Math

unknown addend

"I have 5 pieces of fruit in my fridge. Two are apples, and the rest are oranges. How many are oranges?" -goal is to have students visualize 2+?=5 put-together/take apart

Fractions

# expressed as 1 integer written above another integer - represents the quotient of 2 numbers "x divided by y." top# is numerator, part being considered bottom #is denominator undefined if denominator is 0

Perfect square

# that has an integer for its square root 10 from 1-100 1, 4, 16, 25, 36, 49, 64, 81, 100

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2 midpoint of 2 points

Linear equation

an equation in which variables only appear by themselves: not multiplied together, not with exponents other than one, and not inside absolute value signs or other functions x + 1 - 3x = 5 - x is a linear equation x^2 - 5 = 3x NOT a linear equation x + xy =5 not an equation because 2 variables are multiplied together

Linear equations

ax + b = 0. *a not = to 0 I.e. 5x +10 = 0 *solution is called a root

building number sense

understanding there is more than one way to solve a problem -asking students to make their calculations mentally and rely on their reasoning ability. -having class discussion about solutions the students found using their minds only and comparing the different approaches to solve the problem. have students explain their reasoning in their own words -modeling the different ideas by tracking them on the board as the discussion progresses -presenting the problem to the students that can have more than one answer

Parenthesis

used to designate which operations should be done first in multiple operations 4 - (2+1) = 1

constant of proportionality

when 2 quants have a proportional relationship, there exists a constant of proportionality between the quantities. -the product of this constant and one of the quantities is equal to the other quantity. If one lemon costs .25, and 2 lemons cost .50... the constant of proportionality is the unit price of .25/lemon

Solving equations with absolute values

1. isolate the absolute value term 2. consider whether the absolute value is positive or negative POSITIVE: I 2x-1 I + x =5 1. I 2x-1 I = -x +5 2x= -x +6 3x = 6 x =2 NEGATIVE I 2x-1 I + x =5 1. I 2x-1 I = - (-x +5) x= -4 BOTH solutions

Inequalities

Algebraic statements that have ≠, <, >, ≤, or ≥ as their symbols of comparison Conditional: those with certain values for the variable that make the condition true and other values for the variable where the condition will be false Absolute: have any real # as the value for the variable to make the condition, where there is no real number value for the variable that will make it false

Types of fractions

Equivalent fractions- have the same value but expressed differently 2/10, 3/15, 4/20 Proper fraction- denominator is greater than numerator (values less than 1) Improper fraction- numerator greater than denominator (greater than 1) Mixed #- a number that contains both an integer and a fraction.

Solving One-Variable Linear Equations

-Multiply all terms by the lowest common denominator to eliminate any fractions -Look for addition or subtraction to undo so you can isolate the variable on one side of the = sign -Divide both sides by the coefficient of the variable -When you have a value for the variable, substitute this value into the original equation to make sure you have a true equation

Dividing Polynomials

-Set up a long division problem, dividing a polynomial by either a monomial or another polynomial of equal or lesser degree By monomial: divide each term of the polynomial by the monomial

Dividing polynomial by a polynomial

-arrange terms of each polynomial in order of one variable - divide the first term of the dividend by the 1st term of the divisor -multiply 1st term of the quotient of the entire divisor and subtract that product from the dividend -repeat for the 2nd and successive terms until you either get a remainder of a zero quotient + remainder/divisor

Number line

-graph to show distance between numbers -each dashed line between 2 whole numbers is 1/4

Least Common Multiple

-integer increments of a given factor Multiples of 7: 1x7=7 2x7=14 3x7=21 4x7=28 5x7=35 LCM- the smallest number that is a multiple of 2 or more numbers Multiples of 3- 3, 6, 9, 12, 15... Multiples of 5- 5, 10, 15, 20 LCM of 3 and 5 is 15x

Isolating variables

-manipulate the equation so that the variable appears by itself on one side of the equation

Equation with square root example

-x + 2 √x+5 + 1 = 3 1. 2√x+5 = x + 2, square each side 2. 4 (x+5) = x^2 + 4x + 4 20= x^2 + 4 x^2= 16 x= 4 or -4 -4 spurious solution, 4 is solution

Graphing Inequalities in 2 variables

1. Graph the border of the inequality (graphing the equation that we get if we replace the inequality sign with an =) 2. If strict- dashed or dotted line If not strict- solid line 3. test any point not on the border to see if it satisfies the inequality, if it does shade that side. Y > 2x + 2 1.Strict inequality- so we use a dashed line Slope is 2 over 1. First graph +2, then rise 2 and run 1 2. Chose a test point (0,0) 0 > 2(0) + 2 0 > 2 *this is not true, so we shade in the side of the border that does not include the point (0,0)

Graphing Compound Inequalities in 2 variables

1. graph each of the component inequalities *AND: shade in only the parts where they overlap OR: shade in any region that pertains to either of the individual inequalities

Solving inequalities with absolute values

1. isolate the term with the absolute value 2. proceed to treat the 2 cases separately as with an absolute value equation, but flipping the inequality in the case where the depression in the absolute value is negative 3. combined into a compound inequality, if the absolute value is on the greater side of the inequality then it is an or compound if it is on the lesser side, it is an and 2 + Ix-1I ≥ 3 x-1 ≥ 1 and x-1 ≤ -1 (inequality is flipped) x ≥ 2 and x ≤ 0 "x ≤ 0 or x ≥ 2"

Solving equations involving roots

1. isolate the term with the root if possible 2. raise both sides of the equation to the appropriate power to eliminate it 2 √x+1 -1 =3 2 √x+1 = 4, divide by 2 √x+1 = 2 x+1 = 4, square both to remove radical x= 3

Solving equations with roots

1. isolate the variable with the exponent 2. take the appropriate root of both sides to eliminate the exponent 2x^3 + 17 = 5x^3 - 7, subtract 5x^3 -3x^3 + 17 =-7 -3x^3 = -24 x^3 = 8, divide by -3 x = 2

Linear equations graphically

1. plot both equations on the same graph *solution is where both lines cross, if they do not cross (parallel) there is no solution *most useful if the equations are in slope-intercept form

Decimals to fractions

1. put the decimal in the numerator with 1 in the denominator 2. multiply the numerator and denominator by tens until there are no more decimal places 3. simplify the fraction to lowest terms 0.24= 0.24/1= 0.24x100 / 1 x 100 = 24/100 = 6/25

Percentages

3 components- whole (W) part (P) and percentage P= W x :/. :/. = P/W W= P/ :/. I.e. In a school cafeteria, 7 students chose pizza, 9 chose hamburgers, and 4 chose tacos. What percentage of students chose tacos? Whole= 20 (7+9+4) Part= 4 4/20 = 20/100 = 20 percent

Properties of exponents

37^0= 1 1^30= 1 2^3 x 2^4 x 2^x = 2^(3+4+x)= 2(7+x) (3^x) ^3= 3^3x (12/3) ^2= 4^2= 16

Place value

4,546.09 4- thousands 5- hundreds 4- tens 6- ones 0- tenths 9- hundredths four thousand five hundred forty-six and nine hundredths

Subtraction with regrouping

525 -189 ________ 1. - 5 2 5 1 8 9 2. 5 1 15 - 1 8 9 -------------- 6 3. 4 11 15 - 1 8 9 _________________ 3 3 6

Graphing inequalities with absolute values

Absolute values: 1. convert it to a compound inequality 2. graph normally I x +1 I ≥ 4 can be rewritten as x ≥ 3 OR x≤ -5 *Or is Outside

Operations on radical expressions

Add/subtract: -find the common denominator -rewrite each fraction as an equivalent fraction with the common denominator -add or subtract the numerator of the answer, and keep the common denominator as the denominator of the answer Multiply: factor each polynomial and cancel like factors -multiply all remaining factors in the numerator to get the numerator of the product -multiply all the remaining factors to get the denominator of the product Divide: take the reciprocal of the divisor and multiply by the dividend

Applied/Pure mathematics

Applied: application of math to natural science, engineering, medicine, and social sciences -inspires and makes use of new mathematical discoveries and sometimes leads to the develop of new disciplines Pure: mathematics for its own sake, with no application in mind

Elimination

Begin by rewriting both equations in standard form Ax + By = C 5x + 6y = 4 x + 2y = 4 If we multiply the 2nd equation by -3, we can eliminate the Y terms. (6 and -6 cancel out) -3x -6y = -12 (cancels out w/ the positive 6y in the first equation) 5x = 4 -3x = -12 __________________ 2x= -8 x= -4 Substitute this value back into 1st equation and solve for Y 5 (-4) + 6y =4 -20 + 6y =4 6y = 24 Y= 4 Solution: (-4, 4)

Rational numbers

Can be expressed as ratio or fraction *rational if and only if it can be represented by a fraction a/b where a and b are integers and b does not = 0.

Common denominator

Common denominator is the LCM of the 2 original denominators 3/4 and 5/6= LCM of 4 and 6 is 12 9/12 and 10/12

Adding and subtracting fractions

Common denominator: can be added or subtracted by simply adding or subtracting the numerators MUST have same denominator 1/2 + 1/4 = 2/4 + 1/4 = 3/4

Classification of #'s cont.

Decimal #- any # that uses a decimal point to show the part of the # that is less than 1 Decimal point- a symbol used to separate the ones place from the tenths in decimals or dollars Decimal place- position of a number to the right of a decimal (0.123- 1 in the 1st place to the right of the decimal, indicating tenths, 2 is the hundredths, and 3 is thousandths) Rational #s- all integers, decimals, and fractions. Any terminating or repeating decimal # is a rational #. Irrational #s- can not be written as fractions or decimals because the # of decimal places is infinite and there is no recurring patterns of digits within the #. I.e. pi Real- set of all rational and irrational numbers

Adding and subtracting decimals

Decimal points must always be aligned 4.5 + 2 = 6.5

Converting between percentages, fractions, and decimals

Decimal to percentage- move decimal TWO places to the RIGHT Percentage to decimal- move it TWO places to the LEFT *percent is always larger than equivalent decimal number 0.23 = 23 percent 700 percent= 7.00 5.34= 534 percent 0.007= 0.7 percent Fraction to decimal- divide numerator by denominator in the fraction Decimal to fraction- put decimal in the numerator with 1 in the denominator, multiply the numerator and the denominator by tens until there are no more decimal places. 0.24 = 0.24/1 = 0.24 x 100/ 1 x 100 = 24/100 = 6/25 *Fractions can be converted into a percentage by finding equivalent fractions with a denominator of 100 7/10 = 70/100 = 70 percent To convert a percentage to a fraction, divide the percentage number by 100, and reduce the fraction to the simplest terms. 60 percent= 60/100 = 3/5

Dividing decimals

Divisor must be converted to a whole number (in the problem 14/7, 7 is the divisor) -move the decimal to the right until the divisor is a whole #. then move the dividend the same number of decimal places 4.9 into 24.5 would become 49 into 245 245/49 = 5

Factors and GCF

Factors- numbers that are multiplied together to obtain a product Common factor- a number that divides exactly into 2 or more numbers Factors of 12: 1, 2, 3, 4, 6, and 12 Factors of 15: 1, 3, 5, 15 Common factors: 1 and 3 Prime- also a prime number Prime factors of 12: 2, 3 15: 3, and 5 GCF: largest number that is a factor of 2 or more numbers GCF of 12 and 15 = 3

FOIL method

First, Outside, Inside, Last Multiplying (AX + BY)(CX + DY) 1. First terms- ACX^2 2. AX x DY- ADXY 3. BY x CX- BCXY 4. BY x DY- BDY^2

Lines on a graph

If slope is POSITIVE- line slopes upward from left to right if NEGATIVE- line slopes downward from left to right If y-coordinates are the same for 2 points on a line- slope is 0 and the line is horizontal If x-coordinates are the same for 2 points, there is no slope and line is vertical Two or more lines with same slope- parallel Perpendicular lines- slopes with negative reciprocals of each other. a/b and -b/a

Evaluating student solutions

Important concepts: sufficient instruction Procedures: using appropriate procedures when faced w/ various tasks and they are executed correctly Vocabulary: ensure students are able to read, understand, and communicate their mathematics thoughts at age and grade appropriate levels

Flipping inequality signs

Inequality sides can be reversed by swapping the 2 sides and changing the inequality sign x + 2 ≥ 2x -3 is the same as 2x-3 ≤ x +2 *only flipped when multiplied/divided by a negative number -2x ≤ 6, divide by -2 x≥ -3

One-Variable Linear Equations example

Kim's savings: X Y 2 1300 5 2050 9 3050 11 3550 16 4800 A constant rate of change, or slope, of 250. Y2 (2050) - Y1 (1300) = 750 X2 (5) - X1 (2) = / 3 _______________________________ =250 Y = mx + b Y= 250x + b *subsitute X and Y values into the equation (2, 1300) 1300 = 250(2) + b B= 800 Savings equation: 250x + 800

Small/Whole group instruction

Less complex tasks- whole-group instruction, in which the entire class is introduced to a new concept together -includes a connection to prior knowledge, direct instruction, and some forms of media More complex tasks- small-group instruction, students are grouped on

Polynomials

Monomials: a single variable or product of constants and variables I.e. X, 2x, or 2/x *never addition/subtraction symbols Polynomials: algebraic expressions which use addition and subtraction to combine two or more monomials Degree of a monomial is the sum of the exponents of the variables Degree of a polynomial is the highest degree of any individual term

Multiplying and Dividing fractions

Multiply numerators AND denominators 1/3 x 2/3 =2/9 Dividing- flip the numerator and denominator of the 2nd fraction and then proceed as if it were multiplication 2/3 divided by 3/4 = 2/3 x 4/3= 8/9

Multiplying decimals

Multiplying- work as though the numbers were whole rather than decimals -count the # of places to the right of the decimal in both the multiplicand and multiplier -count the number of places from the right of the product and place the decimal in that position 12.3 x 2.56 = 123 x 256 = 31488 3 decimal places, move to the left 31.488

Classifications of numbers

Numbers- basic building blocks of mathematics Integers- any positive or negative number, including 0. Prime- any whole number greater than 1 that has only 2 factors, itself 1. *can only be divided evenly only by 1 and itself Composite- any number greater than 1 that has more than 2 different factors *any whole number, thetas not prime (8 has the factors of 2, 4, 6, 8) Even- can be divided by 2 Odd- can not be divided by 2

Slope

On a graph with 2 points (x1, y1) and (x2, y2) the slope is found with the formula m= y2 - y1 / x2 - x1 (m = slope) If value is POSITIVE- line is UPWARD if value is NEGATIVE- line is DOWNWARD Given 2 points from a graph (3,7) and (1,3)= 7-3 =4 3-1 =2 Slope = 2

Systems of Equations

Set of simultaneous equations that all use the same variables *a solution to a system of equations must be true for each equation in the system Consistent: at least 1 solution Inconsistent: no solutions

Patterns for factoring

Perfect trinomial squares: x^2 + 2xy = y^2 = (x + y)^2 OR x^2 - 2xy = y^2 = (x - y)^2 Difference between squares: x^2 - y^2 (x+y) (x-y) Sum of 2 squares: x^3 + y^3 = (x+y) (x^2 - xy + y^2) Difference between 2 cubes: x^3 -y^3 = (x-y) (x^2 + xy + y^2) Perfect cubes: x^3 + 3x^2y + 3xy^2 + y^3 = (x + y)^3

Kindergarten concepts introduced before #s

Position- top, middle, below, before, after, between, under Visual attributes- same and different colors, shapes, sizes, identifying items out-of-place Sorting- by size, color, type, or shape, identifying an equal number, more, or fewer Graphing- the use of picture graphs and using data from graphs Patterns- identifying, copying, extending, and making patterns, finding patters that are like or different, making predictions from patterns Measurements- longer and shorter, how much they weigh, heavier and lighter, how much an item can hold

Format of graphs

Quadrant I- X and Y are greater than 0 Quadrant II- X is less than 0 and Y is greater than 0 Quadrant III- X and Y are less than 0 Quadrant IV- Xis greater than 0 and Y is less than 0

Graphing Simple and Compound Inequalities

Simple: 1. mark on the number line the value that signifies the end point of the inequality 2. is strict - hollow circle *is not strict (less than or equal to)- solid circle Less than- LEFT greater than- RIGHT Compound: 1. plot the endpoints of each inequality on the number line OR: fill in the appropriate side of the line for each inequality *do not overlap, shaded part is OUTSIDE the 2 points AND: we fill in the part of the line that meets both inequalities

Sets of linear equations

Solution set- all of the solutions of an equation Empty set- no true solutions Equivalent equations- identical solution sets Identity- a term whose value is = 1

solving compound inequalities

Solve each part seperately x + 1 ≤ 2 or x +1 ≥ 3 x ≤ 1 or x ≥ 2 1 ≤ 2x ≤ 6, divide each term by 2 1/2 ≤ x ≤ 3

Equations with no solution

Some types of non-linear equation, such as equations involving squares variables, may have no solution. x^2 = -2 has NO solutions in the real numbers because the square of any real number must be POSITIVE IXI = -1 has no solution, because the absolute value of a number is always positive 2(x +3) + x = 3x 3x + 6 = 3x 3x both cancel, leaving 6 = 0 *whenever the variable terms in an equation cancel leaving different constants on both sides, it means that the equation has no solution *if the terms are the same, the equation has infinitely many solutions

Standard Form

Standard: Ax + By = C Slope= -A/B Y-intercept = C/B

Substitution

System of equations: x + 6y =15 3x- 12y =18 1. Solve 1st equation for X X= 15-6y 2. Substitute this value of X into 2nd equation 3 (15-6y) -12y = 18 45 - 18y -12y = 18 30y = 27 Y= 27/30... 9/10... 0.9 3. Plug this value of Y back into the 1st equation X + 6(0.9) = 15 X + 5.4 = 15 X= 9.6 Solution= (9.6, 0.9)

Solving inequalities

The same as =, but reverse inequality sign if you divide or multiply by a negative number To determine if a coordinate is a solution of an inequality, substitute he values into the inequality, simplify, and check if the statement still holds true (-2, 4) a solution to the inequality y ≥ -2x +3? 4 ≥ -2(-2) +3 4 ≥ 7 NOT a solution

Work/unit rate

Unit rate: a quantity of one thing in terms of one unit of another * denominator of a unit rate is always 1, I.e. the unit rate of traveling 30 miles in 2 hours is 15 mph

Graphing equations w/ 2 variables

Y= 2x-1 *linear equation because each variables are only raised to the 1st power 1. substitute 0 for X to plot Y 2 (0) - 1 = -1 Plot -1 on the Y coordinate 2. Substitute another value to discover another point, linear equations only need 2 points. 2 (2) - 1= 3 3. Draw a line between the points (0, -1) and (2, 3)

Ratios

a comparison of 2 quantities in a particular order I.e. If there are 14 computers in a lab, and a class has 20 students, there is a student to computer ration of 20:14. 20:14, reduced to smallest whole number relation would be 10:7

Tenth digits

a digit in one place represents TEN TIMES what it represents in the place to its RIGHT ONE-TENTH what it represents in the place to its LEFT

Operations

a mathematical process that takes some values as input and produces an output addition- increases the value of one quantity by the value of another quantity (both called addends) *result is a sum subtraction- decreases the value of 1 quantity (the minuend) by the value of another (subtrahend) *result is difference multiplication- one number (the multiplier) indicates how many times to add the other number (multiplicand) *result is a product *odd number of negative factors=negative *even amount= positive division- one number (the divisor) tells us how many parts to divide the other number (the dividend) into *result is a quotient

Absolute value

a numbers distance away from zero on a number line -always positive I3I

Combining like terms

adding or subtracting like terms with the same variable - and therefore reducing sets of like terms to a single term. 2 (x +3) + 3 (2 + x + 3) = -4 2 (x +3) + 3 (5+ x) = -4 2x +6 + 15 + 3x = -4 5x + 21 = -4 5x = -25 x = -5

Compound inequalities

an equality with two or more inequalities joined together with either "and" or "or" AND: written more compactly by having one inequality on each side of the common part 2x ≥ 1 and 2x ≤ 6, can be written as 1 ≤ 2x ≤ 6 *the two parts of an AND inequality must overlap, otherwise no numbers satisfy the inequality OR: if the inequalities overlap, all #s satisfy the inequality and such is not meaningful

Quadratic expressions

ax^2 + bx + c Create two sets of binomials Ex. x^2 + 8x + 12 (X+ ) (x+ ) Figure out the factors of the last term (12), and find the one, that when added together, equals the coefficient of the middle term (8) (X+2) (x+6)

Cancelling terms

can be canceled if and only if they EXACTLY match each other * must have same variable raised to the same power with the same coefficient 3x + 2x^2 + 6 = 2x^2 - 6 2x^2 appears on both sides of the equation and can be canceled 3x + 6 = -6 *must be independent terms, and not part of a larger term. I.e. 2 (x +6) and 3 (x+4) the x's can not be canceled because they are part of a larger term

2 variable equations

chose the variable that can be more easily isolated a^2 + 2b = a^3 + b +3 a appears squared and cubed, where b is only to the 1st power so it should be the variable solved for *if both variables are easy to isolate, its best to isolate the independent variable. if the 2 variables are x and y, y is the independent variable

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²] *same as the length of the hypotenuse of a right triangle with 2 given points as endpoints, and 2 sides of the right triangle parallel to the x and y axis

Distance Formula

d=rt D- distance R- rate T- time elapsed Ex: A fish tank measures 5 ft long, 3ft wide, and 3ft tall. Each cubic foot of the tank holds 7.48 gallons of water. The fish tank is filled with water, leaving 6 in of empty space at the top of the tank. How much of empty space is at the top of the tank. How much feet is in the fish tank? 2.5 height (6 in, or half a foot, missing from height) x 5 long x 3 wide= 37.5 cubic feet 37.5 x 7.48= 280.5 gallons of water in the fish tank

Rational expressions

fractions with polynomials in both the numerator and denominator *the value of the polynomial in the denominator can not equal 0 I.e. a denominator of x-3 indicates that the expression is not defined when x = 3

Law of exponents

indicates how many times the base number is to be multiplied by itself Properties of exponents: a^1= a 1^n=1 (#1 raised to any power is =1) a^0 (any # raised to the power of 0 is =1) a^n x a^m= a^n+m (add exponents to multiply powers of the SAME base number) a^n / a^m= a^n-m (subtract exponents to divide powers of the SAME base number) (a^n)^m= a^nxm (when a power is raised to a power, the exponents are multiplied) (axb)^n = a^n x b^n (muliplication and division operations inside parenthesis can be raised to a power) a^-n= 1/a^n (a negative exponent is the same as the reciprocal of a positive exponent)

Proportions

relationship between 2 quantities that dictates how one changes the the other changes Direct: relationship in which a quantity increases by a set amount for every increase in the other quantity, or decreases by the same amount for every decrease in the other quantity I.e. the distance to be traveled and the time required to travel is proportional Inverse- a relationship in which an increase in one quantity is accompanied by the decrease in another I.e. the time required for a car trip decreases as the speed increases

Order of Operations

set of rules that dictates the order in which we must perform each operation in an expression so we will evaluate accurately 5+20/4x(2+3)-6 P: (2+3)=5 E: no exponents, 5+20/4 x 5 -6 MD: 20/4=5, 5x5=25 5+25-6 AS: 5+25=30, 30-6=24

Equations with more than one solution

some types of non-linear equations may have more than one solution x^2 = 4 has the solutions 2 AND -2 IXI = 1 has the solutions 1 AND -1 I.e. 2 (3x+5) = x + 5(x +2) 6x + 10 = 6x + 10 this solution is TRUE for any value of X, so it has infinitely amount of solutions If both sides of the equation match exactly, it has infinitely many solutions

Like terms

terms in an equation that have the same variable, regardless of whether or not they also have the same coefficient. I.e. x^2 + 3x + 2 = 2x^2 + x - 7 + 2x Like terms: -2 and -7, constants -3x, x, 2x, variable X raised to the 1st power -x^2 and 2x^2, variable X raised to the 2nd power

Intercept form

x/x1 + y/y1 =. 1 where (x1, 0) is the point at which a line intersects the x-axis, and (0, y1) is the point at which the same line intersects the y- axis

Point-slope form

y - y1 = m(x - x1) M= slope (x1, y1) = point of a line

Two point form

y-y₁=(y₂-y₁)/(x₂-x₁)×(x-x₁) (x1, y1) and (x2, y2) are 2 points on a given line

Slope-intercept form

y=mx+b where m is the slope and b is the y-intercept of the line.