ELEMENTARY MATHEMATICS

All homework must be turned in as a pdf. When turning in homework that requires building, it must be scanned in and turned in as a pdf document. The zombie problem was to get our minds thinking with different solutions to a problem that might not be solvable. It was used to get us to at least try the problem, which is required for all problems within class.Polya's 4 step problem solving; read and understand the given problem, come up with a plan to solve the problem, implement the plan/attempt to solve, double check the answer (make sure answer makes sense).Early number sense; cardinality, one-to-one correspondence, counting, more and less, subitizing. Intro to base ten blocks; units (1) small cube, long (10) a single line of units, flat (100) a large square made up of longs.

Biz/buzz game was the teacher coming up with a pattern and asking the class to get in groups. We then had to figure out what the pattern was, then continue the pattern as a group (multiples of 7 were buzz). A 5 frame is used as a stepping stool for students to get used to a smaller number before moving up to the 10 frame. When a 10 frame is full students will begin to see that a full frame becomes one 10.The intro to other bases was how to determine what the blocks would look like in any base other than ten. eg. base 6; a flat would be 36 (6x6), a long would be 6 units, and a unit would still count as 1.

Homework for W2D1: review for how to BUILD converting numbers to a different base. Problems 1-2: How do you write 8 units in base 5? . . . . . . . . -> 8 units . . This shows one long of 5 (base) and 3 left over units. .. . answer: 13five .. How do you write 18 units in base four?. . . . . . . . . . . . . . . . . . -> . . . . This shows 4 longs of 4, which will convert into a. . . . flat. . . . . 102four. . . . .BUILD this number in the base provided: 23six . . two longs of 6 units & 3 units . . . . .. . .. . .. .BUILDING addition problems: 4six+5six . . .. + . = . . 13six. . . .. . . . . . .24five+33five. . . . .. . . . . . .. . . + . . . . = ◼️I . . = 112five . . . . . . .. . . . . .SHOW each number in their base: 35 -> ||| . . . . .62nine -> |||||| . . SHOW how to add in different bases: 3seven+2seven. . . + . . = . . . . . (5seven)32five+14five ||| . . + | . . . . = [] . (101five)SHOW how to add: 86+64 |||||||| . . . . . . + |||||| . . . . = [] ||||| (150)

Traditional algorithm ex:124+312=? 124 rewriting the problem vertically & solving from + 312 right to left= 436What makes a good algorithm? A good algorithm reinforces prior knowledge, repeatable as problem gets more difficult, and is efficient.ALT algorithms (ADD): expanded: 349+284 300+40+9+ 200+80+4= 500+120+13 = 633left-to-right:534+382 800 110+ 6= 916friendly numbers: 63+27Start by taking 3 from 63 and adding it to 27, this makes the numbers end in 0. 60+30 = 90trading off: 75+28Take 2 from 75, making it 73. Add those 2 to 28, making it 30. Then simply add 73+30 which = 103 (just getting one number to be friendly, as opposed to both)scratch: 354 276 5+4 = 29lattice:25 To solve this create a rectangle in the answer box, +48 create lines from each corner on the box that cross= through the entire thing. Solve as usual, 5+8, write that answer in the two triangles under those numbers and continue that process with 2+4. Then go from right to left, solving each diagonal until the answer is revealed.

BUILDING subtraction:Building subtraction is the same as addition. Although, sometimes you need to borrow. If the number on top is bigger than the number below, you should borrow from the next number. Breaking that number down into units so you are able to remove any and all units not necessary for the answer, out of the problem. SHOWING subtraction: Showing subtraction is also the same as showing for addition. Each 100s place is a flat, each 10s is a long and 1s are a unit. To show 35 it would look like this:||| . . . . .SHOW how to solve the following problems: 35-12||| . . . . . Remove one long by circling and drawing an arrow to show it is removed. Circle two units the same way. This will leave you with 2 longs and 3 units. || . . . or 2354six-14six ||||| . . . .Remove 1 long and 4 units. Leaving you with...|||| or 40sixexpanded form: 56-23 50+6- (20+3)= 30+3 = 3347-28 40+7 (17)- (20+8)= 20 10+9 = 19equal addends: 42-28 Whatever is added to one side must be added to+2 +2 the other, making the top # bigger than the bot.44-30 = 1452six-13six+3 +355six-20six = 35six

3x5 means there are 3 groups of 5ex of group: 3(5)(.....) (.....) (.....)Array: Everything repped in the problem in its own individual points, cannot be touching (introduces the shape of a rectangle)Area model: 12(13)Write out the expanded version of both numbers, one on top of a box and the other on the side. The box drawn will contain 4 separate sections for the answers. Multiply each number to the opposite and so on. Add the answers in the boxes to get your final answer.

BUILDING multiplication:To build multiplication, have an answer board to help show the answers in an organized way. Set up each number with base ten blocks that is being multiplied on the same corner of your answer sheet (on the outside border, as it is not part of the answer). Then build on your answer sheet, fill in the base ten blocks that correspond with the two numbers being multiplied. This should make a large rectangle. Count how many of each type of base ten blocks you have (flats are 100s and so on with the other blocks), add those numbers together and you will get your final answer. array (show):3(5). . . . .. . . . .. . . . .A good order in which to teach math facts from 1-10 is group 1: 1s, 2s, 5s, 10s. group 2: 3s, doubles, 9s. group 3: 6s, 7s, 8s, 4s. ALT ALGO:expanded: Same as the addition expanded set up. Multiply the same as you would double digit numbers in the traditional algorithm. 42x23 40+2 start with 40x20, then continue x 20+3= 800+120x` 40+60= 840+126 = 966left-to-right:37x56DONT FORGET THE 0s when multiplying 37x 56=1500 350 180+ 42= 2072area model: 12(13)Write out the expanded version of both numbers, one on top of a box and the other on the side. The box drawn will contain 4 separate sections for the answers. Multiply each number to the opposite and so on. Add the answers in the boxes to get your final answer.

Solving multiplication using lattice:Draw out a rectangle that corresponds with the amount of digits in each number. Then write the numbers surrounding the box. One number should be on top, the other one the RIGHT SIDE ONLY. Draw in the lattice lines and solve multiplying each number by each other. Add the numbers on the outside of the lattice to get the answer. SUBTRACTION:expanded:425-204 400+20+5- 200+00+4= 200+20+1 = 221equal addends: 53-27 53 +3- 27 +3= 56-30 = 26Add the same amount to each number!! Try to make one number a friendly number/the the top number should be bigger than the bottom (both digits bigger than their corresponding number on the bottom). 

The divisibility rules are 2s - even #s (end 0,2,4,6,8)4s - if the last two digits (together) are divisible by 4, then yes! 8s - if the last 3 digits (together) are divisible by 8, then yes!3s - if the sun of the digits is divisible by 39s - the sum of the digits is divisible by 9 6s - if 2 and 3 work, then 6 works5s - lasy digit is a 5 or 010s - last digit is 0Alternate algorithms for division: Repeated subtraction: 57/3 _3| 57 10 -30 +9= 27 =19 -27= 0 727/6 _6|727 100- 600= 127 20- 20= 7 + 1 - 6 = 1 =121 (1/6)Upwards division:83/5 8 - 5 3 - 30= 3 -------- 3 = 16 (3/5) 5

The numerator of the fraction is called the dividend, the denominator is called the divisor. SOLVING fractions:Addition:8+10(15/34) \ /18(15/34)(3/8)+(5/12)To begin cross out the 8 and break it down to 2x4, then cross out the 12 and break it down to 3x4. Next, add whichever number is missing to each side. In this case do (3/3) on the left side and (2/2) on the right side. This will make the divisor 24, giving you (9/24) + (10/24) = (19/24). Subtraction:10-(15/34)9(34/34) - (15/34) = 9(19/24)Comparing fractions: (7/11) > (15/34) - it is over half. (15/16) > (5/6) - this is missing one piece from the pie (small)

SOLVING fractions:Multiplication:(24/35)x(21/40)To begin break down each number into its factor. The numbers are as follows:3x8 x 3x77x5 8x5The next step is to find funky 1s. The 8s will become a 1, the 7s will become a 1. The left over numbers will multiply across, 3x3 and 5x5. The answer to the problem is (9/25). Division:(10/9)/(16/21)To begin with fractions, first do KCF. Then just like multiplication, you break down each number into its factor. 2x5 / 7x33x3 2x8You will then find funky 1s, the 3s become 1, the 2s become 1. 5x7 is 35 and 3x8 is 24, therefore the answer is (35/24).

There are 3 types of fractions (which is also how you build different fractions as well): Set model - needs to have different characteristics, anything that differs. (2 red blocks, 1 yellow block so 2/3 are red)Area model - comparing how much area there is of an object (1/3 would be built like a Mercedes sign and one section would be filled in)Linear model - compares distance (lengths)*(1/4) of the distance is green (class ex)*(1/4) of the pieces is facing downward Circle vs RectangleCircles are easy because they know what they are trying to make, it is easier to see what they are creating. SHOW addition and subtraction: Addition:(1/2) + (1/3)To solve this problem first draw a rectangle that shows (1/2), do this by splitting the box in half and coloring in one side. Do this same process with (1/3). After the boxes are drawn, fill in the missing lines from each box. Finally, draw a third box that contains all the lines from the previous two boxes and then fill in the coloring. To get your answer count how many boxes are filled in out of the full amount and that will give you the answer. In this case it is (5/6).Subtraction: (2/5) - (1/3)To solve this problem first draw a rectangle that shows (2/5), then a rectangle that shows (1/3). Once the boxes are drawn fill in the missing lines from each box. Use the take away method (drawing a circle around what is being removed with an arrow) to indicate how much is being removed. To determine the amount being removed, take the number of colored-in boxes from your second rectangle and remove that amount from your first. In this case the answer would be (1/15). Multiplication: (1/5)x(2/3)To solve this problem you only need to draw one rectangle. First draw the fraction (1/5) in the rectangle and then draw (2/3) in the same rectangle. It is ok if the two fractions overlap, they are supposed to. To get the answer to the problem, count how many squares inside the rectangle overlap. Take that number and put it over the total squares and that is your answer. In this case it is (2/15).

SHOW addition of decimals: Addition:0.4 + 0.3To begin this problem draw one square and divide it into 10 sections. This is done because 0.4 & 0.3 are in the 10s place. If there was another number after one of those you would need to make 100 sections. The next step is to fill 4 of the sections for 0.4 in one color and 3 for 0.3 in a different color. For addition the next and final step is to count how many sections are filled in. In this case it is 7, so the answer is 0.7. Subtraction:0.7 - 0.2To begin this problem again draw one square and divide it into 10 sections. If the numbers have a 100s place filled you must make 100 sections. In subtraction you only color in the amount of sections that correspond with the first number, so 7 will be colored. To solve you will need to take away the amount be subtracted. So 2 will be removed, circle and draw an arrow to show removal. However many are not circled and are colored in is the answer. In this case the answer is 0.5. Multiplication:(0.3)(0.5)To begin this problem draw one square and divide it into 10 sections vertically and horizontally because both number will be represented on one square. If there is a 100s place filled more sections will be required. Next color in the sections corresponding with the numbers given. Vertically 3 will be filled and horizontally 5 will be filled. The same as fractions, the overlap of the two numbers will determine the answer. In this case the answer is 0.15.

SOLVE: To solve in add/sub/multi of decimals, you should line up whole numbers!!Addition:4.25 + 10.3Start by lining up the numbers vertically in the same order. While lining up them up this way remember to line up the whole numbers as well. Then add in the traditional algorithm. 4.25+10.3= 14.25Subtraction: 42 - 8.14Repeat the same steps as addition for lining up. Then subtract as usual to get the answer. It may be easier for students to see the decimal if you add .00 to the end of 42. 42.00- 8.14= 33.86Multiplication: (hint: estimate)(3.15)(5.2)To multiply the first step is to look at the two numbers given and come up with an estimate of the potential answer. Since there is a 3 and 5 the estimate could be around 15. The next step is to multiply the numbers without the decimal point. Multiply as normal and the answer to that multiplication should be 16380. Add a decimal point in the answer that makes the number the closest to your estimate and this will be your final answer. For this problem the final answer is 16.38. 315x 52= 15000 500 250 600 20+ 10= 16380 ---> 16.38

Two color counters: red means subtract or negative number, white/yellow means add or positive number. To BUILD integers start with first number in the problem built with two color counters (from left to right), add or subtract two color counters to determine the answer to the problem. Create a zero bank (if needed), these are needed if you do not have enough two color counters to remove from your original number. They are mostly used only in subtraction. If you take away positives, the answer will be more negative. If you take away negatives, the answer will be more positive. SHOW integers: Addition3+(-2)+++ =1 --start with 3 positives and add two negatives. The positives and negatives will create a zero bank "canceling out". Whichever amount of the sign (positive or negative) that is left out of the zero bank is your answer.Subtraction: -6-(-4)------ =-2 With this subtraction problem, start with 6 negatives. Then you simply take away 4 negatives by circling and drawing and arrow, to show you are "taking away". Therefore the answer will be -2. -3-(-4) +---- = 1With this subtraction problem, start with 3 negatives. Since there are not enough negatives to take away, you have to add a zero bank in order to remove 4 negatives. After circling the 4 negatives you are left with 1 positive. This is the answer.

SOLVE integers: Addition To solve integers, we use a mini diagram (Hector's method). 35+(-24)++ - -> Subtract35-24= 11Determine which number is bigger. In this case, there will be two +s under the 35 and one - under the 24. Circle one sign in each, if the signs are different subtract, if they are the same then add. Whichever sign is out of the circle is what your answer will be.Subtraction:34-(-16)34+16++ + Add34+16= 50For subtraction the steps are the same, before you start the steps keep change change the signs.

Multiplication:BUILD: -2(3): take away two groups of three0-2(3), if the first number is negative you need a zero bank.SHOW:2(3)(+++) (+++) = 6-2(3)(+++)(+++) = -6 -----For this method, (problem 2); it is telling you to take away 2 groups of positive 3. Therefore, you will circle and take away the 2 groups. Since you have to create a zero bank, left over is 6 negatives. Making your answer -6. SOLVE: using rulesFor multiplication and division, same signs equal a positive and different signs equal a negative.-32(-12)same sign = positive384(-40)/10different sign: negative-4

Order of Operations: GEMDAS grouping, exponents, multiplication, division, addition, and subtraction. multiplication and division, addition and subtraction are interchangeable reading left to right. Addition and subtraction signs mark groups for grouping. 5x4/2x6/5x3 -> start at mult/div and go left to right.20/2 10x6 60/5 12x3 36Scientific Notation:Scientific:_____x10 raised to an exponent-3.42x10 raised to -40.000342When there is a negative exponent the number will be small2.5x10 raised to 152,500,000,000,000,000Standard: 0.0000021672.167x10 raised to -62,143,000,000,000,0002.143x10 raised to 15