Elementary Mathematics

Image: https://drive.google.com/file/d/1hWBUEWGeSOe8JFSyHveOb-lC4_IXP8Jl/view?usp=sharingTraditional methods of division, long division, can be confusing for kids as they have to use place value and carrying values. Instead, there are alternate methods of dividing including repeated subtraction and upwards division. All methods are shown in the photo linked. Both alternative methods make it easier for students to keep organized and allows them to work with easier numbers.

Part One of the mind map

https://drive.google.com/file/d/15gHHta5GB0yPkVJxmmtX0ZUDRrNm83kZ/view?usp=sharingDivisibility rules are important to know whether or not a number goes into another number. It also helps students break down numbers and simplify fractions. There are divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10.

https://drive.google.com/file/d/10BcpiukhX4z4PHhTjdvGaT90OU_Gle7W/view?usp=sharinghttps://drive.google.com/file/d/1efHE7CCMeMyNLmHkl3fZAEoszioz0k_X/view?usp=sharinghttps://drive.google.com/file/d/19ys6uo_pnPYWbkhgyEXRhz5h1YRZV4Iz/view?usp=sharingBuilding and showing integers is done using counters. These are manipulatives that can be really helpful for showing positive and negative numbers because they are color coded, one color on each side. When teaching students how to use them, it's important to show that red is negative and yellow is positive so they understand the meaning of the colors. It's also important to teach that negatives go on the bottom, positives go on the top, and you can build an infinite "bank of zeros". Adding one positive and one negative makes a zero, so if you have to subtract 5 negatives and you don't have that many, building zeros will allow you to have the number you need.

https://drive.google.com/file/d/1cVM89KMuLm9WgRk2V5QHqfQwDSpUu-bC/view?usp=sharingWhen adding integers, you can build mini diagrams as shown in the image to determine what sign the final solution will have. When adding integers using counters, as the video shows, you add the number of counters corresponding with the number you're adding. If you're adding 2 negatives, add two negative counters.Using the mini diagram is going to be especially helpful when adding large numbers whereas counters are for small numbers.

https://drive.google.com/file/d/1bCLQg_L84V-t3XroQG8-jJkoIBNfmOA4/view?usp=sharingWhen building integer problems, this refers to the counters. You show the act of taking away by circling the ones you are taking away and drawing an arrow outward to indicate that you are taking them away. When showing, this refers to using positive and negative symbols. The same take away rule applies, you just circle what you're taking away and draw the arrow. The difference between showing addition, subtraction, and multiplication is that when you working with addition and subtraction, you circle the whole amount you're taking away. With multiplication, you're working in groups. So instead of taking away one large group of 12, you're maybe taking away 3 groups of 4. You still circle them and use the arrow.

This week, we created lesson plans targeted towards students who were working on fractions with unlike denominators, finding the LCD, and making common denominators, which led into mixed numbers and improper fractions. https://www.youtube.com/watch?v=CoCmsyFQ_Xc

https://drive.google.com/file/d/1i2NU-KZZtQb457W9ofY7LYXrJZWaM0u-/view?usp=sharingWhen working with decimals, there are various ways to show them, including exponential form and fraction form. When working in decimal form, it's easier to work with them if we show it. In the picture, there are examples of sort of area models that you can use to show decimals. When multiplying, you do one number horizontally and one vertically, and use the part that overlaps as the answer.

PEMDAS is the acronym that we heard growing up, but a better one to use is GEMDAS. It stands for grouping, exponents, multiplication/division, and addition/subtraction. The reason that M/D and A/S is because you move left to right. Multiplication doesn't come before division, it just moves left to right. Order of operations can also be a way to break down really large problems and doing them in steps.https://drive.google.com/file/d/1h3Ei1w5kHiPoVAMHmZETiImuzpqmWYq1/view?usp=sharing

Scientific notation is a way to represent really big or really small numbers like the depth of the ocean or an atom. The first step is to determine if the number is big or small. This will determine which way the decimal is moving, and in turn, whether the exponent is positive or negative. If the number is big, such as 3,400,000,000 then the exponent will be positive. You have to move the decimal to be between the first two numbers. If the number is small, such as .00000000045, than the exponent will be negative.https://drive.google.com/file/d/13FYwDNjdBqlwQkj4a5_fCu3I1e6l7hbk/view?usp=sharing

Polya's problem solving strategy consists of four steps and is a way to break problems down so they are easier to understand. The first step is to understand the problem, the second step is to devise a plan, third is to carry out the plan, and the last step is to look back and reflect on the problem. A few examples are drawing a picture, finding formulas, looking for patterns, or working backwards.Five frames and ten frames are ways to help kids with their one-to-one correspondence and transition them into using base 10 blocks. They help kids when learning to count.

Base 10 blocks are tools that can be used to help kids have a visual representation of problems and they help with learning place values. Units represent ones, longs represent 10s, flats represent hundreds, and cubes represent thousands. When working in base 10, 10 units become a long, 10 longs become a flat, and 10 flats become a cube.https://drive.google.com/file/d/1vX_YpvcK0dd0fMXRM3OJs1w6w3j7_RfX/view?usp=share_link

Each number has a base. If the base isn't shown, it's safe to assume that we're working in base ten. A good way to help kids understand adding in different bases is to draw it out in units, longs, and flats. For example, if we are working in base 10, 4+5 is 9. However, if we are working in base 6, 4+5 is 13 in base 6. This is because numbers cannot be larger than their base. Since 9 is larger than 6, it gets converted to one long and 3 units.https://drive.google.com/file/d/1rxjT_KZuZSeo7_qmndX-QmAoW1l0h9CQ/view?usp=share_link

When subtracting in bases, the same principles apply as with adding bases. The numbers cannot be larger than the bases. So when subtracting 156 in base 7 from 243 in base 7, the answer is 54 in base 7. An easy way to see this would be to sketch it out in base blocks. https://drive.google.com/file/d/1rZkcFi72-yrZudZgjmfAel8JCc8fzrR1/view?usp=share_link

When converting from one base to another base, it can also help to sketch it out in base blocks to see how many blocks you have in total. The numbers you have still cannot be more than the base of the number you're converting to. So if you're converting 432 base 6 to base 12, all of the digits need to be 11 or lower since none of the numbers can be 12 or above. So the answer would be 328 base 12. This means you have 3 flats, 2 longs, and 8 units, the flats being 144 units, and the longs being 12 units each.https://drive.google.com/file/d/1-MkBMfYkF_4lZyEmhhowpzddQ3iB6oNB/view?usp=share_link

If you're converting bases, the easiest way to explain what you mean is to use base blocks. So look at the base that you're starting in, and draw it out. If you're working in base 5 and the number is 134, that means you have 1 flat, 3 longs, and 4 units. That gives you 25 +15+4, leaving you with 44 units total. So if you're converting to base 10, know that you have 44 blocks to work with total. So in base 10, the new number would be 44. It helps to know how many units you have total and redistribute them accordingly to match the parameters of the new base.https://drive.google.com/file/d/1kHz7Rv3dHx43juFMcNQ4A74p2Z_dTP9k/view?usp=share_link

The traditional way of doing addition can be difficult for children because they may issues carrying numbers or knowing their place values. Alternate algorithms are ways of doing addition that make the numbers easier for kids to approach. The traditional method involves lining the numbers up and adding down the rows. Friendly numbers are when instead of lining the numbers up, you carry some of the numbers over to make the numbers end in zeros. So an example of this would adding 24 and 16. Instead of lining them up, carry the 4 over to 16 to make it 20. Now you're looking at 20+20, and that's much easier to solve. Another example would be 275 + 135. Move the 3 over to the 7 and the 5 over to the 5. Now you're left with 310+100, which leaves you with 410. Trade off is really similar to friendly numbers, but it doesn't always line up perfectly. So the method of solving is the same, but you are only taking part of the number away. An example of this would be 36 + 98. Move 7 from the 9 over to 36, and 4 from the 8 over to the 6. You're left with 110 + 24 which gives you 134. The scratch off method is arguably the most similar to the traditional method since you're lining up all the numbers. However, it differs once they are lined up. You still add down the columns, but once you reach 10, or the base you're working in, you scratch off that number, and each scratch correlates to a long that carries over to the longs category. Then, you do the same in the next categories, and those numbers carry over accordingly.https://drive.google.com/file/d/1KhxyBuwUCJ-nhtlQLpa7JTETLATFs2Vz/view?usp=share_link

These are similar to the other alternate algorithms mentioned in the way that they eliminate the need for carrying and make the numbers friendlier for kids to work with. With expanded form, this is another way to make the numbers easier. So an example would be with 25+146. You expand them like this: 20+5+100+40+6. This makes the numbers friendly. So the answer would be 171. With left to right, you line the numbers up, and add the columns left to right. So when adding, an example would be 537 + 265. You add left to right leaving you with 700+90+12, giving you 802. With the lattice method, it gives kids a spot to write each digit and helps them understand place value. So you line up the numbers, and add them into the lattice, which is a sort of grid that you use. You make one box for each place value, and put a diagonal line through each box going to the left, and when you add, if you get a double digit number, instead of carrying that number, you put it into the designated box and add diagonally.https://drive.google.com/file/d/1o-v6-PRiAPlYfhbP8fnxjnXQmSoA8DB2/view?usp=share_link

"Showing Subtraction" is the method of depicting subtraction problems in a visual way in order to make it easier for students to understand. You draw out (or "show") the subtraction problem using units, longs, flats, ect, and show that some of them are being removed. So if you are subtracting 22 from 125, you start by drawing 1 flat, 2 longs, and 5 units. Then, you circle what you taking away and draw an arrow to show you're removing them. So you'd circle 2 longs and 2 units, and draw arrows showing they're being removed.https://drive.google.com/file/d/100_BQreZuaa-8ETWJbhi_PgWSy9QtEkW/view?usp=share_link

Building multiplication is a visual way of depicting multiplication. You start by building a column and a row, these are the numbers you're multiplying, and then you sort of fill in the gaps. So if you're multiplying 3 by 5, you use 3 units on top and 5 units down, and then, not touching those units, fill in the space. You'll find that you have 15 units.https://drive.google.com/file/d/1CbFjDYUCfh4ZkI2ZT0VvlkUVxbjcH40D/view?usp=share_link

Showing multiplication and area models are another visual way of depicting multiplication. One way of showing multiplication is using groups. So if you have 7 times 3, you make seven groups, and put 3 dots in each one. The total number of dots, 21, is your answer. Another method of showing multiplication is a similar concept but instead of using circles, you just use units. So with the same example, 7 times 3, you'd draw 7 groups of 3 and count the units, which gives you 21. You can also use an area model. This is where you draw a box that is the same number of boxes as the amount of units you have. So if you have 132 times 43, you expand it into 100, 30, and 2 on the top, making a box that's 3 long by 2 wide since you break 43 into 40 and 3. Then you multiply each number into it's corresponding box. This can help students build foundational skills for algebra and later math.https://drive.google.com/file/d/1NTyroXZekuKtFpYepncn-amZ5TMlBW03/view?usp=share_link

Over time, it has been shown that the classic method of timed multiplication tests are ineffective and often induce more stress than results. Instead, we can use other methods of teaching to improve their automaticity. We should teach multiplication in a certain order to make sure that students work from easiest to understand to hardest to understand. This order is 1, 2, 5, 10, then 3, 9, doubles, and then 4, 6, and 8. This is the easiest way for them to learn. They should use flashcard to improve their speed and automaticity, and should stay away from tricks that don't always work and won't help them learn in the long run.https://drive.google.com/file/d/1wW7OFAhNu5MFrKxFk6IIiq16q5BZvg_z/view?usp=sharinghttps://drive.google.com/file/d/1ioZOTft43XLyOljnps3YYSCdkKcFMmXq/view?usp=sharinghttps://drive.google.com/file/d/1TInq2gLgFTqpJhWQ4xQmpLdDTcR_RFVx/view?usp=sharing

Focusing on standards is important when teaching. We completed an activity that aligned to a grade level standard.For my activity, I wanted to focus on second grade. I chose the math standard 3.OA.C.7 Multiply and divide within 100 which states that students should be able to fluently multiply and divide within 100. By the end of grade 3, know from memory all multiplication products through 10 x 10 and division quotients when both the quotient and divisor are less than or equal to 10. I think a domino activity could be really good to help them practice multiplication through 10s. Dominos have a line in the middle, so they could multiply across the line, 2 times 6, 5, times 4, ect. They could also practice division if you specially select the dominos, like 6 divided by 2, 4 divided by 2 ect. You could also print the dominos on paper to reduce the need for equipment and make sure they get the numbers they need to practice. This could also be done with dice, you could roll one number, then the next number and multiply. If you want bigger numbers, use two dice and add them up so you get things like 10 times 9 ect.