MTE 280

-Ploya's Problem-Problem solving strategies Ex: Pigs/chicken problem4 Step problem solving process------------------------------------------------Step 1- Un Dev Car LoUnderstand the problemre-read the problemexplain the problem to someonebreakdown the parts of the problemStep 2- Develop a planRelate the problem to prior knowledge Identify what is similar / differentbrainstorm ideasStep 3- Carry out plantry the planStep 4- Look back to see if your answer makes senseCheck your answer for reasonablenessSolve it differently to see if you get the right answerwork backwards using your answerMethods--------------------------Guess/check EX: pigs and chickens 56ft of land and 18 animals pigs: 5x4 chickens: 13x3 ------> guessed the # of animals and checked for the # of fact eventually getting the right answer.Diagrams-Drawing out the problem using basic diagrams such as dot or circles.Lists-make lists of the number of animals this will help set up patterns to get the right answer.Algorithms -p= pigs c= chickens p+c=18 4(p)+2(c)=56Video:https://www.youtube.com/watch?v=TVhOFPm7lbc

Intro to building Different bases--------------------------------------------------- key: / = longs .= units []=flats1) 8 to base five -groups of 5. . . . . . . . .-----> exchange 5 units for one long / . . . = 13 five2) Base 12// . . . 2 longs and 3 units = 23 twelvenumbers in the base cannot be bigger then the number the base51 four , 5 needs to be exchanged for a flat then. =1 flat 1 long 1 units in base four.3) 142 base six //// . . =1 flat 4 longs and 2 unitsVideo:https://www.youtube.com/watch?v=iIhSNgZqdJE4) 247+185= 2 flats + 4 longs + 7 units + 1 flat + 8 longs + 5 units= 4 flats 3 longs 2 units = 4325) 6 eight + 5 eight. . . . . . . .= 1 long . . .= 3 units = 13 eight6) 22 five + 14 five= // . . + / . . . .= //// . = 41 five

Building addition in Base ten blocks---------------------------------------------------------EX: 6 + 7 =. . . . . . + . . . . . . . = 1 ten and 3 units = 13 23 + 42= // ... + //// .. = 6 longs and 5 units 74 + 29= 7 tens and 4 units + 2 tens and 9 units = 1 flat 0 longs and 3 units = 103-------------------------------4six + 1six= . . . . + . = 5 sixno exchange because 5 is less then 66 eight +5 eight=. . . . . . + . . . . . = 13 eightup to eight units needs to be exchanged for a long because the numbers are in base eight22 six + 44 six = 2 long 2 units + 4 long 4 units= 1 flat 1 long 0 units = 110 six-------------------------------------------------------------------------Showing addition5+3.....+...=13 - Simplified diagramsexchanging shown in diagrams using arrows to show that the group of longs,units, or flats are exchanged out -Friendly Numbers32 + 58 add the units from the left number to the right number to make it end in 0 +230 + 60 = 90 -Trading off26+67 ---> 23+70moved 3 from 26 over to the 67 to make it a friendly number8+4+7+6+24+6 = 10 8+2= 10 ---> 10+10+7= 27-Scratch methodScratch out the number to show its converted into a long6 +7 = 13 --> 7 would be scratched out to make it a long but I cant draw over the numbers on the notes here. -Latticedraw a lattice box to signify where the tend and units gohelps show how carrying worksshows exactly what numbers to addnot very efficient, is expandable, and is based in mathVideo:------------------------------------------https://www.youtube.com/watch?v=tYmF2GW0wwQ

Building and showing Subtraction---------------------------------------------------------1) 7-3=7 take away 3 =4. . . ( . . . .)> - taken away2) 46 - 15 /// (/)> . (. . . . .)> = /// . = 313) 24 - 8 / /--> exchange for units / . . . . . . (. . . . . . . .)> = / . . . . . .= 16Solve subtraction & alt algorithms---------------------------------------------------------1) 57 2) 374 -24 -146------------ -------- 33 228 -borrowing from the other digitsExpanded form:46-24 ----> 40 + 6 -(20+4) ------------ 20+2 =22numbers too big to subtract from should be broken down further.43 - 27 -----> 40+3 -(20+7) ------------------- 20---> 10+10+3 -7 ----------------------- 10+3+3= 16Left to right:1) 96 - 35----> 60+1=61 -take away2) 256 - 132------> 100+20+4= 1243) 53 seven - 25 seven-------> 20+7-5 20+2+3= 25 sevenEqual Addends:1) 64-38+ 2 +2------------------- 635-350+ 50 +50------------------685 - 400 =285friendly numbersif u add to one you have to add to the other number2)

Building Multiplication-----------------------------------Base ten3 x 4| \# of inside those groupsgroupsthree groups of 4Show multiplication-------------------------------using arraysusing the box methodMultiplication order-------------------------------Flash cardsgrouping flashcardsgroup 1-1,2,5,and 10group 2-2", 3,9 doublesgroup 3-4,6,7,8avoid competitive games involving mathtime tests to help then get the math down fastSolving multiplication-------------------------------------1) 36 x 28 = 288+ 720= 1008Expanded:23 x 45 20+3 x 40+5 ------------- 800 + 120 + 100 + 15= 1035left to right:1) 38 x 26 38x 26----------600 + 180 + 160 +48=988Area model: can't include the real drawing but it will be a rectanglehelps show which numbers will be flats, longs, and unitsVideo:https://www.youtube.com/watch?v=3zRPty9n6fg

Division--------------------12 / 3 12- is the # we are dividing up 3- could be the number of groups we have or the number of things inside the groupwith remainders breaking the remainder up into the groupsthe remainder will be a fractionex:17/5 is 3 with a remainder of 2 will be 3 2/5Showing division:using diagrams Solving division:using long division know how to write the equationwrite it correctlyestimatemultiply/subtractrepeatknow how to write the final answer and how to write the remainderVideo: https://www.youtube.com/watch?v=_rzq8U78e6MUsing alt algorithms----------------------------------long divisionarea model for division -similar to the one used for multiplication upwards division -write the equation like you say it -makes it easier to write remaindersDivisibility rules:2-even3-sum of digits /34- look at last 2 digits5-# ends in 5 or 06-if 2 and 3 work then 6 works7-none8-look at last 3 digits9-sum of digits /910-# ends in 0prime factorization -factor treesupside down division -uses boxes to factor out the numbers

Multiplying / Dividing integers^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ex) 6 x 46 groups of 4 positivesex) 5 x (-2)5 groups of 2 negativesex) -3 x (2)0 take away 3 groups of 2ex) -2 x(-4)0 take away 2 groups of 4 negativesDivision:Drawing diagrams with groups using zero banksneg times neg= posDifferent signs = negVideos:https://www.youtube.com/watch?v=FeP-EVRW-RA

Order of operations________________________________PEMDAS XGEMDAS is betterParenthesis are not the same as groupsG - groups E - Exponents M/D - Lft/Rt A/S - Lft/Rtex) -12-4(2)+5Separate groups -12 - 8 + 5 -20 + 5 = -15groups are done at the same time instead of re-writing ex) -24 / 2 x 4-12 x 4 = -48Video:https://www.youtube.com/watch?v=e8fIo7CQe70

Percentages~~~~~~~~~~~~~~~~~~~~~What is a percent?30% is a ratio / just a fraction just like miles per hour ex- 50mph/1hr30% ---> 30/100Diagrams:using boxes and filling in the number10 parts in one boxthe box = 100% ex- 40% = 4 out of 10ex- 20% of 40 = 8 using a box put 4 in each box because 10% of 40 is 4 in each piecewhen percents don't end in 0 cut one piece of the box in halfex- 35% of 80 10% = 8 30% = 24 + 4 = 28 5%=4ex- 27% of 6210% = 6.2 20% = 12.4 5% = 3.112.4+6.2+3.1 = 16.74video:https://www.youtube.com/watch?v=WYWPuG-8U5Q

Fractions---------------------parts of a fractionnumerator is the amount we have denominator is the size of the whole piece2/3 vs. 2/5 -2/3 is bigger piece-Common denom. -Multiply denoms. togetherex)2/3 + 3/4-changing denom. will change numer.multiplication-----------------------------2/3 - number of groups times3/4 - inside the groupBuilding fractions----------------------------------using diagrams and manipulative fraction circles-linear models-color counters-cutting line through the diagrams to show fractions-can use whole parts of fractions with manipulativesvideo:https://www.youtube.com/watch?v=c0aMoMaiquA

Module 7 ------------------------Building fractions:using area models building fractions circlesusing fraction diagrams and shading in modelsusing linear models for problems using different fraction manipulatives to build fractionsVideo:https://www.youtube.com/watch?v=Mz_cPFgu6UM

Module 8------------------------Subtraction:ex) 2/3 - 1/3 using take away termstaking away 1/3using fraction circleseasier to demonstrate using circle diagramshave to change the diagram to the same size piece ex) 2/3 - 1/6 breaking up 2/3 into 6 thswhen doing so have the ability to take 1/6 awayAddition:ex) 2/3 + 1/6= 5/6used manipulatives to solveex) 2/5 + 1/2= 9/10can't be simplifiedtry to use problems that can be simplifiedMultiplication:ex) 4/1 x 2/1 = 6/1ex) 1/2 x 4 half group of 4 unitshalf of the four groups is shadedex) 4 x 1/2 four groups of 1, 1/2 are redex) 1/2 x 1/4 half a group of 1/4 is redVideo:https://www.youtube.com/watch?v=Mz_cPFgu6UM

Module 9-------------------------------------Showing fractions:using diagrams to solve fractions equations circles are great when they know the pieces need to be the same sizecan use rectangles/linear models when denoms. get bigger or harderShowing adding:ex) 1/5 + 2/5= 3/5using rectangles to shade in pieces being added togetherex) 1/3 + 1/2 when problems have different denoms. they're should be three boxes used2 different sized pieces laid over top each other in the third boxShowing subtraction:ex) 3/4 - 1/4 = 2/4using one box for common denoms. two boxes for not common denoms. still using take awayex) 2/3 - 1/2 = 1/6Show multiplication:groupsgroups insideex) 2(1/3)= 2/31/3 in each two groupswhole #(Fraction) = smallerfraction (whole #)= smallerwhole # (whole #)= biggerfraction(fraction)= smallerex) (2/3)(1/4)= 2/12Video:https://www.youtube.com/watch?v=WrvDWD9HvOs

Module 10----------------------------Solving fractions:ex) 20 + 7/820/1 (8/8) + 7/8 = 160/8 + 7/8 = 167/8ex) 20+7/8 = 20 7/8 don't have to draw it all out ex) 13 - 1/512 5/5 - 1/5 = 12 4/5cross out the whole number to make a common denoms. ex) 14 - 6 3/8 8 - 3/8 = 7 8/8 - 3/8 = 7 5/8finding common denoms. w/o saying common denoms.factoring the fractions to find common denoms.ex) 7/10 - 5/127/ 2x5 - 5/ 2x6 = 7x6/10x6 - 5x5/12x5= 42/60 - 25/60 = 17/60keep numbers small leaves less room for errorsMultiplication:using funky onesfactoring the fractions out for faster simplificationnot cancelling only can create a funky one from numer. to denom. simplify then multiply Dividing:keep, change, flipbackwards C's ex) 4 2/3 / 5 5/6multiply the denom with whole # then add numerator14/3 / 35/6 k,c,f14/3 x 6/35 use funky ones = 4/5fact familiesboth multiplication, division and addition, subtractionVideo:https://www.youtube.com/watch?v=96NOYcnmThUhttps://www.youtube.com/watch?v=nMZJKGyu-Kk

2 Color countersex) build 5 0 0 0 0 0Yellow is positivered is negativeex) building -2using the terms 2 negativesBuilding Zero banks to build problems with multiple countersZero banks cancel out, are equal to 0Addition^^^^^^^^^^^^^^^^ex) 2+50 0 + 0 0 0 0 0 = 0 0 0 0 0 0 0 (7)ex) -5 + (-3)= -8(Using diagrams, I can't make the dots to represent the negatives)Using parenthesis to help read the problems better for studentsDon't use perfect rowspositives go on top and negatives on bottom for zero banksSolving integers------------------------------------Using diagrams:ex) show 4 + + + +ex) show -3 _ _ _ Showing Additionex) 3 + 4+++ ++++= 7make sure the diagrams are organizedVideo: https://www.youtube.com/watch?v=S_-C5csujgA

Solving Integers~~~~~~~~~~~~~~~~~~~~~Subtraction:Using Take away using zero banks in diagramsusing as many zero banks as neededBuilding subtraction:what is the best way to say each problem?ex) 5 take away 3= 2circling and taking away number for diagramex) -5 - 8----- +++++++++_ _ _ _ _ _ _ _take positive 8 away-13 is left overHector Method:mini diagraminverse when the same sign addsopposite signs subtractex) -18 - (-15)-----> -18 + (+15) _ _ sub + = -3The bigger number gets two signs The leftover sign from the bigger number goes to the answerVideo:https://www.youtube.com/watch?v=CfkaifC7tGY