by Rachael Mertes 2 years ago
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Comparing Fractions: 3/4 > 5/16: closer to a whole
Youtube: https://youtu.be/dIc_CD6KTuo
Simplifying Fractions: 60/210= 6*10/21*10=6/21= 2*3/7*3= 2/7 (simplified fraction)
Equivalent Fractions: 2 fractions that can be simplified to equal one-another ex: 3/6= 9/18
Improper fraction: a/b where it is greater or equal to 0 ex: 6/4
Proper fraction: rational # where a/b is less than 1 ex: 3/4
Set Model: https://youtu.be/E0KQauFeJo8
Number Line: https://youtu.be/fLY9yRdBObQ
Bar Model: https://youtu.be/5NsUu9s_yGc
denominator: the number below the line in a common fraction; the whole
numerator: the number above the line in a common fraction showing how many of the parts indicated by the denominator are taken
rational #: a/b
Ordering Integers
equal to: a=b
more than: a>b
less than: a
Integer Division
Using Chips and Number Line: https://youtu.be/lDR-B_OhUQo
Integer Multiplication
Patterns of Multiplication
zero: a*0=0=0*a
identity: a*1=a=1*a
distributive:a(b*c)= a*b+a*c
associative: (ab)c=a(bc)
communicative: ab=ba
closure: ab is a unique umber
Chip Model & Number Line Representation https://youtu.be/fW3FWuLfpFc
Properties of Integer Addition:
Identity: 0+a=a=a+0
Associative: (a+b)+c=a+(b+c)
Communicative Property: a+b=b+a
Closure: a+b is a whole unique number
Number Line Model: https://youtu.be/3LUTYhmltQY
Chip Model: https://youtu.be/_77vO0uzBfA
Absolute Value: the distance between the point corresponding to an integer and zero. ex: |-3| = 3
Integer: a whole number that is not a fraction
Slide Method: https://youtu.be/IM7ToCYo-_A
Least Common Multiple LCM
of 2 whole numbers a& b is the least non-zero multiple of both a&b
Greatest Common Factor GCF
Prime Factorization Method: 180 & 168 ex: GCF 180 & 168180: 2*2*3*3*5= 2^2*3(3*5)168: 2*2*2*3*7= 2^2*3(2*7)GCF= 2^2*3 or 12
of 2 whole numbers a &b not both 0, is the greatest whole # that divides both a&b
Prime Factorization: containing prime and composite numbers to find, where we rewrite the #s as a product of two whole numbers, where we continue to process until the numbers/factors are prime.
Youtube: https://youtu.be/tW97UU01ShY
ex: 260= 26*10= (2*13) (2*5)= 2*2*5*13= 2^2*5*13
Composite Number: any whole number that has factors that make up the whole. ex: 4,6,8,10,12 etc
Prime Numbers: any whole number that is only divisible by 1 and itself. ex: 1,2,3,5,7,11,13 etc
Factorization: breaking a # down into smaller #s, that when combined back together gives you the original #
Odd #: a whole # with a remainder of 1 when divided by 2
Even #: a whole # that has no remainder when divided by 2
Divisibility Rules:
Youtube: https://youtu.be/E4yzE5NumV8
10: a number is divisible by 10 when the units digit ends in a 0 ex:100 or 250 or 490
9: a number is divisible by 9 if the sum of the digits of the number is divisible by 9 ex: 998
8: a number is divisible by 8 if the last 3 digits represent a number divisible by 8 ex: 234800
6: both rules for 2 & 3 work, then the number is divisible by 6 ex: 12 (12/2=6 or 12/3=4)
5: a # is divisible by 5 if the units digit is divisible by 0 or 5 (last digit ends in a 0 or 5) ex: 100 or 155
4: a # is divisible by 4 when the last two digits of the number make a # that is divisible by 4 ex: 4520
3: a number is divisible by 3 when the sum of its digits is divisible by 3 ex: 876
2: a # is divisible by 2 only if the units digit is even ex: 542
Youtube Video: https://youtu.be/wT8tGc-SwKk
Scale Drawing: ratios and proportions: scale= the ratio of the size of the drawing to the size of the object
Unit Rate Strategy: for solving problems involves finding the unit # compared of the 1 ticket to comparing the unit cost
ex: 12 ticket= $15=1=$1.2520 tickets= $23=1=$1.15
Constant Proportionality: x and y are related by equality y=kx or k(y/x), then y is proportional to x+k to the constant of proportionality between y and x
Example: https://docs.google.com/document/d/1m5Li_NSH6CWH6B3OgqMu-c0CfM5MAbqYP3twmmuo0cA/edit?usp=sharing
Ratio Word Problem Ex in
Part to Part: ratio of boys to girls would be 1:3 or teacher to students would be 24(class size): 1 (teacher)
Whole to Part: ratio of all children(whole) to boys(part) is 4:1
Part to Whole: 1:3 boys to girls- ratio of boys(part) to children(whole) is 1:4 ratio
Definition: a/b or a:b where a and are rational numbers is a comparison of 2 quantities
Dividing Decimals: Ex in https://docs.google.com/document/d/16prYEmOk5CthA6BWWTwYq06CCbtNITChDLYimYKR4DA/edit?usp=sharing
Long Division: https://youtu.be/LGqBQrUYua4
Algorithm for multiplying decimals: if there are n digits to the right of the decimal point in 1 # and m digits to the right of the right of the decimal point in a second #, multiplying the 2 #'s, ignoring decimals, then placing the point so there n+m digit places from the right of the decimal point of the product
Multiplying Decimals using the Grid model: Ex https://docs.google.com/document/d/1rcqLDmTGWJdTRJLtC7jzEkHy7y9aj2CJHb6unl6-kMc/edit?usp=sharing
Multiplying decimals using expanded base 10 blocks model: Ex in https://docs.google.com/document/d/1DY2ACD7pCF-BgSESnaqlLOPpodz2KCaEDyLEo8JQtkQ/edit?usp=sharing
2.63 *8.2 21.566
Youtube video for Multiplication: https://youtu.be/Dm028SSei88
Mental Compution:
Breaking and Bridging:1.5 + 3.7 +4.48(1.5+3=4.5) & (4.5+.7=5.2) & (5.2+4=9.2) & (9.2+.48=9.68)
Subtraction: is doen using the grid by representing the first number in grid form, then by taking away the second number.
Ex https://docs.google.com/document/d/12A5k4AWCaVIcGLKU2dlCPJQiM5-ZjnIqPKVvDFDSe74/edit?usp=sharing
Addition: is done by placing the decimal reps of the 2 addends in the same decimal grid. May need to regroup.
Ex https://docs.google.com/document/d/1Y6oTRgnuOuOuLj6nXQahKVGO-wL4XVufRLl8MTE_CIo/edit?usp=sharing
Rational Fraction Division
Grouping Model: EX in https://docs.google.com/document/d/1VlcZqwTud3pKK_vNSRqVh1eVV8QWaU99iS2_d7dYCyU/edit?usp=sharing
Unlike Denominators:
(a/b)/(c/d)= (ad/bd)/(bc/bd)=(ad/bd)(bd/bc)=ad/bc
Equal Denominators:
(a/b)/(c/b)=(a/c)/b
"Invert and Multiply"
https://youtu.be/e1gcBP2TmPk
ex: (2/3)/ (5/7) is equivalent to (2/3)(7/5)
EX in https://docs.google.com/document/d/1HRqwoUUTjGpPNoVFZ9Im2hC75JW8M39RP0nyEStMmIc/edit?usp=sharing
Rational Fraction Multiplication:
YouTube Video: https://youtu.be/mUQbh_chhQQ
Using different models: https://docs.google.com/document/d/1GC1CVPsIkMCS-T6zhgTKf3tldmk4dFu9111-yPtug9U/edit?usp=sharing
Multiplication with Mixed #s : Convert the mixed number into an improper fraction. Then multiply the numerators of the fraction and multiply the denominators of the fraction. Last, onvert it into simplified form if required.
Fundamental Law of Fractions: a/b= (an)/(bn) if b doesn't equal 0 and n doesn't equal 0
Identities:
Distributive: a/b(c/d +e/f) = (a/b)*(c/d) + (a/b)*(c/d-e/f)= (a/b)*(c/d)-(a/b)*(e/f)
Multiplicative: rational number 1 is the unique # where every rational # a/b= 1* (a/b)=(a/b)=(a/b)* 1
Estimating Rational Numbers: estimating fractions to see if they are closest to 1, 1/2, or 0; and in some cases, estimating bigger fractions to see if they would most likely be reduced closest to 0, 1/2, 1/4, 1/3, 3/4, 1, etc. https://youtu.be/UBFyCnSKfBE
Is 52/100 closest to 1, 1/2, or 0?- Closest to 1/2 because 52/100 is closest to 50/100= 5/10=1/2. 52/100 is 2 away from 1/2 vs being 48 away from 1 whole
Addition&Subtraction
Youtube Video: Start at 2:15 time mark for fractions https://youtu.be/pZD5jxgHit0
Subtracting Rational Numbers: a/b-c/b= (a-c)/b
Adding Mixed Numbers: for any rational # a/b, there is a uniquerational # -a/b thats the additive inverse of a/b
EX: in https://docs.google.com/document/d/1qbhIR5Q_Ws4rXuRXJLTzWq08q8LoPNLbibM1dt34TXo/edit?usp=sharing
(a/b)+(-a/b)=0=(-a/b)+(a/b)
+ of Rational #s with unlike denominators
if a/b & c/d are rational numbers then a/b+c/d=(ad+bc)/bd
+ of Rational #s with like denominators
Bar Model using addition- https://docs.google.com/document/d/1RN7_b-bTjvBtb-bxHWyDGNOCM8QcelbD63LZfQX4k_E/edit?usp=sharing
Number Line using addition- https://docs.google.com/document/d/1i_JGxpessqGWUQip_6ibJkOv1kSHjrMRoGFInyQKiNk/edit?usp=sharing
Pie chart using addition-https://docs.google.com/document/d/1OO_pzeMQL_mAVNpSKnfgrNRO7eYN0SqrDKM6_FwHDJw/edit?usp=sharing
If a/b & c/b are rational numbers then a/b+c/b =(a+c)/b
