MTE280🧠🗺

Content: Add, Subtract, Multiply, DivideProcess: Problem-solving 12 SticksMove 3 sticks to make 3 boxes Trial & Error => Problem solving => NOT very effective method Move 4 sticks to make 3 boxes Bc you know the first one you tend to do the second one faster. Tug-of-war: Acrobats, Grandma, & Ivan R1: 4A vs. 5G = Draw * Use colored blocks to R2: I vs. 2G = Draw demonstrate R3: I + 3G = ????I + 3G WIN OR R1: 4A = 5G A=5 4(5) = 5(4) => 20 = 20 (20 = 20) I=13 I = 2(4) + 1 (5) => 13 = 8 + 5R2: I = 2G + 1A ````` G=4 13 + 3(4) = 4(5) => 13+12= 20 R3: I + 3G < 4A Discard the Old Books ProblemMr.Smith is getting rid of books by class period1st prd. could have 1/6th of the books 2nd prd. could have 1/5th of the books left3rd prd. could have 1/4th of the books left4th prd. could have 1/3rd of the books left 5th prd. could have 1/2 of the books leftThis left 14 books6th prd. took to leave no books. How many books did he have before?

Day 1: Numeration Systems :Ways to record quantity -Problem-solving is a process -US system is a Base-Ten (Decimal systems) 1:10 Relationship -Position system: where a number sits indicates it's value Ex: 1,111-The number gets bigger when the number moves left x10 -The number gets smaller when the number moves right x10Decimals -Separate the function -ALWAYS sits right of the unitExpanded Notation375= 300+70+5 = (3x100)+ (7x10)+ (5x1) =(3x102)+ (7x101)+ (5x100)Base-5 Base-10Ones : 50 Ones : 1005's : 51 Tens : 10125's : 52 Hundreds : 102125's : 53 Thousands : 103Ex: 2325= (2x25)+(3x5)+(2x1)=(2x52)+ (3x51)+(2x50)= 50+15+2 = 67Digits usedBase-10: 0,1,2,3,4,5,6,7,8,9,10Base-5. : 0,1,2,3,4,105,115,125,135,145,205Example: Use Base-10. a) 305 b) 2,315 c) 203= 300+00+5 =2000+300+10+5 = 200+00+3=(3x100)+(0x10)+(5x1) = (2x1000)+ (3x100)+ (1x10)+ (5x1) = (2x100)+(0x10)+(3x1)=(3x102)+(3x101)+(3x100) = (2x103)+ (3x102)+ (1x101)+ (5x100) = (2x102)+ (2x101)+ (2x100)Example: Use Base-5a) 333 b)43 c)1010=(3x25)+ (3x5)+ (3x1) =(4x5)+(4x1) =(1x125)+ (0x25)+ (1x5)+ (0x1) =(3x52)+ (3x51)+ (3x50) =(4x51)+ (3x50) =(1x53)+ (0x52)+ (1x51)+ (0x50)=75+15+3 =93 =20+ 3 =23 =125+ 0+5+0 = 130

Base-3ones : 303's : 319's : 3227's. : 330,1,2,103,113a) 10123=(1x27)+ (0x9)+ (1x3)+ (2x1)=(1x33)+ (0x32)+ (1x31)+ (2x30)=27+0+3+2 +32 b) 2213=(2x9)+ (2x3)+ (1x1) =(2x33)+ (2x32)+ (1x31) =18+ 6+ 1 = 25c) 11013=(1x27)+ (1x9)+ (0x3)+ (1x1) =(1x33)+ (1x32)+ (0x31)+ (1x30)=27+9+0+1 =37Base? w/ Decimals 21.15 21.115 21.13 21.113 =1/5 =1/25 = 1/3 = 1/9Base-8 Ones : 808's : 8164's : 82Compare (< > =) a)2113 ? 31015(2x9)+(1x3)+ (1x1) (3x125)+ (1x25)+(0x5)+(1x1)=18+3+1 =375+25+0+1=22 > =401b) 10213 ? 10215(1x27)+(0x9)+(2x3)+ (1x1) (1x125)+ (0x25)+(2x5)+(1x1)=27+0+6+1 =125+0+10+1=34 > =126130 into a base-5 number 1,5,25,125 130 there are one 125's in 130 so there would be one 5's. in that case it would be 10105-125 519 into a base-3 number 1,3,9,27 19 there are two 9's which equal 18 and 1 left so 2 9's zero 3's and 1 one's = 2013-.910-9= 1

American Standard right to left Use this as the last step to teach subtraction student won't fully understand why to cross out the numbers they need to understand where they are getting the numbers from European/Mexican right to left instead of adding carried numbers to the top add it to the bottom numbers Reverse Indian left to right start with subtracting the hundreds then subtract the tens and the ones cross out as your going down then add the last numbersLeft to rightleft to right start with hundreds and write then in the 000 then do tens 00 and ones 0. cross out the underside if they need to be reduced Integer subtraction Right to left not 4 elementary start from the ones then to tens, then hundreds. then reverse subtraction with negative numbers ex: 300-10-3 and you will get a positive number Expanded notation place-value explicitinstead of subtracting in the standard form you would have to change the numbers and change them and give them their value Ex: 576 to 500+70+6

2 x 5 =10 |_ _| |_ Product FactorsStandardmultiply like normal and then addWe add because the top number is 176 five times and the bottom is 176 twenty timesex: 176x25Place value 176x25 5xx6=30`` 20x6=1205x70=350 20x70=14005x100=500 10x100= 2000=880 =3520 Add 880+3520 to get 4,400Lattice almost like a punnet squareadd diagonally and read left to right

Types of symbols division signsdivision bar aka vanulum or the fraction bar ( ?/?)There are multiple things apart of a division quotient is the answer dividen is the number inside the vanculum remainder is what is left over and can't be divided anymoreEx: 11 cookies and 3 plates 11/3 is 3 remainder 2. The cookies can be divided onto 3 plates and have 3 cookies on each plate. There would be 2 cookies leftLong division -Standard just divide the American way most common way that is taught alot of kids don't know why they do somethings -Place Value more exphasise on the places and where they are coming from Alternate algorithm just do multiples of 10 instead of having to do all the work much faster

a is divisible by b if there is a number c that meets this requirement c x b = a10 is divisible by 5 ... 2x5=10Terms: 10 is divisible by 5 10 is a multiple of 5 5 is a divisor of 10 5 is a factor of 10 Divisibility rule: Ending 248-by 2: 0, 2, 4, 6, 8 -by 5: 0, 5-by 10: 0sums of digits 248 = 2+4+8=14-by 3: if sum of digits is divisible by 3-by 9: if sum of digits is divisible by 9-by 6: if it is divisible by both 2 & 3 228 ends w/ 8; 2+2+8=12 12/3= 4Last digits-by 4: # is divisible by 4 is last 2 digits form a number that is divisible by 4 344: 44/4=11 228: 28/4=7-by 8: # is divisible by 8 is if the last 3 digits are divisible by 8 have to do long division -by 7: double the last digit : subtract the doubled digits with the remaining number : check to see if it is divisible by 7-by 11: AKA chop method : chop last 2 numbers and add it to the remainderg number and repeat until the smallest number is divisibly by 11

24: 1, 2, 3, 4, 6, 8, 12, 24, (composite: Even #) 7: 1, 7 (prime: Odd #)Why do we use factors? -To express the product of numbers of prime -Used for factors like GCF or LCF 36-multiples of of 36 are 9 times 4factors of 9: 3 & 3 (3x3=9)factors of 4: 2 & 2 (2x2=4)=2x2x3x3=22x32Prime Factor Method: 36= 6x6 24= 12x26=3x2 26=3x2 12= 6 (3x3) x236: 2*2*3*324: 2*2*2*3GCF: 2*2*3 =12LCF: 12 (2X2X3) *2*3List method: 24: 1,2,3,6,8,12,1436: 1,2,3,4,6,9,12,18,36Multiples: 24, 48, 72,... 36, 72,...⭐️We need to know this for when doing fractions Fractions GCF is used when we simply LCM is used to find the common denominator 11/12 is the same as 22/24Greatest Common Factor (GCF) The Prime Factor Method find two multiples keep finding multiples until you end with an even number or odd number ex: 36 6-3 & 2 6- 3& 2 ```` 24 2- 2 12- 6-3&2 2 36=2x2x3x3 24= 2x2x2x3 -There are two sets of 2 and one set of 3 -So we multiply 2x2x3 which equals 12= GCF -To find the lowest common factor we multiply the GCF and the ones that don't pair so 3x2x12 which equal to 72 so that is the LCM

1.) Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took 1/4 of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get? So I did 1/4 * 1/3 and got 1/12, this told me that I have to split it into 12ths to find how many bars there were. I then made a diagram split into 4ths and in each 4th I divided it into 3rds. So with this 11/12 bars were shaded. So the last box not shaded was worth 4 bars. So all the boxes are worth 4 bars and so 12 * 4= 48. So there was 48 bars in the bag originally. Jim got 12 bars, Kim got 16 bars, Len got 16 bars and max got 4 bars.2.) Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took 1/4 of all the bars. Then Ken took 1/3 of the remaining bars. Next, Len took 1/3 of the remaining bars, and Max took the remaining 8 bars. How many bars were in the bag original? How many bars did Jim, Ken, and Len each get? How is this problem different from problem 1? So we drew a diagram once again with 4 squares and shaded 1/4 of the bars because Jim took them. Then I did there it was split into 3rs and Ken took a 3rd. Then I divided what was left into sections of 3rds. After since 1 section represents 2 bars then we multiple by the 12 sections and we get 24 bars together. After we count the shaded parts and Jim took 6 bars, Kim took 6 bars, Len took 4 bars, and max took 8. The difference is in one case we are working with parts of the same whole and the other we are working with parts of what's left. The video attached is an example video of another problem explaining it mathematically with numbers and using a diagram.

One to 10 base system: everything RIGHT of the decimal is part of a whole everything LEFT of the decimal is a whole getting 10x bigger0.123- one hundred twenty three thousands 0.003- three thousandthsthe 0 holds the ones & tenths place$1.03 = 3/100 = 3 pennies = 0.003 money is a manipulative for decimals 3/10 is 3 dimes = 0.3 or 30 pennies the 0 at the end of the decimal doesn't change anything 0.8 or 0.95 0.9 or 0.85 8/100 < 95/100 0.90 > 0.85⭐️when comparing decimals use a grid, money, or drawingsthey should know place values72/100: 0.72 -Seventy two hundredths 31/1000: 0.031 - thirty one thousandths the 0 is important bc it holds the place value of the tenths5/10: 0.5 -five tenths 2/5 = 4/10 easier if fraction has 10, 100, or 1000find equivalent fractions w/ 10 or 100-Terminating Decimals : DO NOT repeat -Repeating number: repeats the same number -Irrational number: never ends -ex: 3.14 (pi) not a decimal -doesn't repeat or terminate

Our common number line is the line that is horizontal⭐️The best way we can help students with integers is if we use a vertical line that way students can make a connection with it like stairs or the floors on a building. "chip method" r= (-) Y=(+)R&Y= zero pair/ They cancel outAddition(+5)+(+1)= +6(-5)+(-1)= -6(+5)+(-1)= +4 (see notebook for diagrams)(-5)+(+1)= -4Subtraction(+5)-(+1)= -4(-5)-(-1)= -4 (see notebook for diagrams)(+5)-(-1)= +6(-5)-(+1)=-6multiplication(+3)x(+2)= +6(+3)x(-2)= -6(-3)x(+2)= (+2)x(-3)= -6 (COMMUNITIVE PROPERTY)(-3)x(-2)=(-2)X(-3). Not possible sooo... read (-3)x(-2) as the opposite of (-3)x(-2) which is (+3)x(+2) which equals +6

Inverse operation3+4=7 2x3=67-3=4 6/3=27-4=3 6/2=3Division +6/+2=+3 (see diagram in notebook)+6/+3=+2

Polya's 4 Steps to Problem-Solving 1.Understand the problem What do you need to find? Restate the problem? 2.Devise a plan Guess & Check -> Trial & Error Draw Diagrams/ Pictures Look for a pattern 3.Carry out the plan This is easier than planning BE PATIENT! -Most problems aren't solved quicklyBE PERSISTENT! 4.Look back (reflect) Does the answer make sense? What did you learn? 7 people are strangers and want to shake hands with one another. They must all shake hands ONCE. Multiplication is REPEATED additionthree groups of two2 + 2 + 2 = 63 x 2 = 6🚩When kids use the first 2 number in a word problem it means they do not understand the task/ problem.

Four Operation: concept and properties Addition Identity property -When you add zero to any problem and nothing changes a, a+0= a ex: 7+0=7 Community property: (AKA order property) -For any two numbers (a,b) the order does not matter a,b a+b = b+a ex: 3,4 3+4 = 4+3Associative Property (Grouping) -The way you group any number does not matter a,b,c (a+b)+c = (a+)b+c) ex: 1,2,3 (1+2)+3 = 1+(2+3)Subtraction Take away ex: 4 take away 1 =3Comparison -"How many more...?" KEY PHRASE ex: June has 5 cookies and ate 2. How many more does she have? Missing Adden (more of a addition problem) -Evon has 4 cookies, mom gave her more. Now she has 7. How many did mom give her? ex: 4+ ??= 7 -Naturally students will add BUT they should subtract 4 from 7Multiplication (repeated addition) :groups of things 3x2= 3 groups of 2 = 6 2 + 2 + 2 (repeated addition) 2 4 6 ( Skip counting) Identity Property a, ax1= a 7x1= 7Community Property a,b axb = bxa 7,2 7x2 = 2x7Associative Property a,b,c (axb)xc = ax(bxc)Zero-Property -Any number multiplied by 0 equals zero a, ax0=0 bx0=0 cx0=0Distributive Property a,b,c ax(bxc ) = (axb)+(axc) 3+3+3+3 ‼️read TOP to BOTTOM‼️

Addition Algorithms-American Standard (R to L) 1 1 576+279 855-Partial Sums (R to L) 5|7|6+2|7|9 1|5 1|4+ 7|. 8 5 5-Partial Sums (B) ⭐️ -emphasis on place value (R to L) -Start with explain what the values are and where they go DO NOT start with the 'standard' way until they know where the values go 5|7|6+2|7|9 |1|5 1|4|0+7|0|0 8 5 5-Left to Right 576 *read as 500+200= 700, 70+70= 140, 6+9=15+279 700 140+ 15855-Expanded Notation -place value explicit 100 10 576 500 +70+6+279 + 200+70+9855 = 800+ 50+5Lattice 5 7 6 * Diagnals add (4+1= 5) +2 7 9 |0/1/1/ /7/4/5| 8 5 5

Fractions: An expression :relationship between a part and a whole 3 <- how many pieces were taken 4 <- how many pieces of a whole is divided intoWays fractions can be used: Part of a whole: 3/4 Quotient: 3/4 Ratio 3:4 (most complicated) Sometimes a fraction is a ration... Vice versa example: 15 boys and 5 girls = 20 total 15/20 is both a ratio and fraction 5:15 Ratio but not a fraction (5) part of a whole Models Area (surface) length (line model) set (groups of things) ⭐️Four Operations before fractions is a NO NO student will add across will look at denominator as a whole 1/2 +2/5 = 3/8 is a NO NO]1/2 < 1/8 will think 1/8 is bigger because 8 is smaller than 2-When numerate and denominator have the same number it is a whole (1) doesn't matter if its a variable a/a=1. xy/xy= 1-Fractional parts are equivalent parts -The more pieces I cut the whole into the smaller the pieces get

Folding Paper Activity 1/2 -1 second -1 twoth students will refer to 1/2 as 1 second or 1 twoth a/a * 2/2 is not multiplying by 2 it is multiplying by 1(2/2) Simplifying 25/100divide by a common factor so 5 =5/20 or 1/4How is the factor 5 different than 25? 25 is the greatest common factor When adding numbers you need to find the common denominator. add numerator and leave the denominator ⭐️Kids will add both numerator and denominator ⭐️multiply numerator by numerator and denominator by denominator -When you multiple fractions they will get SMALLER

2.) Marc opened a pizza box. Inside is 3/4 of a pizza. Marc ate 1/2 of what was in the box. How much did he eat? divide the pizza up into 4ths then divide each 4th into halves which gives us 8th. So we know Marc ate 1/2 of the 3/4th pizza So Marc ate 1.5 3/4th slice of pizza which is the same as 3/8 slice of pizza SO ultimately he ate 3/81.) Janice is prepping a recipe that calls for 3/4 of a cup of oil. If Janice needs to prep 2 and 2/3 servings, how many cups of oil will she need? 3/4 * 2/3 = 6/12 = 1/2She will need 1/2 cups of oil 3.) If **** represents 2/7 of a whole, draw what the whole looks like. **** = 2**= 1So, **************= 7/714 * equals a whole ⭐️Use a rectangle instead of a circleThe video attached is an example video of another problem explaining it mathematically with numbers and using a diagram.

A student takes a test with 45 questions and gets 37 questions right. What percent of the test did she get right? you would do long division and divide 37 by 45you would get .8222 repeating and so you would round up but since the 3rd 2 is not higher than 5 it would stay at a 0.82. to turn 0.82 into a decimal you move the decimal 2 times to the right causing you to get 82%. So the student got 82% on the testA factory makes sandals. If they produce 820 sandals a day and 32% of them are blue. How many are blue? For this problem you would multiply 820 by 32%firs you change the 32% into a fraction by moving the decimal two times to the left so it would be 0.32Then you just multiply 820 X .32 which equals 262.40So since you can't have 262.40 sandals you would have 262---------------------------------------------------------------------------------------There are three types of problems: a) 8 is what percent of 22% 8=nx22n=8/22= 0.36=36%b) 8% of 22 is what number0.88x22=nn=1.76 c) 8% of what number is 220.08xn=22n=22/0.08= 275"is" means = "of" means to multiply "what" means the variable of n change % into decimals and"what percent" write the decimal as a percent