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1-1: A plan for Problem Solving

1. The steps of a 4-Step plan:

A. Understand- Read the problem carefully.

B. Plan- How do the facts relate to each other?

C. Solve- Use your plan to solve the problem.

D. Check- Does your answer fit the facts given in the problem?

1-2: Powers and Exponents

1. factors- two or more numbers that are multiplied together to form a product

2. base- common factors

3. exponent- tells you how many times the base is used as a factor

4. powers- numbers expressed using exponents

5. evaluate- find the value

6. standard form- numbers written without exponents

7. squared- a number to the 2nd power

8. cubed- a number to the 3rd power

9. exponential form- numbers written with exponents

10. Look at page 30 at the red and blue print.

11. The base is what you write over and over.

12. The exponent tells you how many times to write the base over and over.

13. Look at page 30 at the examples at the bottom of the page.

14. Look at page 31 at the examples at the top of the page.

15. Look at page 31 at the examples in the middle of the page.

1-3: Squares and Square Roots

1. square- the product of a number and itself

2. perfect squares- squares of whole numbers (They come out with no decimals)

3. square root- the factors multiplied to form perfect squares.

4. radical sign- the symbol used to indicate a square root (√).

5. Look at the examples on page 34 in the middle of the page.

6. Look at the key concept at the top of page 35.

7. Look at the examples on page 35 in the middle of the page.

1-4: Order of Operations

1. numerical expression- a combination of numbers and operations

2. order of operations- rules that help you solve numerical expressions

3. order of operations:

A. Do all grouping symbols first (parentheses).

B. Do all powers.

C. Multiply and divide in order from left to right.

D. Add and subtract in order from left to right.

4. Remember: Please Excuse My Dear Aunt Sally

5. Look at examples on pages 38-39.

1-5: Problem-Solving Investigation: Guess and Check

1. To use Guess and Check---guess a reasonable answer and then work the problem:

A. If the answer is too big, guess again with a smaller number.

B. If the answer is too small, guess again with a larger number.

1-6 Algebra: Variables and Expressions

1. variable- a symbol used to represent an unknown quantity

2. algebra- the branch of mathematics that involves expressions with variables

3. algebraic expression- an expression that contains variables, numbers, and at least 1 operation

4. coefficient- the numerical factor of a multiplication expression that contains a variable (the number in front of a variable)

5. Look at the examples on pages 44-45.

1-7 Algebra: Equations

1. equation- a sentence that contains two expressions separated by an equal sign

2. solution- a numerical value for the variable that makes a sentence true

3. solving an equation- the process of finding a solution

4. defining the variable- choosing a variable to represent an unknown quantity

5. Look at the examples on pages 49- 50.

1-8 Algebra: Properties

1. equivalent expressions- expressions that have the same value

2. properties- statements that are true for all numbers

3. distributive property- a property that combines multiplication and addition OR multiplication and subtraction


5. Study the Concept Summary on page 54.

1-9 Algebra: Arithmetic Sequences

1. sequence- an ordered list of numbers

2. term- each number in a sequence

3. arithmetic sequence- when each term is found by adding the same number to the previous term


1-10 Algebra: Equations and Functions

1. function- a relationship that assigns exactly one output value for each input value

2. function rule- the operation performed on the input

3. function table- a table in which you can organize the input numbers, output numbers, and the function rule

4. domain- the set of input values (the x coordinate)

5. range- the set of output values (the y coordinate)


2-1: Integers and Absolute Value

1. integer- (whole numbers and their opposites) any number from the set {…-4,

-3,-2, -1, 0, 1,2,3,4, …} where … means continues without end

2. negative integers- integers less than zero. They are written with a - sign.

3. positive integers- integers greater than zero. They can be written with or without a + sign.

4. Zero is neither negative nor positive.

5. graph- to draw a point on a number line to show its location.

6. absolute value- the distance a number is from zero on a number line (LEFT OR RIGHT DOES NOT MATTER).

7. Look at the absolute value bars on page 81, Key Concept, examples.


2-2: Comparing and Ordering Integers

1. When 2 numbers are graphed on a number line, the number graphed to the left is always less and the number graphed to the right is always greater.

2. Negative numbers are to the left of 0 and positive numbers are to the right of 0 on a number line.

3. When dealing with negative numbers the bigger they look; the smaller they are.

4. < means “less than” and > means “greater than.”


2-3: The Coordinate Plane

1. coordinate plane- a plane in which a horizontal number line and a vertical number line intersect at their 0 points

2. quadrant- the four regions that a coordinate plane is separated into

3. ordered pair- a pair of numbers used to locate a point on a coordinate plane

4. x- axis - the horizontal number line in a coordinate plane

5. y- axis - the vertical number line in a coordinate plane

6. origin- where the x-axis and y-axis cross at their zero points

7. x-coordinate – the FIRST number in an ordered pair that tells you TO GO LEFT OR RIGHT

8. y-coordinate- the SECOND number in an ordered pair that tells you TO GO UP OR DOWN

9. Look at the Key Concept on page 88.

10. Look at the examples on page 89.

2-4: Adding Integers

1. opposites- numbers that are the same distance from zero but are on opposite sides of zero

2. additive inverses- opposites

3. Additive Inverse Property- the sum of any number and its opposite is 0

(ex. 5 + -5=0)

4. To add integers with different signs:

A. subtract their absolute values

B. the answer is positive if the absolute value of the positive integer is greater

C. the answer is negative if the absolute value of the negative integer is

Greater

5. To add integers with the same sign:

A. add their absolute values

C. if both integers are negative, the answer is negative

6. Study the Key Concept box on page 95.


2-5 Subtracting Integers

1. To subtract integers add its opposite.

2. To subtract integers:

A. Change the subtraction sign to addition

B. Make the number to the right its opposite

C. Now follow the rules for lesson 2-4.


2-6 Multiplying Integers

1. To multiply integers:

A. Ignore the signs and just plain multiply

B. If the signs are the same the answer is positive

C. If the signs are different the answer is negative


2-7: Problem Solving Investigation: Look for a Pattern

1. Find a pattern then use the pattern to solve the problem.

2-8: Dividing Integers

1. To divide integers:

A. Ignore the signs and just plain divide.

B. If the signs are the same the answer is positive.

C. If the signs are different the answer is negative.


3. Study the concept summary on page 116.

3-1: Writing Expressions and Equations

1.

+ - X ÷Increased by Difference Each divideMore (than) Less than Product Quotient

Sum Decreased by Multiplied PerPlus Minus Twice SeparateAdd Subtract Times SharedIn all Less Per

2. To write phrases as expressions:

A. Decide if you need a variable.

B. Look for the key words above to decide what operation symbol to use.

C. Put in the numbers.

3. Look at the example on page 128.

4. EXPRESSIONS DO NOT CONTAIN EQUAL SIGNS BUT EQUATIONS DO CONTAIN EQUAL SIGNS.

5. The word IS means put an EQUAL sign.


3-2: Solving Addition and Subtraction Equations

1. Think of an equation as a balanced scale. If you add weight to one side, you must add that exact same amount of weight to the other side. If you take weight off one side, you must take off the exact same amount of weight from the other side. You must do the same thing when dealing with an equation.

2. On an equation the equal sign is the middle.

3. Subtraction Property of Equality- If you subtract the same number from each side of an equation the two sides remain equal

4. To solve equations with addition signs:

A. Put a dashed line through the equal sign.

B. Now ask yourself what number is on the same side as the variable.

C. Put the opposite of this number underneath it and also on the other side of the dashed line.

D. The numbers on the same side as the variable cancel out.

E. Now use the rules from lessons 2-4 and 2-5 to do the math on the other side of the equal sign.


6. Addition Property of Equality- If you add the same number to each side of an equation, the two sides remain equal.

7. To solve equations with subtraction signs:


B. Change the subtraction sign to an addition sign and make the number to the right its opposite.

C. Now ask yourself what number is on the same side as the variable.

D. Put the opposite of this number underneath it and also on the other side of the dashed line.

E. The numbers on the same side as the variable cancel out.

F. Now use the rules from lesson 2-4 and 2-5 to do the math on the other side of the equal sign.


3-3: Solving Multiplication Equations

1. formula- an equation that shows the relationship among certain quantities

1a. (Look back at note #1 lesson 3-2)

2. Division Property of Equality- If you divide each side of an equation by the same nonzero number, the two sides remain equal.

3. Remember when 2 numbers are written like a fraction it can also mean divide:

4/2 means 4÷2

4. To solve multiplication equations:

A. Put a dashed line through the equal sign

B. Ask yourself what number is on the same side as the variable

C. Put the exact same number underneath this number and also put it on the other side of the dashed line ( You will divide to solve these problems).

D. The numbers on the same side as the variable will cancel out.

E. Now use the rules from lesson 2-8 to do the math on the other side of the equal sign.

5. Look at the examples on page 143-144.

3-4: Problem Solving Investigation: Work Backwards

1. Sometimes it is easier to solve a problem if you work backwards, like solving a maze.

3-5: Solving Two-Step Equations

1. two-step equation- an equation that requires two steps to solve

2. To solve two-step equations:


B. Now ask yourself “Does my equation have an addition sign?”

C. If you said yes skip steps D & E and go straight to step F.

D. If you said no go straight to step E.

E. Change the subtraction sign to addition and make the number to the right its opposite.

F. Then ask yourself “What 2 numbers are on the same side as the variable?”

G. Now decide which of these 2 numbers are farther from the variable.

H. Put the opposite of this number underneath it and also put it on the other side of the dashed line.

I. The number you picked in step G cancels out. Now use the rules from lesson 2-4 & 2-5 to do the math.

J. Bring all the numbers you did not use straight down.

K. Now ask yourself “What number is on the same side as my variable?”

L. Put the exact same number underneath this number and also put it on the other side of the dashed line.

M. The numbers on the same side as the variable will cancel out.

N. Use the rules from lesson 2-8 to do the math on the other side of the equal sign.

3. To solve two-step equations:

A. undo addition or subtraction first.

B. Then undo multiplication or division.


3-6: Measurement: Perimeter and Area

1. perimeter- the distance around a geometric figure

2. The perimeter (P) of a rectangle is twice the sum of the length (L) and width

(W) or you can just add up all the sides.

P=2L + 2W or P= L+L+W+W

3. area- the measure of the surface enclosed by a rectangle

4. The Area (A) of a rectangle is the product of the length (L) and width (W)

Multiply the length and the width A= L X W


6. Remember to square your answer when you find area (write your answer as the unit to the second power).

3-7: Functions and Graphs

1. linear equation- an equation whose graph is a straight line

2. When given a function, ordered pairs are in the form (input, output) or

(x, y) and provide useful information about the function.

3. In an ordered pair x always comes first.

4. A graph of ordered pairs shows where the data is.

5. x tells you to go left (negative) or right (positive).

6. y tells you to go down (negative) or up (positive).

7. One way to graph a linear equation is to make a table of solutions

8. To make a table of solutions:

A. Choose the numbers you wish to use for your x-values (We usually use

-2,-1, 0, 1, 2).

B. Plug the numbers into the place of the x in the equation.

C. Solve for y.

9. Study the Key Concept at the bottom of page 165.


4-1: Prime Factorization

1. prime number- a number that has exactly two factors one and itself

2. composite number- a number that has more than two factors

3. prime factorization- when a composite number’s factors are all written as prime numbers

4. factor tree- a tool you can use to find the prime factorization of a number


4-2: Greatest Common Factor

1. Venn diagram- uses overlapping circles to show how common elements among sets of numbers or objects are related.

2. Venn diagrams also show common factors.

3. Greatest Common Factor (GCF)- the biggest of the common factors of two or more numbers

4. Notice there are two different methods to find the GCF:

METHOD 1:

A. List the factors of each number.

B. Circle the largest common number in both lists.

METHOD 2:

A. Use prime factorization.

B. Write the prime factors that are in both lists.

C. Multiply these numbers together.


4-3: Problem-Solving Investigation: Make an Organized List

1. Sometimes if you organize your data in a table or list, it will help you solve the problem.

4-4: Simplifying Fractions

1. equivalent fractions- fractions that have the same value

2. simplest form- when the GCF of the numerator and denominator is 1

3. To write a fraction in simplest form:

A. Find the GCF of the numerator and the denominator

B. Divide the numerator and the denominator by the GCF

4. If you have really large numbers, use lesson 4-2 to find the GCF and then divide the numerator and the denominator by the GCF.


4-5: Fractions and Decimals

1. terminating decimals- a decimal whose digits end

2. repeating decimals- decimals that have a pattern in their digits that repeats forever

3. Bar notation is used to indicate that a number pattern repeats indefinitely.

4. To write a fraction as a decimal divide the numerator by the denominator (divide the top number by the bottom number)(the top number goes in the division box and the bottom number goes outside the box).

5. To write a mixed number as a decimal:

A. Ignore the whole number

B. Change the fraction to a decimal

C. Now put the whole number to the left of the decimal


4-6: Fractions and Percents

1. ratio- a comparison of two quantities by division

2. percent- a part to whole ratio that compares a number to 100

n/100= n%

3. To write a fraction as a percent:

A. The fractions denominator needs to be 100

B. Now just write the numerator with a percent sign.

4. To write a percent as a fraction:

A. Put the percent over 100 and take off the percent sign

B. Reduce if possible.

5. To compare fractions and percents:

A. Either make them both fractions or both percents.

B. Now compare.


7. Study the Key Concepts on page 208.

4-7: Percents and Decimals

1. To write a percent as a decimal:

METHOD 1 A. Move the decimal two places to the LEFT and remove the percent sign (If there is no decimal it is

all the way to the right of the number)

METHOD 2 A. Write the percent as a fraction.

B. Write the fraction as a decimal.

2. To write a decimal as a percent:

A. Move the decimal two places to the RIGHT and add a percent sign.

3. Study the Key Concept table on page 208.


4-8: Least Common Multiple

1. multiple- the product of a number and any whole number

2. Least Common Multiple (LCM)- the smallest of the common multiples (does not include 0)

3. To find the LCM:

METHOD 1 A. List the multiples of the numbers

B. Circle the smallest numbers they have in common

METHOD 2 A. List the prime factorization of each number

B. List the smaller numbers factors

C. Only add the numbers from the larger numbers list that aren’t in the smaller numbers list.

D. Now multiply the numbers in the list together.


4-9: Comparing and Ordering Rational Numbers

1. rational numbers- numbers that can be expressed as a fraction

2. common denominator- a common multiple of the denominators of two or more fractions

3. The Least Common Denominator- the LCM of the denominators

4. To compare fractions:

METHOD 1 A. find the LCD

B. now compare the numerators

METHOD 2 A. Use cross products

5. Study the Fractions-Decimals-Percents table on page 217.

6. To compare and order fractions, decimals, and percents they must first all be in the same form.


5-1: Estimating with Fractions

1. compatible numbers- numbers that are easy to compute

2. To estimate with mixed numbers:

A. Ignore the whole number and look at the fraction.

B. If the fraction is ½ or more add 1 to the whole number.

C. If the fraction is less than ½ add 0 to the whole number.


4. To estimate with fractions:

A. Decide what the fraction is closest to: 0, ½, or 1.

B. Round the fraction to : 0, ½, or 1 and then do the math.


6. Sometimes after you round the problem may not work out evenly; then you must use compatible numbers.


5-2: Adding and Subtracting Fractions

1. To add or subtract like fractions (fractions with the same denominators):A. Add or subtract the numeratorsB. Put this answer over the common (same) denominator and reduce if

possible.


3. To add or subtract unlike fractions ( fractions with different denominators):A. Find the LCD of the two denominators and do the backwards zB. Now follow note #1


5-3: Adding and Subtracting Mixed Numbers

1. Work the addition problems just like lesson 5-2. Just remember to figure the whole numbers.

2. Look at the examples on page 242.3. To work the subtraction problems:

A. First change the mixed numbers to improper fractions.B. Now work these problems just like lesson 5-2.


5-4: Problem-Solving Investigation: Eliminate Possibilities

1. Sometimes it is easier to eliminate answers you know are incorrect before you figure the problem.

5-5: Multiplying Fractions and Mixed Numbers

1. To multiply fractions, multiply the numerators and then multiply the denominators.

2. Look at the examples on pages 252-254.3. If the number is mixed make it improper first and then follow note #1.

5-6: Algebra: Solving Equations

1. Reciprocals ( multiplicative inverses)- two numbers with a product of 1 (just flip the fraction over)

2. Look at the examples on page 258.3. Multiplication Property of Equality- If you multiply each side of an equation by

the same nonzero number, the two sides remain equal4. To solve division equations:

A. Put a dashed line through the equal signB. Ask yourself, “What number is on the same side as the variable.”C. Now multiply each side by this exact same numberD. The numbers on the same side as the variable cancel outE. Just multiply the other side

5. Look at the examples on page 259.6. Remember note section 3-3 note #4 to solve multiplication equations.7. Look at the examples on page 260.

5-7: Dividing Fractions and Mixed Numbers

1. To divide by a fraction:A. Multiply by its reciprocal

2. To divide fractions:A. Flip the 2 nd fraction and multiply

3. Look at the examples on pages 266-267.4. To divide mixed numbers:

A. Change the mixed numbers into improper fractionsB. Now follow not #1.

6-1: Ratios

1. Ratio-a comparison of two quantities by division2. Equivalent ratios-ratios that express the same relationship between two

quantities 3. When you write a ratio whatever the problem asks for 1st you must list first, or

on top if writing the ratio as a fraction. Remember to write it in simplest form.4. A ratio may be written 3 ways:

1:3 1 to 3 1/35. Look at the examples on page 282-283.6. Use cross products to determine if 2 ratios are equivalent.

6-2: Rates

1. Rate- a ratio that compares two quantities with different kinds of units2. Unit rate- when a rate is simplified so that it has a denominator of 13. To write a rate it should look like this “14 days per 2 weeks”

14 days/2 weeks

4. To write a rate as a unit rate divide the 1st number by the 2nd.5. To find the unit price divide the dollar amount by the other number (the dollar

amount goes in the box).6. To find the best buy find the unit price of each and choose the amount that is

the smallest.7. To use a unit rate:

A. Find the unit rateB. Then multiply

8. Look at the examples on page 287-289.

6-3: Rate of Change and Slope

1. Rate of change- a rate that describes how one quantity changes in relation to another

2. To find the rate of change from a table:A. What they ask for 1st goes on top (the difference in the pattern)B. What they ask for 2nd goes on bottom (the difference in the pattern)C. Then write it as a unit rate

3. Slope- the rate of change between any two points on a line4. Slope=change in y vertical change

change in x horizontal change5. Look at the examples on pages 293-295.

6-4 Measurement: Changing Customary Units

Customary Units

Type of Measure Larger unit smaller unit

1 foot (ft) = 12 inches (in)

Length 1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5,280 feet

Weight 1 pound (lb) = 16 ounces (oz)

1 ton (T) = 2, 000 pounds

1 cup ( c) = 8 fluid ounces (fl oz)

1 pint (pt) = 2 cups

Capacity 1 quart (qt) = 2 pints

1 gallon (gal) = 4 quarts

2. Unit ratio- a ratio in which the denominator is 1 unit3. To convert from a larger unit to a smaller unit:

A. Multiply by the conversion factor4. To convert from a smaller unit to a larger unit:

A. Divide by the conversion factor.

6-5: Measurement: Changing Metric Units

1. Metric system- a decimal system of measures2. Meter (m)- the base unit of length3. Gram (g)- the base unit of mass4. Liter (L)- the base unit of volume5.

(km)Kilometer

(hm)hectometer

(dam)Dekameter

Base unitMeter (m)

(dm)Decimeter

(cm)Centimeter

(mm)Millimeter

(kg)Kilogram

(hg)Hectogram

(dag)Dekagram

.Gram (g)

(dg)Decigram

(cg)Centigram

(mg)Milligram

(kL)Kiloliter

(hL)Hectoliter

(daL)Dekaliter Liter (L)

(dL)Deciliter

(cL)Centiliter

(mL)Milliliter

6. To convert metric units:A. Use the table aboveB. First, find the unit where you are starting.C. Then find the unit where you are going.D. Count how many boxes you moved.E. Now decide what direction you moved.F. Finally, move the decimal the same number of places and the same

direction.7. Study the table on page 306.8. To convert customary and metric units:

A. Multiply by the conversion factor (use the table on page 306 to find the conversion factor).

6-6: Algebra: Solving Proportions

1. Proportional- when two quantities have a constant rate or ratio (they are equal)

2. Proportion- an equation stating that two ratios or rates are equivalent3. Cross products- when you multiply across diagonally4. The cross products of any proportion are equal.5. Look at the example (method 2) on page 311.6. To solve a proportion:

A. Multiply across diagonally when there are two numbers.B. Now divide this answer by the number you haven’t used yet.

6-7: Problem-Solving Investigation: Draw a Diagram

1. To solve problems that involve measurements, it is sometimes easier to see and understand if you draw a picture.

6-8: Scale Drawings

1. Scale drawing (scale model)- objects that represent items that are too large or too small to be drawn or built to actual size

2. Scale-this gives the ratio that compares the measurements of the drawing or model to the measurements of the real object

3. Scale factor- a scale written as a ratio without units in simplest form4. Look at the examples on page 320-322.

6-9: Fractions, Decimals, and Percents

1. Remember to write a percent as a fraction:A. Put the percent over 100B. Take off the percent signC. Reduce if possible

2. To write a fraction as a percent:A. Multiply by a number that will make the denominator become 100B. Remember if you multiply the denominator by a number you must multiply

the numerator by that exact same number.C. Now write the numerator with a percent sign.

3. If the problem does not come out even work the problem as a proportion (set the fraction equal to a variable over 100).

4. Another method to write a fraction as a percent is to:A. Divide the numerator by the denominator(divide the top number by the

bottom number)B. Now change this decimal to a percent.

5. Remember, sometimes you will have to round your answer.6. Study the table on the bottom of page 330.7. Look at the examples on pages 328-330.

7-1: Percent of a Number

1. To find the percent of a number:

Method1 Write the percent as a fraction and then multiply

Method 2 Write the percent as a decimal and then multiply


7-2: The Percent Proportion

1. Percent proportion- one ratio or fraction that compares part of a quantity to the whole quantity

2. Percent proportion

Part = %

Whole 100

3. The number that comes after the word of is the whole.4. The percent always has a percent sign or the word percent.5. Study the table at the bottom of page 352.6. Look at the examples on pages350-352.

7-3: Percent and Estimation

Omit

7-4 Algebra: The Percent Equation

1. Percent equation- an equation that describes the relationship between the part, whole and percent. Part= percent times whole

2. Sometimes it is easier to write an equation to solve percent problems.3. To writ an equation look for key words:

A. Is means =B. Of means multiplyC. What number, some number, etc means use a variable

4. Look at the examples on pages 361-3635. Study the Concept Summary table on page 363.

7-5: Problem-Solving Investigation: Determine Reasonable Answers

1. Make sure you really read the word problem to see if the answer is reasonable.

7-6: Percent of Change

1. Percent of change- a ratio that compares the change in quantity to the original amount

2. Percent of change = amount of changeOriginal amount

3. Percent of increase- when the original quantity is increased (gets bigger)4. Percent of decrease- when the original quantity is decreased (gets smaller)5. To find the percent of change:

A. Subtract to find the amount of changeB. Divide the amount of change by the original amount (the answer you get

when you subtract goes inside the box).6. Look at the examples on page 370.

7-7: Sales Tax and Discount

1. Sales tax- an additional amount of money charged on items that people buy2. Discount- the amount by which the regular price of an item is reduced3. To find sales tax:

A. Multiply the purchase price by the tax rate4. To find the total cost: (Method 1)

A. Find the sales taxB. Add the sales tax to the purchase price

5. To find the total cost: (Method 2)A. Add the tax rate to 100%B. Now multiply this new percent by the purchase price

6. To find the sale price: (Method 1)A. Multiply the discount rate by the original priceB. Subtract this product from the original price

7. To find the sale price: (Method 2)A. Subtract the discount rate from 100%B. Multiply the difference by the original price.

8. To find the original price:A. Subtract the discount rate from 100%.B. Now set up your equation new price=percent times original priceC. Solve the equation


7-8: Simple Interest

1. Principal- the amount of money borrowed or invested2. Simple interest- the amount paid or earned for the use of money3. Use the formula listed on page 379 (it is listed in different colors) to figure

simple interest.4. Just plug in the numbers you know and solve the problem.5. Look at the examples on pages 379-380.

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