Addition refers to a merger or the process of joining two or more things. Alternatively, it can be described as the process of combining parts of a whole. The relationship between multiplication and addition concepts enables students to comprehend the process of multiplication much easily. In this essay, I will explain the relationship between these two concepts.
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I will elaborate on how the associative, commutative, and distributive properties are related and how grasping these principles apply to later concepts.
Teaching the Concepts of Multiplication
The process of multiplication involves three main concepts. The first concept relates to partitioning. This concept is in turn related to division. For instance, four groups of six students can be represented as 4×6 and this translates to 24 students. The rate or price forms another multiplication concept. For instance, a car is taken to travel at 70 miles per hour for five hours; it is taken to travel a total of 70×5, which translates to three hundred and fifty miles.
Besides, the process of comparison is part of the multiplication concept. For instance, John has three pens while Peter has four times as many as John. Joseph, on the other hand, has half that of Peter. This implies that Peter has 3×4 pens translating to 12 pens. Joseph, on the other hand has six pens.
The repeated addition process involves adding a given value or digit repeatedly, for instance 4+4+4+4 = 16. This simply means 4 × 4 = 16. This implies that when four is added four times, it yields 16. Students should be made to know that it is much easier and faster to represent the repeated addition as multiplication. Grouping is also a method that can be used to teach the concept of multiplication by making use of addition. As a way of elaborating on this concept to the students, the teacher can come up with three circles on the board.
The teacher should then draw two dots in each circle. This facilitates interpretation of the multiplication process by explaining that if each of the three circles encloses two dots, then all the three circles enclose a total of six dots. The addition concept is put into use by stating that two dots added to two dots and added again two dots gives a total of six dots.
Last but not least, another way of teaching the multiplication process is with the help of a line of numbers. For instance, 0—1—2—3—(4)—5—6—7—(8)—9—10—11—(12)—13. Counting four three times, starting from 0, this gives you 12 as the answer. This implies that 4 × 3 = 12.
Through continual use of the above mentioned multiplication concepts, students also reinforce their addition skills. Similarly, application of the number line assists students in reinforcing the correspondence between each other while enhancing their addition skills. Grouping assists learners to put their addition facts into practice since it is easier to recognize a group of objects.
It is critical for students to comprehend the associative, commutative and distributive properties. The associative property is derived for the term “associate or group”. For instance, x (yz) = (xy) z or 2(3×4) = (2×3)4. In this property, one obtains the same answer even after regrouping the numbers. The commutative property emanates from the term “commute” for instance, ab = ba or 3×4 = 4×3.
In this case, moving a number round still gives the same answer. The distributive property is derived from the term “distributing numbers” for example, a (b×c) = ab×c = b×ac or 3 (2×4) is equal to 3 (2) × 4 or 2 × 3 (4). In this case, one distributes a number and still gets the same answer.
The properties specified above are instrumental in thinking strategies explored by students as they perform computations. The commutative property can be linked to grouping. In the same way, they can also sketch five circles and insert four stars in them. In the distributive property, students are in a position to reflect and create sets out of the groups. For instance, 2(3) ×4 is equal to 3×2(4). They can multiply 2×3 to get 6 and then making four groups of six to obtain 24.
Besides the commutative strategy, students could relate the multiplication concept to the descriptive property by splitting array. Splitting array is significant since learners may apply known facts to study difficult facts.
For instance, learners may split a 6×9 assortment further into 6 sets of 4 and 6 sets of 5. They may then proceed to adjoin the cumulative sum of the sets. Besides, the students may decompose cumulative sum into sets. Reflectively, the product of 3×9 may be obtained through decomposing 9 into summation of 3 and 6.
The computation will take the form 3× (3+6). In addition, the associative property may be used by the student to mentality simply computation. For instance, 3× 5 × 2 = 3 × 5 = 15 thus, 15 × 2 = 30 besides other forms of clustering sets of numbers with the same multiplication result. Students may apply this property to carry out series of mental arithmetic for a single multiplication problem.
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It is natural that learners are able to perceive some mistakes made in the concepts, though such mistakes can be largely ignored when the relationship between addition and multiplication is explained to the students. Among the misconceptions that students may perceive is summing the numbers other than multiplying.
One of the measures to avoid these misconceptions is to constantly remind students on the difference of the two signs. Another misconception is the order in which to multiply numbers in the event that the multiplication factor is in parenthesis. As such, absolute care should be taken to avoid misconceptions during the teaching process.