Elementary Mathematics

Went over Ploya's four steps for problem solving:1) Read and understand the problem2) Make a plan to solve the problem3) Solve the problem (carry out plan)4) Double check your work (look back)These steps can also be known as UnDeCaLo (Understand, Develop Plan, Carry Out Plan, Look Back)Went over terms corresponding with a child's early number sense:One-to-One Correspondence is when a child associates one unit with a number, and is able to count with one number value assigned to each unitCardinality is when a child understands that the last number they counted one-to-one is the exact number of units on their matSubitizing is when a child can look at a group of units and know the number of them without counting one-to-oneFive and Ten Frames are used to help develop a child's one-to-one correspondence. They can help a child understand the number of boxes in a frame is what the number physically looks likeWent over Base 10 Blocks:100 units in base 10 blocks is known as a flat10 units in base 10 blocks is known as a longIndividual cubes are known as units

Went over what makes a flat, long, and unit in base 10There can be many other bases, not just 10-For example, base five is where there are five units in a long, and 25 units in a flat. Things written in different bases will look something like this: 5seven. The number will be written as a digit, but the base will be spelled out and written in subscript

We learned that building means to build with manipulatives and showing means drawing a diagramBuilding AdditionWe learned that in order to build addition with base 10 blocks, we must first show the first number with our blocks on our mat, then the second number, then combine them together. (For example: 24five + 13five would have two longs and four units, as well as one long and three units. When we combine them together [in base five], we count out four longs and two units, or 42five)Showing AdditionFirst off, we learned the "shorthand" version of drawing base ten blocks in a diagram (a rectangle for a flat, a vertical line for a long, and a dot for a unit. Then, we learned how to show addition. It is very similar to building addition, where we put to two numbers separate first and then combine them together to get our sum. However, we must learn to combine units into longs, and longs into flats. (For example: 14seven + 25seven will look like one long[vertical line] and four units[dots] as well as two longs[vertical lines] and five units[dots]. Now, when we combine these two terms, we end up having a fifth long[vertical line] from our units[dots]. This gives us an answer of 52seven)

We covered the three things that make good algorithms:1) Efficiency2) Expandable (can it be done with harder numbers?)3) Supports Basic Mathematical ConceptsWe also covered four alternative algorithms for addition:Expanded Form: This algorithm allows students to expand the numbers given to them in an addition problem to help reinforce place value. For example, instead of 27 + 46, we show to problem as 20 + 740 + 660 + 13 = 73Expanded form supports basic mathematical concepts, is expandable, but is not very efficientLeft to Right: Left to right also reinforces place value by having students add like numbers starting at the left, and moving towards the right side of the term. For example, the problem 3,426 + 573 would be solved as 3,000 + 900 + 90 + 9= 3,999. We added 400 + 500 to get 900, 20 + 70 to get 90, and 3 + 6 to get 9. This method is expandable, supports math concepts, but isn't efficientFriendly Numbers: This is mostly used in a multi-number addition problem. The goal is to move numbers in the ones place to other terms in order to make a "friendly" number (that is, a number ending in zero). For example, in the addition problem 42 + 58 + 77 + 23, we would move the two from forty-two to fifty eight, making 40 + 60 + 77 + 23. We would do the same thing with the three in twenty-three, and seventy-seven, making the problem 40 + 60 + 80 + 20. This way, we have a much "friendlier" addition problemTrade Off: Trade off and Friendly Numbers are very similar algorithms. The only difference is that Trade Off allows you to move just a couple of numbers from different terms in order to create a "friendly" number. The problem 45 + 17 + 39 allows us to move just three of the forty-five over to the seventeen in order to make a twenty. Now, we have 42 + 20 + 39, where we can move one from forty-two over to thirty-nine, making the problem 41 + 20 + 40. We can now answer the problem much easier than we could before.

At the beginning of class, we went over the expectations for the Mindmap assignment and when it was dueWe then went over two more alternate algorithms for addition:Scratch: This method (my personal favorite), is specifically tailored towards multiple-digit and multiple-term addition problems. This allows the child to count up to 10, and when they do so, scratching out the number they are on and then repeating the process. For example, the problem 28 + 43 + 14 + 57 would look something like this:22 84 3 11 4 5 4 7 2 142This helps with the confusion that goes with "carrying" in the traditional algorithm.Lattice: Lattice is used only in base ten, and consists of making a lattice-type box under the two terms you are adding. After adding up your terms, you add across the diagonal line in order to help eliminate the "carrying" fiasco. Lastly, we went over how to build subtractionIn order to build subtraction, we must build the first number in a subtraction problem with our base ten blocks on our mat. Then, we put the number we need to take away off of our mat. From there, we are able to take away the number of units needed off of our mat in order to get out answer.

We went over how to show subtraction in diagramsShowing subtraction should be used with multiple different colored pens/pencils in order to keep on track. When showing subtraction, we first draw out the first number (AKA the number we will be taking away from) in flats, longs, or units. From there, with a different colored pen/pencil, we circle what we will be taking away from, and drawing an arrow away from the problem (giving the illusion that we are taking it away). If borrowing is needed, we cross out one flat or long and rewrite it in longs or units.

Before going into multiplication models, we discussed what it meant to multiply, and what we want students to know about multiplicationIt was concluded that 3(4) means three groups of four units, not four groups of three units, or any other variation. It is important that our students know what exactly we are asking and in which variation. We went over multiplication modelsGroups are similar to what was stated above, they are a certain number of groups with a certain number of units in said groups. They can be drawn like circles (groups) with dots (units) in the middle.Arrays are when we lay out the units (drawn as dots) into rows and columns based on the factors of the problem. The dots are spread apart and it is important to note that the array resembles a rectangle.Area models are similar to arrays in that the units will make a rectangle. This is to help students understand that the area of a rectangle (which will be covered as they go into future grade levels) is basically a multiplication problem: the length multiplied by the width.

We covered how to build a multi-digit multiplication problem using base ten blocksIn order to build a multiplication problem, we set the factors up outside of the mat vertically and laterally (like a corner). Then, we fill in the space on our mat inside the corner with the appropriate blocks. If a ten is stacked vertically and laterally, the best way to fill that in is with a flat, and so on. We then learned how to show multi-digit multiplicationThe way we show multiplication is very similar to the way we build it. First we make a "corner" out of our factors, but leave the fourth side of our drawn blocks blank, to show that it is not a part of our answer. Then, we fill in accordingly by drawing lines to make flats, longs, and units. By the end, it looks similar to a rectangle, which calls back to the area model we talked about last class.

Firstly, we talked about what it means to "know our multiplication facts"It is vital the our students know multiplication because they will need it in order to understand any higher math concept. It is also just as important that students know 4(3)=12, but ALSO that 12=4(3). This will help them later on when learning concepts like greatest common factor, etc. Then, we went over how to teach multiplication factsIt was established that we should teach multiplication facts 1-10 in three groups: Group One: 1s, 10s, 2s, & 5sGroup Two: 3s, 9s, & doublesGroup Three: 4s, 6s, & 8sIt was also established that the "nines trick", or a hand strategy used to help students learn their nines in multiplication, was inadequate. It doesn't teach the kid 9(3), it just teaches them to use their fingers. The appropriate "nines trick" is to know that a product of a nine multiplication problem will always add up to nine. Lastly, we discussed alternate algorithms for multi-digit multiplicationExpanded Form: This is very similar to addition expanded form, only multiplying. You must be sure you multiply the correct amount of times, however. Left to Right: Left to right multiplication is very similar to Left to Right addition. It helps kids be sure of their place value, because when you multiply 300(2), it is 6, but you add the amount of "zeroes" in the problem; making the answer 600. Area Model: This is used to help kids understand once again that area and multiplication are very closely related. Students use expanded form to write out the numbers multiplied on the top and right side of the rectangle, and then put the products in the divided squares.

We went over the final alternate algorithm for multiplication: LatticeLattice Multiplication is very similar to lattice addition, except for the different set up of the problem. You draw a rectangle, and write one term on top of the rectangle and the other term on the right side of the rectangle. Then, you draw lateral and vertical lines according to the amount of digits in both of your numbers. Draw a diagonal line going left through each section, and then multiply accordingly. Then, we covered alternate algorithms for subtractionExpanded Form: Very similar to expanded form in other functions. The only difference is the issue of borrowing. When borrowing in expanded form, we cross out the number we need to borrow from, like so:50 + 1 320 + 83020 + 5= 25Equal Addends (Subtractends): Equal addends follow the math concept that as long as you add the same numbers to both terms you're working with, the answer will be the same. For example, the problem 53-27 looks difficult. But, if we add three to both sides, we will have 56-30, which is a much easier subtraction problem. The term 30 is also known as a "friendly" number, calling back to the friendly numbers strategy for addition.

Went over Divisibility Rules via video:2: If the number ends in an even number, then it is divisible by 2 (Ex: 24,536)3: If the sum of the digits is divisible by 3, the whole number is (Ex: 1,425 --> 1+4+2+5=12. 12 is divisible by 3)4: If the last two digits of the number is divisible by 4, the whole number is (Ex: 24,736-->36 is divisible by 4)5: If the last digit is a 5 or a 0, the number is divisible by 5 (Ex: 23,6090)6: If the number is divisible by two and three, it is divisible by 6 (Ex: 342 is divisible by 2 because it is even. it is divisible by three because the sum of the digits is 9, which is divisible by 3)8: If the last three digits of the number is divisible by eight, the whole number is (Ex: 723,648)9: If the sum of the digits is divisible by 9, it is divisible by 9 (Ex: 2,736-->2+7+3+6=18. 18 is divisible by 9, therefore the original number is divisible by 9)10: If the last digit of the number is 0, the number is divisible by 10 (Ex: 303,670)Alternative Algorithms for Division:We discussed how our original algorithm, long division, is difficult for kids to learn; it is efficient and expandable, but doesn't teach kids how to work with remainders, and is hard for kids to guess how much one number goes into another. We then discussed two other alternate algorithms to teach division: Repeated Subtraction: Repeated Subtraction is set up very similar to long division, except a vertical line goes down the furthest right hand side. When solving a division problem, a student that has more difficulty with long division can repeatedly subtract a multiplied number from the dividend, until they eventually get the quotient. For example, in the problem 4 ÷ 238, a student can subtract 40 (4 · 10) multiple times until they reach an answer of 59 2/4 Upwards Division: Upwards Division is a strategy to help students understand remainders. It involves setting the problem up like a fraction, with the divisor in the denominator and the dividend in the numerator. (Ex: 532 ÷ 3 would be written 532/3)

We wrapped up division by further discussing the problem with remaindersWe then discussed fractions by comparing them:Is 6/11 <,>,= 13/27?We then determined that 6/11 was bigger than 13/27 because elevenths are bigger size pieces than twenty-sevenths. It was also established that the numerator stands for the number of things, while the denominator symbolizes the size of the pieces. Anchor fraction: using a fraction to further compare two other fractions

Went over how to solve fractions via video:Adding Fractions: When fractions have the same denominator (same size pieces), they can add "straight across". However, when the denominators are different, students must "multiply by what's missing". For example, in the problem 5/12 + 2/9, students must see that 3·4 is 12 and 3·3 is nine. After seeing this, students will then multiply 2/9 · 4/4 and 5/12 · 3/3. By doing this, we now have the expression 15/36 + 8/36. The denominators are now the same size pieces, so students can add straight across. It is important to note that when mixed numbers are involved, students should add the whole numbers first before beginning to add the fractions. Subtracting Fractions: We use the same approach to subtraction as we do to addition. Our goal is to get the same size pieces on the bottom, and if mixed numbers are involved, subtract the whole numbers first. In the problem 7/18 - 1/6, we notice that 3•6 is 18, so we multiply the 1/6 by what is missing: 3/3.

Went over how to multiply and divide fractions:Funky Ones: When multiplying and dividing fractions, we look at the factors of each numerator and denominator in order to find the same numbers. When we do, we make a "funky one". For example, in the multiplication problem 8/1 · 5/12, we see that the factors of 8 are 2,4 and the factors of 12 are 3,4. From here, we make a "funky one" with the 4s we see. What is left is 2/1 · 5/3, which we can multiply across in order to have a product of 10/3.Multiplication: Multiplying fractions goes in three simple steps:1) look for common factors2) find the funky ones3) multiply acrossIt is important to note for both multiplication and division that all improper fractions should be changed into mixed numbers.Division: A common strategy for dividing fractions is the concept of KCF, which stands for Keep Change Flip. This means we Keep the first fraction, Change the division sign to a multiplication sign, and Flip the last fraction's numerator and denominator. Another common name for this strategy is "multiply by the reciprocal". From there, we use the three step process we used in multiplication.

We went over how to build and show fractionsFraction Manipulative Basics: Leave at least one blank side for discovery (don't have the name "1/12" on both sides of the manipulative). This way, the less information we give students, the more room they have to think for themselves. Also, fraction rods can be helpful when teaching fractions. Lastly, two color counters can be used as fraction manipulatives.Building Fractions: When we build fractions, there are typically three models we use:Set Model: The Set Model uses different color or differently shaped objects in order to convey fractions. For example, we can have a triangular block, a circular block, and a rectangular block and tell students "1/3 of the objects is circular" Area Model: The area model is built with one "whole" rectangle, with a smaller rectangle on top. This is typically used with rectangle fraction manipulatives. However, it is easier to start with circular manipulatives for area model because circles have a distinct shape and it is easy to see how much is missing or filled in. Linear Model: The Linear Model compares lengths of objects. For example, you can set your rectangular fraction manipulatives in a lateral line and say things like "this is 1/3 the length of the line" Showing Fractions: We show fractions in the same three models as we build them. However, we exclusively use the area model to show addition, subtraction, and multiplication of fractions. Showing Addition: For example, in the problem 1/3 + 1/2, we use three rectangles-one to show 1/3, one to show 1/2, and one to show our answer. We show 1/3 by drawing three sections (vertical or lateral) in our rectangle and color one section in. Showing 1/2 is a bit different because whatever orientation you drew thirds in, we must draw one half the other way and color one portion in. Then, we draw the opposite lines in the two different rectangles (draw the one half line in the one third rectangle and vise versa). We then see how many portions of our new size pieces are colored in. Lastly, in the third rectangle, we draw a combination of both lines in how we drew them in the original two boxes and color in the number of spaces we observe in the first two rectangles.

We finished learning how to show fractions:Showing Subtraction: We use two rectangles to show how to subtract fractions. The first box is where we show the first fraction. The second box is where we show what is being subtracted. We then draw both fraction lines in both boxes to see how much will be missed, and then x-out this amount from the first box. For example, in the problem 3/4 - 1/3, we draw 3/4 as an area model in the first box, and 1/3 in the second box (but in the opposite direction). We then divide both boxes into the "same size pieces" (3/4 in the 1/3 box and vise versa). Now, we can how much will need to be missed from the original (3/4) box, and x said amount out. Showing Multiplication: We only use one box to show multiplication of fractions. For example, in the problem (2/3)(1/2), we divide the box in half laterally and the box into thirds vertically. From there, we color in how many thirds are listed in the original problem, and how many halves as well. Now, we can see how many pieces are double shaded. The double shaded portions will be our answer. We then went over how to show addition, subtraction, and multiplication of decimals:Showing Addition of Decimals: In problems where the only things being added are tenths, we draw one box with nine vertical lines (ten pieces) and color in what is being added in different colors. We count all the shaded areas, and the number we get is our answer. When hundredths are being added, we have nine vertical lines and nine horizontal lines in order to divide the box into one-hundred pieces. We color in the amount of tenths being added in two different colors, and then place circles in the hundredth-boxes in two different colors in order to see what is being added. By counting all the shaded and dotted areas, we can then see our answer. Showing Subtraction of Decimals: Showing subtraction is similar to showing addition, only we color in the first number in the problem. From there, we circle and draw an arrow away from the diagram (this shows that we are taking away the second number value). Showing Multiplication of Decimals: When multiplying anything, it is vital to remember that the first number value is the number of groups, and the second value is how much is inside each group. In the rectangle, we make sure and divide it into hundredths. From there, we color in the first number value in one color (vertically), and the second one in a different color (laterally). From there, we can see which units are double shaded. The number of double shaded hundredths is the answer (or product).

We went over how to solve addition, subtraction, and multiplication of decimals: Solving Addition and Subtraction: We use the same method to add and subtract decimals. The first step is to estimate what the answer would be. Secondly, we line up the whole numbers. We shouldn't "line up the decimals", because kids have trouble with place value. For example, in the problem 32-4.2, we estimate first. 32-4 is 28, so that will be my educated guess. After that, we line up the whole numbers, because the whole numbers are the most important in the problem. We do this like so:32 4.2From there, we subtract as normal and get an answer of 27.8. Now, we can look at our estimation, which was 28, and our answer, which was 27.8. These two are pretty close, and make sense as an answer. Solving Multiplication: We multiply decimals in three steps:1) Estimate First!2) Multiply Without the Decimals3) Place Decimal LastFor example, in the problem 2.5(4.1), we know that 2(4) is 8 so that will be our estimate. From there, we multiply without decimals, which will look like this:2541This will give us an answer of 1,025. This is nowhere near close to our estimate, but we can place our decimal in a way that does. 10.25 makes sense and is close to our original estimate. A caveat of this strategy is when we multiply with hundredths and tenths in the same problem (Ex: 0.02 · 0.6) In these cases, we will have to count decimal places.

Learned About Two-Color Counters (Introduction to Integers):It was discovered that the red side of the counters represented negative, while the yellow/white side represented positive. Building Integers:We use the two-color counters to represent adding and subtracting integers. All red (negative numbers) with line up laterally under the positive numbers. When negative and positive counters are stacked together, they represent a zero bank. When the number of negative counters is more than the positive integers, the answer is a negative number. This works the other way around as well, when positive counters outweigh the negative counters.

Showing Addition and Subtraction of Integers:Showing Addition: Showing adding integers is similar to how we build adding integers. When showing integers, we use the plus sign (+) to signify positive values, and the negative sign (-) to represent negative values. For example, in the problem 5 + (-2), we first double-check how we say the problem out loud, which would be "Five positives take away two negatives". Then, we draw out the problem like so:+++++--The highlighted blue values create a zero bank, and are therefore no longer a part of our diagram. This leaves us with three positives, which is our answer. Showing Subtraction: We use the same showing system as addition for subtracting integers. The only difference is we create a zero bank if we are taking away a value we do not already have. For example, in the problem 4 - 6, we do not have six positives to take away, so we make more zero pairs like so:+++++++ ---After making a zero bank, we circle six positives and draw an arrow away from the diagram, to signify that the value is gone. What we have left is two negatives, which is our answer.

Solving Addition and Subtraction of Integers:Hector's Method: This alternative algorithm, created by a student, is rooted in previous math content, catches, mistakes, and is expandable. It allows students to add larger integer values without drawing out a large diagram. For example, in the problem -14 + (-30), we first find the bigger pile of the two, which is negative thirty. We draw two negative signs under (-30), and one under -14. We circle one of the two negatives under (-30) and the negative under -14. Because both are the same sign, we know it is an addition problem. The remaining negative sign allows us to know that our answer will be negative. From there, we add fourteen and thirty, which is 44. But, we know our answer will be negative, so our answer is -44. Hector's Method works for subtraction of integers as well, but a first step must be added before carrying out this algorithm. This first step is called "Keep Change Change", meaning we keep the first sign of the first number in the problem, change the middle sign, and change the sign of the second value in the problem. From there, we can carry out Hector's Method. For example, in the problem -41 - (-30), we keep the negative forty one, but change the subtraction sign to an addition sign, and change the negative thirty to a positive thirty. This way, we can get the correct answer by using Hector's Method to solve the rest of the problem.

Went Over Multiplication of Integers: How to Show and Solve: Showing Multiplication: Showing multiplication reinforces the use of addition (+) signs for positive units, and subtraction (-) signs to signify negative units. For problems such as 2(-3), we would draw two groups, with three negative units inside of these groups. however, in problems such as this one: -2(-3), we would make many zero pairs (enough in order to do the problem). Then, we would circle a group of three negatives, as well as another group of negative three (as the problem is take away two groups of three negatives). We then draw arrows away from each circle to show that we are taking that value away. Solving Multiplication: Solving multiplication problems involving integers is actually very simple: if the two numbers being multiplied have the same sign, the product will be positive. If the two numbers being multiplied have different signs, the product will be negative.Went Over Dividing Integers: When dividing integers, we use the same rules as when we multiply integers. If the two values are of the same sign, the answer will be positive. If the two values are different signs, the answer will be negative.

Went Over Order of Operations: We decided that PEMDAS (the typical system for order of operations) was archaic and made students more confused than they needed to be. Then it was established that this diagram is a better way to teach kids order of operations: GroupingExponentsMultiplication/Division L-->RAddition/Subtraction L--> RThis diagram is tiered, so it helps students know what comes first. A "Group" can be a numerator or denominator, parenthesis, brackets, or separated by addition and subtraction signs. The different groups of a certain problem is highlighted below: -42 + 3 · 2 - 6 ÷ 2 + 3 - 32After grouping effectively, one can complete the problem starting with any group they want. As the groups simplify, we can add/subtract from left to right, like so:-16 + 6 - 3 + 3 - 9Went Over Scientific Notation: The form of Scientific Notation looks like this:__________________ • 10exponent1-9 (one-digit #)A positive exponent means the number has a lot of zeroes on the right side (big number), while a negative exponent means the number has a lot of zeroes on the left side (small number). We convert to scientific notation because it is easier to calculate without having the multiple zeroes in the number value.