Elementary Mathematics

https://www.youtube.com/watch?v=_c-3QrR8LEQWhen addressing a mathematical situation, try this four step approach to problem solving:1) Make sure you understand the problem! -Do I know what I read? Can I break this down into parts? -Can I explain this to someone and it make sense?2) Develop a plan on how to tackle the problem. -What do you already know about this problem from previously taught math? -Do you see anything similar to another problem you have solved? If so, how did you handle that? -What tools in my tool belt can work here? More explained in a subtopic.3) Carry out the plan. -Did it work? Do I need to fix something?4) Look back. -If you can solve it more than one way, or solve it backwards, you've probably got it. -It has to make sense in the context of the problem.

There is more than one way to possibly solve the problem. What works for one learner may not be the best fit for another.As a teacher, encourage students to share their thought process as it may help the student fully explore their thoughts and make those meaningful connections to the math problem. For the other learners, they may see a different approach to help them solve the problem differently in the future. Good educators will expose students to more than one strategy when possible for a mathematical problem. We can help students build their math tool kit. Strategies we used in week 1 to solve the same problem about chickens' and pigs' feet:1) guess and check 2) Diagrams and drawings -tape diagrams -arrays -representations using circles, boxes, tally marks, etc3) lists4) algorithms

This is the first time we used the manipulatives to build numbers. This exercise helps students really understand the representation of units in place value, starting the with the individual units. Students will begin to solidify their understanding of the base ten system, and could easily apply this knowledge to other bases.We would introduce the idea of multiplication using the manipulatives. The students will build the first number and then add the units for the next number to the original number built. For example, if they had the number 12, they would build it with one long and two units. To add 5 more, they would count out five more individual units. The students would then count all of the pieces laid out to solve the expression of 12+5. Later in the module, we practiced having the students move away from the physical manipulatives and using diagrams to show and solve addition problems. A box would represent flats (100s), a long line for the longs (10s) and dots for the units (1s). Students would be encouraged to use more than one color to track the original number and what was added to it. The student will then practice counting each flat, long and ones (remembering to exchange as the number in the place value can never exceed the base) to solve the expression.

This lesson discusses the test for determining if an algorithm is good. These three criteria should be applied to each algorithm:1) is it expandable (able to be used on harder problems)2) is it efficient3) is it based in math conceptsWe explore how many students of my generation were mostly taught the traditional algorithm; however, this method can cause problems for some learners as it is not rooted in solid math concepts when we "borrow" from tens and hundreds. Students can also find confusion in how it is solved from right to left when that is not how we write or read numbers. It is a good idea to be well versed in alternate ways to solve problems so that students have different strategies that may work better for their needs. We will present expanded form first and then move the student to more efficient algorithms, such as left-to-right, scratch method and lattice method (which I had never seen applied anywhere outside of multiplication). Once the students have mastered the math concepts, we can teach the majority how to calculate quickly using the traditional algorithm. Some learners may need additional time to build their number sense and may prefer other strategies in the interim, while some learners may simply opt for these strategies in lieu of the traditional method. Our jobs as educators is to build up their understanding of the math concepts and the strategies to help them solve mathematical problems.

Students will start learning the basics of subtraction by using ten frames to build a number and practice "taking away" from each place value. Students learn how to exchange units for tens, tens for flats, and so on. Each unit subtracted is removed from the board until they have the answer of what is left. Students can then "add it back" to understand that they have the right answer. Once students have mastered this, they can gain a better understanding of the math concepts of subtraction by learning the alternative algorithms rooted in math solid bath concepts. These are decomposing (expanded form), left-to-right and equal addends. Eventually, they will move to the traditional algorithm for solving subtraction that is not based on solid math principles, but they will have a general understanding of what they are doing so they just quickly calculate.

In this module, we are introduced to how to teach multiplication. 1) expose students to creating multiplication problems with base ten blocks to think of multiplication as best represented as the area of a rectangle.2) introduce students to diagrams and arrays to demonstrate the idea of equal groups (i.e. 5x4 means 5 groups with 4 each)3) help students solve multiplication using the expanded form, solving left-to-right, area models and lattice4) we are shown the order to teach the multiplication facts for automaticity starting with the easiest facts students learn to skip count and build on prior knowledge to help them move through the harder facts1's, 10's skip-counting, 2's skip-counting, 5's skip-counting, 3's, 9's without the finger trick, final 12 -as long as commutative property is understood first

In this module, we are introduced to how we will teach division to our students with a focus on expressing the remainder as fractional units. 1) remember to make sure that students know how to set up a division problem correctly. They should understand the relationship between the number that goes inside of the division symbol as the whole and the outside as what it is being divided by. It does not always need to be a smaller number on the outside. 2) when moving from building to diagramming, help them think about the easiest way to make the drawings/ less drawings and how you can manipulate the problem to achieve that goal3) make sure the students can follow the four steps a. know how to write the problem correctly b. estimate c. multiply and subtract correctly d. write the answer correctly 4) the algorithms we were introduced to: a. long-division b. repeated subtraction c. area model d. upwards division (great for helping students see the relationship of the remainder as a fraction)5) divisibility rules: 2: even numbers 3: sum of the digits divisible by 3 4: last 2 digits divisible by 4 5: # ends in 5 or 0 6: if 2 works and 3 works, divisible by 6 7: no rule exists 8: last three digits is divisible by 8 9: sum of the digits is divisible by 9 10: ends in 0 720 is the smallest number divisible by all!!!

This lesson introduced us to the different fraction manipulatives students can use. My classroom uses fraction tiles as a part of our curriculum; however, I should invest in the circular/pie fraction pieces as well. Students should have a chance to play with the materials to learn how the pieces fit to the whole so that they can begin to conceptualize the relationship between the numerator and denominator. We will teach them to think of the numerator as the number of pieces and the denominator as the size of the pieces. We will build upon their knowledge that a smaller number piece like 1/3 or 1/2 means less pieces/larger piece sizes. Then students will be given an opportunity to play with the fraction manipulatives to find different ways to build equivalent fractions.

Adding/Subtracting:Our answer will have one color with smallest number of pieces.

Students will naturally use circles to represent easy denominators. We just need a shape to represent 1 whole unit. In cases like these, encourage students to represent trickier denominators with rectangles instead of circles. When students are adding together fractions, remember we are adding pieces to get to a whole. The bottom represents the size of the piece. Students should not be adding the bottoms together, but first drawing the unit and breaking it into size pieces. Then students will shade the numbers they add across the top. When students encounter different size pieces, encourage them to draw both fractions out so they can see the pieces are different sizes. Rectangles are ideal and one unit should be divided vertically and one horizontally. Then show them how to draw one unit with both size pieces represented. The students will go back and transpose the lines from each unit over to the other to see the number of pieces needed to add from each equivalent fraction. Encourage students to use different colors to help them visualize making the fractions equivalent. Students will follow similar steps when working subtraction problems. Students will need not need to draw the third box. They will "take away" the same size pieces from the original box.When teaching multiplication of fractions to students, model that we will start with a group and only take the parts that we want. We will model multiplication using only one box, starting with the drawing of the group and shading the part we are concerned with. We will then break up the group by the fraction of what is inside the group. The answer is the only the part of what's inside the group we originally shaded (double-shaded).

When adding a whole number plus a fraction, model it for the students that they just need to put the two numbers together. When subtracting a fraction from a whole number, take the whole number down by one and change the final unit into the fractional form of what fraction you need to subtract. Ex 13- 1/5, change the 13 to a 12, then take the final unit borrowed and convert into 5/5. Now you can subtract the same size pieces. Remind students that we are working with the size of the pieces, and we are only adding or subtracting the number of same size pieces that we need.Students will need to know their factors of a number in order to work with fractions that do not have same size pieces. We should encourage them to look for the numbers that have the least number of factors.When working with mixed numbers, do not teach converting to an improper fraction. They have to remember the steps and they will be working with huge numbers. We have other ways to teach solving this based in sound math principles.Multiplication of fractions is easier because we do not need to find common denominators. There is a strategy that set students up for success with harder math concepts down the line. They should not multiply first, then simplify. We should teach them to simplify first! He teaches the "Funky Ones" where we do not "cancel," rather use factors that create ones to reduce the factors down.We only put a one under a whole number when we multiply or divide fractions. We also only make improper fractions when multiplying and dividing fractions. Think of the "backwards C." Division- Keep, Change, Flip. Why behind the division algorithm?-fact families: groups of numbers that are related in some way based on operationsex: 35\5=7, 5X7=35, 7x5=35

When building integers using two color counters, it is important to remember that red equals negative integers and non-red is positive. We will use the language "build 5 positive or build 5 negative" initially when introducing this concept to students.When adding integers, we teach them to put positive values on top and negative on the bottom. This mirrors what we see in real life which activates prior knowledge and helps with retention. Model lining up the positives and negatives so that they can visualize and understand the concept of the "zero bank" by making pairs of zero (cancelled out). What is left over when building or using diagrams is the answer and the corresponding color or symbol determines if the answer is a positive or negative integer.Diagrams are demonstrated by placing the plus sign for each number on the top row and the negative sign on the bottom. When they show how to solve, it is important that they do not use any symbols to try to demonstrate how they are adding... this will create confusion and the wrong answer. Students may put everything (positive) in a line on top or chunk it if it helps them visualize the problem. Same for the bottom.

You cannot do Hector's Method with subtraction. This model doesn't compute mathematically as it is. If you draw diagrams and then solve Hector's, the answers will not match.In order to use Hector's Method, you need to change the operation to addition and add the inverse of the number. Once you do the inverse operation, you can then use the model to finish solving.Remember when drawing diagrams, we do not need to create a zero bank unless we are trying to account for something that is missing.

Draw out the equal groups with the positive and negative symbols to represent the numbers as they appear in a problem.If you have a negative number of groups, create a zero bank to create groups. This is because the problem is really saying 0 take away x number of groups of y. The zero bank will help demonstrate this for the student. When multiplying integers, If the signs are the same = positive If the signs are different = negativeNow just multiply5 x -8 = -40 Division: the rules are the same

I was taught PEDMAS many years back, but as he pointed out in the lecture, this method is problematic. P means parenthesis, not multiplication. He then introduced us to GEMDAS. This is fine until students get confused about the order of solving. They think they must multiply before dividing and add before subtracting in every case.Then he introduced us to what we should teach starting with finding groups, working exponents and moving from left to right on the multiplication/division before moving left to right on addition/subtraction which will be at the bottom of the V you make when you solve this.The goal is for students to find groups based on the addition and subtraction signs, as well as the numerator and denominators. Students will solve all of the groups at once and then add/subtract from left to right at the end. This limits the amount of rewriting students will need to do leading to possible errors like the professor made on the very first problem. Lastly, he talked about the right way to think of squaring negative numbers meaning the the negative sign is on the inside of the parenthesis and the exponent falls on the outside,

When thinking about what percent means, we need to teach students that percentages are really just ratios which means they are fractions. This is a number divided by (per) 100 (cent). For example, 30% means 30 per 100. He helped demonstrate this by creating a diagram for the fraction with ten boxes (equal to tenths) and had us shade in three boxes to represent 30% or 3/10. We worked through a few additional examples of this concept before we began to mentally calculate percentages. Before moving on, we need to check for the student's understanding and mastery of thinking about what 10% of any given number will be. Once they can calculate 10%, they can think about how to multiply that number by the percent we need to calculate. He also showed us how to break apart percentage problems that were not clean 10% values. I opt to solve mostly by finding what 1% is and then multiplying that number by whatever actual percent I need to find (with the exception of problems in increments of 5%).