weeks 1-3

-Lattice Addition-Partial Sum-Left to Right-Expanded Sum-Partial Difference-Equal Addition-Traditional Algorithms-Scratch Method-Compatible Numbers- "Friendly Numbers" (estimation)-Front end Estimation-Friendly Numbers (compatible numbers)

-Units-Longs-Flats-Cube-Manipluations

UnDevCarLoUnderstand the problemDevelop a planCarry out planLook back at work

Conjecture: Ones values will always be less than base?FalseExample: 6 base to four | .. = 1 long 2 units= 12 four *Must always add units even if a number is not presented* Example: 15 to base five||| = 3 longs 0 units= 30 five*You can never have your numbers (units, longs,flats, etc.) be bigger than your base*

Example:Convert 23four to base ten Step 1: Separate them to longs and units 23 four 2 longs 3 UnitsStep 2: Multiply the number of longs with the base given 2(4) 3(1)Step 3: Add the values given after multiplying 8+3= 11Step 4: Answer is the value after addition 11 Units

ExampleA rectangle with 4 longs and 4 units. Length * Height __ 4(10+1)__ 40+4__ 44 __....

Traditional 47-12=35Partial Difference 47-10=37 37-2= 35Left to Right 40+7-(10+2)----------30+5= 35Equal Addition (we want to make the number in the zero state so it is easier to subtract) 47+8 55-12+8 -20 ----- 35

Adding Algorithms- The best method for children to learn about the value of ones and tenth are as follows: 1. Expanded Sum 2. Partial Sums 3. Left to Right 4. Lattice Addition 5. Traditional Examples: Traditional Algorithms 24+15=39Partial Sums (Left to Right) 24+20=34 34+5= 39Expanded Sum 20+4+10+5---------30+9=39Lattice Addition(helps with knowing where to place the tenth value) 24+39-------|0/3|0/9|----------- 39

-Scratch Method- Compatible Numbers "Friendly Numbers" (estimation)Groups of tens to add easier-Front-End Estimation (start to learn how to round) Example: 31+24+15+42+39 30+20+10+40+39=130-"Friendly Numbers" Compatible Numbers

Example(23 base five) + (21 base five)= 134 units4 longs 3 units 4 longs 1 unit|||| ... + |||| . 1 flat 3 longs 4 units

Area Model: Create blocks and use the area of each to add to get the entire area.Lattice Form:Same format as the addition lattice form. Only difference is multiplying the numbers instead of adding them. Expanded Form: Expansion of numbers 27x31(20+7)x(30+1)

Long Division: traditional division*note:to help a student that is having trouble setting up the format of a problem from this 68/2 tell them to create house. When they have trouble with what number to put outside and inside tell them the first number goes IN and the the second number goes OUT. In-&-Out!*Repeated Subtraction: see YouTube link Upwards Division: you format the problem like a fraction. You place the first number on top and the second number at the bottom.*Note: Upwards division helps students with making fractions from the remainder*

Prime Factorization Methods*Important Note: To find the GCF/GCD you use the common numbers and use the smallest exponent to multiple them.**Note: To find the LCM/LCD you use all the factors that have the biggest exponent and they do not have to be common numbers*Factor Trees: you stop factoring once a all the numbers are primeUpwards Division: All the numbers on the side have to be primeVenn Diagram and Double Bubble: Places all equal numbers in the middle and the different numbers for each set on a different side.

Divisible by:By 4: Look at the last two digitsBy 8: Look at the last three digitsBy 5: Look at the numbers ending in 5 or 0By 10: Numbers ending in 0By 3: Add numerals= needs to add up as a number to be divided by threeBy 6: If number is divisible by 2 and 3 then its divisible by 6By 9: If number is divisible by 3 then it's divisible by 9

Greatest Common Factor/ Divisor: Finding the greatest factor that are in common with both number sets. Note: GCF are the smaller numbers. Factors: numbers that multiply equally to each other. Example the GCf of 2 and 3 is 6. Least Common Multiple/ Denominator: a common multiple of is a number that is a multiple of the set of numbers given. Note: LCM are the bigger numbers. Multiples: number that can be divided equally without a remainder. Example 6 is a multiple of 2 and 3. Prime Numbers: a number that is greater than 1 and the only two factors are 1 and itself. The number 2 is the only even number that is prime.Composite Numbers: numbers that can be divided evenly by numbers that are not only 1 and itself.

Adding IntegersDrawing: Use if the number is less than 10. Note: Use norms when drawing out the problem. Keep all positive on top in a line and all negative on the bottom and in a line as well.Ex. 2 add -4++- - - -Diagram: Use if the number is more than 10. Ex. -15 add 436Step 1. Place the negative or positive on the top of the numbers. You add two signs to the number that will create the longer row and one sign that will create the shorter row. - ++-15 +436Step 2. Circle one sign of each number. If the signs in the circle are a positive and negative you subtract and if the signs are two negatives or two positives you use addition.- sub.++-15+ 436Step 3. After you add or subtract, bring down the sign that is left over since that will determine if the number is positive or negative. 436-15=+421Number Line*Note: Some problems will need zero pair/bank to solve.*Example: -4-1=-5(Color Key: orange is zero bank, red is eliminated, black is answer)+++++- - - - - - - - -Subtracting IntegersKeep Change Change Example. -39-(-156)= +117Step 1. -39+(-156)Step 2. -39+(+156)Step 3. -39+156Step 4. Do regular steps that you would do for diagram

Integers: All whole positive and negative numbersZero-Pairs: A positive and negative number (usually 1) that cancel each other out to make zero.Zero-Bank: Grouped zero-pairsKeep Change Change: used for subtractionAbsolute Value: measurement of distance away from zeroExample. |-3|=3Example. -|-5|=-5 *The sign in front of the absolute value says "the opposite off"

+ and + is + + and - is -- and - is +- and + is -Rules: when signs are the same it's positive. When signs are different it's negative. *same rules as multiplication apply in division**When doing a problem if the coefficent is negative, use a zero bank**if the number is bigger than 10 don't show. if it is less thant 10 draw out*

Properties: makes doing problems easierAssociative: order doesn't change who you "associate" with change(Parenthesis Exponent Multiplication Division Addition Subtraction) ex. (4+2)+5 -5+(5+3) 4+ (2+5) (-5+5)+3Commutative: the order can change*doesn't work for subtraction* ex. 5+7 7+5Distributive: multiplication *multiplying 2 numbers means finding area**multplying 3 numbers means finding volume*

Distance FormulaGeneric Rectangle

Scientific Notation: number that is bigger or equal to one. It is also multilpied by the power of ten.Standard Form: numbers that are written in standard form Example: 300, 1, 0.17, 45

Rational Numbers: numbers you can write as a ratio (fraction) ex. 3, 4.12 Irrational Numbers: numbers that cannot be written as ratios (fractions) Note: when a number is being squared (inside the house) and is negative it has no solution as it leads to imaginary numbersNote: x^2=5 5= +/-5

Factoring Rule: Be careful where the sign is placed; it goes with the digitDifference of Square*be careful of where the sign goes in the box**A B C terms: used to describe the numbers of the problem*

Area Model: show a visual representationof the product of two fractions. Helps with multpliying and reducing.

Linear Model: Lenght of whole is divided into equal lengths

Set Model: Using individual shapes to match fractionsCan use: Two colored counters Cubes Foam Shapes Etc. *create a "1" fraction such as 2/2 or 9/9*

Equivalent Fractions: have same value although they look different.

Simplifying Fractions: fraction is in the simplest form when the top and bottom cannot be any smaller.

Why do we need to find common denominators when we add/subtract fractions, but not when we multiply/divide?Makes it easier to easier if they have same numbers

1.)We need to find the LCM/LCD of the fractions. An easier way to find them is to write the factors that equal the number below it. 1/6 + 1/42*3 2*22.)After, multiply the fraction with the number that isn't the same (2/2) 1/6+ (3/3) 1/43.)This equals=2/12+3/12=5/12*This rule applies when subtraction with unlike denominators. The goal is to find the same denominator*

*This rule for subtraction and addition can be applied in basic mathematics along with upper mathematics such as algebra* Basic Math:Problem: 3/5+1/6Find LCM: (6/6)3/5+(5/5)1/6Add: 18/30+5/30=23/30Problem:18- 4/7Find LCM: 17 7/7- 4/7Subtraction: 17 3/7Problem: 27+4/9Addition: 27 4/9Algebra: Example: 3/5x+ 1/6x(6/6) 3/5x+ (5/5) 1/6x18/30x+ 5/30x= 23/30xExample: 7/(x+3) + 2/(x+5)(x+5/x+5) 7/(x+3) + (x+3)/(x+3) 2/(x+5)7x+35/x^2+8x+15 + 2x+6/x^2+8x+15= 9x+41/x^2+8x+15

Find the ones is an alterntaive way for students to simplify the fractions before multiplying. You are able to simply them as a fraction and acrosso from one anotherExample: 22/5 *65/11 2/1*13/1= 26

*Note: If you have an example as 1/3(1/2) to find the answer you do the rectangles and shade them in as we did in last week's lesson.The box that has two colors shadedin is the numeritor.*

10/2- What is it?- How many times 2 goes into 10 evenly-Total number divided by # of groups= # in each (FULL) groupKCF ( Keep Change Flip)1.)When dividing numbers you need to change the sign to multiplication and flip the fraction.*Also if you have a mixed fraction you need to turn it in an improper fraction before continuing*2.) Find the ones before cross multiplying3.) Cross multiply*if you successfully found the ones then your answer is in the simplfied form *

Back wards CMultiply the whole number and denominator and then add the product to the numerator.You can also show the fractions as indicated above and in the modeling subtopic .

Parenthesis Exponent Multiplication Division Addition Subtraction*There are four levels in order of operation Groups of number, exponents, division/multiplication, subtraction/addition*GEM DA S