Elementary Mathematics

Addition can be defined in many ways: finding the sum, increasing, all together, etc.Base Ten Blocks:Base ten blocks are particularly helpful with helping students understand one-to-one correspondence (understanding that when counting, one number is used for each item in a sequence)The blocks are made up of groups of one, ten, and one hundred. They are made so as to be as visually accurate in comparing these three amounts as possible.https://youtu.be/p6bvKB4jUf8 < < < How to use base ten blocks.

Showing:“Showing” is very similar to the base ten blocks when used to complete addition problems. It follows the same shape concept as base ten blocks to represent values, where a square equals 100, a line equals 10, and a dot equals 1.Solving a problem by showing means that you will begin by redrawing the first number in these base ten drawings. To add, you will simply draw the second number alongside the first one, and then count them altogether. Plainly, this idea reinforces the one-to-one correspondence and number sense as well.23 + 34l l . . . (23) + l l l . . . . (34) = l l l l l . . . . . . . (57)

Algorithms:There are many different algorithms besides the standard algorithm that we can use to solve an addition problem. It is best practice to teach all of these strategies to be able to best reach ALL students, and provide a way that makes sense to all of them.Strategy #1: Expanded formExpanded form is essentially decomposing each problem to make it much easier to solve. Example: 144 + 234100 + 40 + 4+ 200 + 30 +4__________________300 + 70 + 8 = 378Strategy #2: Left to RightLeft to right allows the student to begin the problem in a more natural direction than going from left to right, starting instead at the largest place value.Example: 264 + 321Step 1 : 500Step 2 : 80Step 3 : 5Step 4 and final answer : 585Strategy #3: Friendly Numbers/ Trading offThis algorithm allows the student to create numbers that are “easier” to deal with. Example: 13 + 17 + 24Step 1 : 13 and 17 combine to make an even 30.Step 2 and final answer: 30 + 24 = 54Strategy #4: Scratch MethodThe scratch method is helpful for regrouping. It involves drawing a slash through a number when you have reached a full group of the next place value. After adding all the numbers in that place value, the scratches will then be counted and added to the next place value upExample: 13 + 24 +54 +78  Example: 13 + 24 +54 +7811 1 32 40 5 4 1+ 7 8—————1 6 9 

Algorithms:There are many different algorithms besides the standard algorithm that we can use to solve an addition problem. It is best practice to teach all of these strategies to be able to best reach ALL students, and provide a way that makes sense to all of them.Strategy #1: Expanded formExpanded form is essentially decomposing each problem to make it much easier to solve. Example: 144 + 234100 + 40 + 4+ 200 + 30 +4__________________300 + 70 + 8 = 378Strategy #2: Left to RightLeft to right allows the student to begin the problem in a more natural direction than going from left to right, starting instead at the largest place value.Example: 264 + 321Step 1 : 500Step 2 : 80Step 3 : 5Step 4 and final answer : 585Strategy #3: Friendly Numbers/ Trading offThis algorithm allows the student to create numbers that are “easier” to deal with. Example: 13 + 17 + 24Step 1 : 13 and 17 combine to make an even 30.Step 2 and final answer: 30 + 24 = 54Strategy #4: Scratch MethodThe scratch method is helpful for regrouping. It involves drawing a slash through a number when you have reached a full group of the next place value. After adding all the numbers in that place value, the scratches will then be counted and added to the next place value upExample: 13 + 24 +54 +78  Example: 13 + 24 +54 +7811 1 32 40 5 4 1+ 7 8—————1 6 9 

Showing:Similarly, showing involves drawing the symbols that represent base ten and either circling or crossing out the amount you are taking away. Count up what is left and that will be your answer.https://youtu.be/vyxPCV8hnHs < < < Shows this process on a whiteboard.Algorithms:There are not as many algorithms for subtraction as there are for addition, but there are still several strategies students can learn to find which one fits best. Strategy #1: Expanded FormThis is similar to expanded form in addition. The larger numbers are decomposed to simplify the subtraction process.Example: 346 - 221300 + 40 + 6— 200 + 20 + 1_____________________100 + 20 + 5 = 125Strategy #2: Left to RightLeft to right is again similar to the addition strategy, instead of beginning on the right side, the student begins subtracting from the largest place value.It is not necassary to rewrite the problem at all, but you can if you waStrategy #3: Equal AddendsEqual addends simplifies a subtraction problem by creating simple numbers to deal with. To complete the problem, you will add an equal number to each side.Example: 34 - 163 4 - 1 6+ 4 +4______________3 8 - 20 = 18https://youtu.be/lIY8rQQqgpw < < < Equal Addends

Multiplication is often described as repeated additions. While this is not incorrect, it is also important to think of multiplication in terms of “groups of”. For example 4(9) does not just mean 4 × 9, it means 4 groups of 9.Building:Building is more complicated with multiplication. It is important that we keep in mind the “groups of” concept. This will help us to lay out base ten blocks in appropriate rows.The above image has 12 groups of 14, which can be shown as 12(14). The total amount of all the base ten blocks pictured is the answer to the problem. Showing:Showing is much like the base ten method, only you will be drawing your own representation of the blocks.It will look very similar to the above picture, but to show the thought process and make the comparison to base ten blocks more clear, you can also close off all the edges and label each unit as 1, 10, or 100.

AlgorithmsAlgorithms to solve multiplication should be based in mathematical principles.Strategy #1: Expanded FormThe expanded form involves decomposing the numbers first, multiplying, and then adding them together at the end for the final answer.The above problem demonstrates how this can be used for two digit numbers.Strategy #2: Area ModelThe area model algorithm is an organized representation for multiplying (after decomposing) and adding at the end. It is somewhat similar to the expanded form method but much easier to follow.Strategy #3: LatticeThe lattice algorithm is another organized model, which emphasizes place value.The student begins by multiplying the adjoining numbers of each box. They will then add these answers diagonally, to get their final answer.https://youtu.be/ZCqtT9B0NZs > > > This video includes the expanded form and the area model.https://youtu.be/Lhy9-ITGqjw < < < Lattice Multiplication

Week 4 Day 1: Division with Alternative AlgorithmsThe first thing to consider when we are teaching division to our students is how we want them to phrase and think about a problem. For example, what is the best way for students to think of 15/5? We can think of 15 divided into 5 groups. This phrasing is better for conceptual understanding. * * * * * * * * * * * * * * * This image could be described "After dividing 15 equally into five groups, there are 3 in each group." With any division problem, after you have divided equally into groups, however many are in each group is the answer to the problem. _______________________________________________________________In class, we learned about two main algorithms to solve a division problem, aside from the typical long division. 238 / 4Long Division: 5 9 r. 2 ________ 4 l 2 3 8 -2 0 l _____ l 3 8 -3 6 ______ 2Long division can be confusing for a few reasons: The way the problem is organized. It is presented initially as 238 / 4, which looks more like a fraction. When they set up the problem for long division, they put the "groups of" 4 on the left side of the schedule, draw a little half-box, and write the "bigger" number inside. Students may be able to learn to divide this way simply out of practice if they are taught to, but it is unintuitive and will never be as easy for them to understand as a problem that is set up in such a way that makes clear the underlying concept. Remainders exist in real life, so students can discuss what they are. But moving forward, it will be easier for the student to understand that the remainder be presented as a fraction of 2 / 4. After all, in the context of the problem, there is not an arbitrary leftover 2, but an incomplete group of 4.Repeated Subtraction: ____________ 4 l 2 3 8 | _ 4 0 | 10 _________ | 1 9 8 | _ 4 0 | 10 _________ | 1 5 8 | _ 4 0 | 10 __________ | 1 1 8 | _ 8 0 | 20 ___________ | 3 8 | _ 2 0 | 5 __________ | 1 8 | _ 1 6 | + 4 ___________ | _______ 2 | 59 2/4 Answer= 59 2/4Repeated subtraction is a really good tool for students who don't know or have a hard time remembering their math facts. The organization can still be confusing.Upwards Division: 532 / 3 5(-3) 3(-21) 2(- 21)= 1 _______________________ = 177 1/3 3 Step 1: How many times does 3 go into 5? Once, so we write the 1 on the opposite side of the equal sign and subtract 3 from 5. Step 2: How many times does 3 go into 23? 7 times, so we write down the 7 and subtract 21 from 23. Step 3: How many times does 3 go into 22? 7 times, so we write down the seven and subtract 21 from 22. This gives us 1, so then we know we have an incomplete group of 3, which we write as 1/3.

What are integers?Integers are any whole numbers with either a positive or negative value. They cannot be fractions or decimals.We use two-color counters to represent integers and problems involving integers. No matter what the other color is, the red side should always be used to represent negative integers. This way there is consistency across color counters and in the future when they are taught how to show these problems.One yellow counter will represent one positiveOne red counter will represent one negativeA red counter and a yellow counter together will represent a 0 pair, because they "cancel" each other outSeveral zero pairs grouped together can be referred to a zero bankExample: Show 3 using seven tiles. There will be three yellow tiles standing alone, and then two zero pairs.When showing integers, you can use + and - symbols to represent the tiles rather than drawing the circles.Example: Show 3 using seven tiles. + + + + + - -It is important that students begin with these problems to get used to the tiles, and especially get used to and understand zero pairs.

Solving Fractions Numerator: The number on top of the fraction, the number of "pieces" the fraction hasDenominator: The number on the bottom of the fraction, the number of total pieces that will make a value equal to 1.To add fractions, they must have the same denominator. 6 2 8 ---- + ---- = ---- 12 12 12If you are adding mixed numbers, or whole numbers with fractions, then you can add the whole numbers separately from the fractions. 1 1 7 + 2 ---- = 9 ---- 6 6 If the denominators are different, then we will need to convert them to fractions with equivalent denominators. 5 3 ---- - ---- 12 8 We will think of factors for 12 and 8, with one being the same as the other. 4 x 3 is 12, and 4 x 2 is 8. 4 is on both sides, but to make an equivalent fraction we need to multiply the opposite side to the factor they are missing. ( 2 ) 5 3 ( 3 ) 10 9 --- ---- - --- ---- ----> ---- - ---- ( 2 ) 12 8 ( 3 ) 24 24 Answer: 1 ---- 24

Subtraction with IntegersWe can show subtraction with integers using the tiles the same way that we would using base ten blocks. We begin with the integer that we start with on the board.Example: 2 - 5We start by putting two positives on the board. Clearly, we do not have enough pieces to represent the problem, and we can't change it to make it easier. What we can do is show the zeros on either side of the 2 positives. Like we practiced earlier, this will be a positive on top and a negative underneath it. Once we have it set up like this, we can then take away our five positives, including three positives which are a part of the zero bank. This will leave us with 3 negatives. Answer: - 3Showing follows the same principles as building, but instead using the + and - signs.Example: -4 - 3 - - - - Do I have what I am taking away? + + + - - - - - - - No, so show zero pairs. ( + + +) ----> - - - - - - - Take away 3 positives - - - - - - - 7 negatives are left Answer: - 7

Decimals:Decimals are easy at this point because we have learned how to do them already through what we have learned leading up to this point. To show decimals, when adding, subtracting, multiplying, we can simply use the same strategies as we did with fractions. This is because decimals are fractions, they represent a value less than one.To solve decimals, it is the same. We can use any of the algorithms that are used for adding, subtracting, multiplying, and dividing.

Subtracting Fractions:To subtract fractions, they must have the same denominator. This is easily shown through the same message as showing adding subtractions. To show the subtraction of 1/3, we draw the 1/3 reference box and then cross out a third, doing the same with the other box. What is left shaded is the numerator, and the total number of boxes, 12, is the denominator.Multiplying Fractions:To show multiplying fractions, we only need to draw one box. We begin with the 1/3, and shade it in. After this, we draw horizontal lines to represent 2/3, and shade those in. The two boxes that are double shaded are the numerator, and the total number of boxes, 9, is the denominator.

Building and showing Fractions:The biggest thing we want students to understand about fractions is that they always represent an absolute value less than 1, and that they show pieces that make up a 1, or a whole. Manipulatives are useful for students to see visually how fractions that have a smaller denominator, like 1/2, will have larger pieces. It can be confusing initially because it somewhat goes against what they have been taught to believe about number value. The circle fraction manipulatives show this well because they are all made up of the same size circle, but have different-sized pieces. When showing fractions, it is easier for students to draw rectangles than circles, because they are easier to divide into pieces.

Order of Operations:Order of operations is the process through which we solve equations with multiple steps. In the past, we have learned the acronym "PEMDAS" to help us remember what order to complete the problem in. However, this is not necessarily accurate, because the reason there is a specific order to follow in the first place is considering the context of a math problem. For example, if we have a math problem that states "Tricia puts $200 in the bank. Two years later, it is worth twice as much. She uses the money to buy a computer for $300. How much money will she have left over? The problem looks like this: 200 x 2 - 300 If a student subtracts 300 from two before multiplying, they will get the question wrong. It is helpful for students to understand the actual value that the numbers represent. When students do not have access to this information, they can remember that multiplication does not necessarily come before division, and addition does not necessarily come before subtractions. Parentheses (or groups) should be followed by exponents, and following this the problem can be solved by moving from left to right to get the correct answer.

Subtraction is decreasing, literally taking away. The smaller number is always being taken away from the larger number.Building:Because the larger number is always the starting number, building with base ten blocks is an especially helpful visual. Begin by laying out the larger number. Then, take away as many blocks as the smaller number. What you are left with is your answer.https://youtu.be/Q_k0rWfSZTs < < < Video showing this subtraction process.

Division: To understand division, students should have a good understanding of prime factorization. There are a couple ways this can be represented. Factor Trees: 2 4 / \ 6 x 4 / \ / \ 3 x 2 2 x 2 24: 23 x 3Downwards Division: 3 | 2 4 |______ 2 | 8 |_____ 2 | 4 |_____ 2 24: 23 x 3 Divisibility Rules: After students have a strong conceptual understanding of division, it is helpful to teach them the "divisibility rules", or ways to help you see if you are dealing with a prime number or not without doing all of the above work. 1: Everything is divisible by 1!2: Any even number.3: Add the digits of the number, if the sum is divisible by 3 then the original number is as well. 4: Look at the last 2 digits of the number. If they are divisible by 4, then so is the number. 5: Any number that ends in 0 or 5 is divisible by 5. 6: If the rules for both 2 and 3 are applicable, then the number is divisible by 6. 7: No rule works for 7.8: Look at the last 3 digits, if they can be divided equally into 8 then the original number is divisible by 8.9: Add the digits of the number, if the sum can be divided by 9 then the original number is divisible by 9. 10: Any number that ends in 0 is divisible by 10.

Solving Adding IntegersWhen we solve adding integers by showing, it can be represented with the + and - signs used to show values in the previous lesson.Example: 3 + 4 + + + + + + + Answer: 7 Example: -2 + (-3) - - - - - Answer: - 5Example: 3 + (- 7) + + + - - - - - - - Answer: - 4 ( A positive above a negative is always a zero)When adding numbers greater than 10, it is not feasible to show by drawing the diagram. So we use the Mini Diagram method!Example: 15 + (-22) ( + - ) - 1 5 + (-2 2)We will determine how to solve the problem using this mini diagram. The larger "pile" will have two of the correct sign above it, and the smaller "pile" will have one. We will draw a circle around the + and the - in this case, to represent our zero bank. The sign left outside of the circle will be sign that the answer to the problem will have. Answer: - 7

Multiplying IntegersBuilding multiplication problems with integers can be confusing because there is a lot to remember, depending on whether the numbers are negative or positive and whether they represent the groups of or the number in the groups.Example: 3 ( 2) *3 groups of 2* + + + + + + 3 groups of 2 Answer: 6Example: 3 (-2) *3 groups of 2 negatives* - - - - - - Answer: - 6Example: - 3 ( 2 ) * Take away 3 groups of 2* 1: + + + + + + - - - - - - 2: - - - - - - Answer: - 6Example: - 3 ( - 2) * Take away 3 groups of 2 2 negatives* 1: + + + + + + - - - - - - 2: + + + + + + Answer: 6Algorithms for Multiplying IntegersOnce students have conceptual understanding of multiplying integers, we can teach them multiplication "rules".If we multiply numbers with the same sign, the answer is positive.If we multiply numbers with different signs, the answer is negative.**The rules are the same for dividing integers**

Multiplying Fractions: When we multiply fractions, we can multiply directly across the numerators and directly across the denominators. But there is also a way to simplify the fraction as we multiply. 3 8 24 --- x --- = ---- 7 11 77Especially with larger numbers, this way of simplifying first is much easier. (2 x 8) (3 x 9) 16 27 ----- x ----- 21 22 (3 x 7) (2 x 11)Once we have figured out the factors for each number, we can simplify by crossing them out diagonally. The 2 above the 16 and the 2 above the 22 will cross each other out. The 3 above the 27 and the 3 above the 21 will cross each other out. Then, the problem will look like this: 8 9 72 ---- x ---- = ----- 7 11 77You can multiply across, and then you have your answer.