K-8 Math for Teachers

The base tells us how many units make a long - how many longs make a flati.e. Base 6 is 6 unites make a long and 6 longs make a flatwritten as a subscriptwe usually work in base 10. 10 units make a long, 10 longs make a flatNo digit in a number should ever be equal or bigger than it's basei.e. 356 base 4 is impossible but 356 base 7 is ok

UnDevCarLoMethod for solving problemsUnderstand the problemDevelop a planCarry out the planLook back to see if your answer make sense

Attempt all the problems in the homeworkCheck the answer key and submit your workAdd a reflection in the comment sectionTests use Respondus LockDown BrowserDownload the secure browser - show ID during each test

Additionuse a long as your 10 framefill the 10 frame with your unitsi.e. 3 + 2 would have 3 units + 2 units = 5 units (you did not fill a long)when you fill a long have students write the results as 1 long and 3 units = 13manipulatives help visualize the place values. 0 holds a place value. If you have flats and no longs, then that place value is 0In non-base 10Remember the base tells you how many units make a longchange to a long when you get to that base number and a flat when you get to the base number in flatsSubtractionTAKE AWAYbuild first number on the board and the second number OFF the boardi.e. 17-9In non-base 10the base tells you how many units make a long12 base 8 take away 6 base 8 would need to exchange a long for units. A long equals 8 units (NOT 10)

squares = flatslines = longsdots = unitsAddition36 + 173 longs, 6 units + 1 long, 7 unitsDraw a line through 10 units to make a new long5 long, 3 unitsif you have enough longs to make a flat, close the 10 longs with a ZSubtractioncircle and arrow what you are taking awayX what you are exchanging36 - 173 longs, 6 units - 1 long, 7 unitsTake away what you cancircle 1 long and arrow it awayX 1 long to exchange for 10 unitsCircle and arrow away 7 unitsYou are left with 1 long and 9 units

What makes a good algorithm?Is it expandable, efficient, and based on solid math principles?Addition Expanded Form237+185200+30+7 + 100+80+5300+120+12=422Left-to-Right237+185add from left to right300+120+12=422Friendly Numbers = numbers that end in 032+58move the 2 from 32 to the 8 from 5830+60= 90Trading Off - like Friendly numbers but only moving part26+67Move 3 from 26 to the 7 in 6723+70=93Subtraction - take awayExpanded Form46-2440 + 6 - (20 + 4)20 + 2 = 22Left-to-right96-35take away left to right90 take away 30 is 606 take away 5 is 161Equal Addends - this works because subtraction is the measurement, amount, distance between two numbers.64-38add +2 to each number, usually best to make the second number the Friendly number (ends in 0) so it's easier to take away.(64+2) - (38+2) = 66 - 40 = 26

3 (4) - 3 groups of 4 unitsBuildmake 3 circles, fill with 4 units eachuse base 10 blocks, set up as a rectangleset a guide base 10 blocks off of your board 3 x 4 would have 3 units on one side and 4 along the topOn the board, Show - drawing, no blocks, flats, longs, units - 3 x 43 circles with 4 units each3 rows of 4 units eacharray model - 10] = long .] = unitmake the lines go all across where your brackets endTeach multiplication facts in this order:1s, 2s, 5s, 10s, 3s, 9s, doubles, 4s, 6s, 7s, 8sExpanded Form - good for keeping place valuesLeft to Right - make sure you keep track of what you already multipliedArea Model - box method, this is my favorite - good for showing place valueLattice - Setup the second number on the right side, not the left like usual.

Build - 12/312 is the number of units - 3 is number of groups OR number of items in each groupnumber of groups is better for numbers under 10number of times in each group is better for numbers bigger than 10Show - 129/25use longs and units to make groups of 255 full groups of 25 plus 14 left overnumber left over becomes your numerator and 25 is the denominator5 14/25Repeated Subtraction good because you can use numbers you are comfortable withDraw a "house" with a long right sidePut your factor on the right and subtract under the houseadd up your factors to get your answerRemainder is the numerator, divisor is your denominator832/7 = 118 6/7Subtraction Area ModelYou can use numbers you're comfortable withwrite the dividend in expanded formdivisor goes on the left649/5600 40 9 write your factor on top of the column and subtract undermove your remainder to the next box and continue. KEEP THE PLACE VALUEAdd the factors at the end to get your answerUpwards DivisionWrite it the way you say it and keeps the denominator at the bottom for the remainderfactors go on the right next to each other37 2/9=41 3/99 goes into 36 4 times, 37-36=1, move the 1 next to the 2 in 3729 goes into 12 1 time, 12-9=3, 3 is your remainder

Divisibility Rules2: even3: sum of the digits can divide by 34: last two digits divide by 45: ends in 5 or 06: #2 and #3 works7: NONE8: last 3 digits can divide by 89: sum of the digits can divide by 910: number ends in 0Prime FactorizationPrime number - only factors are 1 and itself (i.e. 2, 3, 5, 7)Factor Tree48 < 4 x 124< 2 x 212< 3 x 44< 2 x 22x2x2x2x32^4 x 3Upside down division - use an upside down "house", use only prime numbers40 < 5 x 88< 2 x 44< 2x22x2x2x52^3 x 5Use prime factorization to find LCDuse whichever is bigger2^2 x 3 x 5^4 x 11 and 3^3 x 5^2 x 7^2LCD = 2^2 x 3^3 x 5^4 x 7^2 x 11

Numerator: number of pieces you haveDenominator: tells me the size of my pieceInverse relationship17 is bigger than 5 BUT 1/17 is smaller than 1/5 - the pieces are smaller in 1/17 than in 1/5LCD for adding or subtraction fractions but you don't need to for multiplication, why?Multiplication is _____ groups of ______so 2/3 x 3/4 is, 2/3 groups of 3/4, and does not need a LCD

Which is bigger?[3/8] or 3/11 - same number of pieces but the smaller denominator means the pieces are bigger [9/17] or 7/15 - 1/2 anchor fraction, 9 is more than half of 17 and 7 is NOT more than half of 1513/23 or [20/23] - more pieces of 23 - same denominator15/17 or [27/29] - both missing 2 pieces to become whole, 29 is a smaller piece so more of this one is filled in

Area Model circle cut into 4 parts with one section colored in 1/4 of the area of the circle is colored inLinear ModelRectangle cut into 4 parts with one section colored in3/4 of the line are not colored inSet Model groups of items, they do not need to be the same

Build Addition2/5 + 1/2Use two 1/5 tiles and one 1/2 tilePut them next to each other in a lineFind a different tile that fits the to whole like of 2/5 + 1/2In this case 1/10 tiles will give us the least number of tiles.You will use nine 1/10 tiles to equal the 2/5 + 1/2 line.The answer is 2/5 + 1/2 = 9/10 Show AdditionArea Model use rectanglesDraw one less line than your denominator - if you need 4ths, then you will draw 3 lines.I like to make my bigger number vertically since that seems to be easier to draw than the horizontal linesWith addition, you will have three rectangles3/4 + 1/5First rectangle has three lines going horizontal for a total of 4 pieces. Color in 3 parts with red to represent the 3/4Second rectangle has 4 lines going vertical for a total of 5 pieces. Color in one 1 part green to represent the 1/5Make the missing lines in both your first and second rectangles. You now have the same size pieces in both rectangles.Make a new rectangle with all the lines. Fill in the total number of colored in boxes from both of the rectangles (this is addition!)The third rectangle is your answer3/4 + 1/5 = 19/20

TAKE AWAYBuild subtraction3/4 - 1/3build the first fraction on the board and the second on off the board3/4 take away 1/3check if the take away number fits into the first numberConvert both fractions using the same size piecesWe would use 1/12 tiles3/4 would convert to 9/121/3 would convert to 4/12We now have 9/12 - 4/12 and we can do this because they are now the same size piecesWe take away 4/12 and are left with 5/123/4 - 1/3 = 5/12Show SubtractionArea Model use rectanglesDraw one less line than your denominator - if you need 4ths, then you will draw 3 lines.I like to make my bigger number vertically since that seems to be easier to draw than the horizontal linesWith subtraction you will have two boxes2/3 - 1/2First rectangle will have 2 vertical lines to make 3 pieces. Color in 2 pieces red to represent 2/3Second rectangle will have 1 horizontal line to make 2 pieces. Color in 1 piece green to represent the 1/2.Make the missing lines in both your first and second rectangles. You now have the same size pieces in both rectangles.From your first box (2/3=4/6) take away what is in your second box (1/2=3/6)The red piece left is your answer2/3 - 1/2 = 1/6

Use color countersBuilding Multiplication(2/3) * (1/4)2/3 groups of 1/4 are redmultiply the denominators12 pieces totalooooooooooooOnly need 2/3 of the groupooooooooooooof the 2/3 we only need 1/4 to be redoooooooooooo(2/3) * (1/4) = 2/12Showing MultiplicationArea Model use rectanglesDraw one less line than your denominator - if you need 4ths, then you will draw 3 lines.I like to make my bigger number vertically since that seems to be easier to draw than the horizontal linesWith multiplication you will only have one rectangleThe answer is the part that will have both colors shaded in(2/3) * (1/4)First draw 2 horizontal to make 3 pieces. Color in 2 pieces red to represent 2/3On the same rectangle, draw 3 vertical lines to make 4 pieces. Color in 1 piece to represent the 1/4.You now have a total of 12 pieces in the rectangleThe pieces with both red and green in them are the answer(2/3) * (1/4) = 2/12

Solving Fractions - use logic!20 + (7/8)Just add them together20 + 7/8 = 20 7/813 - 1/5Remember that a number over the same number is 112 5/5 - (5/5) is the same as the original problem= 12 4/514 - 6 3/8take away your wholes first14 - 6 = 88 - 3/87 8/8 - 3/8 = 7 5/8Unlike denominators, use fraction trees7/10 - 5/1210 - 2 * 512 - 2 * 62 in common to bothMultiple the fraction by 1, using what it is missing from the other fraction (REMEMBER THAT ANY NUMBER OVER THE SAME NUMBER IS 1)7/10 (6/6)5/12 (5/5)42/60 - 25/60 = 17/60FUNKY ONES, simplify first, then multiply - I love this!!14/36 * 7/21factor each number - try to see if you can find numbers in common2*7/9*4 x 9*3/3*7Remember that a number OVER the same number is 1I can make a 1 with the two 7s (numbers can't both be on the top or both on the bottom numerator/denominator)1 with the two 9s1 with the two 3sI now have (2*1*1*1)/(1*4*1*1) = 2/414/36 * 7/21 = 2/4 = 1/25 * 8/15You'll want to make the whole number into a fraction and then do Funky Ones5/1 * 8/15 3 3/5 * 2 2/4same as Funky Ones but make it into an improper fraction first.denominator times the whole number, add that to the numerator, that total goes over the original denominator18/5 * 10/4continue with Funky Ones(8/15) / (4/5)Keep Change Flip - Fact Families is why we can do this8/15 * 5/4Funky Ones

BuildUse two color countersRED is negative1 positive with 1 negative is a ZERO PAIRBuild -5 using 9 tiles OOOOOOOOOThe two zero pairs make a ZERO BANK4 + (-3)OOOOOOOZero bank cancel each other outThe answer is 4 + (-3) = 12 + (-5)OOOOOOOZero bank cancels out2 + (-5) = -3

Instead of color counters, use + or --5 + 7 +++++++- - - - -zero bank leaves you with 2-5 + 7 = 2Hector's Method - only works for addition-30 + 27 - - +The bigger pile gets two of whatever it's sign isCircle together one sign from each sideIf the signs are different, we subtractIf the signs are the same, we addThe sign outside of the circle will be the sign for the answer30-27=3-30 + 27 = -3-320 + (-430) - - -circle, same sign means add320+430=750a - is on the outside of the circle so the answer will be --320 + (-430)= -750

5 - 3 5 take away 3OO [OOO]OO5-3=2-6 - (-2)6 negatives take away 2 negativesOOOO [OO]-6 - (-2) = -44 - (-3)OOOOWait, we have no negatives, how can I do this??ZERO BANK!OOOOOOO [OOO]4 - (-3) = 73 - 6OOOI don't have enough positives to take away 6. I need a zero bank[OOOOOO] O OOO O3 - 6 = -3

Showing Subtraction - use + and - instead of color counters5 - (-2)+++++++[--]5 - (-2) = 7Solving SubtractionAdd the inverse - this is the same as making a zero bank-18 - (-15)-18 + (+15)Now I can use Hector's Method-18 + (+15) - - +subtract the two numbers18-15=3a negative is still left outside the circle-18 + (+15) = -3-53 - 20-53 + (-20) - - -add53 + 20 = 73negative sign left outside the circle-53 + (-20) = -73

Building Integer Multiplication3 (-5)3 groups of 5 negativesOOOOO OOOOO OOOOO3 (-5) = -15-4 (3)0 TAKE AWAY 4 groups of 3 postives0 - 4 (3)ZERO BANKOOO OOO OOO OOOOOO OOO OOO OOOtake away 4 groups of 3 positives12 negative is left-4 (3) = -12-3 (-2)zero take away 3 groups of 2 negativesOO OO OOOO OO OOtake away 3 groups of 2 negatives6 positives are left-3 (-2) = 6Solving Multiplication and DivisionIf the signs are the same, the answer is positiveif the signs are different, the answer is negative-25 (-30) same sign, answer will be positive-25 (-30)=750-100/20signs are different, answer will be negative-100/20 = -5

GroupsExponentsD/M divide/multiple, left to rightS/A subtract/add, left to rightgroups are made with an add or subtract sign-12 - 4(2) + 5bring down the - and +-12-8+5add the inverse-12+ (-8)-20 + 5-15-24 / 2 x 4-12 x 4-48Remember to go LEFT TO RIGHT20 - 4^2 x 3 / 220 - 16 x 3 / 220 - 48 / 220 - 24-4-4^2 + 8 - 3^2 - 4-16 + 8 - 9 - 4-8 - 9 - 4-17 -4-21(-7)^2 = 49-(7^2) = -49(-7^2)= -49-7^2 = -49-(7)^2 = -49

% symbolizes a ratio. amount per 10030% is 30 / 100Using the logic that it is out of 100, we can draw 10 frame boxes as a way to show percent. Each box is equal to a tenth of the total (100).40% would have four boxes colored in70% would have seven boxes colored in35% would have 3 and one half boxes20% of 4010% of 40 is 4 (move the decimal to the left one spot)A 10 frame that represents 40 would have 4 in each box20% would be two of the boxes4 + 4 = 820% of 40 = 845% of 8010% of 80 is 840% of 80 would be 8x4 = 325% of 80 would be 4 (half of 10%)32 + 4 = 3645% of 40 = 3642% of 12010% of 120 is 121% of 120 is 1.2 (move the decimal one more spot to the left of the 10% total)40% is 12x4 = 481.2 + 1.2 = 2.448 + 2.4 = 50.442% of 120 = 50.4