Learning and Memory Aides

 

Learning aides for math-- some I have developed as well as some from colleagues.

I am assuming on a lot of these that you have seen examples elsewhere and just need some general instructions or ways to remember steps in working exercises. Click the topic to name to jump to a particular topic.

Absolute Value Equations and Inequalities

AC Method for Factoring

Algebraic Properties

Completing the Square

Conic Section Flowchart

Consecutive Integer Problems

Cost Problems (Mixtures)

Divisibility rules

Exponent Rules

Factoring

Factoring Flowchart

Factoring Guidelines

Functions and relations

Graphing inequalities on a number line

Graphing quadratic functions and parabolas

Imaginary Numbers

Inequalities and absolute value inequalities

Intercepts

Linear Equations

Linear Equations, Forms of

Lines, linear functions, equations, slope

MARFF

Mixture Problems

Mixture Problems Revisited

Order of operations

Parallel and Perpendicular Lines

Properties of Real Numbers

Rate, Time, Distance Problems

Solving equations

Solving word problems

Square Roots

Translating English to Math

Triangle Trigonometry

 

Order of operations

Please Excuse
My Dear
Aunt Sally
People Eat
Mc Donalds
And Smile
People Enjoy
Mountain Dew
At Seven

Parentheses Exponents
Multiplication Division
Addition Subtraction

Generally, work everything from left to right, doing the operations in the order indicated above. Work form the inner-most set of grouping symbols, outward.

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A simple algorithm (set of procedures) for solving equations in 1 variable.

1. Is the equation a proportion?
YES – Cross Multiply
NO  –  Question 2
2. Are there grouping symbols?
YES – Distribute
NO  –  Question 3
3. Can the equation be simplified on either side of the equal sign?
YES – Combine all like terms on either side of the equal sign
NO  –  Question 4
4. Are there variables on both sides of the equal sign?
YES – Isolate all the variable terms to the side with the largest coefficient by
using inverse operations
NO  –  Question 5
5. Are there numbers added or subtracted on the same side as the variable term?
YES – Use the inverse operation to move these numbers to the other side, simplify
and collect like terms
NO  –  Question 6
6. Are there numbers multiplied or divided on the variable?
YES – If multiplied to the variable, then divide to get variable alone and equation is
solved.  If divided on the variable, then multiply to get the variable alone and
equation is solved.
NO  –  Question 7
7. Did the variable term disappear altogether?
YES – Is the equation an identity (true statement), meaning it has an infinite number
of solutions, or is it a contradiction (false statement), meaning the equation has no solution.
NO  –  This should be your answer.
This algorithm was adapted from a web page by Jean Coffman.

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Five Step Plan for Solving Word Problems

adapted from various sources, including Algebra Structure and Method Book 1,
by Brown, Dolciani, et. al. published by Houghton Mifflin, copyright 1994.

Step 1 Read the problem and determine what is unknown. You might make a drawing or table to organize your information. Determine what is asked.
Step 2 Pick one or more variables for the unknown part(s). If you choose only one variable, then you might have to have an expression for other unknown parts.
Step 3 Read the problem again and write an equation that represents the information in the problem or connects the facts.
Step 4 Solve your equation and find the value(s) of the unknown part(s).
Step 5 Check your answer in the equation and make sure that you have answered the question(s) asked in the original problem. Is your answer reasonable?

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English to Math Translation

adapted from someone at St. Charles Community College, St. Peters, Missouri

I apologize, I do not remember which instructor there gave me this.

This statement... often means...

plus

more

more than

added to

increased by

sum

total

sum of

increase of

gain of

addition

minus

less

less than

subtracted from

decreased by

subtract

fewer

difference

take away

loss of

subtraction

product

double

triple

times

of

twice

twice as much

half

multiplication

quotient

divided equally

divided by

divided

per

divided into

goes into

division

is

was

will be

the same as

equals

equal to

yields

results in

are

is equivalent to

equals

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Consecutive Integer Problems

If the problem asks for... then the variables should look like...
Consecutive integers n, n+1, n+2, n+3...
Consecutive evens n, n+2, n+4, n+6,...
Consecutive odds n, n+2, n+4, n+6,...

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Solving Word Problems Involving Mixtures of Solutions

Jump to a video of the seesaw method for solving mixture problems

1 variable method: Make a chart to organize your information and one equation to represent the problem.

 

  Volume of solution % strength Amount of pure substance
Strong      
Weak      
New Mixture      

The top two cells in the first column add up to the bottom cell. The last column is also related by a sum; the first two cells add up to the bottom cell.

Usually, only four cells in the first two columns will be filled with information that will be given to you. One additional cell will have your variable in it, and yet another cell will have a variable expression that you have to develop. The right-hand column is filled in by multiplying each row across. Remember to distribute if there is a sum or difference in any cell!

Your equation to solve is then the right hand column; the top plus the middle equals the bottom.

 

2 variable method: Write two equations to represent the problem.

Pick a variable for each quantitiy that is being combined; usually x for one strength and y for the other, or maybe w for the weak solution and s for the strong one.

One equation is just x + y = the total amount of solution, or w + s = the total.

To get the other equation, you multiply each term of the first equation by the corresponding percent for each solution.

You must solve this resulting system of two linear equations. This can be done in various ways.

 

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Solving Word Problems Involving Mixtures of Products that

Cost Some Given Amounts

1 variable method: Make a chart to organize your information and one equation to represent the problem.

 

  Amount of product unit cost Total cost of product
Product 1      
Product 2      
New Mixture      

The top two cells in the first column add up to the bottom cell. The last column is also related by a sum; the first two cells add up to the bottom cell.

Usually, only four cells in the first two columns will be filled with information that will be given to you. One additional cell will have your variable in it, and yet another cell will have a variable expression that you have to develop. The right-hand column is filled in by multiplying each row across. Remember to distribute if there is a sum or difference in any cell!

Your equation to solve is then the right hand column; the top plus the middle equals the bottom.

 

2 variable method: Write two equations to represent the problem.

Pick a variable for each quantitiy that is being combined; usually x for one product and y for the other, .

One equation is just x + y = the total amount of the mixed products.

To get the other equation, you multiply each term of the first equation by the corresponding cost for each product.

You must solve this resulting system of two linear equations. This can be done in various ways.

 

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Solving Word Problems Involving Rate, Time, and Distance

1 variable method: Make a chart to organize your information and one equation to represent the problem.

  Rate Time Distance
Situation 1      
Situation 2      

Usually, only two cells in the first two columns will be filled with information that will be given to you. One additional cell will have your variable in it, and yet another cell will have a variable expression that you have to develop. The right-hand column is filled in by multiplying each row across. Remember to distribute if there is a sum or difference in any cell!

Your equation to solve involves the right hand column; depending on the situation, they may be set equal, you might have to subtract them and set that difference equal to some given number, you might have to add them and set that sum equal to some given number, etc.. This is where a drawing usually helps.

 

2 variable method: Write two equations to represent the problem.

Usually, you pick a variable for time, t, and one for distance, d.

One equation is often just some rate you are given, times t =d. The other equation is often another rate you are given, times some expression involving t =d.

 

Occasionally, the variables are rates, maybe for wind speed, w, and the speed of an aircraft, p, in no wind. In such cases, the first equation is something like a given time times (p+w) = some given distance, while the second equation is another given time times (p-w) = some other given distance.

You must solve such a resulting system of two linear equations. This can be done in various ways.

 

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Slope of a Line

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What are Intercepts of a Line?

The x intercept is the point (a, 0); it’s where the line crosses the x-axis. The y intercept is (0, b); it’s where the line crosses the y-axis.

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What is a Linear Equation?

linear equations can be written in the form Ax+By=C
no exponents other than 1
no variables in the denominators of fractions
no xy terms

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What are some of the Forms of Linear Equations?

Name Form Details
Standard Form Ax+By=C
x intercept C/A
y intercept C/B
slope -A/B
Slope Intercept Form y=mx+b
slope m
y intercept b
Point Slope Form y-y1=m(x-x1)
slope m
a point on the line (x1,y1)
Point Slope Form (y2-y1)/(x2-x2)=(y-y1)/(x-x1)
a point on the line
another point on the line (x2,y2)
Intercept Form x/a+y/b=1
x intercept a
y intercept b

 

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Parallel and Perpendicular Lines

Parallel lines

Perpendicular lines

Lines have equal slopes. Lines have slopes that are opposites AND reciprocals.
The slopes are the same. If you multiply the slopes, you get –1 as your answer.

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Graphing Quadratic Functions, Parabolas

f(x)=Ax^2+Bx+C f(x)=A(x-h)^2+k
Vertex: (-B/(2A), f(-B/(2A))) vertex: (h,k)
axis of symmetry: x=-B/(2A) axis of symmetry: x=h
x intercepts are found using quadratic formula x intercepts are h plus or minus square root of -k/A
y intercept is at C y intercept is at A(h)^2+k
A is like the attitude: If A is positive, the parabola opens upward, like a smile, Whereas if A is negative, the parabola opens downward, like a frown.
The Vertex is the highest or lowest point on the graph. It is where the graph changes direction.
x intercepts are of the form (a, 0) it’s where the parabola crosses the x-axis.

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A Caution About Solving Inequalities

solve like equations, but remember to change the direction of the inequality if
you multiply or divide by a negative. multiplying or dividing a negative by a
positive doesn’t count-- it’s only when YOU multiply or divide BY a negative
that you change the inequality symbol.

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Graphing Inequalities with a Single Variable on a Number Line

< or > is graphed with an open circle or parentheses ( )
"> or = to" or "< or = to" is graphed with a closed circle or brackets [ ]
IF the variable is on the left side of the inequality symbol, THEN shade the number line in the direction that the inequality symbol is pointing. IF the variable is on the right side, then shade the OPPOSITE direction the symbol is pointing.

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Working with an Equation or Inequality with Absolute Value

If the absolute value is by itself on the left side of the equation, then the statement is an OR statement, a disjunction.
If the absolute value is by itself on the left side of the inequality and the symbol is < or , then the statement is an AND statement, a conjunction. Remember that the symbol points to the left, toward the beginning of the alphabet, where AND is located.
If the absolute value is by itself on the left side of the inequality and the symbol is > or Ž, then the statement is an OR statement, a disjunction. Remember that the symbol points to the right, toward the end of the alphabet, where OR is located.

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Factoring

  advfactor

Advanced Factoring Checklist

1. Is there a GCF?
example: 12x + 18xy = 6x(2+3y)

2.
Are there only 2 terms? See if it’s a difference of two squares, a sum of two cubes, or a difference of two cubes.


examples: 49b2 - 16c2 = (7b + 4c)(7b - 4c)


8x^3+27y^6=(2x+3y^2)(4x^2-6xy^2+9y^4)


64x^12-125y^3=(4x^4-5y)(16x^8+20x^4y+25y^2)

3.
Are there only 3 terms?

3A. If the first or last term is a one, then factor directly.
examples:

x2 - 19x + 60 = (x - 4)(x - 15)

x2 + x -6 = (x + 3)(x - 2)

a2 - 3a - 40 = (a - 8)(a + 5)

a2 + 8a + 16 = (a + 4)(a + 4)

3B. If neither the first nor last term is a one, then MARFF.
examples:
shows marff process example 1

 

shows marff process example 2

4.
Are there only 4 terms? Try to factor by grouping (use RFF).
example: ab - 2b + ac - 2c = b(a - 2) +c(a - 2)
= (a - 2)(b + c)


5.
Check by FOIL!

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MARFF?

What is that MARFF thing? Many books call it the "ac" method.
M multiply multiply the first and last numbers (assuming the trinomial is in descending order)
A add find two numbers that add to the middle number but multiply to the result of the M step
R rewrite and regroup rewrite the first and last terms while regrouping the middle term by using the two numbers from the A step as coefficients
F factor begin factoring by grouping, factor the GCF from the first two terms and the GCF from the last two terms the remaining parts that are written in parentheses should match-- if they don’t you have probably made a mistake
F factor finish factoring by grouping, factor again, copying the binomial that appeared in the parentheses and the GCF’s go in another set of parentheses as the other factor

Click here for another factoring method that is called Red Mustang

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Functions and Relations

A relation is just a set of ordered pairs, (x,y). It can be written in a variety of ways:

 

set, written in set notation, using braces, { }

table of values
mapping, two blobs, each containing numbers with arrows running from each first coordinate to each second coordinate
graph of points representing each ordered pair
an equation relating the numbers in each ordered pair

 

A function is a relation in which each y value is assinged to exactly one x value. There are several ways to think of this:

If no x values are repeated, then the relation must be a function.
If any x value is repeated, it must have the same y value each time in order for the relation to be a function.
Any vertical line will cross the graph of a function in, at most, one place.
No x value is assinged to more than one y value.
Several different y values can't be paired with the same x value.

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Tests for Divisiblity

A number is divisible by... If...
2 it ends in 0, 2, 4, 6, or 8.
3 the sum of all its digits is divisible by 3.
4 the last two digits make a number that is divisible by 4.
5

it ends in 0 or 5.

6 it is divisible by both 2 and 3.
8 the last three digits make a number that is divisible by 8.
9 the sum of all its digits is divisible by 9.
10 it ends in 0.
11 the difference of the sums of alternating digits is divisible by 11.

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Triangle Trigonometry

Sine (sin) opposite over hypotenuse
Cosine (cos) adjacent over hypotenuse
Tangent (tan) opposite over adjacent
Many people remember these above ratios as SOHCAHTOA.
If you can't remember that acronym, try remembering one of these sentences:
Some old hag cracked all her teeth on asparagus.
Some old hag cut all her toes on asphalt.

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Properties of Real Numbers

Think of these as Laws. Much like our own laws, we know not to steal or kill, but probably can't name the exact paragraph of legal code that says certain crimes are illegal. Most of these properties are common sense, since we've dealt with real numbers for so much of our lives. You are about to become a Mathematical Lawyer!
commutative property of addition a+b=b+a you can add any two numbers in either order you want
commutative property of multiplication ab=ba you can multiply any two numbers in either order you want
associative property of addition a+(b+c)=(a+b)+c you can change the grouping of three numbers when you add

associative property of multiplication

a(bc)=(ab)c you can change the grouping of three numbers when you multiply
identity property of addition a+0=0+a=a zero plus a number is that number
identity property of multiplication a(1)=1(a)=a one times a number is that number
inverse propterty of addition a+(-a)=-a+a=0 a number plus its opposite is zero

inverse property of multiplication

a(1/a)=(1/a)a=1 a number times its reciprocal is one
distributive property of multiplication over addition a(b+c)=ab+ac and (a+b)c=ac+bc adding two numbers to get a sum and then multiplying a certain number by that sum is the same as multiplying that certain number by each of those numbers that make up the sum and then adding at the end (this lets you go around order of operations, opting to multiply into the parentheses)

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Exponent Rules and Like Terms

Like terms must have 2 things in common: the same combination of variables and the same combination of exponents on those variables. Only like terms can be added and subtracted (you can multiply or divide just about any monomials or polynomials you want to). When you add or subtract like terms, do not change the exponents. When you add three two door cars and five two door cars, you don't get seven four door cars, you just get seven two door cars! The things you are adding don't change, just the number of those things that you have! when you add like terms, leave the exponents alone when you add like terms, leave the exponents alone
When you multiply two terms (monomials) with the same base, add the exponents. When you write the terms, often the exponents are horizontally across from one another, this may remind you to add the exponentssince you can draw a line between the exponents and then another vertical line to make an addition "+". Remember that the coefficients, the numbers in the front, are still multiplied! when you multiply items with exponents and matching bases, keep the base and add the exponents when you multiply items with exponents and matching bases, keep the base and add the exponents
When you raise a term (monomials) to some power, multiply the exponents. When you write the terms, often the exponents are diagonally arranged from one another, this may remind you to multiply the exponents, since you can draw a line between the exponents and then another diagonal line to make a multiplication "x". Remember that the coefficients, the numbers in the front, are still raised to the power! when you have something raised to a power and then that is raised to a power, keep the base and multiply the exponents when you have something raised to a power and then that is raised to a power, keep the base and multiply the exponents
When you raise anything to the zero power, the answer is 1. The 0 power is telling you that you have none of that base. Consider it to have a coefficient of 1 and then the only thing there is that coefficient. anything raised to the 0 power is 1 anything raised to the 0 power is 1
Any base raised to a negative exponent will lead to a reciprocal of that base, raised to the positive exponent. Think of the negative exponent as a negative attitude. When your mom told you you had a negative attitude, you were probably sent to your room. Let's imagine that the base has a room in the basement. In that case, the negative attitude sends it to the denominator and then it has a positive attitude. when something is raised to a negative exponent, we simplify by taking the reciprocal of the base and changing the sign of the exponent when something is raised to a negative exponent, we simplify by taking the reciprocal of the base and changing the sign of the exponent
When you divide two terms (monomials) with the same base, subtract the exponents. Look at the exponents and see where there are more factors shown. Compare the exponents. If there are more factors in the numerator, then write that base upstairs. If there are more factors in the denominator, then write the base downstairs. Figure out how many more you had in one place than the other in the original problem and that is your new exponent in your answer. Remember that the coefficients, the numbers in the front, are still divided or put in lowest terms! when you have something to a power divided by the same base to another power, we keep the base and subtract the bottom power from the top one when you have something to a power divided by the same base to another power, we keep the base and subtract the bottom power from the top one

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More Factoring Thoughts

FACTORING GUIDELINES

1) GCF

2) TWO TERMS -
DIFFERENCE OF TWO SQUARES
x2 - y2 = (x + y) (x - y)

DIFFERENCE OF CUBES
x3 - y3 = (x -y) (x2 + xy + y2)

SUM OF CUBES
x3 + y3 = (x + y) (x2 - xy + y2)

3) THREE TERMS - (ax2 + bx + c)

FACTOR DIRECTLY IF “a” EQUALS 1
1x2 + bx + c

TRINOMIAL SQUARE PATTERN
x2 ± 2xy + y2 = (x + y) (x +y)
(x - y ) (x - y)

M.A.R.F.F.

4) FOUR TERMS - REGROUP BY PAIRS
( “+” between )

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A Factoring Flowchart

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A flowchart for identifying conic sections

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How to solve a quadratic equation by completing the square

completing the square example 1

completing the square example 2

completing the square example 3

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Working with imaginary numbers
First we need a definition:
square root of -1 is i.
i squared is -1.
i cubed is -i
i to the fourth is 1
To simplify any power of i larger than 4, just take the exponent and divide it by 4. Determine the remainder from dividing.
If the remainder is 0, then i to the original power is 1. (Remember, anything to the 0 power is 1.)
If the remainder is 1, then i to the original power is i.
If the remainder is 2, then i to the original power is -1.
If the remainder is 3, then i to the original power is -i.Whenever you have to simplify the square root of a negative number, you will get a result that is not real-- it is imaginary or complex. Follow the horror film rule-- always pull the i’s out first.
For example, change
root -8 to
root -1 root 8 to
i root 8
i root 4 root 2
i 2 root 2
2i root 2

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Simplifying square roots.
Look for the largest perfect square you can find that is a factor of the number inside the radical (root symbol).
Make a small factor tree from this original square root, having the square root of the perfect square you found times a square root of the other factor. Simplify the square root of the perfect square and place that result in front of the other square root.

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Solving a mixture problem using one variable and pictures
1. Draw a bucket for each solution being mixed or gained.
2. In each bucket, write the amount used or gained, including a variable or variable expression, as appropriate.
3. Below each bucket, label the percent strength for the corresponding solution. You can use a percent or decimal, just be consistent throughout the problem.
4. For each buckiet, multiply the amount times the percent or decimal and write that result on the next line of the problem.
5. Solve this resulting equation.

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