MTE 280 Elementary Foundations

Juggling Mr. Milt

Class expectations and grading(see syllabus in canvas)

Converting BasesDiagramsusing longs and units to represent numbersone long equals whatever base you're inif there are the number of long that equal what base you're in, it turns into a flatflats = 100Examples:24six --- 2 long and 4 unitseach long = 6 and each unit = 1so...6+6+4=1613 to base eight --- 1 long and 3 unitseach long = 10 and each unit = 3so...13 will make 1 long and 5 units = 15

CountingCounting and how to works in different bases practice the bizz/buzz game to practice how to count helps with counting in different basesBasic counting tools10 framesflatslongsunitsOne to one correspondencedifficult for younger students to start to comprehendit is easy for children to count from 1-10, but they don't know what that means and how it is representedpractice one-to-one counting to get this idea in their head

Alt. Algorithms for AdditionFriendly numbers:adding/subtracting to make both numbers end in zeroexample: 28+62 --- 30+60 = 90Trade off:adding/subtracting to make one number end in zeroexample: 46+25 --- 50+21 = 71Left to right:adding from left to right instead of right to leftreinforces place valueexample: 37+42 --- add do 30+40 then add the 7 and 2 = 79Expanded form: expanding each number so they're multiples of 10reinforces place valueexample: 748+165 --- 700+40+8 +100+60+5 = 913 Scratch:scratch when numbers make up the baseadd remainders then carry to the next columnhelps students add multiple numbers without having to keep track of everything in their headsexample: 34+12+15you would scratch when the ones place adds up to 10 then carry over

How to Write ProblemsShow:convert from base ten to other bases34 to base 6convert from other bases to base 10207eight to base tenSolve: convert from base ten to other bases33 to base fourconvert from other bases to base ten3,425six to base ten

Introduction to Alt. Algorithms for Multiplication Area Model:somewhat like expanded form additionExample:24(28) 20 + 820 +4multiple each number by its boxed counterpart (20x20 and 20x4, 20x8 and 4x8)then add each row to get the answer this helps reinforce place value and multiplying with numbers ending in 0

Multiplication1st number: # of groups2nd number: what is inside the groupsWhat order to teach times tables in:teach first:1s2s10s5sTeach second:3s4s9sTeach 3rd:everything elseMULTIPLICATION = AREA OF A RECTANGLEshowing using base 10 blocks:draw a square for 10x10add sections and lines for each additional unitexample: 14(12)have a 10x10 flat then have 4 units added on one side and 2 on the other than add everything to get the answer = 168

see notes

81/82 on exam 1

Alt. Algoritms for MultiplicationMultiplication = Area Area/base 10 block expanded formexample: 27(36)20+730+6------- then you multiple each number by its vertical and diagonal counterpart= 600+210=120+42 then add them all togetherArray MultiplicationExample: 3(4)o o o oo o o oo o o o =12Lattice MultiplicationExample: 25(15)see notes for set up create box with lattice lines then multiple each boxafter than add the lattice lines (diagonally) then you'll get your answer= 375

Alt. Algorithms for DivisionTips: don't say "divided by", say "goes into"spend the first day of lessons showing students simply how to set up problemsdoing multiples that students already know and are comfortable with is the best way to start Repeated SubtractionExample: 146/8see notes for how to set upstart by slowly subtracting multiples of 8 and that will go off to the sideit allows students to practice their multiples has a margin for error, but is a good way to introduce divisioneventually you'll take 8 away a certain amount of times until you find the answer

Alt. Algorithms for SubtractionSubtraction = the distance between two numbersshow using longs and units24-12 will go to be two longs and 4 units...then you'll take away one long and 2 units to find the answer24-1834-28 =6 the same distance between 2 #33-27show using tilesshow 5 using 9 tiles +++++++_ _show 5+(-4)+++++_ _ _ _ = 1show -3(-2)_ _ _ _ _ = -5

Adding IntegersTip: when talking to students about whether the positive or negative number is bigger...not "which is bigger" should be "which one would have a bigger tile pile"Example: 3+(-5)3 is the "bigger" number because it is positive, but for our purposes -5 is actually bigger because it would have the bigger pile of red tiles then 3SHOW: 3+(-5) Mr. Milt's student's algorithm+++_ _ _ _ _ = -2(there is a bigger pile of negatives, so the answer must be negative)SLOLVE: 24+(-35)24 + (-35) = -9+ - - so...it will end up being a negative number

Subtracting IntegersKEEP, CHANGE, CHANGEShow:-5-(-2) = - - - - - take away 2 = -3-4-2 = - - - - - -take away 2 ++ = -6Making zeros: a - and a + go together to create a "zero"Solve:-35-(-15)K C C = -35 + 15 = -20(use Mr. Kilt's student's algorithm shown in adding integers)5-9+ -- so it's going to be a negative number = -4

Intro to Fractionswhich fraction is larger? 4/7 or 5/7whichever fraction in closer to "1" is largerthink of it like pieces of a pie 5/7 pieces is more than 4/7 because it is closer to 1 5/8 or 5/9 5 out of 8 pieces of pie is more then 5 out of 9 pieces of pieEasy tip: when doing a problem like 32+8/11it equals 32and8/11 so don't make it more complicated numerator= # of piecesdenominator= size of pieces

More Fractions Important tip:the fractions have to be equal to add them correctly, but not to multiple or divideWays to show fractions example: 4/6set model: xxxx~~area model: [] [] [] [] [] []linear model: -+-+-+- -+-+-+-+-+

Adding FractionsTip: if you're given 5+3/8, all you have to do is add it, not convert 5 into a fraction5and3/7 + 4and1/7 goes to be 5+4 then 3/7+1/7 because they have the same denominator 25-4and3/16 is more complicated 25-4=21 but then you have to take 1 out and make it a fractionso then you have 20and16/16-3/16 then you get 20and13/16What fractions mean:3(5) = 3 groups of 53(2/7) = 3 groups of 2/7(1/3)12 = one shirt of a group of 12

see notes

Divisibility Rules2: even #s3: sum of digits divide by 34: last 2 digits divide by 45: any # ending in 5 or 06: even and sum of digits divide by 38: last 3 digits divide by 89: sum of digits divide by 910: any # ending in 0Least common multiplebigger number and shared multiplesfind prime factorization then multiply the prime factors togetherGreatest common factorwhat the two prime number groups share then multiplysmaller numberExample: GCF and LCM of 60 and 45prime factorization for 60=2^2, 3, 5prime factorization for 45=3^2, 5so...LCM=3x5=15GCF=2^2x3^2x5=180

Prime Factorization: finding which prime numbers multiple together to make the original numberExample:909 103 3 5 2so... the prime factors= 2x3x3x5

Multiplying Fractions:No different than normal multiplicationExample: (3/4)(2/3)=6/12=1/2Makes the problem much more simple and less prone to simple math mistakes if you try to reduce/ simplify the fractions before multiplyingExample: (24/35)(21/40)can be simplified by doing the funky "1s" example from class24=6x4 and 40=4x10 so the 4's can be crossed out35=5x7 and 21=7x3 so the 7's can be crossed outthen you're left with (6/5)(3/10)=18/50=9/25

Dividing Fractionswhen dividing fractions, always KCF (keep, change, flip)also remember to use the funky 1's as seen in multiplying fractions to reduce to limit mistakesexample: 3and3/5 divided by 1and2/10first convert numbers to a mixed numbers to get 18/5 and 12/10then find out the multiples18 = 3x610 = 2x512 = 6x2the two 6's and two 5's cancel out so you're left with 6/2 which = 3remember: when converting to mixed number, use "backwards c"

Order of Operations