Elementary Mathematics

Week 1/Day1 Homework on the Chicken and the Pigs (refer to submission) Warm Up- 1) 3x + 5 = 6x – 13 2)  3/5 + 1/53) 7.76 – 3.9 4) -17 + 435) Find 60% of 30 5)  (253) (45)7) 24 – 3/7 (refer to notebook)Day 2 Hw- Personality Check & the correct way to submit homeworkTen frames conceptsDay 3- Juggling Hw

Cordinality- Knowing how many objects there is already by counting the first timeOne-to-One Correspondence: the skill of counting one object as you say one numberMore/less: the different amountsSubitizing: is the ability to instantly recognizing the number of objects without actually counting them Bases:32 base seven- | | | : 6 6 6 2 = 20 base 1024 base six- ||:: 6 6 4 = 16 base 10

5 frames and 10 frames are tools for teaching math and help develop number sense. There are 5 spaces on top and 5 on the bottom. We use counters or math manipulative to represent numbers less than or equal to ten on the frame. To convert from a different base to base 10:I would first identify each number by flats, longs, and units. Depending on the base, I would then add each total amounts of flats, longs, and units there are. That would give me the total for base 10.For example: 312 base five to base 103 flats, 1 long, 2 units3 flats= 25+25+251 long= 52 units= 1+1= 82 base tenTo convert from base ten to a different baseI would first draw out the total amountthen I would place them into the base we are trying to convert tothen I would count how many flats, longs, and unitsfor example:8 to base three[] [] [] [] [] [] [] [] 8 in total[] [] [] 3= one long[] [] [] 3= one long[] [] 2= two units22 base three*we use commas when we writs large numbers so we can identify place values.a. 2(6) five- cannot make sense in base fiveb. (5)1 four- cannot make sense in base four

Examples:12 seven + 34 seven 1 long and 2 units plus 3 longs and 4 units is 4 longs and 6 units, 46 seven235eight + 126eight 2 flats, 3 longs, 5 units 1 flat, 2 longs, 6 units is 3 flats, 6 longs, 3 units, 363eight24nine + 41 nine 2 longs, 4 units 4 longs, 1 unit is 6 longs, 5 units = 65nine243 seven + 316 seven 2 flats, 4 longs, 3 units 3 flats, 1 long, 6 units is 5 flats, 6 longs, 2 units= 562 seven

What makes a good algorithm? Based in math principlesRepeatableEfficientExpanded form is based in math principle, is repeatable but not efficientexample: 347+26(300+40+7) (20+6)300+60+13=373Left-to-right is based in math principle, is repeatable, and is efficient example: 473+59473+59400+0= 40070+50=1203+9= 12=532Friendly numbers/trade off- end in zerosexample: 35+26+17+14+5540+20+17+20+50= 147Trade-off28+15=30+13=43Scratch Method is math based, is repeatable, and efficient. (look at notebook)Lattice method is not math based, is repeatable, and is efficient.(look at notebook)

Decomposing (expanded form) is math based, repeatable, and is not efficientexample:38-24 (30+8- (20+4)(10+4)=14317-146 (300+10+7)- (100+40+6)200 turns into 100 & 100100-40= 6060+10=70100+70+1= 171Left-to-right is math based, repeatable, and is efficientexample: 57- 2350-20= 307-3= 430+4= 34734-158700-100=60030-50=x4-8=x600-50=550550-8=542542+34=576Equal Addens is math based, repeatable, and is efficientexample:64-48+2. +266-50=16Subtracting in bases refer to notebook to look at references using diagrams. (w2D4 homework)

Why are multiplication facts so important? they are so important because students can apply their prior knowledge to further math contentBesides a student knowing 4x6=24, what is even more important?it is even more important that 4x6=24 is also 6x4=24 and 24/6=4. It is a fact and relating numbers.Why is it helpful to introduce ( ) as form of multiplication?it is important to introduce ( ) as a form of multiplication because signs such (x) and (-) can be confused in further math content and essentially "( )" refers to "groups of" as multiplication us grouping each factors. Best way for student to say this problem: 5(4) = 5 groups of 4Build2 (5) [] [] [] [] []. [] [] [] [] []3(4). [] [] [] []. [] [] [] []. [] [] [] []Array (refer to notebook)draw out an array lines crossingArea Model (refer to notebook)draw out a box and divide into four equal partsplace each factor left side and topmultiply the flats, longs, unitsthen add all together Expanded Form is math based, repeatable, and is not efficientexample12(16) (10+2)x (10+6)10x10=10010x2=206x10=606x2=12=192Left-to-right is math based, repeatable, and is efficientexample(23) (14)20x10=2004x20=804x3=1210x3=30=322Lattice (refer to notebook)draw out a box and divide draw lines across the box making triangles each factor on top and right side

To show how to multiply fractions we will draw boxes like when adding and subtracting fractionsFor example: (2/3) (1/2)Draw a representation of 2/3 with a boxand then instead of drawing a second box, for multiplication we will demonstrate 1/2 within the same box of 2/3 then shaded a half of the box. then count the boxes that are double shaded only. that amount over the total amount of boxes. Double shaded is the numerator the total of boxes is the denominator.

To build adding decimals we will use lines and dots to represent tenths and hundredthsfor example, 0.42 + 0.12 = |||| .. + | .. = |||||.... = 0.54To show adding decimals, we will draw a box and evenly draw 10 lines vertical and horizontal to make a hundredths. for example, 0.21 + 0.43, shade in 2 tens and one unit then shade in 4 tens and shade in 3 units.then add each shaded tens and units.To show how to subtract decimalswe will draw just like when we are adding, a box with 10 horizontal lines and vertical lines to make a hundredths. for example 0.35 - 0.13. we will take away a tenth from 0.13 by drawing a circle around and arrow away then take away 3 units with a circle and a arrow. then count each shaded tenths and units that's left. 0.22To show how to multiply decimals. we will do the same ^ a box with 10 horizontal lines and vertical lines to make a hundredths. for example 0.2 ( 0.3 ) shade in 2 tenths and 3 tenths one going vertical and another one going horizontal. the boxes that are double shaded is the answer. To solve decimals: 0.2 (0.4) multiply both tenths 2 x 4 = 8 then add the two zeros with the decimal 15 + 3.25 line up the whole numbers then add 4.6 - 1.2 subtract 4 - 1 and 6 - 210.3 (2.4) multiply 10 x 2 = 20 then place the decimal closes to 20.14.2 - 2.53 multiply 14 x 2= 28 then place the decimal close to 28.

To show how to solve subtracting fractions we all draw a box demonstrating the fractionFor example: 3/4 - 1/3 is said as 3 fourths take away one thirddraw a box representing 3/4 then draw another a box representing 1/3then draw the opposite fraction representations to each box- this is called transpose then cross out each the shaded boxes from 1/3 and cross out the same amount from the 3/4 boxesthen count each boxes from the 3/4 that are not shaded. put that amount that is not crossed out out of the number of boxes in total. 5/12

What is Order of Operations?the order by which we are required to do mathematicsLogical Orderits a processHow can we remember Order of Operations by?Levels order of Operations1. Grouping2. Exponents3. Divide/Multiply L to R4. Subtract/Addition L to R(PEDMAS does not work for all students and is not recommended) For examples: 6 x 2 + 5 x 4 We can create a 'story' like 6 chickens and 5 dogs how many feet? 6 chicken and each have 2 legs, 6 x 2 = 12 5 dogs and each have 4 legs, 5 x 4 = 20 we can add both total of legs 12 + 20 is 32 feetWhen we look at a problem like: 2 2-3 + (8 - 2 (3) ) 7 + 4 - 3 + 10 -2 (3)We will draw lines down for each sign we see and solve each sign that are togetherwe will create a upside triangle throughout the process from left to rightWe can use the mini diagram to solve integers solve the 3 negative with the exponentsolve -2 (3) = -6solve 8 -6 = 2 then bring the 7 down to multiply 2 x 7 =14solve the next exponent -3 2 = -9continue to solve the integers= 4Exponents: 2-5 = -25 2(-5 ) = -25 2-(5 )= -25 2(-5 ) = 25. two negatives will result a positive answer

ADDING FRACTIONSThere are three steps to adding fractions:Ensure that the bottom numbers (denominators) are the same. If they are not, change them so that they are the same (they have a common denominator). To change the denominators, find two numbers that multiply that equals the denominator. Do the same to the second fraction. Circle the same number and the other name, multiply it to the numerator and denominator of the opposite fraction. If there are whole numbers add them togetherOnce the denominators are the same, add the top numbers (nominators) and place the result over the common denominator.Simplify the fraction (if possible). We did not have to do this part yet. THE MORE PIECES I CUT THE WHOLE INTO, THE SMALLER THE WHOLE PIECES GETSFractional parts are equivalent partsSUBTRACTING FRACTIONSIF there are whole numbers subtract them. Like the process of addition, we need to find the common denominators to subtract fractions. It is the same step as addition. Multiply two numbers that equal the denominator and do the same to the other fraction but use at least one of the same factor. Then circle the same factor, the other number will be multiplied to the opposite fractions. If the second fraction is bigger than the first fraction, then the whole number will be changed, and the fraction must be equal to One whole. Upwards division might be involved if the denominator is smaller than the numerator. GREATER THAN> LESS THAN < EQUAL TO =To find which fraction is greater than, less than we need to look at the denominators and see which pieces are smaller or biggerwe also need to think about how many missing pieces are away from the denominators, if there a few pieces away then we can see which pieces are smaller or bigger.If the missing pieces are too much, then we can call it a anchor fraction, and multiply by a 1/2 (x2), this identifies if its less than a half or over a half.

The best way for students to say at this 3(5) is 'three groups of 5'but what happens when there is a negative sign in front of 3 and 5 '-3-(5)'to read this, students should say take away three groups of 5 negatives. to show you are taking away, draw around the signs as a circle with arrows demonstrating there are being taken awayto demonstrate if there is a zero bank, circle all together 0-3(-5)= 15(- - - - -) (- - - - -) (-----)+ + + + + + + + + + + + + + +

Alternative Algorithm Division 15➗5= 15 divided into 5 groups5 circles with three 'dots' in each group equals to the totalThe first algorithm is called Long division, the number to be divided into is called the dividendThe number which divides the other number is called the divisor Steps are : 425/25The first digit of the dividend (4) is divided by the divisor.The whole number result is placed at the top. Any remainders are ignored at this point.The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into.Now we subtract the bottom number from the top number.Bring down the next digit of the dividend.Divide this number by the divisor.The whole number result is placed at the top. Any remainders are ignored at this point.The answer from the above operation is multiplied by the divisor. The result is placed under the last number divided into.Now we subtract the bottom number from the top number.Bring down the next digit of the dividend.RepeatWhich is said this is not the best easiest method to teach and can be challenging for students to do because students can get a wrong estimation. The algorithm that students are learning and are being taught today is called Repeated Division. Repeated addition allows students to use multiplication facts they already know for a more accurate answer. Although the set up is the same as long division, there is an extra line added to this algorithm. steps are: Set up like long division but with a line going down on the right side of the set up.start multiplying with multiplication we already know, most of the time if the number is bigger like three digits, 10 is the easiest number to usekeep using multiplication fact throughout the divisionAfter the process, add all the "outside" number include any remaindersThe algorithm that was just taught to me completely new, is called Upward Division. Which in my opinion is most efficient and less confusion algorithm for division. For Upward division, the set up is how it is said. For example, 532 divided by 3. 532 on top----------3 on the bottomyou solve going 'upward'how many times does 3 go into 5 =11x3=3subtract 5-3 = 2the 2 goes next to the 3 =23how many times does 3 go into 23 =7 times3 x 7 = 2123 subtract 21 = 2 the 2 goes next to the 2= 22how many times does 3 go into 22= 73 x 7 = 21subtract 22-21 = 11 is the remainder the denominator is 3, the way how it is seen.

Divisibility Rules:1: x2: even #'s ex: 2,4,6,83: Add the digits, sum ➗ 3. ex: 4+7+2= 13 ➗3= x4: Look at last 2 digits, ➗ 4. ex: 78324 24 ➗45: Five ends in 0/5 6: Both 2 & 37: None8: Look at last 3 numbers ➗89: add the digits, sum ➗910: ends in zero

Integers: POSITIVE & NEGATIVE NUMBERSBuild -4 using 8 tiles |⚪️⚪️| this is considered a zero bank🔴🔴🔴🔴|🔴🔴|Zero pairsPositive on topNegative on bottomBuild 3 using 9 + + + |+ + +| zero bank | - - -|Tiles METHOD -The best technique, we used the red side to show negative and yellow for positive ex: +5 + (+1) = +6 + + + + + +-5 + (- 1) = -6. - - - - - -+5 +( -1) = +4 + + + + |+| |-| |+|-5 + (+1) = -4 - - - - |-|To show, I first look at the first number and see if it's a positive number or a negative number. Then put the amount of that number. Then look at the sign. If it's adding, then add the amount of the second number, if its negative the - goes at the bottom. This can create a zero bank. If it's a positive number add it to the amount from the first number. For addition, we create a Mini Diagram- this diagram was create by a student The mini diagram reinforces the big pile of tiles with either the positive or negative signs. Then determines which sign the answer will be. + (+. -) different signs means to subtract +3 + (-2) = +1 + |+ +| |--| -(- -) same signs means to add-6+ (-2)= -8 - - - - - - (+) - - +3+ (-4)= -1 | + + +| | _ _ _| _-2+ (-1)= -3 - - -Subtraction Practice- In order to use the mini diagram for subtraction, we must use the concept 'keep, change, change' means to keep the first number, then change the sign to the opposite, then change the second number to the opposite sign positive or negative. k. c. c. +5 - (+1) = +4 + (+. -) +5 + (-1) k c. c-5 - (-1)= -4 - (- +) -5 + 1= -4Multiplcation Practice 2+2+2 = repeated addition (..) (..) (..) = 3 groups of 2 = 3 x 2 = 6POSITIVE x POSITIVE = POSITIVEPOSITIVE x NEGATIVE = NEGATIVENEGATIVE x NEGATIVE = POSITIVEPOSTIVE DIVIDED BY POSITIVE = POSITIVENEGATIVE DIVIDED BY POSITIVE = NEGATIVE POSITIVE DIVIDED BY NEGATIVE = NEGATIVE