NES Subtest II - Math

The greatest common factor of 24 and 18

6

Every natural number is divisible by itself and 1. Example

6 = 1 × 6 and 6 = 6 × 1

Convert 65% to a fraction

65% = 65/100 = 13/20

Multiply 2/3 x 4/5

= (2 x 4) / (3 X 5) = 8/15

Convert -1.45 into a fraction

= -1 (45/100) = -1 (9/20)

Every Rational number can be expressed as...

A decimal that either terminates or repeats in a predictable way

The Number Line

A fundamental concept of mathematics is that the set of real numbers is in one-to-one correspondence with the set of points on a line. That is, each real number corresponds to exactly one point on a line, and each point on a line corresponds to exactly one real number, called the coordinate of the point. A number line is a drawing of a line, usually horizontal, expressing this correspondence. On the number line, the point corresponding to the number zero is called the origin. When oriented horizontally, points to the right of the origin usually correspond to positive numbers, while points to the left correspond to negative numbers. In early counting, children may simply learn how to plot points on a number line and also how to draw one. Later, number lines can be useful when students are learning about adding or subtracting positive and negative integers and about the relative magnitude of decimals, fractions, and square roots.

Divisor

A natural number b is divisible by a natural number a if there is a natural number n so that b = n × a, in which case the number a is a divisor or factor of b.

Negative Exponents

A negative exponent indicates a reciprocal. Example: 2^-3 = 1/2^3 = 1/8

A prime number

A prime number is a natural number with exactly two different divisors (itself and 1).

Diagrams

A variety of diagrams can be used to represent numbers. Area models work well for describing multiplication of integers. The process of division can also be modeled using diagrams. Fractions can be represented by using circles or squares, divided into equally sized regions. Concepts such as equivalent fractions and mixed numbers, along with operations such as multiplication of fractions, can be represented and explored using these figures.

Composite numbers

A whole number greater than 1 that has more than two factors.

Identity property of Addition

Adding zero to a number does not change it.

Compatible numbers strategy.

Adjust the numbers so that they are easier to work with: in division, adjust the divisor, the dividend, or both, so that they are easier to compute. For example, to estimate 67 ÷ 3, observe that 67 is close to 66, which is divisible by 3. So, an estimate for 67 ÷ 3 is 66 ÷ 3 = 22.

Tree Diagrams

Another type of model is the tree diagram. Tree diagrams can be used to model a variety of ideas, such as factoring a number into primes, determining number of combinations, or finding the probabilities of events.

Our Hindu-Arabic numeration system

Base-ten. The symbol 12 represents one collection of ten and two collections of one. The digit just to the left of the decimal point represents the number of collections of one. The next digit to the left represents the number of collections of ten. To represent larger quantities, use larger groupings of tens.

Step 4: Check your answer to make sure it is reasonable.

Check your answer to decide if it is reasonable. Have you answered the question posed in the problem? For example, if a problem tells you the regular price of a pair of shoes and then asks how much money the shoes would cost if they were on sale for 30% below the regular price, your answer should be less than the regular price of the shoes. Use common sense to decide if your answer is reasonable. If your solution does not make sense, ask yourself why, and recheck your plan and your calculations. Try to find an answer that is reasonable.

Three Basic Properties of Addition

Commutative, Associative, and Identity

Properties of Multiplication

Commutativity, associativity, and identity.

Converting a fraction to a percentage

Convert the fraction to a decimal and then convert the decimal to a percentage.

Converting a percentage to a fraction

Convert the percentage to fraction with a denominator of 100.

A proportion can be expressed in the following two ways.

Equal fractions, such as a/b = c/d Odds notation, such as a:b = c:d We read the notations as "a is to b as c is to d.

Estimation

Estimation is finding an approximate answer.

1 is the only natural number that has only one divisor.

Every other natural number has at least two distinct divisors.

Rational numbers and decimal

Every rational number can be expressed as a decimal that either terminates or repeats in a predictable way.

Unique Opposite/Negative

Every whole number has a unique opposite or negative sum with is is 0. For example: 2 + (-2) = 0. or 0+0=0

Why is not every rational number is an integer?

Ex - 1/2 is a rational number that is not an integer

Exponentiation

Exponentiation is repeated multiplication. An exponent is often called a power. For example, the third power of 2 is: 2³ = 2 × 2 × 2 = 8

To determine the greatest common factor of 90 and 84

Factor both numbers into primes. 90 = 2¹ × 3² × 5¹ 84 = 2² × 3¹ × 7¹ The prime factors involved in both are 2 and 3. Select the power of each with the smallest exponent. The smallest exponent of 2 is 1. Select 2¹. The smallest exponent of 3 is 1. Select 3¹. Multiply: 2¹ × 3¹ = 6 Therefore, the greatest common factor of 90 and 84 is 6.

Converting a decimal to a fraction

First, note that the numerator is made out of the digits that fall after the decimal point and that the denominator is the place value of the digit farthest to the right of the decimal point. Note that if a decimal has both integer and fractional parts, keep the integer and convert the fractional part to form a mixed number.

Mental Computation Example

Grouping (clustering) operations can facilitate mental computation. For example, to find the sum 14 + 23 + 16, choose a pair of addends to make the mental computation more manageable: 14 + 16 = 30. So, the sum is 30 + 23 = 53.

If multiplying a fraction by a whole number

If multiplying a fraction by a whole number, turn the whole number into an improper fraction and then follow the steps as shown above for multiplying fractions.

Divisibility by 5

If the last digit is a 5 or a 0, then the number is divisible by 5. For example, 1995 is divisible by 5 since its last digit is 5.

Divisibility by 2

If the last digit is even, then the number is divisible by 2. For example, 158 is divisible by 2 since its last digit is 8.

Divisibility by 8

If the last three digits form a number divisible by 8, then the number itself is also divisible by 8. For example, 1120 is divisible by 8 since 120 is divisible by 8.

Divisibility by 4

If the last two digits form a number divisible by 4, then the number is divisible by 4. For example, 316 is divisible by 4 since 16 is divisible by 4.

Divisibility by 10

If the number ends in 0, then it is divisible by 10. For example, 670 is divisible by 10 since its last digit is 0.

Divisibility by 6

If the number is divisible by both 3 and 2, then it is also divisible by 6. For example, 168 is divisible by 6 since it is divisible by 2 and it is divisible by 3.

Divisibility by 3

If the sum of the digits is divisible by 3, then the number is also. For example, 177 is divisible by 3 since the sum of its digits is 15 (1 + 7 + 7 = 15), and 15 is divisible by 3.

Divisibility by 9

If the sum of the digits is divisible by 9, then the number itself is also divisible by 9. For example, 369 is divisible by 9 since the sum of its digits is 18 (3 + 6 + 9 = 18), and 18 is divisible by 9.

Sometimes you may need to order a list of numbers that includes irrational numbers, such as the square root of an integer.

In that case, you need to estimate the value of the square root using the property that the square root of a number times itself returns the initial number.

A model

Is an object, picture, or drawing representing an idea. For example, the number five may be represented with five fingers (real-world representation), five plastic counters (manipulative model), an illustration of five toys (pictures), the spoken word "five" (oral language), or the numeral 5 (written symbols).

Why is rounding is an important skill?

Its important especially when estimating quantities or determining whether a solution to a problem is reasonable

Special numbers strategy.

Look for special numbers that are close to values that are easy to combine, such as one-half, or the powers of ten. For example, to estimate 51% of 89, observe that 51% is close to 1/2 , and that 89 is close to 90. So, 51% of 89 is approximately one-half of 90, or 45. This estimate is very close to the actual value of 45.39.

Manipulatives

Manipulatives are physical objects that can be used to illustrate mathematical concepts, whether made specifically for learning mathematics, such as Unifix® Cubes, or objects created for other purposes, such as popsicle sticks. Typically, concrete materials can be used to model mathematical concepts, such as number or shape, when introducing students to the concepts. The use of manipulatives enhances mathematical concept formation by developing concrete understanding before moving to the abstract. Many additional concrete manipulatives are available such as plastic counters, geoboards, tangrams, algebra tiles, fraction pieces, spinners, money, and so on.

Mental Computation

Mental computation entails finding an exact answer without the aid of paper, pencils, calculators, or other technology. Mental computation focuses on producing correct answers quickly. Grouping (clustering) operations can facilitate mental computation. Students develop individual strategies for mental computation through extensive practice. The strategies arise from number sense and a solid understanding of the number system, place value, and arithmetic operations.

Converting a percentage to a decimal

Move the decimal point two places to the left and remove the percent sign. If needed, append zeroes to fill decimal places.

Converting a decimal to a percentage.

Move the decimal point two places to the right and add a percent (%) sign. If needed, append zeroes to fill decimal places.

Reciprocal of a number AKA

Multiplicative inverse of the number

Identity Property of Multiplication

Multiplying a number by 1 does not change it.

0^0 = ?

Not defined

Example - convert .45 into a fraction

Note that this number is the same as 45 hundredths, so the numerator is 45 and the denominator is 100, because the 5 is the digit farthest to the right and holds the place value of hundredths (1/100). 45/100 = 9/20

Is 1 prime or composite

Notice that 1 is neither prime nor composite.

Round 2.357 to the hundredths

Notice that the digit to the right of the hundredths place is 7, which is greater than 5. So, round the hundredths place up. Thus, 2.357 rounded to the hundredths place is 2.36.

To round 2.35 to the tenths place

Notice that the digit to the right of the tenths place is 5. Since 5 is not less than 5, round up. Thus, 2.35 rounded to the tenths place is 2.4

Irrational Numbers

Numbers that have a non-terminating and non-repeating decimals.

Probability Equation

P(Event) = number of desired outcomes/total number of outcomes

imagine rolling a fair 6-sided number cube. What is the probability of rolling an even number?

P(even) = desired outcome (3) / total number of outcomes (6) = 3/6 = 1/2

Probability is the branch of mathematics that studies the relative likelihood of events occurring. For instance, when a coin is flipped, what is the probability of the coin landing with heads up? The probability of an event is the ratio of the number of ways that event can occur to the total number of possible outcomes.

P(red) = desired color (1)/Total colors (4) = 1/2

Step 3: Carry out the plan accurately

Perform operations in the correct order. Watch out for error with negative signs. Be careful not to miscopy numbers. Calculate carefully, and check your work as you go along.

Probability

Probability is the branch of mathematics that studies the relative likelihood of events occurring. For instance, when a coin is flipped, what is the probability of the coin landing with heads up? The probability of an event is the ratio of the number of ways that event can occur to the total number of possible outcomes.

Equation for the set of ratio of integers

Q = {m/n} where m & n are integers, and n is not 0.

Converting a fraction to a decimal

Remember that the fraction bar is a division sign and that the fraction is equivalent to 3 ÷ 8 by long division. Divide 8 into 3 to get 0.375.

Divide 2/5 by 4/7

Rewrite the division problem to a multiplication problem by changing 4/7 to 7/4. 2/5 / 4/7 = 2/5 x 7/4 = 14/20 = 7/10

Scientific Notation

Scientific notation allows us to manage very large or very small numbers efficiently. Step 1: Move the decimal point left or right until the digits form a number greater than or equal to 1 and strictly less than 10. Step 2: If you move the decimal point exactly n places to the left, then multiply by 10n. If you move the decimal point exactly n places to the right, then multiply by 10^-n.

How to Round

Step 1: Consider the digit to the right of the rounding place. Step 2: If the digit is less than 5, then round down. Otherwise, round up.

Use the following steps to find the greatest common factor of a set of natural numbers

Step 1: Find the prime factorization of each number in the set. Step 2: For each prime factor appearing in all of the prime factorizations, select the power with the smallest exponent. Step 3: The greatest common factor is the product of the powers selected in Step 2.

Use the following steps to find the least common multiple of a set of natural numbers.

Step 1: Find the prime factorization of each number in the set. Step 2: For each prime factor, select the power with the largest exponent. Step 3: The least common multiple is the product of the powers selected in Step 2.

Word Problem Strategy

Step 1: Read and understand the problem. Step 2: Develop a plan for solving the problem. Step 3: Carry out the plan accurately. Step 4: Check your answer to make sure it is reasonable.

Add 2/3 to 4/5

The common denominator = 3 × 5 = 15. The initial fractions can be converted to equivalent fractions with a common denominator of 15 as follows. 2/5 + 4/5 = 2/3 x 5/5 + 4/5 x 3/3. This can be done since 5/5 and 3/3 each equal 1 and multiplication by 1 does not change the value of a number. This equals: 10/15 + 12/15 = 22/15 = 1 7/15.

The first power of any number

The first power of any number is itself. For example:2^1 = 2

Using the Problem-Solving Method to Answer Problems Example 1 Joan purchased a flash drive for her computer for $16.00, an ink cartridge for $4.75, a package of computer paper for $9.00, and a box of envelopes for $6.25. What fraction of the total price of the supplies was spent on paper?

The first step in answering this question is to read the problem carefully. The problem contains a number of prices that are important in finding the correct answer. The last sentence also contains the words fraction and total. In this problem, the word total indicates that the numbers should be added, and the word fraction indicates that a division will need to be performed. An appropriate plan for this problem should be to find the total price and to use that information to find the fraction of the total price of supplies spent on paper. The fraction of the total price spent on paper can be written as follows. Fraction of total money spent on paper =money spend on paper/total money spent on supplies. The plan is to compute the above fraction. To carry out the plan, work with the numbers given. The money spent on paper = $9.00The total spent = $16.00 + $4.75 + $9.00 + $6.25 = $ 36.00The fraction then becomes Check your answer to determine whether or not it is a reasonable solution to the problem. According to the answer found and the information given in the problem, one-fourth of the total money spent on supplies should equal the money spent on paper, or $9.00. To check this findof $36.00, which is the same as×== $9.00. The solution is correct.

Step 1: Read and understand the problem.

The first step in solving any problem is to make sure you understand the question. Read all the information provided in the problem. Some information may be presented in words, while some may be given as numbers. Diagrams or graphs may also be provided. Be sure you have a clear idea of what the words and the numbers mean and what the question is asking you to find. Ask yourself, "What information am I given, and what do I need to find out?" Use the words in the problem to help you determine what the problem is asking you to find. For example, words such as cost, savings, and discount may indicate that you must calculate an amount of money, while words such as farther, mileage, and length may show that you need to find a distance. Words such as increase and more may mean that you need to use addition to find an answer. The words times and double often indicate that you must multiply to solve the problem. When the words less, remainder, or decrease appear in a problem, you may need to subtract or divide. The last sentence in a word problem often states the question you must answer. Be sure that you read this sentence carefully and understand the information in it. If this is not the case, then locate the sentence that contains the question you must answer.

Front-end strategy.

The front-end strategy focuses on the left-most or highest place-value digits. For example, you can estimate the sum 67 + 91 by adding the highest place-value digits, 6 and 9. Thus, approximate 67 + 91 by 60 + 90 to obtain 150.

Distributive property w/Example

The product of a number with a sum equals the sum of the products of the number with each term of the sum. 2 × (3 + 5) = (2 × 3) + (2 × 5)

Real Number

The set of irrational and rations numbers together. Real numbers is the set of numbers expressed as decimals.

The set of rational numbers

The set of ratio of integers

Notice that solving proportion problems is mathematically equivalent to finding equivalent fractions.

The two ratios are equivalent fractions, since they equal each other. Ex: 2/80 = 5/200

The word percent means "per 100."

Therefore, 25% means "25 per 100" which is the same as 25/100.

Whole Numbers

These are natural numbers, and zero. W = {0,1,2,3,4,5,6....}

Natural Numbers

These are the set of numbers used to count. N={1, 2,3,4,5,6, ...}

Adding and Subtracting Fractions

To add or subtract fractions, first find the common denominator, then convert the fraction to equivalent fractions and add or subtract the new numerators to find the numerator of the answer.

To exponentiate a power

To exponentiate a power, multiply the exponents. For example:(2^3)^5 = 2^3×5 = 2^15 = 32768

To multiply like bases with exponents

To multiply like bases with exponents, add the exponents. For example:2^3 × 2^5 = 2^3+5 = 2^8 = 256

Rounding strategy.

Using the rounding strategy to estimate 67 + 91 would round 67 to 70 and would round 91 to 90 to yield 70 + 90 = 160 as an estimate for the sum.

Associative Property of Addition

When adding three or more numbers, the sum is the same regardless of the way in which the numbers are grouped.

Commutative Property of Addition

When adding two numbers, the sum is the same regardless of the order in which the numbers are added.

Dividing Fractions

When dividing fractions, begin by changing the division problem to a multiplication problem by changing the divisor (the second fraction) to its reciprocal. Then, multiply the fractions and simplify. This is often remembered as "invert and multiply."

Multiplying Fractions

When multiplying a fraction by a fraction, multiply the numerators and multiply the denominators.

Clustering strategy.

When operating with numbers that are clustered close to one another in value, group the operations to make the calculations easier. For example, to estimate the sum 594 + 603 + 622 + 586, notice that each of the 4 terms is approximately 600. So, the sum is approximately 4(600) = 2400.

Associative property of Multiplication

When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped.

Commutative property of Multiplication

When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied.

Signs when you multiply or divide numbers

When you multiply or divide two fractions with the same signs, the resulting sign is positive. When you multiply or divide two fractions with different signs, the resulting sign is negative.

Integers

Whole numbers and their opposites. Z = {...,-3,-2,-1,0,1,2,3,...}

Why is every integer a ration number?

because it is equal to the ratio of itself with 1. Example: 2 = 2/1

Five strategies for estimation:

front-end, rounding, clustering, compatible numbers, and special numbers.

Step 2: Develop a plan for solving the problem

fter you have read the problem carefully and understood it, develop a plan for finding your answer. Use key words or phrases in the problem to form a mathematical plan for solving the problem. Here are a few examples of key words and phrases to help you in your planning.What is the total of . . . ? (Add)What is the sum of . . . ? (Add)What is the difference between . . . ? (Subtract)What is the product of . . . ? (Multiply)What is the ratio of . . . ? (Divide)What is the quotient of . . . ? (Divide)What is the fraction of . . . ? (Divide)What is the percentage of . . . ? (Divide)What is the average of . . . ? (Add and divide)Approximately (or about) how many . . . ? (Estimate) Use the key words and phrases from the problem to write a statement explaining how you are going to solve the problem. Write a clear statement telling what you are going to do. As you prepare your plan, remember that some problems require more than one step. For example, if you need to find the average daily temperature for a particular week, you need to add the daily temperatures and then divide by the number of days in the week. Be sure to include in your plan all the important quantities from the problem. Once you have written your plan, review it to make certain that you are answering the question presented in the problem. For example, if a problem asks you to calculate how much time is remaining in a basketball game, be sure your plan will allow you to calculate the time left in the game, not the time that has already been played.

To express 0.00235 in scientific notation

move the decimal point three places to the right to obtain 2.35. Then, multiply 2.35 by 10^-3 to obtain 0.00235 = 2.35 × 10^-3. Notice that scientific notation is an application of the powers of ten.

Digits to the right of a decimal point correspond to

negative powers of ten. For example: 0.67 = (6 × 10^-1) + (7 × 10^-2)

Exponent of zero

raising any number to the power of 0, the answer is always 1.

The more ways students engage with an emerging idea

the better they will correctly form the idea and integrate it into a web of related concepts. Using multiple representations of numbers and operations develops abstract reasoning skills.

The greatest common factor (GCF)

the largest natural number that is a factor of each number in the set.

The least common multiple (LCM)

the smallest natural number that is divisible by each number in the set.

The numeral 123 represents

three ones, two groupings of ten, and one collection of ten groups of tens,

Find the product of 7 and 3/5

(7/1) x (3/5) = (7x3)/(1x5) = 21/5 = 4 (1/5)

Less than

(<)

Greater than

(>)

An example of a proportionality problem and its solution is given below. If 2 pounds of apples cost 80 cents, how much do 5 pounds of apples cost?

- Choose a variable such as N to represent the unknown quantity. In this case, the unknown quantity is the cost of 5 pounds of apples, so let N = the cost of 5 pounds of apples in cents. - Relate the quantities using a proportion. Note that 2 pounds is to 80 cents as 5 pounds is to N cents. Therefore 2/80 = 5/N. - Use the cross-multiplication property to simplify the equality, and then solve for N. 2 × N = 80 × 5 2N = 400 N = 200 cents = $2.00 So, the cost of 5 pounds of apples is $2.00.

Convert 5% to a decimal

0.05

Convert .123 into a percent

0.123 = 12.3%

Determine the least common multiple of 10 and 12

10 = 2^1 x 5^1 12 = 2^2 x 3^1 Multiple 2^2 x 3^1 x 5^1 = 60

Prime Factorization of 1960

1960 = 5¹ × 7² × 2³

Every none zero integer has a unique reciprocal whose product with it is 1. For Example:

2 * (1/2) = 1, OR 1*1 = 1, OR -1 * -1 = 1 Integers 1 and -1 are the only integers whos reciprocals are also an integer

Associative Property of Addition Example

2 + (3 + 5) = (2 + 3) + 5

Identity property of Addition Example

2 + 0 = 2

Commutative Property of Addition Example

2 + 3 = 3 + 2

Associative property of Multiplication Example

2 × (3 × 5) = (2 × 3) × 5

Identity Property of Multiplication Example

2 × 1 = 2

Commutative property of Multiplication Example

2 × 3 = 3 × 2

List the first prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...

The ratio or fraction of one integer to a nonzero integer is the product of the first integer with the reciprocal of the second. For example, the ratio of 2 to 3 is

2/3 = 2 * (1/3)

Convert 2/8 into a percent

2/5 = .4 = 40%

Expanded form of 2045

2045 = (2 × 10^3) + (0 × 10^2) + (4 × 10^1) + (5 × 10^0)

Expanded form 23.405

23.405 = (2 × 10^1) + (3 × 10^0) + (4 × 10^-1) + (0 × 10^-2) + (5 × 10^-3)

Base and Exponent

2^3 where two is the base, and three is the exponent.