Weeks 1-4

Subtracting in Bases:Exchange when you don't have enough unitsUse the wording of "take away"For typing purposes- values taken away will be represented as non-bolded25six - 12six = 13six|| ***** 34six - 15six = 15six||(******) ****

Converting to Base 10:When converting a number in a base other than ten to base ten, we are simply figuring out how many units are in the number.Show: Show means we must diagram the process we take to get to the answer.24five to base ten = 14||**** Each long is equivalent to 5 because that is the base we are in5 + 5 + 4 = 14Solve: This method may include the diagram if it is helpful, but requires grouping (multiplication)25seven to base ten = 19 Each long is equivalent to seven because it is the base||*****2(7) + 5(1) = 14 + 5 = 19Converting From Base 10:For small numbers we can simply count up all of our units and group them into longs and flats as they appear.For larger numbers, this can be difficult, so we do upside down division.We figure out how many longs will make up our number by using division.37 to base 4: 4|_37 r 4|_9 1 2 1Answer reads from the bottom number up the remainders in order. 211four

Adding Numbers in Different Bases:When you add numbers in bases other than base ten, you follow the same protocol as normal addition.The only difference is if the number that results from the addition has a digit larger than the base.At that point, you must then exchange it for the next value up, whether it would be a long, flat, or cube.Ex:12four + 10four = 22four| ** + | = ||**57nine + 28nine = 86nine|||||******* + ||******** = ||||||||******Scratch Method:Line up numbers verticallyAdd each digit in the ones place, add until a long is formed, and then place a scratch on it. Continue with all columns. (Italices are equivalent to a number being scratched out for typing purposes) 1 2 25six 12 1 4 + 23 3 1 1 3 six

Problem Solving: Look for a Pattern (Math is all about Patterns)Do a similar/easier problem (Identify what aspect you struggle with by isolating each aspect)Identify what makes the problem hard, and get rid of the hard part (This can be done using the mechanics in option 2)Draw a diagram (Not a picture)Guess and Check (It's okay to be wrong sometimes)Write an EquationUnDevCarLo:These are the steps we use to solve a problemUnderstand the problemDevelop a plan to solve itCarry out your planLook back at your answer

Place Value:When looking at a multi-digit number, we look at each digit as a number with a specific place value.We commonly work in Base 10, which means we have 9 digits before we have to add a second digit and a place holder.Once we have multiple digits, we can identify the different place values using blocks.In our blocks, we have small cubes which are equal to one unit each. These units will represent the digit in the ones place of the number.We also have long rods which would represent 10. These are called "longs" and represent the number in the tens place of a number.We also have squares, which represent 100. These are called "flats" and they represent the number in the hundreds place of the number.Finally, we have cubes. These cubes represent 1,000. They represent any digit in the thousands place of a number.After we reach the point of a cube, it becomes our new unit and the pattern of Flat, Long, Unit continues, and we use commas to reflect that the next "unit" is actually a cube.

Introduction to Bases:We most often work out of base 10, which means that one "long" as mentioned before, contains 10 units. The amount of units it takes to make a long is where we identify what base we are in.Example: Base Seven would mean that one long contains 7 units, and one flat contains 49 units. The base you are in is written in word form as a subscript. It is not needed to write the base when you are in base ten.Diagramming Numbers in Different Bases:We can write numbers in different bases the same as we would in base 10: the digit in hundreds place represents how many flats we have, the tens place represents how many longs we have, and the digit in the ones place represents units. For the purpose of notes...D = flat| = long* = unitExample: 23seven = | | *245six = D D |||| *****

Greatest Common FactorSmall #'sFactor: The different #'s that when multiplied give you the number you are factoring30 : 5 x 6 12: 6 x 2 . GCF: 63 x 10 1 x 121 x 30 3 x 42 x 15Least Common MultipleLarge numbers12 : 24, 36, 48, 6030: 60LCM: 60Factors:Prime- Only two factors (one and itself)eg... 2, 3, 5, 7Composite- more than two factorseg... 4, 6, 8, 12, 24Prime Factorization:Tree:12^2 6^2 312: 2^2 x 3Break up the factors of a number until you get down to only prime numbersUpside Down Division: 2| 18 3| 9 32 x 3^2Repeat the tree process, just in a different formGCF Using Prime Factorization: List all of the common factors using the value with the smallest exponent. Multiply them to receive your solutionLCM Using Prime Factorization: List all of the factors for both numbers. Use the values with the larger exponents. Multiply them to get final solution.

Integers:Models: + represents a positive number and - represents a negative numbera + and - stacked represent a zero pairmultiple zero pairs is a zero bankModel:Addition3 + (-4) = +++ - - - - = -1Subtraction (Underline = take away)5 - 2 = + + + + + = 3Multiplication3(2) = + + + + + + = 6Diagram:Addition-23 + -74 - -- add - 23 +74 = 97 -97SubtractionKeep Change Change: Keep the first term, change the sign to its opposite and the final term to its opposite.MultiplicationDifferent signs = Difference (Value is negative)Same signs = (Value is positive)

Alternative Algorithms:Lattice Addition: Line the numbers up vertically. Place a box around where the answer would go and add diagonals to the box. add each digit and place the solution in the box with each place value split by the diagonal. Add down the diagonal to get final solution.Lattice Multiplication: Place one number of the top of a box and one on the right side vertically. Divide the box into small boxes and add diagonal lines. Multiply each number and place the solution in the boxes. Add along the diagonals for final solution.Friendly Number Addition: When adding two numbers, take away some amount from the first number to make the second one a multiple of 10 and then add the numbers.Friendly Number Subtraction: Add the same amount to both terms to make the second number a multiple of 10 and then subtract the number. Left to Right Addition: Add the digits in a multi-digit problem from left to right including their place value. This is a modified expanded form. Area Model: Write one number in expanded form on top and the other in expanded form on the right side. Multiply each section of the expanded number to the corresponding sections from the other number and add all solutions to receive final solution.Scratch: Add each number in a sequence until you get to 10. Place a slash and write how many remain. Write the final remainder at the bottom and carry the number of slashes as a value into the next column. Repeat until a final solution is received.Expanded form: Can be done for +, -, and x143 = 100 +40 +3221 = 200 + 20 +1solution = 300 +80 + 4 = 384

Properties of Addition and Multiplication:Commutative: States that the order of terms in a series of repeated addition or multiplication doesn't matter.3 + 4 + 6 = 6 + 3 + 4 3 x 4 x 6 = 6 x 3 x 4Associative: You may group any terms within a series of repeated addition or multiplication.3 + 2 + 5 = (3 +2) + 5 = 3 + (2 + 5)Distributive: When you have a term being multiplied by a binomial, you can multiply the outside term by each of the inside terms.3 ( 10 + 4 ) = 3(10) + 3(4)Identity Property: Identifies the value that keeps the number the same (Plus and Minus = 0, Multiplication and Division = 1)5 + 0 = 55 - 0 = 55 x 1 = 55/1 = 5

File Types and TechniquesFile Type: What the problem isMultiplying same baseDivide same baseRaise a power to a powerTechnique: How you handle the exponents (examples correspond with the order above)Add exponentsSubtract exponentsMultiply exponentsShow: Draw out each of the values in the problem so that the largest exponent is one.Negative exponents move to the opposite side of the fraction barExamples of each type of problem:Multiply same base (add exponents): X^2 * X^3 = X^5Divide same base (subtract exponents): X^4/X^2 = X^2Raise a power to a power (multiply exponents): (X^2)^2 = X^4

Order of OperationsDO NOT USE PEMDAS!G (Groups)E (Exponents)M/D (Multiplication/Division Left to Right)A/S (Add/Subtract Left to Right)Finding Groups:Place a group anywhere there is an addition or subtraction symbol between terms (sometimes there will be adding and subtracting within a larger group and that is okay, the larger group is where that group boundary is)Remember: Unless a negative is inside (), the exponent does not apply to it

Square rootOpposite of exponentPerfect Square: When a square root comes out to a whole numberSquare root of 36 = 6Square root of 25 = 5For anything that isn't a perfect square, we use perfect squares to estimate them.Estimate the square root of 53Find the closest perfect square below and above itFigure out which one it is closest to using a number line and/or reasoning skillsUse the whole number for the one below it and add a decimal explained belowUse that relationship to estimate a tenths place decimal value based on how close it is to the two perfect squares.

PercentsTeach students to do multiples of 10 and 5 in their head10% = Move the decimal over 1Multiples of 10 = take the value from 10% and multiply it by the number in the tens place of this number30% of 50 = 10 % of 50 (5) * 3 = 155% and multiples of 5= take half of the 10% value and add it to the total of 10% and the multiple of tenHarder Numbers:37% of 47310% = 47.31% = 4.7337% = 4.73 X 37

FractionsWe look for a common denominator so we can have the same size piecesNumerator: How many piecesDenominator: What size piecesHow to find common denominator: Factor each denominator, multiply each by the factors that are missingTo multiply fractions: cross multiply within the problem to simplify and then multiply straight across25- 3/8 = 24 8/8 - 3/8 = 24 5/813 + 5/11 = 13 5/1114 3/7 + 30 2/7 = (14 + 30) AND 3/7 + 2/7 = 44 5/742 3/5 - 20 2/5 = (42-20) AND 3/5 - 2/5 = 22 1/5Comparing Fractions: Which is biggerIf the denominators are the same, look for the bigger numeratorIf numerators are the same, look for the smaller denominatorLook at the number of wholes in front of the fraction and if one whole is bigger than the other, the fractions are irrelevantIf the amount of pieces missing is the same, look for the smaller denomiatorCompare to an anchor fraction such as 1/2Modeling: Draw a rectangle that has the amount of pieces stated in the denominatorShade in the amount in the numeratorWhen adding, subtracting, or multiplying, add additional lines in the opposite direction for the second fraction Operations with modeling:Adding will use 3 boxesSubtracting will use 2 boxesMultiplying will use 1 box

Absolute ValueDistance away from a certain marker (Usually 0)Can't be negative | ||-3| = 3| 5| = 5| 3- 5| = | -2| = 2When there is an X inside the absolute value, split it into two for a positive and negative answer| x - 2 | = 6x - 2 = 6 x - 2 = - 6x = 8 x = -4

Rational and Irrational NumbersRational Numbers: Can be written as a fraction (Decimal can repeat)Irrational Numbers:Anything that isn't a perfect squarePiDecimal that never ends or repeats

Quadratic EquationsThe way that we multiply binomials is the same way we multiply any numbers, we make an area model.Steps:We place each term along either the top or side of the rectangle.We divide the rectangle at the + and - signsWe multiply the terms on the top and side that correspond with the portion of the box we are looking at.We combine like terms to get the final equationFactoring Steps:We create the area model rectangleWe place the first term in the top left boxWe place the last term in the bottom right boxWe do a diamond problem to find the remaining two boxesThe product of the first and last term goes on topThe middle term goes on the bottomWe find the two terms for the wings that add to give us the bottom number and multiply to give us the top number.We find the values that multiply together to give us each value in the corresponding box and place them on the outside where a binomial would goWe rewrite those binomials in this form (x + 3) (x - 4)