5.1.2:
5-22: show all work including the adding/subtracting/multiplying/dividing like terms from both sides. keep handwriting neat to see the numbers properly.
5-23: y^2-y^1/x^2-x^1 to find the slope. then plug in data points for x and y with the slope to find the b-value and put it all together at the end. to see if a data point does go on the line, plug the values in for x and y to hopefully get a true statement.
5-24: template: __% of the variability in *dependent variable* can be explained by a linear relationship with *independent variable*. the slope is typically what happens to the y-axis/dependent variable as the x-axis increases by one unit.
5-25: create a rule/linear equation using y=mx+b for both situations and solve together to find the x and y. those points will be where the two situations intersect.
5-26: it’s helpful to put all the data in a table. a proportional relationship is where both variables (x and y) increase and decrease together.
5-27: to find if a point belongs in a line, plug the x-value into the equation. if the y-value matches, then yes, it does. but if they do not, it does not appear on the line.
5.1.3:
5-35: to find the ratio, simply divide the rebound height by the drop height and use that decimal to find a percentage. to figure out the rebound height OF a drop height, simply multiply by the rebound ratio. to figure out the drop height OF a rebound height, simply divide by the rebound ratio. always include units!
5-36: put everything into a table, it makes it much easier to interpret/organize. round to the nearest whole number for units since that is what the original data table put.
5-37: the slope/m is how much the y-values increase by as the x-value increases by one. the y-value of the y-intercept/b is where the line intersects with the y axis.
5-38: ASSOCIATION IS NOT CAUSATION! be creative but also logical with your answers.
5-39: multiply/divide both sides by the same number to get whole numbers (its easier to work with whole numbers). 3^2 is the same thing as 9 by the way.
5:40: an isosceles triangle is a symmetrical triangle. a rectangle is like a square but squished up teehee.
5.2.2
5-66: plug in the term value number for t(n) and solve. if n=a whole, positive number, then that t(n) is a part of the sequence. if it is like a decimal, repeating or negative value, then it is not part of the sequence. when in doubt, use a calculator.
5-67: you can’t have negative term numbers, only negative term value numbers.
5-68: t(n)=mn+t(0) is surprisingly similar to y=mx+b. once you know the m/slope, all you have to do is find the term value number when the term number equals 0. like finding b. then you add it all together to get the equation, it’s really simple.
5-69: exponential growth means that it is not growing in a constant fashion. exponential growth means increasing by a non-fixed amount, like adding 10% of the previous term number value or multiplying it by 1.2 each time. It will be different each time.
5-70: just be smart when turning a residual plot back into a regular old graph. when the little data dots are closer to/on the line, then that means the predictions were accurate when compared to the actual data.
5-71: you learn something new every day: the pythagorean theorem is a^2=the square root of b^2+c^2. i didn’t know that but there you go.
5.2.3
5-77: t(n)=mn-t(0) is very similar to y=mx+b. make a table for everything and remember that n=term number and t(n)=the value of the term number. if you need to find the value of a term number, plug the term number in for n and solve for t(n). don’t mix the two up and remember to add the value of t(0).
5-78: explicit equations are close to y=mx+b and has a constant rate of change, basically linear and also arithmetic. recursive equations are basically asking how to find the next value of the term number based on the previous value of the term number.
5-79: when in doubt make one of those multiplication squares i forgot what they are called. a negative number and a positive number cancel each other out if the absolute values are equal.
5-80: if you are finding out if a value of a term number is in the sequence, set it equal to t(n) and solve the equation. if it’s a whole number, then yes, it is in the sequence, but if not, then it is not in the sequence.
5-81: r-values close to 1 or -1 are strong but r-values farther are weak in terms of association. positive r-values equal a positive association and vice versa. the r^2 value determines the percentage of the variability in the dependent variable that can be explained by the independent variable.
5-82: set two equations equivalent to each other to find out where they intersect (in a situation and in a graph).
5.3.1
5-86: geometric sequences multiply by a constant amount each time. arithmetic sequences add a constant amount each time. a multiplier is what it multiplies by and the common difference is what it adds each time.
5-87: double check addition/multiplication.
5-88: to find the angles, set both equations equal to each other and solve for x. the little circle above the number is not an exponent of 0, it is a degrees symbol.
5-89: to solve for an unknown value, use the information provided and make an equation. the area of a trapezoid is A=½(b^1+b^2)h.
5-90: to fully simplify an exponential expression, make sure there are no parentheses, negative exponents, or zeroes. to divide exponents that are in fraction form, simply subtract the denominator’s value from the numerator’s value.
5-91: input the data point values for x and y and also add in the slope to solve for b and there you have your equation yayyy.
5.3.2
5-102: when you are increasing, the multiplier is 1.*whatever % you are increasing by*. when you are decreasing, the multiplier is 0.*whatever % you are decreasing by*.
5-103: arithmetic=adding constant amounts geometric=multiplying by constant amounts. reminder that arithmetic graphs are linear and geometric graphs are exponential.
5-104: same reminders as 5-91.
5-105: it’s helpful to set all values in an equation to whole numbers. it’s easier to work with whole values. for example, if you have the square root of x, set all values to the exponent value of 2. double check all work!
5-106: to justify a linear association/relationship, it’s helpful to have a scatterplot with a corresponding LSRL, a residual plot, r-values, R^2 values, and an explanation.
5-107: be situational! be realistic! and don’t be like mathias’ mean teacher!
5.3.2
5-108: convert units into the fractions and find the multiplier. unit rate is typically how many of y-value fits/goes into one unit of the x-value (for example, miles as y and time in hours as x, miles per hour mph).
5-109: arithmetic sequences add a constant amount each time and geometric sequences multiply by a constant amount each time. to find the next number in the sequence based on the previous number, it’s essential to have the equation for t(n+1).
5-110: an area model is always helpful. when the two sides of the equation do not match/solve, it’s likely that either you need to double check work or it is x=no solution.
5-111: to find the slope of a line, take two points and y^2-y^1/x^2-x^1 to get the m of the equation. from there, take the b-value and there you go.
5-112: i don’t really have any info for this one, just double check work! and you can’t divide any number by 0, it comes out as undefined.
5-113: area comes in units squared. square area: A=s^2. triangle area: A=½(bxh). trapezoid area: A=½(b^1+b^2)h. yuppers.
5.3.3
5-120: a domain of a sequence is limited to whole, positive integers since there can only be 1st 2nd 3rd 4th and so on terms but not -9.25th and 0.33rd terms. when the x-value in the linear equation is set as the denominator, it can be any number except for 0 because you can’t divide a number by zero, it’s undefined.
5-121: functions and linear equations are allowed to have almost any x-value but sequences are limited to whole/positive integers. x can equal any number as long as it is not tied to a situation where it requires whole and positive integers (such as shirts and cars).
5-122: in recursive equations, you need the previous term value to get to the next term value. in explicit equations, you can plug in a number for y (or x) to see where and if and the value of a number into the sequence.
5-123: remember what the initials stand for.
5-124: don’t forget units!
5-125: if the line passes through the origin (0,0) then there is no y-intercept in the equation (+0).
6.1.1
6–7: make sure to show all work to not make any mistakes! isolate the variable you want to solve for on one side of the equation and make sure they are a whole, singular variable.
6-8: area models are helpful in solving equations and remember that x times x is not 2x, but x to the power of 2.
6-9: when the r-value of an association is closer to 0, then the association is weaker. for example, a 0.9 is strong, and a -0.4 is weak.
6-10: arithmetic sequences add terms constantly, linear. to figure out if a number is part of the sequence, plug it in for t(n) and solve for n. if it is a decimal number or a negative number, it is not a part of the sequence.
6-11: f(x) is a function. double check work always!
6-12: create an equation to solve for unknown variables and show work. don’t forget units.
6.1.2
6-15: make sure to read the question to find out which variable you are solving for in a multi-variable equation.
6-16: write clearly to differentiate between t’s and +’s. when, for example, 5x=0, x=0 because 0 multiplied by any number is always and forever 0.
6-17: set two equations equal to each other to find the POI (point of intersection) and solve for x. then plug x back into the equation and solve to get a true statement (which is the y).
6-18: t(n)=tn+t*0 is similar to y=mx+b. make sure to differentiate between arithmetic sequences and geometric sequences.
6-19: remember what you are scaling by and any transformation like x-x y-y means the shape stays in the same place.
6-20: don’t forget that a POI is a coordinate point. remember scaling and double check work.
6.1.3
6-25: set two equations equal to each other to find the POI and to find the y-value when x= 12 for example, plug in x for 12 and solve for y. don’t forget units.
6-26: when you get a false statement, that means x=no solution. it’s always helpful to use whole numbers (multiply fractions by their denominators on both sides to get whole numbers).
6-27: only one POI exists when they are two linear lines because then they will just continue off into opposite directions after their intersection point.
6-28: association does not mean causation! (will say it till i die).
6-29: use the formula t(n)=t x *multiplier to the power of n* to solve for geometric sequences/ exponential sequences. a simple arithmetic sequence can use the t(n)=tn+t*0
6-30: multiply fractions by their denominators on both sides of the equation to solve easier.
6.2.1
6-49: make sure to define your variables (with units if applicable)
6-50: remember units and don’t make it too difficult.
6-51: isolate the variable you want to solve for and make sure that variable is a singular one. simplify as much as possible (especially with fractions).
6-52: just use common sense and double check work.
6-53: how many weeks are in a year? oh, 52? divide the number of weeks provided by 52 to find out how many years are in those amount of weeks.
6-54: a multiplier in a sequence means that every other number will be negative (and keep switching because a -6 multiplied by -2 is positive 12 and a positive 12 multiplied by -2 is -24)
6.2.2
6-61: the substitution method has one equation that solves for a variable and one equation in standard form. you take the solution from the solved variable equation and plug it in for the same variable in the standard form equation. solve from there and remember units. you can’t just solve for one variable, plug it back into the equation to get the second variable to get the whole answer. the answer has to be a full sentence if its a word problem.
6-62: it’s easier to make a rule first and then fill in the table and graph. refer back to 5.3.2 on how to take two points and find the slope and later the y-value of the y-int.
6-63: geometric sequences multiply by the same number each time, constantly. they have curved or exponential graphs.
6-64: if a variable is solved, you should plug in the solution for the variable in its spot.
6-65: anything to the power of 0 is 1. simplify fully and remember that scientific notation is written as *number less than 10* multiplied by 10 *to the power of an exponent*.
6.1.4
6-38: any let statement has to include units. make sure to show all work when solving equations.
6-39: consider the context of the units and don’t forget to answer in complete sentences if the question is attached to a real life situation.
6-40: show all work and plug it back into the equation to check your answer.
6-41: geometric sequences multiply by a constant amount each time and remember that an explicit sequence follows the formula of y=mx+b.
6-42: perpendicular lines have opposite and inverse slopes.
6-43: when trying to find the length of a line using slope triangles, use the pythagorean theorem- the length of the line squared equals the height and base of the slope triangle squared so basically the length of the line is equivalent to the square root of height/base of slope triangle. use whole numbers (round).
6.2.3
6-73: utilizing a table helps to find where the value is- in between _ and _ would be the answer if the two values would match somewhere in the middle of two numbers (x-values).
6-74: answer in complete sentences when the question is tied to a real life situation and include units. if one variable is solved for, use the substitution method.
6-75: if y=an expression, then adding y to both sides of the equation would make it equivalent since they are still equal to each other.
6-76: show all work and multiply both sides of the equation by the same number to get whole terms which are easier to work with.
6-77: geometric systems require a recursive equation which is basically like y= b x x^n
6-78: show work and use area models if necessary.
6.3.1
6-84: when setting two equations equal to each other and solving for a variable, make sure to plug that variable back into the equation to find the value of the second variable. answers should be in the form of a coordinate point (x,y)
6-85: when a true statement occurs when two equations are set equal to each other, that means there are infinitely many solutions/all real numbers are a part of the solution. this also means that the two equations are the same line.
6-86: when you define your variables, make sure to include units. answer in complete sentences when the question has a real life situation tied to it.
6-87: include units. also, geometric systems have exponential (curved) lines on a graph so the equation has to be recursive.
6-88: units and answer in full sentences.
6-89: simplify as much as possible and remember that scientific notation is a number less than 10 multiplied by ten to the power of an exponent.
6.3.2
answers for solving systems of equations should be written in an (x,y) format. when it is tied to a situation, the answer should be in a full sentence. exponential sequences should have an explicit equation of t(n)= *0th term* x *multiplier**to the power of n*.
6.3.3
use area models when multiplying multivariable terms. functions have exactly one output for each input. when a graph has a very weak association, the r-value is 0 or close to it. use fraction
busters when solving for a variable that is in a fraction form.
6.4.1
define your variables and don’t forget to include units. answers for solving systems of equations should be written in an (x,y) format. utilize area models for simplifying equations with parentheses. there is no solution to a system of equations when it produces a false statement.
ch6 closure
when both variables are solved for, use equal values method. if one variable is solved for, use substitution. if neither variables are solved for (are in standard form) line them up to use elimination. define variables, include units, answer in full sentences. multipliers of sequences merit an exponential equation where it is raised to the power of n in an explicit equation.
7.1.1
when making a congruence statement, there should be a symbol representing the triangle before the angles and make sure that they correspond. you can determine if the triangles are congruent to each other (same angles same size) by using the Pythagorean Theorem or Triangle Angle Sum Theory. scale factors should be in fraction form and geometric sequences should have an exponential graph. when dividing a number by two each time, it will never be zero or a negative number since it represents an asymptote. a true statement produced by a system of equations means that there are infinitely many solutions.
7.1.2
same as the previous lesson’s notes. also, an equation with the same slope but different y intercepts are parallel to each other and will never intersect, meaning that a system of equations will produce a false statement.
7.1.3
a flowchart should include reasoning for each bubble and have a final congruence statement in the last bubble. there should be a triangle symbol before the statement and the angles must correspond. use area models for simplifying expressions and a scientific notation equation should have a value less than 10 multiplied by 10 to the power of an exponent. R^2 percent of the variability in the y can be explained by a linear relationship with the x. also, a r-value farther from zero has a stronger association.
7.1.4
filling out a triangle congruence statement should have corresponding angles. a straight, flat, horizontal line measures to 180 degrees. a recursive equation involves having to find the next term number in the sequence and does not require a zeroth term. an explicit equation is similarly formatted to y=mx+b and follows the same function. an equilateral triangle means all angles are equal to each other. to find the point after rotating the og point 90 degrees, the relationship between the og point and the second point should be a 90 degree angle as well.
7.1.5
reminder that AAA is not a valid congruence condition because even if the angles are the same, the side lengths could be different. don’t forget to put reasoning for each bubble in a flowchart. the answer to a system of equations should be in (x,y) format, show all work (using either the equal values method, substitution, or elimination). if an association is weak/moderate, that’s a sign that there may be a lurking variable leading to the association. association is not causation btw.
7.1.7
the triangle angle sum theorem states that all interior angles of a triangle add up to 180 degrees. to find the length of 2 points, use the pythagorean theorem: y2-y1=x2-x1 square rooted. the midpoint and the other given point on the line is half of the overall length of the line segment between two points. a true statement from a system of equations means that they are the same line and have an infinite number of solutions. also, even if the association/r values between two variables are really strong, because of the upper and lower bounds, they may not be the most accurate.
8.1.2
exponential equations are y=m x a^x. they do not grow in a linear fashion. valid congruence conditions are SSS ASA AAS. don’t forget units when stating area and perimeter (area should be units squared). use pythagorean theorem to find the length of a line segment. when solving systems of equations, the answer should be in an (x,y) format.