Elementary MathematicsRachael Mertes

1.

Unit 1

3.1 Numeration

Base 10 system:

Includes #s 0,1,2,3,4,5,6,7,8,9

Vocab:

Face Value: numerical value of a #

Expanded form: is the sum of the values of each digitex: 356,039: 300,000+50,000+6,000+30+9

Base 10 Blocks: counts of units(1), longs(10 units), flats(10 longs), and cubes(10 flats)

How to write numbers using base 10 blocks: https://youtu.be/sGHolGT3ieA

Bases in other numbers

Bases in other #s include digits from 0 up to the digit before the number of the base

examples:

Base five: includes digits 0,1,2,3,4

Base two: includes digits 0,1

Base six: includes units 0,1,2,3,4,5

3.2 Addition of Whole Numbers

Vocab:

communicative property: moving numbers around: a+b=b+a

associative property: "switching groups" (a+b)+c=a+(b+c)

identity property: any number plus 0 keeps its numerical identity ex: 4+0=4

closure property: any number plus a number equals its own unique number ex: 3+5=8 *no numbers can be the same in this equation for it to work

partial sums algorithm: place value columns can be added in any order working from left to right and from the largest place value to the smallest.

set model: a way to represent addition of whole numbers.

mental computing: producing an answer without any physical aid

computational estimation: forming an approximate answer to a numerical problem

regrouping: we use this word instead of the word "trading"

carrying: we use this word instead of the word "borrowing"

Addition of Whole Numbers Using Base 10 Blocks https://youtu.be/wOyRTh87bCE

3.3 Whole Number Subtraction

Vocab:

Missing Addend: models subtraction and addition ex: 8-3=5 where 8 is the missing addend.

this gives students an opportunity to use algebraic thinking

Counting up algorithm: "making change" method ex: 100-31= 31+9=40 and 40+60=100. Then 60+9= 69 making 100-31=69

Subtraction using blocks: https://youtu.be/F63kWrYg6fY

Comparison Model: finding the difference between numbers using manipulatives

Subtraction in Base 5 using illustration: https://youtu.be/EEdTcnerc5c

3.4 Multiplication

Repeated Addition: 5*4= 4+4+4+4+4=20

Properties:

Communicative Prop: a*b=b*a

Associative Prop: (a*b)c=a(c*b)

Identity Prop: a*1=a=1*a

Anything times 1 keeps its identity ex: 3*1= 1+1+1=3

Distributive Prop: 5(3+4)= (3+4)+(3+4)+(3+4)+(3+4)+(3+4) or 5*3+5*4

Multiplicative Prop of 0: Anything times 0 is 0

Front End Multiplication:

*mental compution* ex: 64*5= 60*5= 300 & 4*5= 20, so 300+20= 320 making 64*5=320

Partial products:

Multiplying in bases other than 10:https://youtu.be/7rb6ewezE3k

Area Model Multiplication: https://youtu.be/MVZRD4Fa1OY

3.5 Division

Repeated Subtraction Model: Find how many GROUPS

"If Susie has 8 cookies, if she eats two cookies for dessert each day, how many days will she be able to have cookies for dessert?"

Sharing Model: Find how many in EACH group

"If Susie has 4 friends and brings 6 cookies to school, how many cookies will each friend get?"

Division Property of 0: Any # times 0= 0

Brain Pop on Repeated Subtraction: https://youtu.be/k_e-pgiqqYo

Division using Partial Quotients: https://youtu.be/fb2XsYU0o8M

2.

.

Unit 3

Rational Fractions

Addition&Subtraction

+ of Rational #s with like denominators

If a/b & c/b are rational numbers then a/b+c/b =(a+c)/b

+ of Rational #s with unlike denominators

if a/b & c/d are rational numbers then a/b+c/d=(ad+bc)/bd

Adding Mixed Numbers: for any rational # a/b, there is a uniquerational # -a/b thats the additive inverse of a/b

(a/b)+(-a/b)=0=(-a/b)+(a/b)

Subtracting Rational Numbers: a/b-c/b= (a-c)/b

Youtube Video: Start at 2:15 time mark for fractions https://youtu.be/pZD5jxgHit0

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

Rational Fraction Multiplication:

Identities:

Multiplicative: rational number 1 is the unique # where every rational # a/b= 1* (a/b)=(a/b)=(a/b)* 1

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)

Fundamental Law of Fractions: a/b= (an)/(bn) if b doesn't equal 0 and n doesn't equal 0

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.

Rational Fraction Division

"Invert and Multiply"

ex: (2/3)/ (5/7) is equivalent to (2/3)(7/5)

Equal Denominators:

(a/b)/(c/b)=(a/c)/b

Unlike Denominators:

(a/b)/(c/d)= (ad/bd)/(bc/bd)=(ad/bd)(bd/bc)=ad/bc

Operations of Decimals:

Addition: is done by placing the decimal reps of the 2 addends in the same decimal grid. May need to regroup.

Subtraction: is doen using the grid by representing the first number in grid form, then by taking away the second number.

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)

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

Youtube video for Multiplication: https://youtu.be/Dm028SSei88

2.63 *8.2
21.566

Multiplying decimals using expanded base 10 blocks model: Ex in https://docs.google.com/document/d/1DY2ACD7pCF-BgSESnaqlLOPpodz2KCaEDyLEo8JQtkQ/edit?usp=sharing

Proportions

Definition: a/b or a:b where a and are rational numbers is a comparison of 2 quantities

Part to Whole: 1:3 boys to girls- ratio of boys(part) to children(whole) is 1:4 ratio

Whole to Part: ratio of all children(whole) to boys(part) is 4:1

Part to Part: ratio of boys to girls would be 1:3 or teacher to students would be 24(class size): 1 (teacher)

Ratio Word Problem Ex in

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

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

Scale Drawing: ratios and proportions: scale= the ratio of the size of the drawing to the size of the object

3.

Unit 2

4.1 Divisibility

Divisibility Rules:

2: a # is divisible by 2 only if the units digit is even ex: 542

3: a number is divisible by 3 when the sum of its digits is divisible by 3 ex: 876

4: a # is divisible by 4 when the last two digits of the number make a # that is divisible by 4 ex: 4520

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

6: both rules for 2 & 3 work, then the number is divisible by 6 ex: 12 (12/2=6 or 12/3=4)

8: a number is divisible by 8 if the last 3 digits represent a number divisible by 8 ex: 234800

9: a number is divisible by 9 if the sum of the digits of the number is divisible by 9 ex: 998

10: a number is divisible by 10 when the units digit ends in a 0 ex:100 or 250 or 490

Even #: a whole # that has no remainder when divided by 2

Odd #: a whole # with a remainder of 1 when divided by 2

4.2

Factorization: breaking a # down into smaller #s, that when combined back together gives you the original #

Prime Numbers: any whole number that is only divisible by 1 and itself. ex: 1,2,3,5,7,11,13 etc

Composite Number: any whole number that has factors that make up the whole. ex: 4,6,8,10,12 etc

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.

ex: 260= 26*10= (2*13) (2*5)= 2*2*5*13= 2^2*5*13

4.3

Greatest Common Factor GCF

of 2 whole numbers a &b not both 0, is the greatest whole # that divides both a&b

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

Least Common Multiple LCM

of 2 whole numbers a& b is the least non-zero multiple of both a&b

5.1 Integer

Integer: a whole number that is not a fraction

Absolute Value: the distance between the point corresponding to an integer and zero. ex: |-3| = 3

Number Line Model: https://youtu.be/3LUTYhmltQY

Properties of Integer Addition:

Closure: a+b is a whole unique number

Communicative Property: a+b=b+a

Associative: (a+b)+c=a+(b+c)

Identity: 0+a=a=a+0

5.2

Integer Multiplication

Chip Model & Number Line Representation https://youtu.be/fW3FWuLfpFc

Patterns of Multiplication

closure: ab is a unique umber

communicative: ab=ba

associative: (ab)c=a(bc)

distributive:a(b*c)= a*b+a*c

identity: a*1=a=1*a

zero: a*0=0=0*a

Integer Division

Using Chips and Number Line: https://youtu.be/lDR-B_OhUQo

Ordering Integers

less than: a<b

more than: a>b

equal to: a=b

6.1

rational #: a/b

numerator: the number above the line in a common fraction showing how many of the parts indicated by the denominator are taken

denominator: the number below the line in a common fraction; the whole

Proper fraction: rational # where a/b is less than 1 ex: 3/4

Improper fraction: a/b where it is greater or equal to 0 ex: 6/4

Equivalent Fractions: 2 fractions that can be simplified to equal one-another ex: 3/6= 9/18

Simplifying Fractions: 60/210= 6*10/21*10=6/21= 2*3/7*3= 2/7 (simplified fraction)

Comparing Fractions: 3/4 > 5/16: closer to a whole