(Notes) 240>Math>#s & Algebra>Operations (& flashcards from # Theory Concepts)

**Simplify the following expression [using distribution/aka: applying the distributive property [of multiplication]]:

-7 + 9x (or 9x - 7) Work/Steps: 5 + (9x - 12) ---> 5 + 9x + -12 ---> 5 + 9x - 12 ---> -7 + 9x Notice how^ steps 1 and 3 are the same, just with the parentheses dropped. If it's just a plus sign outside the parentheses we really just drop/cancel out the parentheses (and then continue simplifying the expression).

*Steps for for solving for x:

1) Try simplifying equations or expressions first. Remember to use order of operations. 2) Once simplified, try isolating the variable using inverse operations. You should be able to "solve" for a number value. 3) Last, substitute your answer back in to check.

4 basic math operations:

1. Addition 2. Subtraction 3. Multiplication 4. Division

*In most cases, simplifying an equation/expression involves: [2]

1.using the distributive property and/or 2. combining like terms.

Combine like terms to simplify the following expression: 12x² - 7y + 8y

12x² + y

Which type of math model used to visualize multiplication, known as _____ , is useful for multiplying two digit numbers or higher. It uses the expanded form of a number to break down the problem. Then, each number is multiplied together and then the products are totaled to find the final answer.

Area Model(s)

Repeating decimals are Irrational Numbers; T/F

False; Repeating decimals are considered rational numbers because they can be represented as a ratio of two integers (excluding ratios/fractions with zero in the denominator).

T/F: The distributive property applies to numbers &/or variables directly before parentheses, not to + or - signs.

False; it applies to anything directly before the parentheses, including + and - signs

*T/F: 3/2 is a mixed number.

False; it doesn't contain a non-zero whole number. But, the 1 and 1/2, which is equal to 3/2, is a mixed number.

0 is a prime number; T/F

False; prime numbers are natural numbers that are greater than 1

*T/F: The following examples are all valid examples of combining like terms: (a.) 2x + 3y + 3x = 12z ---> 5x + 3y = 12z (b.) -3y + 2x + 1y = 4z ---> -2y + 2x = 4z (c.) 12 + 3x - 8 = 2y ---> 4 + 3x = 2y (d.) 2z² + 3z + 6z² = 2x --> 8z² + 3z = 2x (e.) 2xy + 2x + y + 2xy = 11 -> 4xy + 2x + y = 11

True; *2 or more terms can be combined for the following reasons: 1./(a.) Both terms have the same variable, ex: 'x' (note that the coefficients do not need to match) 2./(b.) Both terms have the same variable, ex: 'y' (note that the signs do not need to match) 3./(c.) Both terms are constants, ex: '12' 4./(d.) Both terms have the same variable and exponent combination, ex: 'z²' 5./(e.) Like terms can have more than one variable, as long as both terms have the same combination of variables, ex: 'xy'

**Explain the difference between: -Natural Numbers, Whole Numbers, Integers & -Rational Numbers

Natural/Whole Numbers/Integers are either 0 or can be expressed exactly without the use of ratios, fractions or decimals. Rational Numbers do include all Natural Numbers/Whole Numbers/& Integers, but they also include numbers that can only be expressed exactly by using ratios, fractions, &/or decimals.

T/F: Inverse operations are helpful as students solve word problems.

True; Example 1: If a student owes a friend $12.00 and has $5.00, how much more would she need? 5 + ? = 12 and the inverse is 12 - 5, therefore the answer is $7.00. Example 2: If there are 24 students in a class and the teacher would like to separate them into equal groups of 4, how many groups will there be? 24 ÷ 4 = 6 because the inverse is 6 × 4 = 24.

T/F: You can manipulate expressions to create simplified, equivalent expressions.

True; In the same way that fractions can appear to be different, but still have the same value [ex: 1/2 = 2/4] expressions can look different, but still have the same value [ex: 5y = 2y + 3y ]

***T/F: 4x * 3y = 1 can be simplified as 12xy = 1

True; *While unlike terms cannot be added or subtracted, such terms can be multiplied and divided.

The distributive property is a form of multiplication; T/F

True; The distributive property is a form of multiplication so it must be applied before adding or subtracting to combine like terms.

T/F: If something is outside the parentheses, apply it to everything inside.

True; even though, if it's just a plus sign outside the parentheses we really just drop/cancel out the parentheses. ex: 5 + (9x - 12) ---> 5 + 9x + -12 ---> 5 + 9x - 12 ---> -7 + 9x Notice how^ steps 1 and 3 are the same, just with the parentheses dropped.

T/F: In addition to keywords, drawing pictures that represent the problem can also help identify the correct operation to use.

True ex: A drawing of 4 baskets with 25 written on each basket to represent the number of cherries in the basket would look a lot like the repeated addition problem 25+25+25+2525+25+25+25, and would signify that multiplying 25 × 425×4 is the correct operation to use to solve for how many cherries there are in the baskets.

T/F: Addition and subtraction are matched together, as well as multiplication and division, because they are opposites of each other.

True ex: If adding 7 + 3 then the answer is 10 and inversely 10 - 3 takes it back to 7 and 10 - 7 returns to 3. ex 2: The same can be said for multiplication and division: 6 × 2 = 12 and 12 ÷ 6 = 2 or 12 ÷ 2 = 6.

T/F: Zero is an integer.

True; ((Sometimes it is called a Neutral Integer, since it is not Positive or Negative. this is from a non-240 source; source= a google search))

The following words or phrases, when found in word problems, are associated with which basic math operation: -product of -times

Multiplication

The following words or phrases, when found in word problems, are associated with which basic math operation: -triple -twice

Multiplication

Which Math Property is described below: If the quantities on each side of an equal sign are both multiplied by the same amount, the resulting statement will still be equal. Symbolically, if a = b, then ac = bc.

Multiplication Property of Equality

_____ Numbers are aka as the [positive] counting numbers.

Natural Numbers

numbers that can be pointed to in nature; one tree, two people, three dogs, four leaves, etc.

Natural Numbers

What process does a factor tree use to find factors?

Prime Factorization

What process is shown in the image below: https://miro.medium.com/max/592/1*2wIjQH7NdYAmMI9nQa8BJw.png

Prime Factorization

the process of writing a number in terms of its prime factors

Prime Factorization

a number that is divisible only by itself and 1 (e.g. 2, 3, 5, 7, 11).

Prime number

for 12 and 15, the _____ is 3: A GCF B LCM

A

The following phrase, when found in word problems, is associated with which basic math operation: -how many more

Subtraction

***T/F: In other words, when isolating a variable, inverse operations should always be applied in the reverse order of operations, starting with undoing adding and subtracting, and ending with undoing grouping symbols.

True

**To combine like terms that have exponents, the exponents must match/be the same number; T/F

True

34 and 15 are relatively prime;T/F

True

_____ Numbers are the set of natural numbers and zero;

Whole

_____ Numbers start at 0 and go up by 1s to create the infinite set 0, 1, 2, 3, 4,... A Natural Numbers B Whole Numbers C Integers D Rational Numbers

Whole Numbers

When two fractions have the same number on the bottom they have

common denominators

***While there are many types of equations that can be solved and many approaches that can be used to do so, an essential first step in solving most algebraic equations is to:

completely simplify each side of the equation as much as possible.

represent partial numbers.

fractions

_____ operations are used to solve equations

inverse

*Solve for n: -4n + 3 = 5n - 6

n = 1

a straight line where each number is equal distance from the next one

number line

Using base-10 numbers with exponents in expanded form or writing the prime factorization of a number using exponents gives you your number in _____. ex: 5,232 = 2x2x2x2x3x109 = 2^4x3x109

exponential form

All non-zero integers that are positive are known as _____ Numbers or _____ Numbers

Natural Numbers or Counting Numbers

*****Whole Number

Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...

All positive integers are known as the: A Natural Numbers B Whole Numbers C Rational Numbers

A Zero is not a positive integer. It is neither negative or neutral.

What is often used for teaching students about skip counting (by 2s, 3s, 5s, and 10s), counting forward, and counting backwards.

A Number Line A number line is a straight line with numbers marked at equal intervals.

What is the opposite of a prime number?

A composite number

**mixed number

A number made up of a whole number that is not zero and a fraction.

The distance a number is from zero; always a positive number. ex: The _____ of 5 and -5 is 5

Absolute Value

The following words or phrases, when found in word problems, are associated with which basic math operation: -all together -both -combined -in all -increase by

Addition

The following words or phrases, when found in word problems, are associated with which basic math operation: -more than -perimeter -total

Addition

The following words or phrases, when found in word problems, are associated with with basic math operation: -plus -sum -added [to]

Addition

Which Math Property is described below: If the quantities on each side of an equal sign have the same amount added to them, the resulting statement will still be equal. Symbolically, if a = b, then a + c = b + c.

Addition Property of Equality

a specific set of steps that allows the user to reach the correct answer every time.

Algorithm

Which of the 2 types of math models that can be used to visualize/model multiplication is used [in the image below] to show that 34 x 28 = 952? http://violetfourthgrade.weebly.com/uploads/2/1/1/6/2116637/area-model-1_orig.jpg

Area Model

The following words or phrases, when found in word problems, are associated with which basic math operation: -divided -percent

Division

The following words or phrases, when found in word problems, are associated with which basic math operation: -half -quarter

Division

The following words or phrases, when found in word problems, are associated with which basic math operation: -percent -quotient of

Division

The following words or phrases, when found in word problems, are associated with with basic math operation: -how many each -out of

Division

Which Math Property is described below: If the quantities on each side of an equal sign are both divided by the same amount, the resulting statement will still be equal. Symbolically, if a = b and c ≠ 0, then a ÷ c = b ÷ c.

Division Property of Equality

T/F: The following example is a valid example of combining like terms: (a.) 3y + 5y + 2z = 10x --> 8y² + 2z = 10x

False; Although 3y and 5y were correctly identified as like terms and their coefficients were correctly added, it is wrong to change the variable from y to y²

*In 4x²: -what is the mathematical term for the '4' (other than "power" or "squared")

Exponent

In the number 4x²: -what is the mathematical term for the '2' (other than "power" or "squared")

Exponent

pi (π) is a rational number; T/F

FALSE

*T/F: The following example is a valid example of combining like terms: (a.) x + x² = 2z ---> x³ = 2z

FALSE; to combine like terms that have exponents, the exponents must match. For example, 3x³ + 2y + 5x³ = 8y can be simplified into 8x³ + 2y = 8y

The image below is an example of a _____: 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

Factor Tree

a _____ is a visual tool that is used to find the factors of the number

Factor Tree

Values that are multiplied to get another number.

Factors ex: some _____ of 200 are 100 and 2 because 100 x 2 = 200

T/F: -3 is a Natural Number.

False

T/F: 0 is a Natural Number.

False

T/F: 12/2 is a mixed number

False

T/F: 2.3 is a mixed number

False

T/F: 2/5974 is an Improper Fraction

False

T/F: Fractions can have 0 in the denominator.

False

T/F: Decimal numbers that repeat the same pattern forever, such as 0.7272727272..., are not Rational numbers.

False, if they can be written in fraction form using integers as the numerator and denominator, they are rational. For example, 0.72727272... can be written as 8/11

*T/F: 1, 3, 5, and 7 are prime numbers.

False; *Prime numbers are natural numbers greater than 1

T/F: 1,2,3,4,5,6,7,8,9, and 10 are all the Natural Numbers.

False; All counting numbers, starting from 1, going until infinity, are the Natural Numbers

**T/F: 1.5 is a mixed number.

False; Mixed Numbers are always expressed as non-zero integer and one fraction.

Which visual model for multiplication uses the expanded form of a number, where each number is multiplied together and the products are added to find the final answer

Area Model

**Explain the difference between: Real Numbers, Rational Numbers, & Irrational Numbers

**Real numbers have a specific value. Rational numbers can be expressed without symbols. Irrational numbers must be approximated to avoid symbols.

A way to represent part-to-whole relationships; the part out of 100.

percent

**Simplify the following expression [using distribution/aka: applying the distributive property [of multiplication]]: 2 - (3x + 4)

-2 - 3x (or -3x - 2) Work/Steps: 2 - (3x + 4) ---> 2 - 3x - 4 ---> 2 + -3x + -4 ---> -2 - 3x

Combine like terms to simplify the following expression: 4x - 8y + 5y

4x - 3y

**Name the coefficients in the following expression: 5x² + -8y + 12 + y = 3y

5, -8, 1, 3

Use of Parentheses [in an expression/equation] means you need to

Distribute Parentheses are used to group part of an expression together. Parentheses are used to indicate that an operation should be applied to the entire sum or difference.

Any number in front of a group of terms will multiply all terms in the grouping individually ex: a(b+c)=ab+ac

Distributive Property/Distribution

When an expression features terms/numbers inside parentheses, apply whatever is directly before the parentheses to: A everything inside the parentheses B the first # inside the parentheses C the last # inside the parentheses D everything outside the parentheses

A

[In general] when simplifying, you generally do each operation, in order [PEMDAS], from: A left-to-right B right-to-left

A

****Combine like terms to simplify the following expression: 2n - 4m + 3m

2n - m Work/Steps: 2n - 4m + 3m ---> 2n + -4m + 3m ---> 2n + -1m ---> 2n - 1m ---> 2n - m

**Simplify the following expression [using distribution]: 4 + (2x - 6)

2x - 2 Work: 4 + 2x - 6 --> 2x - 2

Simplify [4(3+5) + 2]/{6[3+ 7(-8)] + 1}

34/-317 Start with the inner groupings and work left to right.

the set of _____ Numbers start at 0 and goes up by 1s to create the infinite set 0, 1, 2, 3, 4,.. A Natural Numbers B Whole Numbers C Integers D Real Numbers

B

***What system is defined the following way: each place location for a number has a value that is a power of 10 ex: 10, 100, 1000, 10000

Base 10 Number System

an easily remembered fraction that can be used to make problems simpler ex: 1/10, 1/4, 1/2, etc.

Benchmark Fraction

*In 4x²: -what is the mathematical term for the number '4'

Coefficient

What is the name of the Addition algorithm/method I typically use to do addition?

Column Addition There are different ways to go about doing it though. For ex: https://www.google.com/search?tbs=sbi:AMhZZivZLmICa1l0KoS_1QtJIx7NZQ8CGGc7o6OiOtG4XzXImeQMFRxSp93wIls2eIuKCAYdMkc2lxczFNESAEoOf3HN0L2Twr9vURKafUasZk9xZpBEVQOl1WiIvVI2rKGC2xktltiTP2wBaat0Rc6mmrWN4q-s7IG0lONTe4V6CX3XZ7N5NjZgAg927YeSzKDEjQM1HX8SPmgC-AJ6nS_1Id2BVNzTgNSUezr9RlJ5ijkUdZw2P1CDq36-VISat60DIpaIrlLsZ-2aCpJPZyw74z0tvNUbEqkuF84XIvc4k_1H3yCNnI7xClrNV06djg8ZWZZ4uSiDeUvEcaHHGIn9eZ6Kn3hmL8Rdg

What is the name of the regular Division Algorithm/method I use:

Column Division

A method of simplifying an algebraic expression by adding or subtracting the coefficients of like terms. ex: 2x + 4x = 6x

Combining like terms

when 2 fractions share the same total parts of whatever item or items are being represented

Common denominators

natural numbers that have numbers that divide into them. ex: 4, 6, 8, 9

Composite Numbers

fractions with a denominator of 10//or, a fraction where the denominator is a power of ten such as 10, 100, 1000, etc.) ex: 1/10 = 0.1

Decimal Fractions You can write decimal fractions with a decimal point (and no denominator), which make it easier to do calculations like addition and multiplication on fractions.

last 30-60ish cards are the flashcards from 'Number Theory Concepts' section; i accidentally added them to this study set.

Following this card are the flashcards for the correct section for this study set, 'Operations'.

Adding fractions to come up with a larger one. In other words, showing a non-unit fraction visually by showing the unit fractions that add up to make it. ex: ¼ + ¼ + ¼ = ¾.

Fraction Composition

Breaking down a fraction into smaller fractions that total to the original. In other words, visually show what unit fractions are added together to makeup a non-unit fraction. ex: 2/3 = 1/3 + 1/3

Fraction Decomposition

***Explain the difference between GCF and LCM

GCF is the largest number that goes into two numbers. LCM is the smallest number that two numbers can each go into [without creating a decimal/fraction in the quotient]

****[this trick is for me personally,] WHEN SIMPLIFYING EXPRESSIONS (by combining like terms, by distributing, etc.) ALWAYS REMEMBER TO

GET RID OF ALL MINUS SIGNS, and then when done, bring them back. That is, change all minus signs to a plus sign and make the number following those minus signs negative. Then when done, take all the negative signs for numbers that come after an addition symbol/plus sign and remove them from those numbers and change the those plus signs back to minus signs. i.o.w., distribute each minus sign to the number immediately following it (making it negative) which leaves you with only plus signs between the numbers/variables. **MUCH EASIER TO DO THIS (BECAUSE OF OLD HABITS) IF I WRITE IT OUT ON PAPER BEFORE COMBINING LIKE TERMS. THEREFORE, FOR ANY TEXES QUESTION THAT ASKS ME TO COMBINE LIKE TERMS, WRITE IT OUT (and of course, remember this rule^^^^ about getting rid of all minus signs Ex: 2n - 4m + 3m ---> 2n - m

the largest number that will divide evenly into two or more numbers.

Greatest common factor (GCF) / Greatest Common Divisor (GCD)

a fraction where the numerator is larger than the denominator

Improper fraction

all positive and negative counting numbers and zero; the infinite set ...-3, -2, -1, 0, 1, 2, 3,...

Integers

**Explain the difference between: Integers, Whole Numbers, & Natural Numbers

Integers all positive counting numbers, all negative counting numbers, and 0. (By counting numbers I mean non-zero integers).

math operations that are the opposite of each other (addition+subtraction, multiplication+division)

Inverse Operations

_____ Numbers are Real Numbers that CANNOT be represented exactly as a ratio of two integers.

Irrational

_____ Numbers cannot be written [exactly] in fraction form, and when expressed as decimals, they have infinitely many, non-repeating digits.

Irrational

______ numbers can be approximated; π ≈3.14 or π ≈ 22/7; e ≈2.72, and √2 ≈1.41. Only after being approximated, can they be positioned on a number line.

Irrational

real numbers that CANNOT be represented exactly as a ratio of two integers.

Irrational Numbers

The image below shows the hierarchy of number systems. It uses a box, that contains two main circles (or, circles that are not contained within any other circles). The main circle on the left, the smaller of the 2 main circles, represents _____ Numbers. https://ecdn.teacherspayteachers.com/thumbitem/Real-Numbers-Venn-Diagram-Project-2059898-1459920007/original-2059898-3.jpg

Irrational Numbers http://userscontent2.emaze.com/images/4139aa01-05e0-4f97-85fd-24ee6d68432b/89ce3c10-adef-4d8f-844a-43266a7b1371.png

_____ numbers are real numbers that, when expressed as a decimal, go on forever after the decimal and never repeat.

Irrational numbers

the smallest number two or more numbers will divide into evenly. ex: for 12 and 15, 60 is their _____

LCM (Lowest Common Multiple)

~What Multiplication Algorithm/method is used in the images below: https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQZibl6FIQ20xFbByoY0w_ymx7stsiw7JD-6P5VFP_Pqk66B9B4 https://i.pinimg.com/originals/5c/6d/af/5c6daf68eaca2cbc68edcedab50b67b3.png

Lattice Multiplication

a whole number with a fraction

Mixed Number

Terms with the same variable and exponent combination.

Like terms ex: 2x and 3x

the size of a number

Magnitude

The following words or phrases, when found in word problems, are associated with which basic math operation: -area -rate

Multiplication

The following words or phrases, when found in word problems, are associated with which basic math operation: -of -per

Multiplication

~Addition algorithm where you add a number to one addend to make a number ending in zero and subtract the same amount from the other addend. In other words, you change one number to make it easier to add with the other addend. If you add a # to one addend, you subtract that # from the other addend, and vice-versa [with subtraction/addition].

Opposite Change rule

Order of operations refers to using _____(/\/\)

PEMDAS

~Which addition algorithm sums the parts [#s in each of the place values] of each addend and then adds all those sums together

Partial Sums Addition

natural numbers greater than 1 that have no numbers that will divide into them without a remainder

Prime Numbers

the value of where a digit is in the number. in other words, the value represented by a digit in a number on the basis of its position in the number. Example: In 352, the 5 is in the "tens" _____[partial], so its _____[same] is 10

place value, place, place value

See the image below that shows the hierarchy of number systems. https://ecdn.teacherspayteachers.com/thumbitem/Real-Numbers-Venn-Diagram-Project-2059898-1459920007/original-2059898-3.jpg Excluding the independent circle in the left side of the box, list what set of Numbers each circle represents, starting with the largest and moving in. (if that didn't make sense here's a lengthier version of the question/challenge) The image contains a box [representing Real Numbers] that contains two main circles (or, circles that are not contained within any other circles). List, in order, what Numbers each circle represents, starting with the main circle on the right (the larger one of the 2 main circles, aka, the main circle that contains 3 circles within in), then the largest circle it contains, then the next largest circle it contains, then finally, the smallest circle it contains.

Rational Numbers Integers Whole Numbers Natural Numbers http://userscontent2.emaze.com/images/4139aa01-05e0-4f97-85fd-24ee6d68432b/89ce3c10-adef-4d8f-844a-43266a7b1371.png

_____ Numbers are numbers that can be expressed as a ratio (fraction or comparison) of two integers; a/b where b ≠ 0 (because zero can't be the denominator of a fraction). All rational numbers can be written exactly in both fraction and decimal forms, and can be graphed on a number line.

Rational; Note that exact is a formal term in mathematics meaning a value that has not been rounded or approximated.

If all the circles in the following image were contained in one large circle, that large circle could be labelled//would represent [all] _____ Numbers. data:image/png;base64,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

Real

*numbers that have a specific value. ex: π, -2, 3, 4, 1/2

Real Numbers

numbers that have a specific value

Real Numbers ((~~~As opposed to imaginary numbers, which have no specific value.))

numbers that represent a value on a number line

Real Numbers -You may wonder how irrational numbers represent a value on a number line, since when they are expressed as decimals they go on forever and never repeat, but that still indicates that it represents a value, just a very specific one (one so specific that we would have to infinitely get more specific about which exact value on a number line it indicates)

(Talking through hierarchy of number systems. Goes from largest/most-inclusive sets to the smallest/least-inclusive sets) _____ Numbers contain all _____ Numbers and all _____ Numbers. One of sets, the _____ Numbers, contains all _____ Numbers, which contains all _____ Numbers, which contains all _____ Numbers.

Real Numbers contain all Rational Numbers and all Irrational Numbers. One of these sets, the Rational Numbers, contains all Integers, which contains all Whole Numbers, which contains all Natural Numbers.

Which of the 2 types of math models that can be used to visualize/model multiplication is used [in the image below] to show that 4 x 3 = 12? https://lh3.googleusercontent.com/grnJSa9-Wl-xGycEyJOnMaZLoxJbMTdAwIaAvcJM4jKoh_iInD3eIJvNKU06fJGf5ejJbQ=s170

Rectangular Array

dots or objects that show the end product of 2 numbers; the objects are arranged so there are an equal number of items in each row and column

Rectangular Array

Which type of math model used to visualize multiplication, known as _____ , use dots or objects to show the end product of 2 numbers. The objects are arranged so there are an equal number of items in each row and column.

Rectangular array(s)

If two numbers share no common factors, they are:

Relatively prime

~Which Subtraction Algorithm/Subtraction rule is described: adding or subtracting the same amount from both numerals to make the numbers are easier to subtract.

Same Change rule

Numbers expressed as the product of a base-10 number and a number between 1 and 10 ex: 2.56 x 10 = 25.6 4.32 x 10^-4 = 0.000432

Scientific Notation

How to approach problems/expressions with complex parentheses (multiple uses of parentheses &/or brackets '[ ]' & ' { }' )

Start with the inner groupings and work left to right.

The following words or phrases, when found in word problems, are associated with which basic math operation: -decreased by -fewer than -left

Subtraction

The following words or phrases, when found in word problems, are associated with which basic math operation: -difference -less -less than -minus

Subtraction

The following words or phrases, when found in word problems, are associated with which basic math operation: -remaining -take away

Subtraction

Which Math Property is described below: If the quantities on each side of an equal sign have the same amount subtracted from them, the resulting statement will still be equal. Symbolically, if a = b, then a - c = b - c.

Subtraction Property of Equality

T/F: Rectangular arrays are used for visually modeling multiplication and division.

True Rectangular arrays use dots or objects to show the end product of 2 numbers. The objects are arranged so there are an equal number of items in each row and column. It can also be used for division as students can rearrange the objects into equal rows or columns to find the dividend.

T/F: [Regarding the order in which numbers are taught to children,] Children learn to count starting at 1 and going up by 1's; "1, 2, 3, 4, 5,..." etc.

True (I made this question. It's not supposed to teach about some instructional theory/method, it's to clarify what is meant by counting numbers to help define natural numbers, whole numbers, and integers.)

T/F: In order to be expressed exactly (without rounding), irrational numbers are often written with symbols, like π, e, or √2​.

True When irrational numbers are written using only "ordinary" numerals, they must be rounded.

**In math, when a number is said to be _____, it means that the number has not been rounded.

exact Similarly, the term exactly is used. For instance, "Real Numbers that can be represented exactly as a fraction/ratio with 2 integers (excluding fractions with 0 in the denominator)..." , means numbers that can be represented as a fraction/ratio with 2 integers without rounding either of the numbers.

Multiplication can be thought of as repeated addition; T/F

True

Prime numbers must be greater than 1; T/F

True

T/F: -2 is a constant term

True

T/F: -21 is an integer

True

T/F: 0 is a Whole Number.

True

T/F: 10 is an integer.

True

T/F: 2 and 1/2 is a mixed number

True

T/F: 2 ways to model multiplication are using: 1. rectangular arrays, and 2. area models.

True

T/F: 2, 3, 5, 7, and 11 are prime numbers.

True

T/F: 2/1 is an improper fraction.

True

T/F: 3/2 is an improper fraction.

True

T/F: Any operation can be applied to an equation, as long as the same operation is applied on both sides of the equal sign.

True

T/F: Fractions can have 0 in the numerator.

True

T/F: In word problems, students need to identify the mathematical operation needed to correctly solve the problem. Understanding keywords and phrases that are identified with each operation can help a student select the correct method to use when solving.

True

T/F: Irrational Numbers are Real Numbers.

True

T/F: Many numbers are rational numbers: -natural numbers, whole numbers, and integers - any number divided by 1 stays the same; 3/1 is still just 3 -mixed numbers - like 4½ which have both a whole number part, like the "4," and a fractional part, like the "½" -decimal numbers that end, such as 0.25 -repeating decimals//decimal numbers that repeat the same pattern forever, such as 0.727272... (also written as 0.72 with a line over the '72', to show it's repeating), which is 8/11 when written in fraction form

True

T/F: Students can learn various algorithms for each operation that will allow them to more easily master concepts.

True

T/F: The distributive property is a form of multiplication, so this property should be applied before combining like terms (adding/subtracting), according to the order of operations.

True

T/F: Understanding the application of algebraic properties, knowing how like terms can be combined, and having a firm grasp on the proper use of the order of operations and properties of equality enables one to solve algebraic equations for x.

True

T/F: While unlike terms cannot be added or subtracted, such terms can be multiplied and divided.

True

T/F: You can combine like terms by adding or subtracting the coefficients of two or more like terms, and keeping the variables and exponents the same.

True

T/F: You can use equivalence to simplify expressions. This is done for lots of reasons, but often when you are asked to solve for x.

True

pi (π) is an irrational number; T/F

True

1 over any rational number. The inverse of a whole number (excluding 0). ex: 1/2, 1/3, 1/19

Unit Fraction

addend

a number that is added to another When you add two addends, you get a sum.

What is a constant term [regarding expressions/equations in Math]?

a term without a variable; ex: 3, 15, and -523 are all constant terms

The four basic math operations are paired: ______ with ______, and ______ with _____.

addition w/ subtraction & multiplication w/ division

**Students can practice counting forward or to the right on a number line to build skills to learn _____. They can also practice counting backwards or towards the left which builds skills for _____.

addition, subtraction

Combining like terms in one way to simplify:

an Expression

All non-zero integers are also known as

counting numbers (both negative and positive counting numbers)

the bottom term in a fraction

denominator

[Regarding simplifying expressions,] Distribution refers to applying the ______ property of _____.

distributive property of multiplication

How to reduce fractions [to simplest form]

dividing the numerator and the denominator by any common factors to put the fraction in lowest terms

The word _____ is a formal term in mathematics meaning a value that has not been rounded or approximated.

exact

If you break apart each digit in a number and show the digits true value you get _____. ex: 4,358 = 4,000 + 300 + 50 + 8

expanded form

Organize the following sets/types/systems of Numbers into a hierarchy: It helps to use the following visual: One large circle that contains two medium circles that do not overlap. One of the medium circle contains 3 smaller circles (with each smaller circle being contained in the last one), and the other medium circle containing no smaller circles. -Integers -Irrational Numbers -Natural Numbers -Whole Numbers -Real Numbers -Rational Numbers

http://userscontent2.emaze.com/images/4139aa01-05e0-4f97-85fd-24ee6d68432b/89ce3c10-adef-4d8f-844a-43266a7b1371.png

**Learning to skip count on a number line begins the basic idea of _____ & _____.

multiplication & division

When a number is immediately next to a parenthesis such as 4(5), it means to

multiply the numbers together so 4 × 5 = 20.

the top term in a fraction

numerator

PEMDAS is the

order of operations

A factor tree is a tool that breaks down any number into its _____ factors.

prime

When we say we are breaking a number down to it's factors, we really mean we are breaking it down to factors that can be not be broken down anymore. This means we are really breaking a number down to it's _____ factors.

prime

a prime number or term that can be multiplied by another to get a number. ex: In 2 x 6 = 12, the number '2' is a

prime factor

a number is simplified to its closest multiple of 10, 100, 1,000, etc. ex: 26 is changed to 30

rounding

Combining like terms and distribution are both methods of:

simplifying an expression

Once each side of an equation has been simplified as far as possible, then the process of _____ can begin.

solving Once each side of an equation has been simplified as far as possible, then the process of solving can begin, using inverse operations.

In math, a _____ can be defined as a fraction whose numerator is 1. It represents 1 shaded part of all the equal parts of the whole.

unit fraction

Solve for x: 3x - 2 = 13

x = 5

Name the variables in the following expression: 5x² + -8y + 12 + y = 3y

x, y, y, y, (or simply, x, y)

(*In other 240 Math sets, I may have incorrectly described numbers as being whole numbers, integers, or whatever. I think I went back and corrected them all but I may have missed some. In case of any confusion, or conflicting info found, see the Instructional Content/Material for "240>Math>#s & Algebra>Operations" and go about half way down the page to see the hierarchy and the official/legit definitions of each type of Number (Real, Whole, Integer, etc.)

ya

Name the exponents in the following expression: 5x² + -8y + 12 + y = 3y

²