Combo Praxis 5001 Math

Area (Definition)<br />

"<b>Area</b> is the two-dimensional measure of how many square units can fit inside the interior of an object<br /><br /><img src=""paste7hksh1.png"" /><div><br /></div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 acre = 43,560 ft²</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">640 acres = 1 mile²</div><div><br /></div>"

Properties of Number System Operations

"<b>Properties of Number System Operations</b><br><img src=""pastehp_zce.png"" />"

probability equation

#possibilities of what I want -------------------- total # possible outcomes

Order of Operations

PREMDAS 1. Simplify inside parentheses/brackets/etc. 2. Simplify any expressions with roots 3. Simplify any expressions with exponents 3. Perform multiplication / division from left to right 4. Perform addition / subtraction from left to right PREMDAS or Please really excuse my dear Aunt Sally

Parentheses

Parentheses are a way to group numbers. Other grouping symbols are: braces { }, square brackets and the vinculum or fraction bar. To remove parentheses, we distribute the number immediately outside the parenthesis (with its sign). We distribute by multiplying by the number. 3(3x +1) = 9x + 3 -(2x - 5) = -2x + 5

radius

The distance from the center of a circle to any point on the circle. It is half the diameter. r= d/2

Whole Numbers

Whole numbers = the counting numbers and zero (0, 1, 2, 3, 4, ...). Positive numbers that have no fractions or decimals, including zero.

Factor

numbers that divide evenly into other numbers - without a remainder. For example, 5 divides into 40 evenly so 5 is a factor of 40. We often create a factor tree or a prime factorization of numbers to help us recognize the factors of a number See pic for factor tree which shows that the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

Counting Numbers

same thing as natural numbers whole numbers 1 and up **DOES NOT INCLUDE 0**

Natural Numbers

the counting numbers (1, 2, 3, 4 ...) whole numbers 1 and up **DOES NOT INCLUDE 0**

circumference

the distance around the outside of a circle C = (π)d = 2(π)r

divisor

the number you divide BY -the 2nd number in a division problem when it is across for example, 3 is the divisor in 2 ÷ 3 -It is the number outside the bracket in long division

Decimal Number Division

"<span style=""font-weight: 600; ""><font color=""#0000FF"">Division is not defined for decimal numbers</font></span><font color=""#0000FF"">.</font> In order to divide by a decimal number, we change that divisor into a whole number: First multiply each number by powers of 10 - multiply by whatever is necessary to make the divisor a whole number. Then divide as you would with whole numbers. Wherever the decimal point is in the dividend, it floats directly up to that position in the quotient (answer).<br /><br /><img src=""pastex0nanh.png"" />"

Rates

"A *rate* is a ratio between two measurements with different units. In addition to the three ways to write a ratio, rates may also use the word "per". Rates are usually simplified to a one in the denominator (second measurement) Examples: 13 miles per gallon $4.59 per pound 12 inches per foot

Cube (Definition, volume, and surface area)<br />

"A <b>cube</b> is a three-dimensional solid where all angles are right angles and all faces are squares. A cube is also informally called a square box.<br /><br /><img src=""pastexwl_ol.png"" /><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = a³</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">SA = 6a²</div>"

Cylinder (Definition, volume, and surface area)<br />

"A <b>cylinder</b> is a three-dimensional solid where the top and bottom are circles.<br /><br /><img src=""pastebdvock.png"" /><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = πr²h</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">SA = 2πr²&nbsp;+ 2πrh</div>"

Decimal System Place Value

"A number in standard form is marked into groups of three digits. Each of these groups is called a period. Within each group, the place values are always the 100's place, the 10's place, and the 1's place. <b>Decimal numbers simply extend the place values</b> to the right and use "ths" to identify the places (e.g. 100 millionths place). There is not a oneths place.<br /><br /><img src=""paste6wllx9.png"" />"

Comparing Decimal Numbers

"Line up the <b>decimal</b> <b>numbers</b> according to place value (as though you were going to add them). Starting at the left-most place value, <b>compare</b> the numbers in each place value to find the largest, next largest, etc.<br><br><img src=""paste6lle1v.png"" />"

Ordering Whole Numbers

"When you are asked to order whole numbers, write them above one another with the place values lined up. Then, starting from the *left*, look for the largest value. For example, if you are asked to order: 5,139 986,733 3,950 77,922 Write them above each other with the place values lined up as you would if you were going to add the numbers. Looking at the place values from left to right, the largest number is 986,733. The next largest number is 77,922. Both the first and third numbers start in the same place value but 5 is larger than 3 so 5,139 is larger than 3,950

Division of Fractions & Mixed Numbers

*Division of Fractions & Mixed Numbers: is not defined.* As such, we use the properties of our number system to *change the division problem into a multiplication problem*, which *is* defined. (see pic) -Change mixed numbers to improper fractions. -Change whole numbers to improper fractions with a "1" on the denominator. -Write the two fractions horizontally beside each other. -Write the reciprocal of the divisor (flip the second fraction upside down) and change the operation to multiplication. Expand each numerator and each denominator into a prime factorization. "Cancel" any ones such as 3/3 or 5/5. Multiply what is left straight across. A reciprocal is the inverse of a fraction. Multiplication is the inverse of division. The inverse of an inverse is the same as the original problem. As such, multiplication of a reciprocal is the same as dividing by the original divisor. *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Do NOT need common denominators* *Answer should be MORE than the original factors* (ex: ½ ÷ ¼= 4/2 = 2) <--2 is more than 1/2 or 1/4*

Equivalent Fractions

*Equivalent fractions* are fractions which simplify to the same simple fraction When two fractions are equivalent, their cross products are equal (see picture) If have a variable, cross multiply to solve. (Each side will now be a linear equation, no longer in fraction form. See example) ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2 *Remember on test, if answers are in fraction form, correct answer will always be reduced*

LCM

*Lowest Common Multiple (LCM)* of two or more numbers is the smallest number that is a multiple of all the numbers. One way to find the LCM is to count by each of the numbers and find the first number that is a multiple of all. For example, find the LCM of 9, 12, and 18 Then make a Venn Diagram with all the factors, the shared factors in the center. Lastly, multiply all the numbers together. This is the LCM. (see pic) example LCM word problem: One day, Edward and his friends had lunch while sitting at tables of 15. Another day, they had lunch at tables of 10. What is the *smallest number of people that could be in the group?* (30 people)

Simplifying Fractions

*Simplifying fractions* is sometimes (incorrectly) called reducing a fraction. Expand the numerator and denominator into prime factorizations (see pic). "Cancel" any ones such as 3/3 or 5/5. Then multiply straight across those numbers that are left. The resulting (simplified) fraction is equivalent to the original fraction but is in simplified form. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

solve multi-step mathematical and real-world problems using multiplication of rational numbers MULTIPLYING FRACTIONS

*When we multiply fractions, we are saying "(fraction 1) OF (fraction 2)* (Ex: 1/2 * 2/3 means 1/2 OF 2/3) https://learnzillion.com/lesson_plans/7906 Just multiply across, then simplify

Proper Fraction

*proper fraction* has a numerator smaller than the denominator and indicates a fraction less than one whole *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Rate Conversion

1 Euro =$1.25 Clock costs 55 Euros. How much would that cost in dollars? (1.25 USD/1 Euro) * Price $1.25/Euro *55 Euros = $68.75

First 10 Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

4 Ways to Indicate Multiplication

1. Using a small "×", such as 3 × 5. 2. Using a small, raised dot, such as 3 • 5 3. Using parenthesis, such as (3)(5) or 3(5) or (3)5 4. Using no symbol, such as 3y (which means 3 times y).

Bob caught and tagged 54 Nene birds, then released them. A week later, he observes 38 Nene birds and notices that 15 of them are tagged. What is the total population of Nene birds?

137 birds (see picture for more detailed steps) x/54 = 38/15 15x =(38)*(54) x =136.8 Round up to 137 because cannot have 0.8 of a bird :)

Prime numbers 1-100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

What number does MMCCXCIV represent in Roman numerals? explain

2,294 In Roman numerals, when a smaller value appears to the left of a higher value, flip those two numbers around and subtract. The number can only be one place value away though to do the flip thing. (ex: cannot do 2,300 - 6 as MMIIVCCC) MM = 2,000 CC = 200 XC = (100-10) =90 IV = (5 - 1) = 4 so, 2000+200+90+4 = 2,294

There is a bag of candy with 8 chocolates, 11 peppermints, 13 gumdrops and 9 pieces of licorice inside. Amy takes one out randomly and gets a gumdrop. What is the probability that the next candy she takes out will be either a gumdrop or a peppermint?

23/40

Divisibility Rules

2= Even numbers (ending in 0, 2, 4, 6, and 8) 3 = If repeated sums of the digits result in 3, 6, or 9 4 = If the last two digits are divisible by 4 5 = If the last digit is 0 or 5 6 = If the number is divisible by both 2 and 3 8 = If the last three digits are divisible by 8 9 = If repeated sums of the digits result in 9 10 = If the last digit is 0

On last night's lottery drawing, 8 of the last 13 numbers were even. What is the probability that the next number drawn will be even?

61% (because 8/13 =0.61...)

real-world problems adding fractions Chad, a pet store employee, wants to fit two fish tanks on one table. One fish tank is 2/5 of a foot wide and the other fish tank is 3/10 of a foot wide. When placed next to each other, what is the total width of the two fish tanks? At the neighborhood block party, Mario served 7/10 of a gallon of hot chocolate and 2/5 of a gallon of apple cider. How much more hot chocolate than apple cider did Mario serve?

7/10 ft

Which equation shows the commutative property of addition? 2+3 = 4+1 8+1 = 1+8 7+2 = 9

8+1 = 1+ 8 move numbers around and they mean the same thing *THINK COMMUTE = move The property that says that two or more numbers can be added or multiplied in any order without changing the result.

Decimal Number Addition

<b>Decimal numbers are added</b> exactly the same as whole numbers: line up the numbers by place value and add each place value from the right to the left.<br><br>When the decimal numbers are lined up by place value properly, the decimal points in each number are also lined up.<br>Any number without a decimal point is lined up so the ones place is right before the decimal point (there is an understood decimal point after the one's place).<br><br>It may help to write zeros in empty places to facilitate addition.<br>

Decimal Number Multiplication

<b>Decimal numbers are multiplied</b> by temporarily ignoring the decimal point. Multiply the two numbers as though they were whole numbers. In the final product, place the decimal point to signify the number of decimal places in both numbers of the original problem.<br /> <br />For example:&nbsp;&nbsp;2.3 (one decimal place) x 1.456 (three decimal places) is the same as 23 x 1456 with the answer having four decimal places (1 + 3 from the original problem)<br><br>Technically, decimal numbers are multiplied by powers of 10 in order to make them whole numbers before multiplication. Then the answer (product) is divided by that same power of 10.<br />

Decimal Number Subtraction

<b>Decimal numbers are subtracted</b> exactly the same as whole numbers: line up the numbers by place value and subtract each place value from the right to the left. <br><br>When the decimal numbers are lined up by place value properly, the decimal points in each number are also lined up. <br><br>Any number without a decimal point is lined up so the ones place is right before the decimal point (there is an understood decimal point after the ones place).<br><br>It may help to write zeros in empty places to facilitate subtraction.<br>

Fractions (Definition, Meaning, and Parts)

A *fraction* represents equal-sized parts of a whole. The top number of a fraction is called the *numerator* because it is the number of parts. The *denominator* is a denominate number (measurement of size) telling how many of the equal-sized parts are in the whole. The line between the numerator and denominator indicates division and is called a *vinculum* Note that the parts MUST be equal-sized. The parts are usually indicated by coloring them or shading them *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Prime Factorization

A *prime factorization* of a number is a list of all prime factors that multiply together to make that number. A factor tree is often used to find the prime factorization of a number. The prime factorization can be written as the product of individual factors or exponents can be used to write the product of repeated prime factors. Ex: 120 = 2 x 2 x 2 x 3 x 5 This can also be written as 120 = 23 x 3 x 5

Ratios (Definition & three ways to write)

A *ratio* is a FIXED relationship between 2 quantities, or in other words, are a comparison of 2 numbers using division. The most common ratios we see are fractions. Write a ratio using a fraction bar, a colon, or the word "to": 3:2 3/2 3 to 2 When we *set two ratios equal to each other,* we say these 2 ratios/fractions are *proportionate.* The comparison between the numerators and the denominators is *the same fixed relationship*. ex: 1/2 = 2/4 "One half is equal to two fourths." ="One half is proportionate to two fourths."

Circle

A circle is a closed figure made up of all points that are equidistant from another point (the center). The distance from the center point to the edge of the circle is called the radius. A = (π)r^2 C = (π)d = 2(π)r

Factor Tree

A factor tree is a method used to find a number's prime factorization. Start by splitting the number into any two factors that multiply to make that number. Then split each of those two factors into two factors each. Keep splitting each "branch" of the factor tree until you reach a prime number, which cannot be split into factors. The final prime numbers at the end of each "branch" of the factor tree are the prime factorization of the number. See pic for factor tree which shows that the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

Whole Number Place Value

A number in standard form is marked into groups of three digits using commas. Each of these groups is called a period. Within each group, the place values are always the 100's place, the 10's place, and the 1's place (from left to right). Understanding place value is key to understanding our number system. Decimal numbers simply extend the place values to the right and use "ths" to identify the places (e.g. 100 millionths place).

dividend

A number that is divided INTO by another number. -the first number in a division problem when it is across for example, 2 is the dividend in 2 ÷ 3 -It is the number inside the bracket in long division

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. A = bh P = 2(b1 + b2)

Rectangular Solid (Definition, volume, and surface area)

A rectangular solid (also known as a cuboid) is a three-dimensional solid where *all angles are right angles* and *opposite faces are equal.* A rectangular solid is also informally called a rectangular box. V = Lwh SA = 2wh + 2hL + 2wL

diameter

A straight line passing from side to side through the center of a circle or sphere. It is radius * 2 d=2r

Trapezoid

A trapezoid is a convex quadrilateral with at least one pair of parallel sides. The parallel sides are called bases and the other two sides are called legs. A = 1/2h(b1 + b2) P = add all four sides

proportion

A true statement that two ratios or fractions are equal. Every proportion is made up of 4 numbers. (2 numerators and 2 denominators) If have a variable, cross multiply to solve. ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2

Proportions (Definition)

A true statement that two ratios or fractions are equal. If have a variable, cross multiply to solve. (Each side will now be a linear equation, no longer in fraction form. See example) ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2 Every proportion is made up of 4 numbers. When we *set two ratios equal to each other,* we say these 2 ratios/fractions are *proportionate.* The comparison between the numerators and the denominators is *the same fixed relationship*. ex: 1/2 = 2/4 "One half is equal to two fourths." ="One half is proportionate to two fourths."

Irrational Numbers

All real numbers except the rational numbers; -numbers that are square roots of non-perfect square numbers -non-terminating, non-repeating decimal s (like pi - just keeps going no end) -most well-known irrational number is pi (π) which is approximately equal to 22/7 or 3.14 The opposite of Irrational numbers are the rational numbers - ALL real numbers are either rational or irrational.

Improper Fraction & Mixed Numbers (Definition)

An *improper fraction* has a numerator larger than (or equal to) the denominator and indicates a fraction that is equal to one or more than one whole An improper fraction can be changed into a whole number or a mixed number by dividing the denominator into the numerator. A mixed number is the sum of a non-zero integer and a proper fraction. *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Improper fractions: denominator might not be most simplified one - make into mixed number THEN SIMPLIFY FURTHER*

Odd vs. Even Numbers

An even number is an integer that is evenly divisible by 2 (without a remainder). Note that the number zero is an even number. An odd number is an integer that is NOT evenly divisible by 2.

Rational Numbers

Any number that can be represented as the RATIO of two INTEGERS (integers are the counting numbers, their negatives, and zero (..., -3, -2, -1, 0, 1, 2, 3 ...).) -integers (because every integer is the ratio of the integer over 1. ex: 2 = 2/1) -fractions -decimals that end (like 12.5) or repeat the same numbers over and over (like 23.666666) -percentages **Square roots of non-perfect squares can NOT be expressed as the fraction of two integers, so they IRrational numbers* All real numbers are either rational or irrational. Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc.

Converting Percentages to Decimals & Fractions

Change a percent to a decimal by moving the decimal point two places to the left and removing the percent sign: 14% = 0.14 Change a percent to a fraction by writing the percent as a fraction over 100 and simplifying: 14% = 14/100, which simplifies to 7/50 *Remember: per = divided by; cent = 100* *On test, all fractions must be in most simplified version*

Classify Numbers in Real Number System

Classify Numbers in Real Number System Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc.

First 10 Cubed Numbers

First 10 Cubed Numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

solve multi-step mathematical and real-world problems using division of rational numbers DIVIDING FRACTIONS

Flip the 2nd fraction upside down, then multiply straight across, then simplify answer *If numerator and denominator are both divisible by the 2nd number, can just divide across* ex: (2/3) ÷ (3/4) = (2/3) * (4/3) = (2*4) / (3*3) = (8/9) cannot divide 2 by 3, so cannot divide across in this problem ex2: 2/3 ÷ 5 Make the whole number into fraction: = 2/3 ÷ 5/1 INVERT the 2nd fraction: =2/3 * 1/5 MULTIPLY ACROSS: =(2*1)/(3*5) =2/15 cannot divide 2 by 5, so cannot divide across in this problem ex3: 3 ÷ 1/4 INVERT the 2nd Fraction: = 3/1 ÷ 1/4 Multiply across: =3/1 * 4/1 = (3*4)/(1*1) =12/1 SIMPLIFY = 12 ex4: 8/10 ÷ 2/5 = ? INVERT the 2nd Fraction: =8/10 * 5/2 = ? Multiply across: (8*5)/(10*2) = ? Simplify (40) / (20) = ? 4/2 =2 On this one, the numerator and denominator are both divisible by the 2nd number, so could also just divide across 8/10 ÷ 2/5 = (8÷2)/(10÷5) = 4/2 = 2 (*When you flip the 2nd fraction upside down, it is called the RECIPROCAL of the 2nd fraction, or INVERTING the 2nd fraction) ♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify!" ♫

Imaginary Numbers

Imaginary Numbers are numbers that contain the imaginary number "i", which is the square root of negative one: Note that if the discriminant portion of the quadratic equation is negative, the function or quadratic equation has no real solutions. The symbol used for the set of complex numbers is C. ***This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics.***

Integers

Integers = the counting numbers, their negatives, and zero (..., -3, -2, -1, 0, 1, 2, 3 ...). No fractions or decimals

distributive property of multiplication

Multiplication outside parentheses distributing over either addition or subtraction inside parentheses does not affect the answer

Multiplying Fractions & Mixed Numbers

Multiplying Fractions & Mixed Numbers: Two ways to do it: 1.) multiply top times top, multiply bottom times bottom simplify if you can 2.) (see pic) -Change mixed numbers to improper fractions. -Change whole numbers to improper fractions with a "1" as the denominator. -Write the two fractions to be multiplied horizontally beside each other. -Expand each numerator and each denominator into a prime factorization (see pic). -"Cancel" any ones such as 3/3 or 5/5. Multiply what is left straight across *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Do NOT need common denominators* *Answer should be LESS than the original factors* (ex: ½ * ¼= 1/8) <--1/8 is smaller than both 1/2 and 1/4*

Composite Numbers

Numbers that are not prime numbers. numbers that have more than two factors or divisors

Unit Rates

Price per item Total Cost divided by number of items Ex: Roses are $24/dozen $24/12 = $2 each

Prime Numbers

Prime Numbers = Integers greater than 1 with exactly 2 factors or divisors; numbers that are evenly divisible by only 1 and themselves. The number 2 is the first prime and it is the only even number that is prime. *The number 1 is neither prime nor composite,* because a prime number is a number with only 2 factors one AND itself, one is itself. Memorize the prime numbers 1-100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Real Numbers

Real Numbers are numbers that can be located on the number line, numbers that can exist in the real world - could be whole numbers, fractions, percents, etc. The opposite of real numbers are imaginary numbers. The symbol used for the set of real numbers is R. Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Rounding Numbers

Rounding a number requires that you understand place value. Rounding a number is a type of estimation. Rounding is also called "rounding off." To round a number, look at the digit to the right of the place being rounded. If that digit on the right is 5 or higher, add 1 to the place being rounded; otherwise, leave the place being rounded as is. Change all places to the right of the place being rounded to zeroes.

solve multi-step mathematical and real-world problems using addition of rational numbers ADDING FRACTIONS

Step 1: Find common denominator -multiply each fraction (both top and bottom) by the denominator of the other fraction - we can do this because any fraction with the same number on the top and bottom is equal to 1, so we are really just multiplying by 1 (look at picture to understand better) Step 2: Add numerators (tops), put the answer over the denominator Step 3: Simplify the fraction (if needed) ex: ¼ + ¼= 2/4 ex2: 1/3 + 1/6 = (6/6) (1/3) + (3/3)(1/6) = (6/18) + (3/18) = 9/18 =1/2

GCF

The *Greatest Common Factor* (GCF) of two or more numbers is the *largest number that is a factor of all the numbers.* One way to find the GCF is to write the prime factorization of each of the numbers. Then make a Venn Diagram with all the factors, the shared factors in the center. Lastly, multiply all the numbers in the middle of the Venn diagram together. (see pic) example GCF word problem: Mei is a dental sales representative who wants to distribute 16 brochures and 10 pamphlets to local dental offices. She wants to deliver *the same combination* of brochures and pamphlets *to each* office, *without having any materials left over.* What is the *greatest number* of dental offices Mei can distribute materials to? (2 offices)

vinculum

The line that separates the numerator from the denominator

identity property of addition/multiplication

The property that states that the sum of 0 and any number is that number AND The property that states that the product of 1 and any number is that number ex: 3+0 = 3 ex2: 3*1 = 3

Addition of Fractions & Mixed Numbers

To add fractions & mixed numbers: 1. Write the two fractions/mixed numbers vertically above each other (lining up place value) 2. Change the fractions to a common denominator. 3. Add the numerators only. 4. Put that sum over the common denominator. 5. Simplify the answer. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Area of Irregular Shapes

To find the area of irregular shapes, divide the shape into regular two-dimensional shapes or picture a regular shape that has been removed:

Subtraction of Fractions & Mixed Numbers

To subtract fractions & mixed numbers: 1. Write the two fractions/mixed numbers vertically above each other (lining up place value) 2. Change the fractions to a common denominator. 3. Subtract the numerators only *be careful to regroup one whole (2/2, 3/3, 4/4, etc.) if you need to borrow* 4. Put that difference over the common denominator. 5. Simplify the answer. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

write numbers using base-10 numerals

Write numbers by showing how many of each place value it has: ex: 1,249 (1*1000) + (2*100) + (4*10) + (9*1) (***NOT 1,000 + 200+ 40 + 9, that is expanded form***)

How would you write 399 as a Roman numeral?

Write the number in expanded form and convert each term to Roman numerals: 300 + 90 + 9 = 399 CCC XC IX CCCXCIX represents 399.

How would you write 99 as a Roman numeral? explain why

XCIX In Roman numerals, when a smaller value appears to the left of a higher value, flip those two numbers around and subtract. The number can only be one place value away though to do the flip thing. (ex: canNOT do IC as 99 (which would be 100 -1), can only do (XC = 100-10=90) + (IX= 10-1 = 9) ) Write the number in expanded form and convert each term to Roman numerals: 90 + 9 = 99 XC IX XCIX represents 99.

reciprocal

a fraction inverted (flipped upside down) *if you multiply 2 fractions that are reciprocals of each other, the answer is always 1 (ex: 2/3 * 3/2 = 6/6 = 1)

Square

a regular polygon made up of four equal sides and four equal angles of 90 degrees each if s = side A = s^2 P = s*4

Percentages (Definition)

a way of expressing a number, especially a ratio, as a fraction of 100 The percent key on a calculator merely divides by 100. If your calculator doesn't have a percent key, hit the divide key and then 100. *Remember: per = divided by; cent = 100*

rational numbers

any number that can be shown as a fraction (decimals, fractions, whole numbers) the proper way to show rational numbers are as fractions (so integers should be written as being over 1)

Rectangle

any quadrilateral with four right angles A = lw P = 2l + 2w

Average

arithmetic mean To average a group of numbers, add all the numbers together and divide by how many numbers there are. For example, the average of 5, 7, 12, and 8 is (5 + 7 + 12 + 8) / 4 = 8.

Which property of addition is shown? 5 + (7 + 1) = (5 + 7) + 1

associative Changing the grouping of numbers will NOT change the value. For example: (7 + 4) + 8 = 7 + (4 + 8) also works with multiplication ex2: as multiplication: (4 × 2) × 1 = 4 × (2 × 1) Grouping numbers together *THINK "associates" means group members/friends

real-world problems diving fractions : ex1: Kevin has 8/10 cups of water. He has to split this water up between dogs' water dishes. He poured 2/5 cups of water into each water dish. How many water dishes can he fill with the water he has? ex2: Cameron has 5/6 yards of string. She needs 1/3 yard of string to make a bracelet. How many bracelets can she make?

ex1: 8/10 c ÷ 2/5 c per bowl = number of bowls This is asking "How many groups of 2/5 fit in 8/10?" 8/10 ÷ 2/5 = ? INVERT the 2nd Fraction: =8/10 * 5/2 = ? Multiply across: (8*5)/(10*2) = ? (40) / (20) = ? 4/2 =? =2 He can fill 2 water dishes with this much water ex2: 5/6 ÷ 1/3 = ? using inverse fraction and multiplying: 5/6 * 3/1 =(5*3)/(6*1) =15/6 both can be divided by 3, to simplify down to =5/2 =2 and 1/2 or dividing across: 5/6 ÷ 1/3 =(5÷1)/(6÷3) =5/2 =2 and 1/2

In Roman numerals, when a smaller value appears to the left of a higher value, _______ _____ _________ _______ and then ________. Ex: XLI = ?

flip the numbers around, subtract M= 1000 D=500 C=100 L=50 X=10 V = 5 I = 1 XLI = XL + 1 = (50-10) + 1 = 41

When the dividend is GREATER than the divisor, the answer should be _________ than one.

greater ex: (3/4) ÷ (5/8) =? 3/4 is GREATER than 5/8, so the answer should be GREATER than one the answer is 1 and 1/5, so this is correct

When the dividend is LESS than the divisor, the answer should be _________ than one.

less ex: (1/2) ÷ (5/8) = ? 1/2 is LESS than 5/8, so the answer should be LESS than one. the answer is 4/5, so this is correct

Triangle

polygon with three angles/vertices and three sides made up of line segments -can be named by its three vertices: ΔABC P = a + b + c A=1/2(bh)

All whole numbers except for 1 and 0 are either _______ or ___________.

prime, composite

round multi-digit numbers to any place value

round 847,256 to the 10,000s place: 850,000 round 847,256 to the thousands place: 847,000 round 847,256 to the hundreds place: 847,300

sample space

what are all the possible outcomes? what are all the things you could pick (in a probability experiment)? The set of all possible outcomes of a probability experiment (all the possibilities NOT the number of possibilities) Usually shown in a tree map, table or chart Lets you see how many possible outcomes there are, so you can figure out probability more easily