Elementary Mathematics

Week 1:This week began with me learning to navigate the course and how to submit assignments. I borrowed materials from my elementary classroom. (Remember to return them at the end of the semester!)I watched the assignment videos and worked on the pigs and chickens math on Playposit.Common methods used to solve equations:(none are wrong)Guess/CheckDiagramListAlgorithmPloya's Method: UnDevCarLo -Mr. MiltenbergerUnderstand the problemDevelop a planCarry out the problemLook back to check answer I like the video inserted in my Mindmap about Guess and Check. This is something I could use with the elementary students I work with as a para.

This week, M2, was about using 5 and 10 frames to assist students with addition and subtraction. Adding say 4+5 could be made easy with putting 4 tiles on a 10 frame, then adding 5 more, (of another color), to the ten frame to get 9. This module introduced me to a new concept to me, bases and converting bases. This became easier for me with practice, though I'm not certain the purpose yet. When drawing a diagram, use dots for units, a viertical line for longs, a square for a flat, and a cube for a cube.The base represents the number of units in one long, and how many longs in a flat, and lastly, how many flats in a long.2120 with base four=1522 cubes1 flat2 longs0 units14 base five, would be one "long," and 4 "units" with base five. This lets us know there are 4 units in the long, plus the 4 units. The sum is 9.

The third week of topics dealt with Alternative Adddition Algorithms. These included writing the equations in Expanded Form, Left-to-right, converting to "Friendly Numbers", and using the Trading Off, Stratch, and Lattice Methods.Example of Expanded Form:349 + 284 =300 + 200 + 40 + 80 + 13500 + 120 + 13 = 633Left-to-Right:2843 + 5139 = 7000 900 70+ 12 7982"Friendly Numbers" and Trading Off:63 + 27 =Take 3 from the left side and move it to 27, making 60 + 30 = 90Scratch Method:When adding vertically or horitzontally, you add in groups of ten, or whatever base you are in. Draw a line through the number your take from and start again with the left over amount adding up to ten again. The remaining amount is the number you write. Each scratch line represents ten. 3 5 4 you scratch out the 4 to make ten, then you start 7 with the 2 left over and add ten again. 6 2 and 7 make 9, then scratch the 6. You took 1 from + 4 6 to make ten, so it is now 5. 5 and 4 are 9. 29 Lattice Method:It is similar to the typical way of addiing numbers. 25+ 48You draw boxes under each place value. Then draw a slash from the upper right corner of each box diagonally down through the bottom left corner. The boxes are then divided into two triangles. The lowe is the ones place, the upper the tens. Then you carry over as necessary in the boxes and add along the diaganol lines. Trading Off with various bases:If using the Trading Off method when adding in bases other than ten, pay close attention, and add in the base you are in!3 base seven + 4 base seven = Take 3 from the left, add it to the 4 to make 7 (which is your base). Now you have 0 units on the left side and 1 long (made of seven units) and no units on the right:0 + 10 = 10 base seven

This week was about Subtraction and various alternative algorithms. On method is to use base ten blocks. For 53-25, you can set 5 ten and 3 ones on the board and 2 tens and 5 ones on the outer edge of the board. Then remove 2 tens and 5 ones from both groups. This should leave you with your answer on the left group, 2 tens and 8 ones, =28. Diagrams: Another method would be drawing diagrams to solve. This would include drawing dots for ones, lines for tens, and sqaures for 100 to solve. Expanded Form with subtraction is a bit trickier than addition. you can not "borrow" or regroup. 47-28Expanded 40 + 7 -20 + 8 20 we can break the 20 down to become 10 + 10. our new equation: 10 + 10 + 7 -8 2 10 + 2 + 7 = 19Equal Addends is the method of creating "friendly numbers" for an easier process. 534-158we can add +2 to both sides to become, 536-160then, we can add +40 to both sides to make, 576-200=376.When using different bases other than ten, be sure that your "friendly numbers" add up to whatever base you're in. 231 base five - 134 base fiveadd +1 to each side. This makes 135 on the right, which becomes, 140 because the numbers can't equal or be more than the base. Remember we only count up to whatever base we're in! (231 becomes 232)Then we can add +10 to each side to make, 242-200=42 base five

This week was about using base ten blocks and alternative algorithms for multiplication and division.Multiplication:One useful method for base ten blocks is to set the two lengths of each number you're multiplying on the outside of the corner of a mat. Then build with squares, tens, and ones to fill in the middle area. This provides a visual of what is actually happening when multiplying. 3x4=The 3 represents the number (#) of groups. The 4 represents the amount inside each group. Division:When dividing, using base ten blocks can be useful. 125 divided by 15=Think of 125 as the number (#) of units. 15 could represent the number (#) of items inside each group. Flash Cards:Not the best methods are always used with flashcards. Mr. Miltenberger suggests a few ways to improve learning multiplication for students.Groups of cards: 1, 2, 5, 103, 9, doubles4, 6, 7, 8Then you can create your own stack of cards with mixtures of these groups depending on which ones students are needing most. Alternative Algorithms for Multiplication: Expanded: 23x45 becomes: 20 + 3x40 + 5 multiply 40x20 & 40x3= 800+120 multiply 5x20 & 5x3 = 100+15 last, add all the red numbers. Left to Right: 273x454 80000 multiply the 200x400 10000 multiply the 200x50 800 multiply the 200x4 28000 multiply the 70x400 3500 multiply the 70x50 280 multiply the 70x4 1200 multiply the 3x400 150 multiply the 3x50 + 12 multiply the 3x4123,942 Area Model:12x26Draw a box with two boxes on the top, one for each digit, 1 and 2.And draw two boxes underneath, one for each digit, 2 and 6. It should look like a box with four squares inside with each digit written on the outside of their box with their place value. In this example, the twelve is written 10 + 2, each above their box. On the left side going down, is written 20 + 6 along side their boxes. (If multiplying three by three digits, then the box would have nine boxes inside).Then muliply the corresponding groups and write answer in each box. Then add the boxes left to right. Lattice Method:When multiplying, it is similar to the lattice method of adding, and you draw the same boxes underneath the equation with the diagonal lines. Only the digits are written as single digits above their corresponding box on the top and down the RIGHT side. Mutliply the corresponding digits, with the digit in the ones place inside the lower triangle, and the tens digit inside the upper triangle. Add along the diagonals for the answer.

This week was about Alternative Division Algorithms and Prime Factors. We went over the concept of using base ten blocks to show division equations. There are a couple of ways to show. One is to put a unit (ones block) into however many groups you have. Making sure to use all of them evenly, and if there is a remainder to put that as the numberator of the fraction with the divisor as the denominator. Alternative AlgorithmsRepeated Subtraction with Division This allows student to estimate how many times a number could go into the number being divided without having to use the greatest possible answer.Area ModelAgian, this is drawing a box diagram for however many digits are being divided. If it is 649 divided by 5, then we draw a rectangle with three boxes inside with the 5 on the left outside. Then we write, 600, on the first left box. Students can guess how many times 5 goes into 600. Let's say, 100. Then 5x100=500. Write 500 under the 600 as a subtraction equation. Carry the answer to the right box, adding the amount the the 40 from 649. Then student will guess again how many times 5 goes into, now 140. Repeat the process. Once there is a number left that 5 can not go into, that is the numerator and 5 is the denominator. Upwards DivisionFor this, we write the division problem "as we say it." 372 9 Then student ponders how many times 9 goes into "37." This is 4 which can be written above the "37" on an angle. Then subtract 37-36 to get 1. This 1 then is written in front of the 2, becoming 12. Now, how many times can 9 go into 12. One time. Write 9 above the "12" and subtract. 3 is left. Our answer is 41 with 3/9.Divisibility Rules:2, even numbers3, if the sum of all digits are divisible by 3.4, if the last two digits are divisible by 4.5, if the number ends in a 5 or 0.6, when both 2 & 3 are factors, 6 is. 7, none, or at least are difficult8, the last 3 digits are divisible by 8.9, the sum of all digits divided by 9.10, the number must end with 0.Factor Trees 42 and 20 /\ /\ 6 * 7 2 * 10 /\ /\2 * 3 2 * 5Numbers that are prime numbers are written in red for clarification. Once you have a prime number, you don't need to keep factoring.Then you write them out: 2 * 3 * 7 and 2 squared * 5 To find the Least Common Denominator, LCD, we need to look at all the factors. If there are duplicate numbers, even with the exponents, we just keep the ones with the highest value. For example, one side has 2, and the other side has 2 squared. We should keep the 2 squared. Our final set of factors and then the LCD would look like this: 2 squared * 3 * 5 * 7 = 420

This week was all about fractions and how to build them using manipulatives. This could be Fractions Bars, Circle Fractions, Linear Rods, Dual Colored Counters, etc.Also, Mr. Miltenberger discussed different visual methods when discussing fractions, including the Area Model, Linear Model and Set Model.One key point to remember is that the Numerator represents the number of pieces. The Denominator represents the size of the pieces. Area Model:This is the typical way to draw a diagram or use manipulates when referring to fractions, often it uses a circle divided into various fractions, 1/3, 1/4 etc. Linear Model:This is demonstrated with using a length of a long piece of a manipulative, with various amounts of fractsion shaded in. Perhaps even using various lengthe of long pieces to better understand how various fractions make a whole.Set Model:This can be demonstrated using various items such as an eraser, marker, sticky notes etc. You could verbalize the fraction by stating, "there are two markers out of the total of 8 objects."Using fractions in such ways can demonstrate to students how fractions work together and how they are related.

In this week, we learned about using manipulatives to add, subtract and multiply fractions.One key element to remember when adding and subracting fractions with UNLIKE denominators, is to find a common denominator for the two fractions. For example, with 1/4 +1/3 we need to find a common denominator, a number in which 4 and 3 will go into evenly. Take time to interchange smaller fraction manipulatives under the origional 1/4 and 1/3 fraction manipulatives in order to find manipulatives which will fit perfectly under both fractions. In this case, it would be 1/12 pieces. Under 1/4 would be 3 pieces of 1/12, and under 1/3, 4 pieces of 1/12. Then it is possible to add the total amount of 1/12 pieces to find our answer. 7/12. Multiplying fractions using manipulatives. Let's look at 2 x 1/2. Remember when we muliply we are saying "2 groups of one half." We can then reach for our 1/2 manipulatives to demonstrate. How many do we need? Two, because it is "2 groups of 1/2." Then we are better able to see that this amount of 1/2 pieces creates one whole. The answer is, 1. Again with Multiplying fractions, such as, 1/2 x 1/4, we should verbalize it as "one half of one fourth." We can multiply the denominators to find a common denominator which would be 8. Let's set our 8 pieces of manipulatives out. Now, let's remember we are needing to find 1/2 of 1/4 of these 8 pieces. First, let's find the 1/4 amount of 8. This would be, 2. Then we find the 1/2 amount of 2 which is 1. This would then be written, 1/8. This could be demonstrated with two color counters and flipping over the counter at the last step to see our fraction, 1/8.

Solve FractionsAdding Fractions:8 + 10 15/34Simply add the whole numbers. 18 15/343/8 + 5/12 Simply factor the denominators8 =2x4 and 12=4x3. Circle the common factors, 4. The remaining ones, 2 and 3 are what you multiply the opposite fraction by, both numberator and denominator.9/24 +10/24= 19/24.Subtracting Fractions:9 7/11 - 5. Simply subtract the whole numbers.4 7/11.10 3/8 - 2 7/10.Simply factor the denominators again.8=2x4, 10=2x5.Circle the common factors, 2.The remaining ones, 2 and 3 are what you multiply the opposite fraction by, both numberator and denominator.Multiply Fractions:2/15 x 7 3/6Convert the mixed fraction to an improper fraction like normal.Then factor BOTH the numerators and denominators.6=2x3, 45=15x3. Then draw a "funky one" through all the common factors, both 15's, 2's, 3's. Multiply the "funky ones" which is still "1." There are no other numbers left in the equation. Answer, 1.Dividing Fractions:This method incorporates the, "Keep, Change, Flip."Convert any mixed fractions to improper fractions. Again, factor BOTH numerators and denominators.Then draw the "funky one" through all common factors. Mulitply the "1's." Multpiply the remaining numerators and denominators straigh across.

Color Counters with Addition.The purpose of the red side of color counters is to represent negative numbers.When showing the number 4 while using 8 color counter chips, use a "zero bank." This would be the use of two additional yellow side up color counters, and two additional red side up colors counters. These cancel each other out, positive and negative, and create a, "zero bank." This would represent 4 while using 8 color counters.-4+5=(+1)Represent -4 with 4 red color counters in a row.Then lay out 5 yellow color counters in a row above and inline with the red counters.This helps show the "zero bank" created with 4 yellow and 4 red. The left over counter is 1 yellow. Anwer is (+1).

Adding and Subtracting IntegersHector's Method:This is a method Mr. Miltengerger observed from one of his past students.Let's look at 37 + (-47)Represent the positive number with a (+) and the negative number with a (-). Use two symbols for the larger number, which is 47. So, it would be represented with (- -).37 + (-47)=+ - -Circle one of each symbol, + and -.Then, use the remaining symbol to represent whether the number is positive or negative once you subtract 47 minus 37. Answer is (-10).Using Color Counters to Solve Subtraction:-8 - (-3) = (-5)Let's represent, -8 with 8 red color counters.Simply remove 3 color counters. 4 - (-2) =6Be sure to use plenty of color counters on this equation. It won't matter, it just needs to "be enoungh" as Mr. Miltenberger states.Let's line up maybe 7 yellow color counters. Underneath, line up 3 red color counters. Now, we can take away our (-2) color counters.This reveals the "zero bank." Circle the "zero bank."The remaining 6 yellow color counters remain.Answer is (+6).Draw digram of Adding and Subtracting Integers.This same method of color counters can be transitioned to writing.Simply write a (+) or (-) for each digit represented being sure to add more when dealing with positive and negative numbers. This will create a "zero bank" which will lead to solving the equation.Here the double negative of subtracting a negative is converted into positive.Circle 3 negative symbols and take them away. This represents, (-3).Then circle the "zero bank."The remaining amount is +2.-1 - (-3)= 2 ++++- - - - -

Show and Solve Multiplying Integers.When signs are negative, the answer will be positive.When signs are different, the answer with be negative.6x4 means "six groups of four."Show this by drawing six circles with four dots inside for example.-3x2 means "zero take away three groups of two." Show this by drawing as many positive signs as it takes, more that 6, which is the 3 groups of 2. Then underneath each positive, draw a negative sign. Then circle the 3 groups of 2 positives with arrows going away. Then circle the remaining symbols as a "zero bank." The remaining negative symbols should be your answer.

PEMDAS isn't a favorite of Mr. Miltenberger. He instead suggests that educators should teach GEM/DS/A"G" stands for "group." The rest being the obvious, Exponents, Multiply, Divide, Subtract, and Add, left to right. There are times when the "P" in PEMDAS doesn't totally make sense or work. We should think of isolating the groups first. This could be a fraction as well. Separate your groups by drawing down arrows under the Subtract and/or Addition symbols. Then perform the operations inside of those groups. Continue with the operations listed in GEMDSA from left to right.

This final week was perhaps my favorite of this course. Percentages were discussed in a clear, useful way that I can even apply in my own life!To figure out the percentage, there are a few simple steps you can take. Let's look at the following example:We want to know what is 35% of 80.First, let's figure out what is 10% of 80. Which is 8.Then, we multiply 8x3=24.Then we figure out what is 5% of 80, which is half of what 10% was, 4.Lastly, add 4 to our 24 to get 28 for our answer.This can also be used to find more difficult percentages like 27% by finding what is 10%, 5% and 1% of 27. Then adding accordingly.