Mathematics Flashcards
Inductive teaching
Deduction approach
- learning through example
- learning step by step
Jean Piaget development stages
Sensorimotor stage: birth - 2 years
Preoperational stage: years 2-7
-symbolic functioning, centration, intuitive thought, and inability to conserve
Concrete Operational: years 7-11
-decentering, reversibility, conservation, classification
Formal Operational stage: years 11- adult
-the ability to use symbols and think abstractly
Issues with preoperational stage
years 2- 7
Centration: focusing on only one aspect of a situation or problem
Conservation: understanding that quantity, length. or number of items is unrelated to the arrangement or appearance of the object of items
Concrete operational stage
2-7 years
Decentering: child can take into account multiple aspects of a problem to solve it
Reversibility: child understands that the objects can be changed and the3n returned to the original state
Conservation: child understands that quantity, length, or number of items is unrelated to the arrangement or appearance of the object
Serialization: child able to arrange objects in an order according to size, etc.
Classification: child can name and identify sets of objects according to appearance
Elimination of Egocentrism: child is able to view things from another’s perspective
The Professional Standards for Teaching Math (NCTM) presents standards for teaching math
Task: projects, questions, problems, construction, application
Environment: the setting for learning
Analysis: the systematic reflection in which teachers engage
Discourse: the manner of representing, thinking, talking, agreeing, and disagreeing
Pre-K
explores concrete models and materials
counts to 10 or higher by ones
begins to describe the concept of zero
identifying first and last in a series
6th grade
Compares and orders non-negative rational numbers, generates equivalent forms of rational numbers
Able to write prime factorizations using exponents, identifies factors of a positive integer, common factors, and greatest common factors
Integer
a whole number includes all positive and negative numbers, including zero
-6, - 5, -4-3, -2, -1, 0, 1, 2, 3, 4, 5, 6,…
Natural Numbers
a positive integer (not zero) or a nonnegative integer (whole numbers includes “0”)
0, 1, 2, 3, 4, 5, 6…
Rational Numbers
a number that can be expressed as a ratio of quotient of two nonzero integers- Fractions and Decimals
Finite decimals, repeating decimals, mixed numbers, whole numbers
Nonrepeating decimals cannot be expressed in this way- said to be irrational
Irrational Numbers
is a number that cannot be represented as an exact ratio of two integers
The decimal form of the number never terminates and never repeats
Ex: pi
Real Numbers
describes any number that is positive, negative, or zero and can be used to measure continuous quantities
Adding and Subtracting Homogenous Fractions
same denominator- add or subtract the numerator and keep the denominator the same
2/5+ 1/5= 3/5
Change improper fraction to Mixed Numbers
divide the numerator by the denominator and represent the remainder as a fraction
5/2= 2 1/2
Adding and Subtracting Mixed Numbers
denominators must be the same- add/ subtract the whole numbers and then add/subtract the numerator and keep the denominators the same
2 5/10 + 1 4/10= 3 9/10
7 9/12 - 5 4/12= 2 5/12
Changing Mixed Numbers to Improper Fractions
numerator greater then the denominator- multiply the denominator by the whole number, then add the resulting numerators
2 3/4= (2x4=3)/ 4= (8+3)/ 4= 11/4
multiplying fractions
multiply horizontally- multiple the numerators together and the denominators together
2/3 x 3/4= (2x3)/ (3x4)= 6/12= 1/2
If the numbers being multiplied are mixed fractions, first rewrite them as improper fractions
Dividing Fractions
take the reciprocal of the second fraction ( the one doing the dividing) and multiply the fractions
reciprocal: a fraction with the numerator and denominator switched
Multiplying Decimals
count the total numbers behind the decimal point in both numbers– this will be the number behind the decimal in the answer
2.3 x 4.56= 3 numbers behind the decimal point
Numbers do not have to be aligned
Dividing Fractions
Number doing the dividing needs to be a whole number so you must move the decimal– the number of places who move the decimal you do the same to the number being divided
1.44/ 0.3 ——– 14.4/ 3.0
decimal carries up to the answer from the number being divided
14.4/ 3.0= 4.8 ( 3 goes into 14 4 times)
Exponential notation
a symbolic way of showing how many times a number or variable is used as a factor
5^3 shows five is use three times (5 x 5 x 5)
Negative exponent indicates a reciprocal, therefore 5^-3 = 1/(5^3)= 1/ (5 x5 x5)= 1/125
Absolute Value
is the distance of a number from zero on the number line
ignores the + and - signs of a number
I-5I = 5 I5I= 5
Expanded Form
shows the place value of each digit
263= 200 + 60 + 3 which equals 2 hundreds 6 tens an d3 ones
Expanded notation
shows place value by multiplying each digit in a number by the appropriate power of ten
523= (5 x 10^2) +(2 x 10)+( 3 x 1) or (5 x 10^2) = (2x10^1) +(3x 10^0)
Scientific Notation
form of writing a number as the product of a power of 10 and a decimal number greater than or equal to 1 and less than 10
2,400,000= 2.4x10^6, 240.2= 2,402x 10^2, 0.0024=2.4x 10^-3
Estimation
is used to make an approximation that is still close enough to be useful
Prime Numbers
natural numbers greater than 1 that are divisible ony by themselves and 1
The first eight prime numbers are : 2 ,3, 5, 7, 11, 13, 17, 19
Composite Numbers
are natural numbers greater than zero that are divisible by at least one other number besides 1 and themselves
have at least three factors
9 is a composite number bc it has three factors: 1, 3, 9
Commutative Property
the order of the addends ( terms being added) or factors (in multiplication) do not change the result
a + b= b + a and a x b= b x a
Associative Property of Multiplication and Addition
the order of addends or factors will not change the sum or product
( a+ b) + c = a + (b + c) and (a x b) x c= a x (b x c)
Property of Zero
The sum of a number and zero is the number itself and the product of a number and zero is zero
8 + 0= 8
8 x 0= 0
Distributive Property
everything within the parentheses needs to be multiplied by the number outside
a (b + c) = ax (b +c) or
a (b + c) = (a x b) + (a x c)
8 (5+ 2) = 8x7= 56 or 8( 5 + 2)= (8 x5)+(8 x 2) = 56
Linear and Nonlinear Functional Relationships
Called the domain of F, a single real number designated as f (x). Variable X is called the independent variable. If y = f(x), we call y the dependent variable. Then, F(0) is the value of the funtion when x= 0.
Linear function is a straight line, nonlinear function does not satisfy the constraints of a linear function.
Function
Functions can be used to understand how one quality varies in relation to (or is a function of) changes in the second quantity.
Algebraic pattern
An algebraic pattern is a set of numbers and/or variables in a specific order.
Algebraic Expression
A mathematical phrase that is written by using one or more variables and constants, but does not contain a relation symbol (e.g., 5y + 8)
Variable
a letter that stands for a number in an algebraic expression.
Coefficient
a number that precedes the variable to give the quantity of the variable in the expression.
Addition and subtraction
like algebraic terms (same variable bases with the same exponents) can be added or subtracted to produce simpler expressions. example 2x³ and 3x³ can be added together to get 5x³.
Multiplying exponential terms
The constant terms are multiplied, but the exponents of the terms with the same variable bases are added together.
Multiplication of Binomials
FOIL stands for “first,outer,inner,last” or First,outside,inside,last. to multiple (x +3) and (2x-5) multiply x by 2x (the first terms), x by -5 (outer terms), 3 by 2x (inner terms), and 3 by -5(last terms).
Factoring
opposite of polynomial, a polynomial means rewriteing it as the product of factors (often two binomials). The trinomial x2 (small 2) - 11x + 28, for instance, can be factored into (x-4)(x-7)
Algebraic Solution
A process of solving a mathematical problem by using the principles of algebra.
Graphs and Symbolic representations
Four kinds of graphs are generally used in grades pre-K to grade 6- pictorial,bar,line,and pie. Pictorial graphs are the most concrete representations of information. They represent a transition from the real object graphs to symbolic graphs.
Bar graphs
Used to represent two elements of a single subject.
Line graphs
Presents information in a fashion similar to bar graphs, but they use points and lines. A line graph tracks one or more subjects.
Pie charts
Used to help visualize relationships based on percentrages of a possible 100%.
Linear and nonlinear functions
A linear function is one whose graph is a straight line. A nonlinear function, is not linear. example quadratic x2 + x-1=0.
Proportional Reasoning
Often they involve two unlike values that are related in a certain way.
Variables,equations,and inequalities
Variables are classified as either independent(free) or dependent (bound). An expression represents a function whose inputs are the values assigned to the independent variables and whose output is the resulting value of the expression. Evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values applied in the context of the expression.
Algebraic Inequality
An algebraic statement about the relative size of one or more variables and/or constants. Inequalities are used to determine the relationship between these values. example- taking x as a variable and saying, “x is less than 5,” may be written as x <5
Multiplying or dividing
Same negative number on both sides and changing the orientation of the inequality sign. The inequality 2x > 6 has the same solution as the inequality x > -2 (dividing by (-2) on both sides and switching “>” to “
Van Hieles levels of Geometric thinking
0-Visualization, develop a mental picture of each shape
1- Analysis, begin to talk and notice the properties of the shapes.
2-informal deduction, stop relying on visualization, and now use relationships to make a conclusion.
3-Formal deduction, Focus is on higher levels of geometry, using various theorems to teach how two traingles are congruent
4-Rigor, Rigor is associated with college level geometry.
Straight line
straight line, lines are one-dimensional. meaning they have infinite length and width but no depth
line segment
any portion of a line between two points on that line. definite start and definite end called the endpoints, from point A to point B is AB.
A Ray
Like a line segment, except it extends forever in one direction.
Angle
Formed when two rays or lines share an endpoint. If two angles have the same size (regardless of how long their rays might be drawn), they are congruent.
Supplementary angles
Two angles that add up to 180°
Complementary angles
Two angles that add up to 90°
Vertical angles
If two lines intersect, they form two pairs of equal angles.
Polygons
Is a many-sided plane (two dimensional) figure bounded by a finite number of straight lines. Described based on the number of sides, which are equal to the number of vertices. Triangles= three sided polygons Quadrilaterals= Four sided Pentagons= Five sided Hexagons= Six sided Octagons=Eight-sided
Properties of Triangles
Three sided polygons
Isosceles traingle
Has two Equal sides and two equal angles
Equilateral triangle
If the measures of all sides of the triangle are equal
Scalene triangle
Triangle has three unequal sides
Acute triangle
All three angles are acute
Right triangle
One of the angles is a right angle
Obtuse traingle
One of the angles is obtuse
Pythagorean theorem
Any right triangle with legs (shorter sides) a and b, and hypotenuse (the longest side) c, the sum of the squares of the legs will be equal to the square of the hypotenuse.
SAS
SAS(side angle side) two pairs of sides of two triangles are equal in length
SSS
SSS ( side-side-side) three pairs of sides of two triangles are equal in length.
ASA
ASA (angle-side-angle) Two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
AAS
Angle-Angle-Side, two pairs of angles of two traingles are equal in measurement and a pair of corresponding sides is of equal length, then the trangles are congruent.
Parallelograms
Four sided polygon with two pairs of equal and parallel sides. the height of a parallelogram is not necessarily the same as the length of the other side.
Rhombus
Special type of parallelogram in which all the sides are the same length
Rectangles
A type of parallelogram with four right angles
Squares
A type of parallelogram with four right angles and four equal sides
Circles
unique shapes in geometry because they have no angles.
Diameter
Straight line of a circle that segments that goes from one edge of a circle to the other side.
Perimeter
Found by adding the measures of the three sides of the triangle.
Circumference
Can be thought of as the perimeter of the circle, the distance around the circle.
Area of the Parallelogram
Found by multiplying the measure of the base by the measure of the height, or a=bh
Area of a rectangle
Similarly found by multiplying the measure of the length or base of the rectangle by the measure of the width of the rectangle.
Area of a square
Found by squaring the measure of the side of the square.
Area of a circle
Found by squaring the length of its radius, then multiplying that product by pi
Congruent
Having the same shape and size
Volume of a cylinder
Pi r squared x height
Area of base x height
Symmetry
Correspondence in size,form, and arrangement of parts on opposite sides of a plane, line, or point.
Tessellations
The arrangement of polygons that forms a grid, a pattern formed by the repetition of a single unit or shape that, when repeated, fills the plane with no gaps and no overlaps.
Coordinate plane
A grid made of two main perpendicular lines, the x-axis and the y-axis.
Origin
Defined by the ordered pair, sets of ordered pairs may be plotted on a coordinate plane to create different lines,rays,shapes,and so on.
Temperature
Water boils at 212F or 100C, and freezes at 32F or 0C
Translation
Also called a slide, simply means moving. Every translation has a direction and distance. Known as a transformation that moves a geometric figure by sliding.
Linear measurement
Customary units of length include inches,feet,yards,and miles
Measurement of mass
Customary units of weight include ounces,pounds and tons
Volume
Customary units of capacity unclude teaspoons, tablespoons,cups, pints,quats, and gallons
Metric units
Units of length include millimeters, centimeters, meters, and kilometers
Measurement of mass
Metric units of weight include grams and kilograms
Volume measurement
Metric units of capacity include milliliters and liters
Deductive reasoning
Requires moving from an assumption to a conclusion.
example: “It is raining so i need to take my umbrella to school”
Inductive reasoning
Involves examining particular instances to come to some general assumptions. This type of reasoning is informal and intuitive.
example: If I do my homework all this week, I think my mother will take me to the concert on Saturday”
Axiomatic Structure
A mathematical rule, This basic assumption about a system allows theorems to be developed
example: The system could be the points and lines in the plane, then an axiom would be that given any two distinct points in the plane, there is a unique straight line through them.
Range of a set data
The difference between the greatest and the least numbers in the data set. subtract these numbers to find the difference, which is the range.
The mean of data
The average of the data values. To find the mean, add all the data values and then divide this sum by the number of values in the set.
The median of data
The middle value of all the numbers, to find the middle value, list the numbers in order from the least to greatest or from greatest to least. Cross out one value on each end of the list until you reach the middle.
The mode of data
The value or values that appear in a set of data more frequently
Probability
A way of describing the likelihood of a particular outcome.
Basics to probability and statistics
Data collection, sampling, organizing and representing data, interpreting data, assigning probabilities, making inferences
Sample space
Is the set of all possible outcomes of an experiment.
Line plot
represents a set of data by showing how often a piece of data appears in that set
