Multiplication and Division situations Assignment | Get Paper Help

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Real-World Connection to Area and Perimeter

Rodregus Garner

Walden University

Dr. Ruth Collins

MATH 6563- The Base Ten Number System and Operations: Multiplication/Division

Multiplication and Division Situations

There are many strategies and techniques students can use to increase their skills with addition, subtraction, multiplication, and division. Students must be allowed to explore and practice these concepts to know which process works best for them. As a teacher, I demonstrate and model how each of the four mathematical operations goes hand and hand. Multiplication is repeat addition, as division and subtraction separate parts and taking away from the minuend. Ideally, the teacher should ensure that the students comprehend each mathematical operation by providing explicit strategy instructions. Van De Walle, Karp, Bay-Williams express that “providing students with explicit instruction is proven to be an effective strategy to help all level learn to succeed academically” (2019).

When understanding the relationship, similarities, and differences in mathematical operations, we must know how each term is connected.  In comparison, addition and subtraction have many things in common. Thinking about both subtraction and addition; subtraction is simply the addition of negative value. They both deal with regrouping and bundling and have two parts that make a whole. Also, they both are comprised of composing and decomposing numbers. Van De Walle, Karp, Bay-Williams states that “Addition and subtraction strategies that build on decomposing and flexibly composing numbers contribute to students’ overall number sense” (p. 239). Another similarity is they both can be solved by using the standard algorithm. Van De Walle, Karp, Bay-Williams suggest that “Three main components are completing the standard algorithm; knowing the step by step procedure and how it is performed, knowing why the algorithm works and how it can be applied, and knowing when an algorithm should be used and comparing the value of usefulness of different algorithms” (p. 249).

On the other hand, there are some differences when it comes to addition and subtraction. Addition and subtraction have an inverse relationship because they both undo each other. Deziel explains that “When considering the operations of addition and subtraction they both cancel each other out. Addition and subtraction are the two operations that act this way”. For example, 4 + 2 = 6, then the inverse will be 6 – 2 = 4.  The inverse of the addition problem 4 + 2 = 6, I just switch the place of the equal sign and put a plus sign and then switch the plus sign to a minus sign. Another example is using a number line to add and subtract, finding the sum and difference will travel in different directions. If I am adding numbers, my point will shift right of the number line. If I am subtracting numbers, my point will shift in the opposite direction. Lastly, the answer to each type of operation has its own name. The answer to an addition problem is called sum and they answer to a subtraction problem is called the difference.

As I stated earlier there are four mathematical operations. The next two operations I will be speaking about is multiplication and division.  There are a few similarities and differences between the two operations. They both work with two equal groups. Also, they are part of the fact family. If the student is fluent in multiplication fact, then division will come naturally to them. Smartick Method expresses that, “Multiplication and division are associated with each other. The inverse operation for multiplication is division. When we view division problem, we think about separate into equal group and multiplication is combining equal groups” (2018).

Students must understand the difference and similarities of mathematical operations. The students should understand that all four mathematical operations start with two numbers that produce a third number. The mathematical operations are basic skills that are learned in the early stage of child education. Children must learn how to add, subtract, multiply and divide early because they will be using these operations throughout their education career. Given the students the opportunity to make a connection with each operation allows them to build a foundation when it comes to problem-solving. MoveItMath discusses that “Although there are four operations, they entail only two actions: combing or separating. Addition and multiplication are both combing actions, and subtraction and division are both separating actions. The difference between addition and multiplication or subtraction and division is thus a matter of style and how the combing and separating are done” (2017).

When teaching the mathematical operation students should be encouraged to make a drawing to justify their process and answer.  Students can make a drawing of grouping, proof drawings, place value charts and more. These drawing help students to visually see what is going on during the process of solving.  The difference when it comes to using mathematical drawing to solve for addition and subtraction is when the student is using addition, they are grouping numbers to get a larger number and subtracting you are regrouping taking 10s from values when needed. Even if the students are using math drawing, they still need to know the rules for each operation.  Learning to use math drawing is an effective practice. The student will begin to do a lot of math drawing when learning geometry. My students are visual learners. They have to visually see what is going on in front of them to grasp the concept of what I am teaching. Math drawing helps aid students with critical and creative thinking.

As a teacher, we should create lessons and activities to help students understand the relationship between addition and subtraction, or multiplication and division. We have to be very explicit when giving instructing. I teach 5^th grade currently and most of my students have a good understanding of how to each operation work but sometimes completing the math is the hard part. Since I teach the upper elementary grade, when introducing the four mathematical operations, I set up my problem using a place value chart. Using a place value chart help student understand why they need the bundle, unbundle, group and regroup. Also, I use the area model as well to compose and decompose numbers. I use both of these methods before introducing the standard algorithm. It is vital that students understand the mathematical operation situations because they connect to math in the real world and also, when help when working on word problems.

Common Core State Standards (CCSS) recommends to “teach compare situations for multiplication and division in Grade 4 and recommends that equal groups and area/array be taught in grade 3” (2020). I do not agree with the CCSS recommendations because having students in grade 3 complete a task that involves equal groups and area/array can be challenging. I can see many misconceptions that students may have completed these strategies. Third grade the students are just learning their math fact. Having the student use an array to solve a problem is way too early for that grade. Students can potentially look at an array and just count all of the objects. Instead of understanding than an array is based on rows and columns and has a separate grouping.  In 3^rd grade, students should be able to compare situations for multiplication and division. At that grade, the students are still adapting to using pictures and manipulatives. In 3^rd grade, students should be understanding multiplicative comparison. For example, if Johnny has 3 balloons and Michael has 3 times many. There should a picture of a boy holding 3 balloons and another boy holding 9 balloons. Then the students will understand that 3 times 3 equals 9. I don’t think it’s appropriate for these strategies in this order because it is working backward. As a fifth-grade teacher, I do see how these strategies align or build up to 5^th grade.

When I teach my students the different comparing statements, I model what each statement means. Using statements like “three times as large and three times larger” has two different meanings. Three times as large is in number. For example, the number 7, three times as larger will be 21. I show my student the factor of 7; such as 7, 14 and 21 or I have them to add

7 + 7 + 7 which equals 21 to get an understanding of what the statement means.

On the other hand, three times larger can be measured in units or objects that can grow or expand three times the size. An example is a square that is scaled at 1 meter by 1meter, it can increase three times larger is 3 meters by 3 meters. Another way a teacher can connect the statement three times as large if saying that the school gym at our school is five times as large as our classroom. The student can see the comparison between the school gym and our classroom.

Math is content that is never going away. I use strategies that connect math to real-world applications and situations. Students are more engaged when they can connect lessons or activities to their daily lives. Providing my students with manipulative and modeling math drawings can also help encourage the habit of mind in my students. Also, promoting math vocabulary and helping them understand the language. Letting them know that even though this is a math class, literacy plays a major fact in this content area as well. Show them how to solve a problem using different methods and allow them to choose the best one fits for them is also good. I currently have students that are having an issue with multiplying multi-digit numbers. Those kids chose to continue using the area model versus the standard algorithm. I am working with those students to move past the area model and shift over to the standard algorithm. It is slowly working but I am now able to see the progression. Practice and experience are also a strategy that can encourage the habit of mind in students. Bryant suggests that “Students immersed in classroom experiences that let them engage in learning mathematics concepts through problem-solving, making and using abstractions, and developing and applying mathematical theories have greater opportunities for developing mathematical habits of mind” (2015).

References

Deziel, C. (2019, March 2). Examples of Inverse Relationships in Math. Retrieved from https://sciencing.com/examples-inverse-relationships-math-8415825.html

Grade 3 ” Introduction. (n.d.). Retrieved from http://www.corestandards.org/Math/Content/3/introduction/

SmartickContent Creation Team. A multidisciplinary and multicultural team made up of mathematicians. (2018). The Relation between Multiplication and Division. Retrieved from https://www.smartickmethod.com/blog/math/operations-and-algebraic-thinking/division/multiplication-division/

(n.d.). Retrieved from http://moveitmaththesource.com/asmdactions/

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and middle school mathematics: Teaching developmentally (10th ed.). Boston, MA: Pearson.