NEW Patterns and Algebra
Transitive Property of Equality
if two quantities are equal to the same quantity, then they must be equal to each other. Symbolically: if a = b, and b = c, then a = c
Dividend
number you divide
Sequence
set of numbers that show a common difference EX: 6, 10, 14, 18, 22, 261 EX: 6, 1, -4, -9, -14, -19
identity Property of Addition
A value added by zero or zero added to a value will yield a sum equal to the original value. Symbolically: a + 0 = 0 + a = a
remainder
always less than the divisor
Equation
A mathematical sentence stating that two expressions are equal.
Reflexive Property of Equality
A quantity is always equal to itself. Symbolically: a=a
Patterns
Can be a design (geometric) or a sequence (numeric or algebraic) that is predictable because some aspect of it repeats.
Addition Property of Equality
If the quatities on each side of an equal sign are both added by the same amount, the resulting statement will still be equal. Symbolically: if a = b , then a + c = b + c
Slope of a Line
A measure of the steepness of a line. It rises M units for every 1 unit moved to the right. M is the slop of the line.
Division Property of Equality
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: is a = b and c ≠ 0, then a ÷ c = b ÷ c
Arithmetic sequence
in order to find a specified nth term (a)n - if the 1st term (a1) and common difference (d) are known the formula a(n) = a(1) + d(n-1) can be used. EX: 6, 10, 14, 18, 22, 26 - if requested to find the 10th term, you can note that the first term (a1) is 6 -the common difference (d) is +4 -and the term number (n) is 10 -Therefore, an= a1 +d(n-1) becomes 6 + 4 (10-1) -which simplifies to 10 = 6 + 4(9) = 6 + 36 = 42
Variable
is a non-numeric symbol used to represent an unknown quantity common choices for letters: x , n but any letter or other symbol can be used - mathematicians avoid using: t, o, e, and i
Factor
is a number being multiplied
Constant
is a number with no variable multiplied with it In the expression 3x + 4, 4 is a constant
Scientific notation
(a X 10^b) For negative exponents: To simplify a negative exponent take the reciprocal (so we're going to flip it) of the number containing a positive exponent 10⁻¹ = 1/10¹ = 1/10 = 0.1 12⁻² - 1/10² = 1/100 = 0.01 ETC... EX: 3 X 10³ = 300 OR you can move the decimal two places to the right (for 2, or 1 , whatever the exponent is) EX: 4 X 10⁻³ = 4 X 1/1000 = 4 X 0.001 = 0.004 (as exponents are decreasing, the decimal place is decreasing by 1, numbers get smaller and smaller/decimals moving to the left each time) - think of as 4. and move three decimals to the left 4. = 0.004
"Like Terms" can be added and subtracted with each other
- "Like Terms" are added and subtracted with each other. - they are added or subtracted by adding or subtracting their coefficients and replacing the variable parts unchanged. - One can use the Distributive Property and Commutative Property of Multiplication to justify this process. 5x + 8x = x(5 + 8) = x(13) = 13x OR one can think of coefficient as a system to track how many of a particular quantity exist, and so "5x + 8x" means that there are 5 of something and then 8 more of that same type of thing, which combine to yield a total of 13 of that thing.
Formal Properties
- Reflexive Property of Equality -Transitive Property of Equality - Symmetric Property of Equality -Closure for Addition - Closure for Multiplication -Substitution Property of Equality - Addition Property of Equality -Subtraction Property of Equality -Multiplication Property of Equality -Division Property of Equality - Commutative Property of Addition - Commutative Property of Multiplication -Associative Property of Addition Associative Property of Multiplication -Identity Property of Addition - Identity Property of Multiplication -Additive Inverse property - Multiplicative Inverse Property -Distributive Property of Multiplication over Addition -Zero product property
Algebraic Terms
- Some variable expressions have a single TERM and some have multiple terms. Algebraic Terms- are numbers , variables, or the products of numbers and variables that are separated from other terms by addition and/or subtraction signs. For EX: 10.2x is considered a single term because its components are connected through multiplication. Meanwhile, -7x² + 3y is considered TWO TERMS b/c the quantities "-7x²" and "3y" are separated from each other by addition and, b/c they do not have identical variable parts, they cannot be combined to become a single term.
Jim wants to walk to Bill's house. To get to Bill's house, Jim walks 3 miles south and then walks 4 miles east. Jim wants to know how many miles he would have walked if he just walked in a straight line. How many miles would Jim have walked if he walked in a straight line to Bill's house? A. 6 miles B. 7 miles C. 5 miles D. 4 miles
- This can be solved by using the Pythagorean Theorem: a² + b² = c². - The Pythagorean Theorem allows students to find the length of any side of a right angle triangle when the length of the other two sides is known. - Because Jim walked south, and then directly east, Jim makes a 90 degree turn. - To find the distance if Jim walked straight, simply take the two lengths Jim walked and fill in the Pythagorean Theorem: (3²) + (4²) = 9 + 16 = 25 = (5²). C = 5.
Rational Function graph
- have 2 branches - one in the 1st quadrant and one in the 3rd quadrant -If you look in the 3rd quadrant its moving left to right it's drecreasing - same with 1st quadrant as we move from left to right it decreases
Degree of a term NOTE: Shortcut for determining the degree of a monomial expression is to simply add the powers of the variables in the term.
-theres times when a degree of term or the degree of a monomial is desired (specifically b/c the degree of a variable expression with one variable has a strong impact on its graph) -the degree of monomial expression is essentially the count of the total number of variables that when multiplied create that term For EX: - the degree of a term "x" is 1 b/c there is one variable multiplied (by 1) to create "x". - the degree of term "8x²" is 2 (b/c there's 2 variable multiplied (by 8) to create 8x². X x X - the degree of the term "-9x²y" is 3 b/c there are three variables multiplied (by 9). (X x X x y) - degree of term 17.62x³y²z , is 6 b/c there's 6 variables multiplied by ( by 17.6) Those are X x X x X x Y x Y x Z
A parallelogram's angles have a sum of -
A parallelogram's angles have a sum of 360°. - By definition, the opposite sides are parallel meaning that side GH and FI are parallel, and sides GF and HI are parallel. - When lines are parallel, their same side interior angles are supplementary: their sum is 180°. - This means that angle H and angle G have a sum of 180°. - So, if the measure of angle H is 80°, then angle G has a measure of 100°. This problem requires an understanding of the properties of parallel lines as well as of the properties of parallelograms
Algebraic Patterns
A set of numbers and/or variables in a specific order that form a pattern.
Algebraic Inequality
A statement that is written using one or more variables and constants that shows a greater than or less than relationship. 2x + 8 > 24
Additive Inverse Property
A value added by its additive inverse or opposite (same number / same absolute value, but opposite in terms of the +/- sign) will have a sum of zero, the identity value for addition. Symbolically: a + ( -a ) = 0 or -a + a = 0
Multiplicative Inverse Property
A value multiplied by its multiplicative inverse or reciprocal ( the numerator and denominator of a fraction are reversed so that a/b becomes b/a) will have a product of 1, the identity value for multiplication. Symbolically: a x 1/a = 1
Identity Property of Multiplication
A value multiplied by one or one multiplied by a value yield a product equal to the original value. Symbolically: a x 1 = 1 x a = a
Simplify: 30 - 2 * 50 + 70 A. 0 B. 570 C. -210 D. -70
A. 0 - To simplify the equation, follow the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (PEMDAS) and work left to right. The steps to simplify the expression would be: 30 - 2 * 50 + 70 = 30 -100 + 70 = -70 + 70 = 0. - PEMDAS can be slightly deceiving. The order of operations is: Parentheses, then exponents, then multiplication and division, then addition and subtraction. - When performing multiplication/division, you perform the actions as they appear left to right; multiplication does not necessarily come before division - they have equal priority. The same goes for addition and subtraction.
A runner is running a 10k race. The runner completes 30% of the race in 20 minutes. If the runner continues at the same pace, how long will it take to complete the race? A. 67 minutes B. 60 minutes C. 85 minutes D. 62 minutes
A. 67 minutes - To find the answer to this question, set up a ratio; remember that 30% = .3 and 100% = 1. (20 / .3) = (x / 1) - When you cross multiply to solve for x, you get the equation .3x = 20. Divide each side by .3 to isolate x and the answer is 66.66666. - The best answer choice is 67 minutes.
What is the 10th term of the geometric sequence 50, 40, 32, 25.6, 20.48,...? A. A(10) ≈ 6.71 B. A(10) ≈ 5.37 C. A(10) ≈ 372.53 D. A(10) = 262,144,000,000,000
A. A(10) ≈ 6.71 - A(10) ≈ 6.71 comes from seeing that the sequence is shown to have a₁ = 50, r = 40 ÷ 50 = 0.8, and n = 10. - The question is answered correctly by keeping an appropriate variable expression in the place of A(n) and inputting all known values so that the formula A(n) = a₁(r)ⁿ⁻¹ becomes A(10) = 50(0.8)¹⁰⁻¹. - When the order of operations is followed correctly, exponents will be simplified before any multiplication occurs. - The resulting equation becomes A(10) = 50(0.8)⁹ and then A(10) = 50(0.134217728) [or, fractions can be used for r so that r = 4/5 and r⁹ = 262,144/1,953,125 so that the equation is A(10) = 50(262,144/1,953,125)]. - The final answer is the product of 50 and the decimal or fraction above, and so A(10) ≈ 6.71 when rounded to the nearest hundredth.
When considering the addition problem 1/3 + 3/8, which of the following statements is true? A. The LCD = 24 B. 3 and 8 are relatively prime C. Both are true D. Neither is true
C. Both are true - Two numbers are relatively prime if they have no common factors except 1. - Because 3 and 8 are relatively prime, there is no whole number, other than 1, that will divide both 3 and 8 evenly without a remainder. - Because 3 and 8 are relatively prime their least common denominator (LCD) can be found using the formula 3 • 8 = 24. - Therefore, 24 is the LCD.
Which of the following is the most appropriate learning progression through the levels of geometric thinking? A. Analysis, Abstraction, Deduction B. Deduction, Visualization, Abstraction C. Abstraction, Analysis, Visualization D. Analysis, Deduction, Abstraction
A. Analysis, Abstraction, Deduction - The progression of geometric thinking is: Visualization -> Analysis -> Abstraction -> Deduction. - Analysis is the ability to identify a shape based on properties (it's a square because it has four equal sides). - Abstraction is where students form abstract definitions about shapes (all squares are rectangles, not all rectangles are squares). - Deduction is the ability to prove statements about a geometric shape are true (geometric proofs). - NOT visualization- is the ability for a student to identify a shape based on its appearance (it's a square because it looks like a square).
If the 5th term in a geometric sequence where r = 3 is 162, then what is a₁? A. a₁ = 2 B. a₁ = 2/3 C. a₁ = 13,122 D. Cannot be solved.
A. a₁ = 2 - a₁ = 2/3 comes from failing to make the exponent "n - 1" and using just the nth power instead, but otherwise simplifying and solving correctly. - a₁ = 13,122 comes from confusing the values for a₁ and A(n) in the formula and solving the resulting erroneous equation: A(5) = 162(3)⁵⁻¹ for the value of A(5). - Someone selecting the option "Cannot be solved" is likely to draw this false conclusion for not understanding how to solve for an unknown that is multiplied by an exponential expression, as is necessary to do in the equation 162 = a₁(3)⁵⁻¹.
The concept of "Like Terms"
Algebraic expressions, like 5x or 6y² are considered "like" each other if they have the same variable parts raised to the same power (sometimes known as the degree) EX's: "5x" and "8x" are "like" terms, while "5x" and "5y" are not. neither are "5x" and "5x³ Similarly, "6y²" and "-1.4y²" are like terms, but "6y²" and "6x²" are not. Neither are "6y²" or "6y⁵" - All constant terms (ordinary Real numbers with no variable parts) are like terms with each other b/c they all have NO variable at all (and therefore do have "the same variable parts raised to the same power")
There is a 15% increase in tuition at UT for next fall. If the current tuition is $3,500 per semester, which equation could be used to find x, the new tuition for the fall? A. 0.15 • 3500 = x B. 1.15 • 3500 = x C. 0.85 • 3500 = x D. (15/100) = (x/3500)
B. 1.15 • 3500 = x - 0.15 • 3500 = x tells us how much the increase alone is but not the total tuition. - The answer from this equation would have to be added to $3,500 in order to determine the new tuition. - 0.85 • 3500 = x would actually result in a decrease since 0.85 < 1, and 0.85 • 3500 < 3500. - When (15/100) = (x/3500) is simplified by cross multiplication, it becomes: 100x = 52500 and x, the new tuition, would be only $525 and definitely not a tuition increase.
A teacher presents a right triangle to her class. She says the hypotenuse can be written as 4x + 30 or 6x + 14. What is the value of x? A. 6 B. 8 C. 10 D. 12
B. 8 4x + 30 = 6x + 14. - This can be simplified to 16 = 2x by subtracting 4x from both sides and subtracting 14 from both sides. 16 = 2x can be simplified to 8 = x by dividing both sides by two.
Which of the following geometric objects has five faces, eight edges, and five vertices? A. An octagonal prism B. A rectangular pyramid C.A triangular prism D. A pentagonal pyramid
B. A rectangular pyramid
What is the 8th term of the geometric sequence -3, 6, -12, 24, -48,...? A. A(8) = 279,936 B. A(8) = 384 C. A(8) = -768 D. A(8) = -384
B. A(8) = 384 A(8) = -384 comes from a sign error. It could result in seeing the ratio as 2 instead of -2, or it could result from raising (-2)⁷ but losing the negative sign, or it could result from a sign error in the multiplication of -3 and -128 at the end of the problem. - A(8) = -768 comes from failing to make the exponent "n - 1" but using just the nth power instead. - A(8) = 279,936 comes from not following the order of operations correctly. Exponents must be simplified before multiplication, but this answer results when the value for a₁ is multiplied with the value of r before the power is applied: A(8) = -3(-2)⁸⁻¹ = -3(-2)⁷ ≠ (6)⁷.
What is the equation for the nth term of the arithmetic sequence -8, -6, -4, -2, 0,...? A. A(n) = -2n - 6 B. A(n) = 2n - 10 C. A(n) = -6n + 6 D. A(n) = 2n - 6
B. A(n) = 2n - 10 - A(n) = -2n - 6 results from misunderstanding the common difference as -2. - This is likely to happen due to subtracting terms in the wrong order while finding d, like -8 - (-6) = -2. - A(n) = -6n + 6 comes from misuse of the order of operations. Because the -8 represents -8 ones and the 2 represents 2 of the quantity "n - 1," it is inappropriate to combine them. - However, some students will erroneously simplify A(n) = -8 + 2(n - 1) into A(n) = -6(n - 1), and get this wrong answer upon distributing. - A(n) = 2n - 6 comes from combining like terms of -8 and -2 together incorrectly. - Though incorrect, some students will erroneously "add the opposite" of the -2 at the stage of A(n) = -8 + 2n - 2, and therefore get -8 + 2 = -6. - However, unless a value is to be moved to the opposite side of the equation's equal sign, like terms should be combined with the signs they are given, and not by adding an opposite.
If -¼ is the 11th term in a geometric sequence where r = -½, then what is a₁? A. a₁ = -250 B. a₁ = -256 C. a₁ = 512 D. a₁ ≈ -0.0002
B. a₁ = -256 - a₁ = -250 comes from setting the problem up correctly, but rounding the decimal form of 1/1024 or 0.0009765625 to the easier-to-use value 0.001. - While 0.0009765625 does round to the nearest thousandth as 0.001, it is inappropriate to round part-way through the process of solving a problem. Rounding should only be performed as the final step in a problem. - a₁ = 512 comes from failing to make the exponent "n - 1" but using just the nth power instead, but otherwise simplifying and solving correctly. (Note that this answer is positive because if n = 11 is used instead of n - 1 = 10, the result of raising r = -1/2 will be negative, and so its quotient with A(11) = -¼ will be positive.) - a₁ ≈ -0.0002 comes from confusing the values for a₁ and A(n) in the formula and solving the resulting erroneous equation: A(11) = -¼(-1/2)¹¹⁻¹ for the value of A(11). (Note that this answer is rounded from -0.000244140625.)
Linear Function Notation
B/C functions are s special to mathematicians, there is a particular notation used only for functions. -notation is the replacement of an isolated "y" with the expression "f(x)" -f(x) is pronounced "f" of "x" -it means the functions value for X. For EX: the equation y=2x + 5 could also be written as f(x) = 2x + 5 then when plugging in a variety of values for "x", such as x=1 or x=3, the process looks like this; f(x)= 2x+5 f(1)=2(1) + 5 f(1)= 2 + 5 f(1) = 7 The sentence f(1) = 7 means that y=7 when x=1, so the point (1,7) makes the equation y=2x+5 true. (Therefore the point (1,7) must be part of the functions graph. another ex: f(x)=2x+5 f(-3)=2(-3) + 5 f(-3)= -6 + 5 f(-3)= -1 - The sentence f(-3) = -1 means that y= -1 when x= -3, so the point (-3, -1) makes the equation y=2x+5 true and will be part of this function's graph. - The final lines of the exaples above show ordered pairs that are said to "satisfy the function" , that is ordered pairs that make the function statement TRUE.
Given a₁ = 512, r = ½, and n = 6, what is the value of A(n)? A. A(6) = 281,474,976,710,656 B. A(6) = 4 C. A(6) = 16 D. A(6) = 8
C. A(6) = 16 - A(6) = 4 comes from accidentally using "n + 1" in the place of "n - 1" in the formula. - A(6) = 8 comes from failing to make the exponent "n - 1" but using just the nth power instead. - A(6) = 281,474,976,710,656 comes from not following the order of operations correctly. Exponents must be simplified before multiplication, but this answer results when the value for a₁ is multiplied with the value of r before the power is applied: A(6) = 512(½)⁶⁻¹ = 512(½)⁵ ≠ 256⁵ = 281,474,976,710,656.
Susan : 5 + 2 = 7 Bob: 3 +6 = 9 Jose: 9-4 = 5 Renee: 7 - 2 = 5 Raul: 7 + 2 = 9 Susan's second-grade class is studying fact families. Each student is given a fact card and asked to find other members of their family. Above are the cards received by 5 students. Which students have cards that belong to the same fact family? A. Susan, Renee, and Raul B. Bob and Jose C. Susan and Renee D. Bob and Raul
C. Susan and Renee - While 5 + 2 = 7 and 7 - 2 = 5 are part of the same fact family, 7 + 2 = 9 is not since it does not contain a 5 and instead contains a 9. - A fact family consists of three numbers which are used in two addition problems with the addends reversed (a + b = c and b + a = c) and two subtraction problems (c - b = a and c - a = b). - Bob's and Jose's two problems do not contain the same three numbers, therefore, they are not in the same fact family. - Bob's and Raul's two addition problems do not contain the same three numbers, therefore, they are not in the same fact family.
Which statement about a rectangular prism and a rectangular pyramid is true? A.Only the prism has a rectangular base B. Both the prism and the pyramid have rectangular faces C. The pyramid has triangular faces D. The rectangular pyramid has one more face than the rectangular prism
C. The pyramid has triangular faces - Both have bases that are rectangles. This option could be mistaken as true if one misses that it is a RECTANGULAR pyramid instead of a triangular pyramid. - The pyramid has a rectangular base, but ALL faces are triangles. - The prism and rectangular pyramid have the same number of faces.
A teacher wants to help her students understand the rule of the order of operations. The student simplified the following expression: 3 * (8 - 2) + 6 (24 - 6) + 6 18 + 6 24 Which of the following best describes the student's work? A. The student performed the order of operations correctly B. The student subtracted at the inappropriate stage C. The student multiplied before simplifying the operation within the parenthesis D. The student simplified the operation from right to left
C. The student multiplied before simplifying the operation within the parenthesis - This is correct. The student should have performed the operations in the parenthesis prior to multiplying. - The order of operations is: Parentheses (left to right), Exponents (left to right), Multiplication and Division (left to right), Addition and Subtraction (left to right). - This can be remembered with the acronym PEMDAS.
If angle DEF is congruent to angle GHI, which of the statements is true? A. ∠E and ∠F are complementary B. ∠D ≌ ∠I C. ∠G and ∠F are complementary D. Side DE ≌ Side EF
C. ∠G and ∠F are complementary - If two triangles are congruent, then corresponding sides (sides that are in the same position) and corresponding angles (angles that are in the same position) are also congruent. - In the figure, this means that DE = GH, DF = GI, and EF = IH and m∠D ≌ m∠G, m∠F ≌ m∠I, m∠E ≌ m∠H. - The symbol in the corner of ∠E and ∠H tells us that E and H are right angles and are each equal to 90°. - Since the sum of the angles in any triangle is 180°, this means that since m∠H = 90°, then m∠G + m∠I = 180 - 90 = 90° and would, therefore, be complementary. - But what about ∠G and ∠F? Well, since ∠F and ∠I are corresponding angles, they are congruent and their measures can be substituted for each other. - So, m∠G + m∠F = 90°. Therefore, this option is a true statement.
Linear function
Can be used to understand how one quantity varies in relation to changes in the second quantity.
A student asks the teacher why the area of a triangle is ½bh? Which of the following would be the most appropriate answer for the teacher to provide to the student? A.The base of a triangle is usually half its height, so dividing the area by two gives an accurate measurement of the area B. The formula ½bh applies only to isosceles triangles C. You can make a parallelogram a triangle by cutting the parallelogram in half, so dividing the area of a parallelogram by two will yield the area of a triangle D. A parallelogram is the combination of two congruent triangles and the area of a parallelogram is simply bh, so taking one half of the area of a parallelogram yields the area of a triang
D. A parallelogram is the combination of two congruent triangles and the area of a parallelogram is simply bh, so taking one half of the area of a parallelogram yields the area of a triangle -Every parallelogram is made up of two congruent triangles. The area for a parallelogram is base x height (bh), so the area of each of the two congruent triangles is one half of the parallelogram, or (½
Given a₁ = 4, r = -5, and n = 7, what is the value of A(n)? A. A(7) = -62,500 B. A(7) = 64,000,000 C. A(7) = -312,500 D. A(7) = 62,500
D. A(7) = 62,500 - A(7) = -62,500 comes from a sign error. - A(7) = -312,500 comes from failing to make the exponent "n - 1" but using just the nth power instead. - A(7) = 64,000,000 comes from not following the order of operations correctly. Exponents must be simplified before multiplication, but this answer results when the value for a₁ is multiplied with the value of r before the power is applied: A(7) = 4(-5)⁷⁻¹ = 4(-5)⁶ ≠ (-20)⁶ = 64,000,000.
A teacher draws many variations of triangles on the board in front of the class. She then asks students to identify which triangles have acute angles, right angles, and obtuse angles. Which of the following is the teacher most likely trying to accomplish? A. Teaching place value using manipulatives B. Applying geometric principles to real-life situations C. Explaining the concept of angle congruence D. Reinforcing geometric definitions
D. Reinforcing geometric definitions - Students already know the definitions of acute angles, right angles, and obtuse angles because the question does not say the teacher is teaching the angles; the question implies the students already know the definition of the angles. - The teacher is allowing students to exercise their knowledge of these definitions by drawing the angles on the board and asking students to identify them.
The coordinate is the
INDEPENDENT VALUE Note: that an independent variable determines the values of other variables, and is not thought to have its value determined by another variable - the X axis of a coordinate plane is placed HORIZONTALLY, like a basic number line - the 2nd value in the pair is commonly represented by "y" and is considered an "output value" The y-coordinate of the pair is the DEPENDENT VALUE (the variable whose value depends upon the value of the independent variable) -y axis on a coordinate plane is placed vertically
Closure for Addition
If 2 Real numbers are added, then their sum will also be a Real number. Symbolically: if "a" and "b" are real numbers and a + b = c , then "c" is also Real (see number concepts doc for further explanation)
Geometric Sequence
If numbers show a common ratio then the sequence is called a geometric sequence EX: the next value is twice as big as the previous value EX: the next value is 1/3 of the previous value To find the specified nth term (an) -if the first term (a1) and the common ratio (r) are known -the formula an= a1(r)^n-1 can be used EX: 18,36,72,144,248 -if need to find the 10th term -can note the first term (a1) is 18 -the common ratio (r) is 2 -and the term number (n) is 10 -a^10= 18(2)^9 = 18 X 512 = 9,216
Commutative Property of Multiplication
If the order of the FACTORS (numbers multiplied with each other) changes in a statement of multiplication, then the prodcut will stay the same. Symbolically: ab = ba and abc=cba , etc.
Commutative Property of Addition
If the order of the addends (numbers added to each other) changes in a statement of addition, then the sum (answer to an addition problem) will stay the same. Symbolically: a + b = b + a and a + b + c = c + a + b
Associative Property of Addition
If the placement of the grouping symbols of a set of addends changes, then the sum will stay the same. Symbolically: a + ( b + c ) = ( a + b ) + c NOTE: the order of the values did not change in this case, as Associativity is only about the grouping of values)
Associative Property of Multiplication
If the placement of the grouping symbols of a set of factors changes, then the product will stay the same. Symbolically: a(bc) = (ab)c
Subtraction property of Equality
If the quantities on each side of an equal sign are both subtracted by the same amount, the resulting statement will still be equal. Symbolically: if a = b, then a - c = b - c
Closure for Multiplication
If two Real numbers are multiplied, then their product will also be a Real number. Symbolically: if "a" and "b" are Real numbers and ab=c, then "c" is also Real. (see number concepts doc for further explanation)
Symmetric Property of Equality
If two quantiites are equal, then it does not matter which quantity is written on the left hand side of the equal sign and which is written on the right hand side. Symbolically: if a = b, then b = a
Substitution Property of Equality
If two quantites are equal, one may replace the other in a true statement and the statement will remain true. Symbolically: if a + b = c and a = x , then x + b = c
Linear Function
One whose graph is a straight line.
Some lines have special relationships to each other based upon their slopes, such as being parallel or perpendicular to one another.
Parallel lines: lines in the same plane that have the same slope and never intersect EX: A line parallel to the line y= 3x-4 would be parallel to any other line with a slope of 3, such as y=3 or y=3x+5 Perpendicular Lines: lines in the same plane as each other that intersect at a right angle (90 degrees). -They have slopes that are opposite reciprocals of each other; they have slopes with a product of -1. - the only exception to that statement is a pair of horizontal and vertical lines -Horizontal lines show no change in Y and so have a slope of 0, while vertical lines have no change in x and so have an undefined slope (b/c we cannot divide by 0 in the slope calculation) - A line with a slope of 3 is perpendicular to any line with a slope of 1/3 so the line y=3x-4 would be perpendicular to a line defined by y= -1/3x +3
Trinomials
Polynomial expressions with THREE unlike monomial terms connected through addition or subtraction. - 2x² + 5x - 8" , " a + b + c" , 3x + 4.9a²b³c-(1/2)y⁵" -polynomials with 4 or more UNLIKE TERMS do not have a special name. They are simply known as polynomials.
Binomials
Polynomial expressions with TWO unlike monomial terms connected through addition or subtraction. EX's: "7x-2" "y+3y⁵" "2a + 2b" "4x + 2x - 5y" (2 like terms so binomial)
B/C veritcal lines have no slope, they can only be written in
STANDARD FORM - The simplest expression for the equation of a vertical line is x = a number (whatever number for X the entire vertical line passes through), such as x=5 as graphed below on the left. -Recall that vertical lines are the only linear equations that are NOT functions
Linear Functions can be described by the equations
Slope intercept form: y = mx + b Point-Slope form: y₁ = m(x-x₁) Standard Form: A× + Bγ = C
Exponential Function
Start by increasing very slowly then start increasing more and more rapidly Has a horizontal asemptote of y=0 asymptote- a line the graph is approaching but doesn't cross
Determine the choice that best describes the characteristics of the slope-intercept form of a line. y = mx + b
The m and b are fixed. The m represents the slope, and the b represents the y-intercept. As x changes, the y changes
Commutative Property
The order of addends or factors do not change the result.
Algebraic solution
The process of solving a mathematical problem using the principles of algebra.
Zero Product Property
The sum of a number and zero is the number itself, and the product of a number and zero is zero. If values are multiplied to yield a product of zero, then at least one of those factors MUT be zero. Symbolically that is: if ab= 0, then either a = 0 or b = 0 , or both a = b = 0
Partial Products
When you multiply by a number with more than one digit - you add partial products together to get product
Distributive Property of Multiplication over Addition
You can add and then multiply or multiply then add. The produt of a factor with a sum ( or difference) is the same ( or differnce) of the product made by that factor and each number involved in the addition pr subtraction. The reverse direction is also true. Symbolically: then the Distributive Property can be represented multiple ways: a(b + c) = ab + ac ( b + c)a = ba + ca ab + ac = a( b + c ) ba + ca = (b + c )a - As subtraction is properly understood as a form of addition, each statemnet above is also true fro subtraction, such as : a(b - c) = ab -ac
Coefficient
a number multipled with a variable, written in front of the variable
Square root function
a parabola turned on its side and just the TOP HALF
The only type of linear equation that is not a function is
an equation of the form x = a number such as x=4 - an equation like x=4 creates an infinite set of points that all use a single input value, in this case 4, paired with an infinite number of unique outputs to make all of the points on the line. ALL other lines/linear equations, however, are FUNCTIONS
Algebraic expressions
are manipulated using the same main rules and restrictions of Real numbers, with the added complexity of how to combine terms correctly. below are the main tools for algebra: - Formal Properties - Order of Operations - Comprehension of "like terms"
Variable Expressions
consist of at least one variable, a coefficient, and sometimes, a power (exponent) or operation. variable expression ex's: 10.2x (the coefficient is 10.2) and -7x² + 3y (coefficients are -7 and 3) and simply "z" (coefficient is understood to be 1 b/c of the Identity Property of Multiplication, 1 multiplied with any number always equals itself, so "z" can be thought of as "1z" without changing its value or meaning)
Convex Polygon
convex polygon - A convex polygon has no angles greater than 180°. - Another way to think of how to identify a convex pentagon is that it has no angles pointing inward. - A pentagon has 5 sides. The shape above is a convex pentagon.
Horizontal Lines
have a slope that = 0 - As such, they can be said to be written in any of the 3 formats - the most common representation, however, is simply y = a number (whatever number for Y the entire horizontal line passes through), such as y=-2
Polynomial/ Monomial
is a variable expression or a sum (or difference) of variable expressions that have whole number exponents, with no division by any variables. - a polynomial is a monomial or the sum of MORE THAN ONE monomial expression. - A monomial is a single term, where the coefficient is connected to one more variables, through multiplication. - A monomial has no addition, subtraction, or division by a variable within it. - polynomial expressions with one term are called monomials (if all powers on the variables are whole numbers) A constant like "6" can be considered a monomial b/c it is the product of 6 and x⁰ (no x at all) A single variable like "z", is considered a monomial b/c it is the product of 1 with z ¹. Other sample monomials: "3x" , "4.9a²b³c," or (- 1/2)y⁵ - The expression "4x + 2x" should be understood to be the monomial 6x b/c the two addends are like terms so can be combined into a single term, the monomial 6x.
Logarithmic function
is the inverse of an exponential function has a vertical asymptote of x=0 asymptote- a line the graph is approaching but doesn't cross
Multiplication Property of Equality
is the quantities on each side of an equal sign are both multipled by the same amount, the resulting statement will still be equal. Symbolically; if a = b , then ac = bc
Divisor
number you divide by
Slope
often referred to as rise/run Formal Slope: γ2-γ1 ------ X₂-X₁ - A horizontal line is perfectly flat and has a slope of 0 - A vertical line is impossibly steep, so we say it has no slope at all or an UNDEFINED slope. ( a slope calculation for avertical line would result in division by zero b/c vertical lines have no "run". -B/C division of 0 cannot be done, slope cannot be calculated for a vertical line.
Percent problems are best understood as proportions
one ratio is a percent (a part compared to 100), while the other ratio compares a part to a whole. EX's: 1. 25% of 72 is x? a. The proportion is set up: 25/100 = x/72 The answer is found: 18 is 25% of 72 2. What % of 58 is 12? x/100 = 12/58 The answer found is: 12 is ∼ 20.7% of 58 3. 85 is 120% of what number? 120/100 = 85/x The answer found is: 85 is 120% of ∼70.8 - It's important that students become familiar with percents larger than 100% and understand what a percent greater than 100% will mean for the value "x" - EX: In the example above because 120% is more than 100%, 85 must be more than the "whole" to which it is being compared.
Quadratic function
parabola
Concept of Ratio:
the comparison of one part to another - 2 numbers are PROPORTIONAL to another pair of numbers, in general, if their ratio is CONSTANT ( if both pairs of fractions reduce to yield the same number) - Equations that are proportions (fraction = fraction) are best solved by cross-multiplication.
If terms are NOT like each other
they cannot be simpliied throuhg addition or subtraction. For EX: "2x + 9y" is considered FULLY simplified. So. is "y + 3y⁵" NOTE: while unlike terms can't be added or subtracted, such terms CAN be multiplied and divided. That material, however is not relevant to a standard elementary school curriculum so won't be explored)