Variable
unknown number or quantity represented by a letter usually x
Dependent Variable
variable with value that is calculated based on the value of other quantities
Constant
number with value that does not change
ex. y=x - 5, 5 is the constant
Coefficient
a constant that is placed in front of a variable in an algebraic expression
ex. 3y+1,3 is the coefficient of y
Strategies for mathematics
-use a manipulative or act out the problem
-draw a picture
-look for a pattern
-guess and check
-logical reasoning
-organized list
-table
-simplify
-work backwards
Steps to solve a word problem
1. read carefully, finding and separating information needed to solve
2. make a plan as to what needs to be solved
3. solve the problem using step 2 plan
4. review the problem make sure answer is correct and makes sense
Commutative Property
product is the same regardless of the order of the factors
ex. 5 x 2 = 2 x 5
Associative Property
product is the same regardless of grouping
ex. (5 x 2) x 3 = 2 x (3 x 5)
Distributive Property
multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products
ex. 2 x (3 + 4) = (2 x 3) + (2 x 4) = 14
Property of Zero
sum of a number and 0 is that number. in multiplication, the product of a number and 0 is 0
ex. 3+0 = 3 and 3x0 = 0
Dividend
number to be divided
ex. 18 is divided by 6 (18/6) then 18 is the dividend
Remainder
surplus value hen one number cannot be evenly divided by another, that is, the number less than the divisor that remains after dividing.
ex. 36 divided by 5 is 7 with a remainder of 1
Divisor
number by which a dividend is divided
ex. 18 divided by 6 (18/6) then 6 is the divisor
Quotient
number that is the result of the division operation
ex. 18 is divided by 6, the outcome is 3, that is the quotient
Additive identity
states that the sum of any number and zero is that number
ex. a + 0= a
Multiplicative identity
property of one, says that product of any number and one is that number a x 1 =1
Additive inverse
property of opposites, the sum of any number and its additive inverse is zero and is represented by the following: a + (-a) = 0
Multiplicative inverse
property of reciprocals, product of any number and its reciprocal is one and is represented by the following: a/1 x 1/a = 1
Angles
congruent- angle that has the same measure as another angle
right- quarter of a full turn 90 deg
straight or flat- half a full turn 180 deg
acute- any angle smaller than a right angle <90 deg
obtuse- angle greater than a right angle >90 deg
reflex- angle
Area of shapes
-rectangle: A = wl, w is width, l is length
-square: A= s^2, s is length of side
-triangle: A= 1/2 bh
-parallelogram: A= bh
-trapezoid: A= 1/2 (b1 + b2) h, b1 and b2 are lengths of two parallel sides or bases
-circle: A= pie R^2
Addition Rule for Probability
P(A or B) = P (A) + P (B) - P(A and B), where P(A and B) is the probability of both events occurring to find the probability of a compound event
Order of Operations
set of rules that dictates the order in which we perform each operation, PEMDAS tells us which one to do first if their are more than one expression
P= parentheses
E=simplify exponents
MD= multiplication and division from left to right
AS= addition and su
Adding and subtracting fractions
if common denominator, they can be added or subtracted by add or sub the two numerators and retaining the same denominator
if no common denominator, one or both must be manipulated to get a common denominator
Multiplying and dividing fractions
multiply from left to right
divide by flipping the numerator and denominator of the second fraction and then multiply form left to right
Complementary Angles
two angles whose sum is exactly 90 deg, may or may not be adjacent, in a right triangle the two acute angles are complementary
Supplementary Angles
two angles whose sum is exactly 180 deg, may or may not be adjacent, two intersecting lines always form two pairs of supplementary angles, adjacent supplementary angles will always form a straight line
Adjacent Angles
two angles that have same vertex and share a side, vertical angles are not adjacent because they share a vertex but no side
Triangle Inequality Theorem
the sum of the measures of any two sides of a triangle is always greater than the measure of the third side
Addition
increases the value of one quantity by the value of another, order does not matter
Subtraction
opposite operation as addition, it decreases the value by the value of another, order does matter
Multiplication
repeated addition, one number tells how many times to add the other number to itself, order does not matter
Division
opposite operation as multiplication, one number tells us how many parts to divide the other number into, order does matter
Improper Fractions
a fraction whose numerator is greater than its denominator is improper, proper have a value less than one while improper fractions value is greater than one
Mixed Numbers
number that contains both an integer and a fraction, any improper fraction can be rewritten as a mixed number
Equilateral Triangle
three congruent sides, will also have three congruent angles
Isosceles Triangle
two congruent sides, will also have two congruent angles opposite the two congruent sides
Scalene Triangle
no congruent sides, three angles of different measures, angle with the largest measure is opposite the longest side, angle with smaller measure is opposite the smallest side
Fraction
number that is expressed as one integer written above another, with dividing line between, represents the quotient of the two numbers x divided by y
Numerator
top number of a fraction, represents the number of parts under consideration, 1/4, the 1 means that 1 part out of the whole is being considered
Denominator
bottom number of a fraction, represents to total number of equal parts, 1/4, the 4 means that the whole consists of 4 equal parts, cannot have a zero as a denominator
Percentages
thought of as fractions that are based on a whole of 100, that is, one whole is equal to 100%, percent means per hundred, the percentage number will always be larger than the equivalent decimal number
Interior Angles
two parallel lines are cut by a transversal, the angles between the two parallel lines are interior angles
Exterior Angles
two parallel lines are cut by a transversal, the angles that are outside the parallel lines are exterior angles
Corresponding Angles
two parallel lines are cut by a transversal, the angels that are int he same position relative the the transversal and one of the parallel lines
Rational Numbers
integers, decimals, fractions, any terminating or repeating decimal number is a rational number
Irrational Numbers
cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring patter of digits within the number, pie is an irrational number
Real Numbers
the set of all rational and irrational numbers
Greatest Common Factor
largest number that is a factor of two or more numbers, factors of 15 are 1, 3, 5, 15, factors of 35 are 1, 5, 7, 35, therefore the GCF is 5
Least Common Multiple
the smallest number that is a multiple of two or more numbers, multiples of 3 include 3,6,9,15 etc. the multiples of 5 include 5,10,15,20 etc. the LCM is 15
Similar Triangle
have corresponding angles that are congruent to one another, their corresponding sides may or may not be equal, but that are proportional to one another
Congruent Triangle
similar triangles whose corresponding sides are all equal, can be made to fit on top of one another by rotation, reflection, or translation
SSS= all 3 sides are equal to another triangle
SAS= two sides and the adjoining angle in one triangle are equal to
Rotational Symmetry
a shape that can be rotated about a point and achieve its original shape and orientation with less than a 360 deg turn
Reflection Symmetry
can be reflected across a line with the result being the same shape as before the reflection, line of symmetry divides the shape into two parts
Volume formulas
pyramid= 1/3Bh, h is the distance between the vertex and the base polygon, measured perpendicularly
prism= Bh, h is the perpendicular distance between the two bases
cube= s^3, s is length of any side
sphere= 4/3pier^3
Kindergarten concepts
position- top, middle, bottom, above, before, after
visual attributes- same/diff colors, shapes, sizes
sorting- size, color, type, identify an equal #, more, or less of a given item
graphing- picture graphs and data to form
patterns- identifying, copying,
Pythagorean Theorem
a^2 + b^2 = c^2
c is the hypotenuse, a and b are remaining sides
Mean, Median, Mode
mean- calculated by summing all of the values in the set and dividing that by the number of values
ex. data has 6 numbers and the sum is 30, mean is 30/6=5
median- putting data set in numerical order, locating the middle value
mode- value that appears mos
Manipulatives in the classroom
-teacher should discuss the purpose
-student should understand that they are intended for use with specific problems, however, give free exploration time so there is less play during tasks
-chart posted with names of manipulative
-give them to students to
How to build number sense
-frequently asking them to make calculations mentally and rely of reasoning ability
-have class discussion about solutions and compare different approaches, explain reasoning in own words
-modeling different ideas by tracking them on the board and discuss
Formulas for volume
cube- a^3, a being side length
rectangular prism- abc, all side lengths
circle- (4/3) pi r^3
Formulas for Area
square = a 2
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b1 + b2)
circle = pi r 2
triangle = 1/2 (bh)
Variable
unknown number or quantity represented by a letter usually x
Dependent Variable
variable with value that is calculated based on the value of other quantities
Constant
number with value that does not change
ex. y=x - 5, 5 is the constant
Coefficient
a constant that is placed in front of a variable in an algebraic expression
ex. 3y+1,3 is the coefficient of y
Strategies for mathematics
#NAME?
Steps to solve a word problem
1. read carefully, finding and separating information needed to solve
2. make a plan as to what needs to be solved
3. solve the problem using step 2 plan
4. review the problem make sure answer is correct and makes sense
Commutative Property
product is the same regardless of the order of the factors
ex. 5 x 2 = 2 x 5
Associative Property
product is the same regardless of grouping
ex. (5 x 2) x 3 = 2 x (3 x 5)
Distributive Property
multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products
ex. 2 x (3 + 4) = (2 x 3) + (2 x 4) = 14
Property of Zero
sum of a number and 0 is that number. in multiplication, the product of a number and 0 is 0
ex. 3+0 = 3 and 3x0 = 0
Dividend
number to be divided
ex. 18 is divided by 6 (18/6) then 18 is the dividend
Remainder
surplus value hen one number cannot be evenly divided by another, that is, the number less than the divisor that remains after dividing.
ex. 36 divided by 5 is 7 with a remainder of 1
Divisor
number by which a dividend is divided
ex. 18 divided by 6 (18/6) then 6 is the divisor
Quotient
number that is the result of the division operation
ex. 18 is divided by 6, the outcome is 3, that is the quotient
Additive identity
states that the sum of any number and zero is that number
ex. a + 0= a
Multiplicative identity
property of one, says that product of any number and one is that number a x 1 =1
Additive inverse
property of opposites, the sum of any number and its additive inverse is zero and is represented by the following: a + (-a) = 0
Multiplicative inverse
property of reciprocals, product of any number and its reciprocal is one and is represented by the following: a/1 x 1/a = 1
Angles
congruent- angle that has the same measure as another angle
right- quarter of a full turn 90 deg
straight or flat- half a full turn 180 deg
acute- any angle smaller than a right angle <90 deg
obtuse- angle greater than a right angle >90 deg
reflex- angle
Area of shapes
-rectangle: A = wl, w is width, l is length
-square: A= s^2, s is length of side
-triangle: A= 1/2 bh
-parallelogram: A= bh
-trapezoid: A= 1/2 (b1 + b2) h, b1 and b2 are lengths of two parallel sides or bases
-circle: A= pie R^2
Addition Rule for Probability
P(A or B) = P (A) + P (B) - P(A and B), where P(A and B) is the probability of both events occurring to find the probability of a compound event
Order of Operations
set of rules that dictates the order in which we perform each operation, PEMDAS tells us which one to do first if their are more than one expression
P= parentheses
E=simplify exponents
MD= multiplication and division from left to right
AS= addition and su
Adding and subtracting fractions
if common denominator, they can be added or subtracted by add or sub the two numerators and retaining the same denominator
if no common denominator, one or both must be manipulated to get a common denominator
Multiplying and dividing fractions
multiply from left to right
divide by flipping the numerator and denominator of the second fraction and then multiply form left to right
Complementary Angles
two angles whose sum is exactly 90 deg, may or may not be adjacent, in a right triangle the two acute angles are complementary
Supplementary Angles
two angles whose sum is exactly 180 deg, may or may not be adjacent, two intersecting lines always form two pairs of supplementary angles, adjacent supplementary angles will always form a straight line
Adjacent Angles
two angles that have same vertex and share a side, vertical angles are not adjacent because they share a vertex but no side
Triangle Inequality Theorem
the sum of the measures of any two sides of a triangle is always greater than the measure of the third side
Addition
increases the value of one quantity by the value of another, order does not matter
Subtraction
opposite operation as addition, it decreases the value by the value of another, order does matter
Multiplication
repeated addition, one number tells how many times to add the other number to itself, order does not matter
Division
opposite operation as multiplication, one number tells us how many parts to divide the other number into, order does matter
Improper Fractions
a fraction whose numerator is greater than its denominator is improper, proper have a value less than one while improper fractions value is greater than one
Mixed Numbers
number that contains both an integer and a fraction, any improper fraction can be rewritten as a mixed number
Equilateral Triangle
three congruent sides, will also have three congruent angles
Isosceles Triangle
two congruent sides, will also have two congruent angles opposite the two congruent sides
Scalene Triangle
no congruent sides, three angles of different measures, angle with the largest measure is opposite the longest side, angle with smaller measure is opposite the smallest side
Fraction
number that is expressed as one integer written above another, with dividing line between, represents the quotient of the two numbers x divided by y
Numerator
top number of a fraction, represents the number of parts under consideration, 1/4, the 1 means that 1 part out of the whole is being considered
Denominator
bottom number of a fraction, represents to total number of equal parts, 1/4, the 4 means that the whole consists of 4 equal parts, cannot have a zero as a denominator
Percentages
thought of as fractions that are based on a whole of 100, that is, one whole is equal to 100%, percent means per hundred, the percentage number will always be larger than the equivalent decimal number
Interior Angles
two parallel lines are cut by a transversal, the angles between the two parallel lines are interior angles
Exterior Angles
two parallel lines are cut by a transversal, the angles that are outside the parallel lines are exterior angles
Corresponding Angles
two parallel lines are cut by a transversal, the angels that are int he same position relative the the transversal and one of the parallel lines
Rational Numbers
integers, decimals, fractions, any terminating or repeating decimal number is a rational number
Irrational Numbers
cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring patter of digits within the number, pie is an irrational number
Real Numbers
the set of all rational and irrational numbers
Greatest Common Factor
largest number that is a factor of two or more numbers, factors of 15 are 1, 3, 5, 15, factors of 35 are 1, 5, 7, 35, therefore the GCF is 5
Least Common Multiple
the smallest number that is a multiple of two or more numbers, multiples of 3 include 3,6,9,15 etc. the multiples of 5 include 5,10,15,20 etc. the LCM is 15
Similar Triangle
have corresponding angles that are congruent to one another, their corresponding sides may or may not be equal, but that are proportional to one another
Congruent Triangle
similar triangles whose corresponding sides are all equal, can be made to fit on top of one another by rotation, reflection, or translation
SSS= all 3 sides are equal to another triangle
SAS= two sides and the adjoining angle in one triangle are equal to
Rotational Symmetry
a shape that can be rotated about a point and achieve its original shape and orientation with less than a 360 deg turn
Reflection Symmetry
can be reflected across a line with the result being the same shape as before the reflection, line of symmetry divides the shape into two parts
Volume formulas
pyramid= 1/3Bh, h is the distance between the vertex and the base polygon, measured perpendicularly
prism= Bh, h is the perpendicular distance between the two bases
cube= s^3, s is length of any side
sphere= 4/3pier^3
Kindergarten concepts
position- top, middle, bottom, above, before, after
visual attributes- same/diff colors, shapes, sizes
sorting- size, color, type, identify an equal #, more, or less of a given item
graphing- picture graphs and data to form
patterns- identifying, copying,
Pythagorean Theorem
a^2 + b^2 = c^2
c is the hypotenuse, a and b are remaining sides
Mean, Median, Mode
mean- calculated by summing all of the values in the set and dividing that by the number of values
ex. data has 6 numbers and the sum is 30, mean is 30/6=5
median- putting data set in numerical order, locating the middle value
mode- value that appears mos
Manipulatives in the classroom
#NAME?
How to build number sense
#NAME?
Formulas for volume
cube- a^3, a being side length
rectangular prism- abc, all side lengths
circle- (4/3) pi r^3
Formulas for Area
square = a 2
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b1 + b2)
circle = pi r 2
triangle = 1/2 (bh)