Introduction
Performing mathematical operations successfully is dependent on the students’ understanding of the relationships between different operations. This paper discusses the relationship between additions and multiplication. It also shows how a good perception of the relationship aids students to understand the operations besides discussing the link between commutative, associative, and distributive properties.
Relationship between multiplication and addition operations
Multiplication is also termed as repeated addition (Reys, Lindquist, Lambdin & Smith, 2012). Good understanding of how to carry out additions can incredibly help students to carry out multiplication successfully and with accuracy. The relationship between the two perhaps explains why addition skills are taught first in the elementary levels (Bassarear, 2008).
Examination of this relationship is perhaps well accomplished through consideration of an example. Consider a solution for 3*4. It can also be expressed as 3+3+3+3, which can be interpreted as adding the number on the left of the multiplication operation sign to itself for the number of times shown in the right of the multiplication sign.
How understanding the relationship between multiplication and addition helps in understanding of the operations
A simpler way of explaining the relationship between multiplication and addition is by considering practical scenarios. For instance, in a class of 10 students, each student may require two books.
If a student is asked how many books are required together, in case the student has good addition skills, the easiest approach is to add up the number of books required by each students for 10 times to get 20 as the solution (I.e 2+2+2+2+2+2+2+2+2+2=20). This operation can be simplified as 2*10=20.
The case shows how multiplication extends addition concepts through multiplication of groups for total products. The relationship implies that students need to learn how to formulate rather than memorize while attempting to learn multiplication from addition principles.
Although this approach is a bit lucid and one that is characterized by many challenges for students with low mathematical skills, it helps to explain the relationship that persists between multiplication and addition thus enabling students to execute multiplication with precision by relating it with addition skills.
Commutative, associative, and distributive properties
As a property of numbers, the term commutative is derived from the word commute, which literally means moving around. In mathematics, it means moving numbers around. When this moving is done, the sum or product is not affected by the changes.
For instance, 2+3=5, the same expression can also be written as 3+2=5. For multiplication, 2*1=2. When the numbers are tuned around, 1*2, the product is the same. Therefore, commutative property holds that the outcome of addition and multiplication remains the same regardless of the order of the digits.
Associative property means that numbers in the mathematical operations can be grouped or associated. In case of addition, the solution to 1+2+3 can be accomplished in two ways. The first approach is to add 1 and 2 first and then add 3 to the resulting sum {(1+2) +3}.
Alternatively, one can add 2 and 3 first and then add 1 to the sum . The total sum for these two approaches is 6. Hence, the operation is said to be associative. When a similar concept is applied in multiplication, 1*(2*3) is expressed as (1*2)*3.
Distributive property underlines the capacity for a multiplication sign to distribute over addition signs. For instance, 2(5*3) means (2*5) + (2*3). Whenever a mathematical question demands application of the distributive property, it simply means taking multiplication sign across parenthesis (brackets).
How commutative, associative, and distributive strategies relate with students’ thinking strategies
Some of the thinking strategies used by students include counting by twos, fives, groupings, or by sets of items and adding several equal groups together (Reys, Lindquist, Lambdin & Smith, 2012). For the distributive case, 2(3*2) would be interpreted as counting items in groups of twos for 3 times and then groups of the sum two times.
In case of associative property, to get the sum of 1+2+3, students can group 6 items in three groups. The first group has 1 item, the second 2 with the third group having 3. Therefore, the order of these groups is not necessary upon applying the concepts of associative and commutative properties.
Conceptual errors in mathematics
One of the common errors in multiplication and addition would arise from erroneous understanding of the application of the addition and multiplication signs especially when operating on large numbers. For instance, 12+12 may be interpreted as 1+2+1+2. To help in avoiding this error, as an instructional strategy, the concept of grouping needs to be developed in students.
Therefore, 12 means a group of 12 items but not two groups with one having one item while the second has two items. Adding 12 to 12 would mean putting twelve items together followed by another group of twelve items with the two groups being separated by some space (representing addition sign) and then counting the two groups.
Students who have poor multiplication skills but good addition skills have probabilities of confusing the signs so that 2*3 is interpreted as 2+3. This case may happen particularly when students are to use addition skills to formulate a multiplication mathematical question. To mitigate this error, the teaching strategy required is an emphasis on understanding the meaning of different signs.
References
Bassarear, T. (2008). Mathematics for Elementary School Teachers. New York: Cengage Learning.
Reys, R., Lindquist, M., Lambdin, D. & Smith, N. (2012). Helping children learn mathematics. Hobokon, NJ: John Wiley & Sons.