard notation: Thirty-two million

1.2 Addition

Most problems are added vertically, even if they are originally written horizontally.

Ex a 34 + 2413 + 222Ex b 782 + 4365 + 28

The result of an addition problem is called a ______

Perimeter – distance around the outside of an object.

Ex c Find the perimeter of each of the objects below:

Practice Problems

1.3 Subtraction

Subtraction is the inverse of addition.

Subtraction is also carried out vertically, even if the original problem is written horizontally.

For each example, subtract and check by adding:

Ex a 85 – 32Ex b 425 Ex c 3000 - 1471

- 86

The result of a subtraction problem is called a ______

Practice Problems

1) 30,008 – 52 2) 5923

- 769

1.4 Multiplication

8 X 3 = 248 and 3 are called ______, 24 is called the ______

If you haven’t memorized the products of single digits (times tables), you should do so.

There are many ways to write products:

5 X 85(8)(5)(8)(5)8(5)X8

When there is no operator shown, the operation which is understood is ______

The purpose of parentheses is ______

(5) + (8) = (5)(+8) =5(+8) =5 + (8) =

Special Products

Multiplying by zero: 0 X 17 =29(0) =

Multiplying by one: 1 X 392 = (53)(1) =

Multi-Digit Multiplication

Ex a Multiply 2 5 9Ex b Multiply 4 5 2 7

X 7 X 3 1

Ex c Multiply 6 5 9

X 4 0 3

Sneaky Multiplication Tricks

Ex d Multiply 1000 X 7834Ex e Multiply 682 4

X300

Area

Ex f Find the area of the rectangle:

1.5 Division

Division is the inverse of multiplication:

can be rewritten as

Ex a Divide and check:

Special Quotients

Dividing by 1:

Dividing by itself:

Dividing 0 by a number:

Dividing a number by 0:

Long Division

Ex b Divide

1.6 Rounding and Estimating

Ex a Round 29 to the nearest 10

Round 22 to the nearest 10

Round 25 to the nearest 10

Rounding Whole Numbers Procedure – for a specific place

1. Find the digit in the specified place.

2. Look at the digit AFTER that place

3. If the digit ______

If the digit ______

4. Replace the rounded digits with ______

Ex b Round 3,682,357 to the nearest:

millionten thousandhundredten

Estimating

Ex c Estimate the following amounts for easier calculations:

Restaurant bill: $ 43.58Truck: $27,875House: $239,995

Ex d Estimate the sum by rounding each number to the nearest ten: 58 + 91 + 37
Ex e Estimate the difference by rounding each number to the nearest hundred:
564 – 238
Ex f Estimate the product by rounding to the nearest hundred: 287 X 726
Ex g Estimate the quotient by rounding to the nearest ten: 476 59

1.7 Solving Equations

Ex a My husband’s brother is 4 years older than he is. If his brother is 59, how old is he?

Solve by Trial (guess and check)

Ex b Solve: x – 9 = 33 Solve: 4x = 36

Solving by Opposite (Inverse) Operations

The opposite (inverse) of addition is ______

The opposite (inverse) of multiplication is ______

We want to isolate the variable.

Ex c Solve each equation, then check your answer:

14 + x = 5218 == p

1.8 Applications

Procedure

Familiarize – understand what is asked for, what numbers are important
Translate – make an equation
Solve
Check
State – Answer the question

Graph from Basic College Mathematics, 12/e, by Bittinger/Beecher/Johnson

Ex a (Problem 1) How much taller would the Aeropolis 2001 have been than the Nakeel Tower?

Ex b (Problem 3) The Willis Tower (formerly Sears Tower) is the tallest building in Chicago. If the Miglin-Beitler Skyneedle had been built, it would have been 551 ft. higher than the Willis Tower. What is the height of the Willis Tower?

Ex c There are 520 seats in an auditorium. If all rows have the same number of seats, and there are 20 rows, how many seats are in each row?

1.9 Exponential Notation and Order of Operations

Ex a Write in exponent form:

Ex b Evaluate:

Simplifying Expressions (Order of Operations for several operations)

Parentheses (and grouping symbols like { } or [ ])
Evaluate all exponential expressions
Multiplication and Division, in order from left to right
Addition and Subtraction, in order from left to right

Ex a 100 – (58 – 21)(100 – 58) – 21

Ex b

Ex c

Average - Add the numbers, divide by “how many”

Ex d Find the average of the test scores: 82, 72, 83

2.1 Factorizations

For the product , a and b are called ______

Dividing , d is a factor of Q if the remainder is _____ .

If d is a factor of Q, Q is a ______of d, and Q is ______by d.

Ex a List all the factors of 24.

Ex b List all the factors of 23.

Ex c List the first 5 multiples of 13.

Ex d Show that 52 is divisible by 4.

Prime and Composite Numbers

1 is ______
A ______has only 1 and itself (2 different factors) as factors
A ______can be “broken down” into other factors besides 1 and itself

Ex e Which of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are prime?

Ex f Is 128 divisible by 7?Ex g Is 128 divisible by 8?

2.2 Divisibility

A number is divisible by 2 if its ones digit is ______

Ex a Which numbers are divisible by 2?

174,201,122380150,000

A number is divisible by 3 if ______is divisible by 3

Ex b Which numbers are divisible by 3?

294,201,122380150,000

A number is divisible by 6 if it is ______

Ex c Which numbers are divisible by 6?

294,201,122380150,000

A number is divisible by 9 if ______is divisible by 9

Ex d Which numbers are divisible by 9?

3874,201,122

A number is divisible by 10 if ______

A number is divisible by 5 if ______

Ex f Which numbers are divisible by 10? Which are divisible by 5?

2953,729,2311620

A number is divisible by 4 if ______

A number is divisible by 8 if ______

Ex g Which numbers are divisible by 4? Which are divisible by 8?

9024387,231420

2.3 Fractions and Fraction Notation

Ex a Shade the portions that represent

Ex b What fraction is represented by the shaded portions?

Ex c Find and

Ratio – a quotient of 2 quantities (can be written as a fraction)

The 2 quantities are often separated by “to”

Ex d A job opening has 97 applicants, and 4 people are hired.

1) Write the ratio of people hired to applicants.

2) Write the ratio of people hired to people not hired.

2.4 Multiplying Fractions

Ex a Multiply

Multiplying a Fraction by a Fraction

Multiply the 2 numerators -

Multiply the 2 denominators –

Do the same rules work for addition (add 2 numerators & keep, add 2 denominators & keep)?

Multiply a Fraction by Whole Number

Ex b Multiply

Applications

Ex c For a training program, 20 out of 71 applicants are accepted. Of the accepted students, 4/5 of the students are hired. What fraction of all applicants are hired?

2.5 Simplifying (reducing)

Fractions that reduce to 1:

Multiplicative Identity (Multiplying identity) - Using it gives the same value (no change)

a _____ = a

Equivalent fractions – have the same value:

We can change fractions to have a new denominator, but the same value

Ex a Find a name for with a denominator of 21.

Ex b Create an equivalent fraction with the new denominator.

Simplifying Fraction Notation (Reducing)

Simplest fractions have NO COMMON FACTOR in the numerator and denominator. To get simplest form, remove fractions that equal 1 (common factors)

Ex c Simplify:

Practice Problems

Simplify:

2.6 Multiplying, Simplifying, and Applications

It’s important to simplify a product before actually multiplying out the numbers

Ex a Simplify and multiply

Procedure

Put numerator and denominator factors together in the num. & denom., but don’t actually multiply out the numbers
Factor the numerator and denominator
Remove factor fractions that equal 1, if possible.
Multiply out the products to get a single number in numerator & denominator.

Ex b Multiply (reminder before multiplying:

Applications

Ex c The pitch of a screw is inch (this is how far it moves with every full turn). How far into a piece of wood will it go when makes 6 full turns?

Ex d Financial aid covers of a student’s expenses. If expenses are $4500, how much is covered by financial aid?

2.7 Division and Applications

Reciprocals - Pairs of fractions whose product = 1. We find a reciprocal by______

Ex a Find the reciprocal of

Dividing Fractions

Ex b Divide:

Solving Equations

Ex c SolveSolve

Applications

Ex b How many 3/4 ounce servings of chips can be made from a 12 ounce bag?

Practice Problems

1. 2. 3.

4. 5. 6. Solve:

3.1 Least Common Multiples

Ex a Find the least common multiple of 9 and 12 by making a list of multiples.

12:

Some common multiples are:

The Least Common Multiple (LCM) is:

Finding LCM’s by Listing Multiples (Method 1)

a) Is the largest number a multiple of the other numbers?

b) If not, list multiples of the largest number until you find one that is a multiple of the other numbers.

Ex b Ex c Find the LCM of 4, 10, and 20

Ex c Find the LCM of 4, 6, and 10:

10:

Prime Factorization - Breaking down numbers to the smallest possible factors

Tree method:Divide Up Method:

Finding the LCM by Prime Factorization (Method 2)

a) Write the prime factorization of each number

b) Create a product – for each factor, use the greatest number of repeats in ONE number

Ex c Find the LCM of 9 and 12

Ex d Find the LCM of 8, 18, and 12 using prime factorization and exponents

Practice Problem: Find the LCM of 25 and 35

We will skip Method 3 (p. 150).

3.2 Addition and Applications

Like Denominators

1. Add numerators

2. Keep same denominator

3. Reduce if possible

Ex a Add: = It doesn’t say reduce – should we?

Different Denominators

1. Get LCD (LCM of denominators)

2. Multiply top & bottom by the needed factor.

3. Add as above.

Ex b Add: =How is this different from multiplying?

Ex c Add: =

Ex d Add: =

Applications

Ex e A tile isin. thick and is glued to a board in. thick. The glue is . How thick is the result?

Practice Problems

1. Add: =

2. Add: =

3.3 Subtraction, Order, Applications

Like Denominators

1. Subtract numerators

2. Keep same denominator

3. Reduce if possible

Ex a Add: =

Different Denominators

1. Get LCD

2. Multiply top & bottom by the needed factor.

3. Subtract as above.

Ex b Subtract: =

Ex c Subtract: =

Order (which is bigger?) - Use < or >

Ex e Use < or > to write a true statement.

Ex f I have ½ lb. butter, and use 1/3 lb. How much is left?

3.4 Mixed Numerals (also called mixed numbers)

What fraction is represented below?

Ex a Write as a mixed numeral: 11 + =9 + =

Ex b Convert to fraction notation (commonly called ______)

Fraction notation (improper fractions) and mixed numbers.

Ex c Convert the improper fractions to mixed numbers:

3.5 Add & Subtract Using Mixed Numerals; Applications

Addition

Ex a Ex b

Subtraction

Ex c Exd

Ex e (# 38) A plumber uses 2 pipes, each of length , and one pipe of length . How much pipe was used in all?

3.6 Multiplication and Division with Mixed Numbers

Convert mixed numbers to fraction notation (improper fractions)

Ex a ()()

Ex b (12)()

Ex c

Ex d

Ex e A space shuttle orbits the earth in hr. How many orbits are made in 24 hours?

3.7 Order of Operations/Complex Fractions/Estimation

Ex a Simplify

Ex b Simplify:

Ex c Find the average of

Ex d Estimate, rounding to the nearest whole number

Compare the numerator to ______

Ex d Estimate each term to the nearest whole number, then perform the operations:

4.1 Decimal Notation, Order, and Rounding

Decimal Values

Ex a Write the value of $178.95 in expanded form.

Place Value Chart (for 1.73205)

Decimal Notation and Word Names – decimal words are similar to the fraction they represent

Number left of decimal point:
Point:
Number right of point:

Ex b Give the word name of 1.73205

Ex c The median age in CA is 35.2 -- write the word name.

Converting Decimals to Fractions:

Count the number of decimal places
Write that number of zeroes in the denominator, with 1 in front
Write the digits in the numerator

Ex Convert 0.357 0.018223.41

Note: Whole number parts on the left of the decimal point make ______

Converting Fractions to Decimals (“simple” denominators with powers of 10)

Count the number of zeroes
Use that number of place values to make the numerator smaller (move left).

Order (state which is larger using > or <)

How to make equivalent decimals:

Comparing numbers in decimal notation

If needed, tack on zeroes to make the decimals equal length
Compare digits beginning starting immediately after the point (if needed, tack on zeroes to make numbers the same length)
When digits differ, the larger number gives the larger amount

Rounding

Look at the specified digit
Look at the next place value (immediately after the specified one)
If the next digit is

0 – 4, keep the desired digit

5 – 9, round up

Ex Round to the nearest

a)thousandth

b)hundredth

c)whole number (unit)

d)ten

e)hundred

Ex Round to the nearest

a)hundredth

b)tenth

c)whole number (unit)

d)ten

e)hundred

4.2 Addition & Subtraction

Procedure:

Line up decimal points! (most important)
Fill in zeroes at the end of decimals if needed

Ex a Add: 2.68 + 11.3 + 0.009

Ex b Subtract: 6 – 4.27

Ex c Solve: x + 3.7 = 9.431

Practice Problems - Perform the operations or solve

1) 7 – 2.381 / 2) 14.843 + 0.34 + 1.9 + 10 / 3) Solve: x – 42.87 = 19.4

4.3 Multiplication

Decimals have fractional equivalents:

2.35 X 0.4

Procedure:

Multiply digits as if they were whole numbers
Move the point the number of decimal places after all points
Fill in zeroes if needed

Ex a Multiply: (6.7)(0.038)

Multiply by 0.1, 0.01, 0.001, etc. (small numbers)

Ex b Multiply 18.47 X 0.001

Multiply by 10, 100, 1000, etc. (large numbers

Ex c Multiply 18.47 X 1000

Large Number Names

Ex d Convert $14.5 million to digits

Dollars and Cents $1 = 100¢ and 1¢ = 0.01$

Ex e Convert 89 cents to dollarsEx f Convert $22.51 to cents

Practice Problems

1) 4.6 X 0.9 / 2) 0.01 X 821.37
3) Convert 530,792¢ to dollars / 4) Convert 192.5 thousand to standard form

4.4 Division

Divide decimals by whole numbers – similar to whole number long division, but put the decimal point in the quotient

Ex a

Decimal divisors (denominators) - make fraction and move point to get a whole number in denominator.

Ex b 2.7320.04

Divide by 10, 100, 1000, etc. (large numbers)

Ex c Divide 128.54 1000

Divide by 0.1, 0.01, 0.001, etc. (small numbers)

Ex d Divide

Solving

Ex e Solve: 2.5t = 300

Practice Problems

1) / 2) 11.2 4 / 3) Solve: 0.3y = 1.38

4.5 Converting Fractions to Decimals - Use Long Division

Write in decimal notation:

Ex aEx bEx c

For each of the decimals above, round to the nearest tenth, hundredth, and thousandth.

tenth:

hundredth

thousandth

Practice Problems

Write in decimal notation and round to the nearest tenth

1) / 2)

4.6 – 4.7 Estimating and Applications

Estimating Sums and Differences– operations are

- Round to the same place value(s) then add or subtract.

Ex a On a shopping trip, Mia buys items costing $38.95, $129.99 and $9.77. Estimate the cost by rounding to the nearest ten. / Ex b A $491.79 tablet is discounted by $109.21. Estimate the final price.

Estimating Products and Quotients – operations are

- Round to one non-zero digit OR round to “easy” digits.

Ex b Dan is paid $892.12 for 11 days. Estimate his daily pay, then calculate the exact amount to the nearest cent.

Practice Problems - Perform the operations or solve

1) Estimate, rounding to the nearest tenth:
1.4368 + 0.1724 – 0.0913 / 2) Coffee costs $3.61 (including tax). How much is spent in a 30-day month?
3) Cole earned $620.80 working 40 hours in a week. What is his hourly wage? / 4) A shipment of seafood costs $88.65, and there are 6.245 lb. Estimate each number, then divide the estimates to approximate the cost/lb.

5.1 Intro to Ratios

ratio – a ______of 2 quantities

There are several ways to write ratios. For example for a TV screen 16 inches wide and 9 inches tall, the width to height ratio can be written as:

Ex a Write the ratios in 2 other formats (without reducing)

3 to 5 14.7:1008½ to 11

Ex b Find the ratio of length to width:

Ex c For the triangle, find the ratios listed and reduce.

height to base ratiohypotenuse to base ratioheight to hypotenuse ratio

5.2 Rates and Unit Prices

rate – a ratio whose numerator and denominator have different units.

Ex a My car travels 500 miles on 15 gallons of gas. What is the rate of miles per gallon (also known as gas mileage)?

Ex b Al earns $30,000 in a year. What is his rate of pay in dollars per month?

Unit rate – ratio where the denominator number is 1

Unit price – ratio of price to number of units, where the number of units is reduced to ______

To find unit rates (including unit price), use ______

Ex c Find the unit cost of the following jars of peanut butter. Which is the better buy?

Brand A is 40 oz. and costs $5.00Brand B is 28 oz. and costs $3.00

Ex d I drive 390 miles in 6 hours. What is the unit rate in miles/hour?

5.3 Proportions

Proportion – 2 ratios that equal each other:

The pairs 1, 2 and 3, 6 can be used to form a ratio:

We can test if 2 proportions are equal if their cross products are ______

Ex a Are the pairs or ratios proportional?

1) 2) 3,5 and 21, 35

3) 4) 2½ , 4½ and 10, 18

Solving Proportions – set cross products ______, then solve for x

Ex b Solve:

Practice Problems:

1) Write fraction notation and reduce:

8 to 122.4 to 6

2) Find each rate:

65 meters to 5 seconds243 miles per 4 hours

3) Are the pairs proportional:

3, 7 and 15, 452.4, 1.5 and 0.16, 0.1

4) Solve

5.4 Applications of Proportions

When to use proportions? You have 2 quantities that are related. One quantity changes, and you want to find the changed value of the second quantity.

Ex a Ravi makes $315 working 21 hours. How much would he make if he works 40 hours?

Ex b Two cities on a map are 2½ inches apart, which represents 300 miles. How far apart are two cities if they are 6¼ inches apart on the map?

Ex c In 2015, 1 US dollar is worth 16.8 pesos. How many dollars is 400 pesos worth, to the nearest dollar?

6.1 Percent Notation

The Butte fire was 15% contained (as of Sept. 12, 2015). What does that mean?

Percent notation:

Fraction notation: or ratio:

Decimal notation:

Percent of a Quantity

What is of 40?

What is 60% of 40?

Proper fractions (less than 1) and “common” percentages (less than 100%) are similar

Amount = Percent of Base (If you have “Percent of”, the next is always base)

Ex b A discount is 20% of the original price. If the item is marked $30, what is the discount?

Converting percent to decimal – Replace % with ______or ______

This causes you to remove ______, make number ______

Ex a Convert to decimal:

58%7.2%150% 0.03%

Ex b Convert to decimal:

Ex c Write decimal values for each of the percentages listed.

Monthly Expenses

If monthly income is $1000, how much spent on transportation?

Convert decimal to percent – multiply by ______Does this change the value?

Ex d Write percent notation for

0.27 0.735 0.4

2.7 0.0009

6.2 Percents and Fractions

Converting a fraction to percent

First, convert fraction to decimal (From 4.5, use ______

Next, convert decimal to percent (From 6.1, multiply by ______

Ex a Convert 7/8 to a percent

Ex b to a percent

Shortcut - Only works when denominator is a factor of 100

Multiply top and bottom to build the denominator to 100.
Change /100 to %

Ex c Convert to a percent:

Converting Percent to a Fraction (postpone repeating decimal to Math 20)

Replace % with ______
Reduce

Ex d Convert to a fraction and simplify

70%12.5%0.4%

Practice Problems

1. A lawn requires 300 gallons of water for every 500 square feet. How much water does a lawn which is 1800 square feet require?

2. Sal burned 200 calories in ¾ hour of walking. How many calories would be burned in 1 ¾ hours of walking?

3 Find percent notation for:

0.70.3891

4. Find decimal notation for:

57%1.5%22 ½ % 240%

5. Find fraction notation for:

57%1.5%22 ½ % 240%

6.3 Solving Percent Problems - Percent Equations

Translating to Equations

of  multiply

is  equals

%  convert number to decimal or a fraction using 1/100

What 

Ex a What is 7% of 45?

Ex b 28% of 30 is what?

Ex c 15 is what percent of 75?

Practice Problems

What percent of 42 is 7?

9 is 25% of what?

70% of what is 35?

6.5 Applications

Ex a From the pie chart below:

Monthly Expenses
/ 1) If a person makes $3000/month, how much is spent on housing?
2) If a person is spending $600/month on transportation, what is their total income?

Ex b A test has 60 questions, and Jan gets 49 correct. What percent are correct (to the nearest whole number percent)?

Percent Increase & Decrease

Ex c Rent was $750/month last month and increased to $800 this month. What is the percent increase?

Ex d A TV cost $400 last year but costs $320 this year. What is the percent decrease?

6.6 Sales Tax, Commission, and Discount

Sales tax and commission increase an original price.

Discount decreases the original price.

Rate of discount or increase is the same as percent of discount or increase.

Ex a Sales tax adds $12.74 to the price of a fire pit. If the sales tax rate is 8%, find the original price.