In understanding multiplication, it is essential for students to comprehend addition rules, since they are interrelated in a big way. Multiplication is sometimes referred to as repeated addition whose applications are quicker and more effective.
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For example, instead of adding 4+4+4=12, multiplication simplifies it to 4*3=12. If there are for instance three students each with four rulers, this can be shown by putting four rulers in three groups and allow the student to count them individually. The result is that same answer will be obtained. This indicate that addition involves using the actual number of times one would multiply the number to add it.
For example, 4*5= 4+4+4+4+4= 5+5+5+5= 20, on the other hand, multiplication involves dealing with many groups with equal size, or groups with same amount of items in every group. Ability of counting the group number and the items is necessary for one to be able to multiply them. This then shows that addition is an instigator for multiplication where one is able to solve a multiplication problem through repeated addition.
This enables the students to understand addition concepts which include manipulation of groups in order to obtain the total product. Moreover, this assists the students in finding out the quantity or the size of items in several groups of equal size.
For instance, if there are eight pencils in a pack and one decides to buy five such packs, one will be able to know the number of pencils he is having. This problem can be dealt with by addition method of 8+8+8+8+8=40 which is a repeated addition, replaceable by multiplication method of 8*5=40. In addition, students are able to realize that math is all about how to formulate and solve problems but not memorizing or reciting.
This enables students to instead of memorizing math such as 4*8=32; they instead consider that the same answer can be obtained through other methods like 4*8= 8+8+8+8= 24+8=32. This is a good method that lucidly shows multiplication and addition relationship. In addition, student can be able to manipulate groups and the item to obtain similar answer, for example 6*7= 3*7+3*7= 21+21= 42 an aspect that gives a creative method of learning multiplication tables, the exact working of math.
The Commutative property is an operation that occurs when one changes the order of the items involved without alteration of the results. For example, 3-2 is not equal to 2-3. Examples of commutative in addition and multiplication are: x+y=y+x or 12+13=13+12; and x*y=y*x, or 9*5=5*9. Associative property is a process where one can regroup numbers in any way without altering the answer. In this property, the answer is not altered by the way the numbers are combined.
For example, (x+y) +z= x+(y+z). or 3+(5+4)= (3+5)+4=3+(9)=(8)+4= 12; and x*(y*z)= (x*y)*z or 3*(5*4)=(3*5)*4=60. On the other hand, in distributive property a number is capable of being multiplied by a sum of two other numbers or be distributed to this numbers separately and give the same answer. For example, x*(y+z) =x*y+x*z or 4*(3+2) =4*(5)=20 or 4*3+4*2=12+8=20.
These properties help the students through supporting memorization, enabling them to understand properties involved through inclusion of patterns and strategies such as fives and nines.
For example, skip counting assists students in finding multiples of two as well as of five by realizing what they already know. For example, an array could be provided to read 2 roes and 6 columns which can be interpreted as 6 columns of 2 or 2 rows of 6 by turning it around. The same applies in three rows of 5 which a student can interpret as 15 put into 3 rows makes 5 columns- or 5 in each row.
In distributive property personal invented strategy is used when trying to recall one of the handfuls of multiplication facts. For example, a student may realize that 7*9 is hard and opt to add 2 more 7’s to already known 7 multiple: 49 to get 63 hence making it easier to know 7*9=63. In addition, it enables using facts of five to get sixes. For example, in 6*3 problems the student can think that this mean 5 groups of 3 and one more group of 3. That is, 6*3= (5*3)+(1*3)=18.
However, there are some conceptual errors that students make, with common errors involving forgetting the previous knowledge taught. For example, it is common for students to forget the addition and multiplication rules. For example, the rule that 4*3 means counting a group of 4 items three times and instead they add the two figures.
Here, it is important for teachers to review the some important information from the previous topic to activate student’s memories in recalling what they learned about these rules. This is an error brought about by the fact that math is a cumulative subject.
Additionally, students get confused by change of signs when dealing with addition or multiplication at the same time. For example, in a problem such as 3*7+3*7, student may solve it by dealing with addition first and multiplying later, that is 3*(10)*7=210 instead of 21+21=42. This is due to confusion of signs application rule as per the order (which one comes first).
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Here teaching the student BODMAS RULE, will ensure they deal with problems containing more than one sign effectively by applying the first sign first. Also illustrating the sign different and cautioning student to ensure that they identify when they are dealing with multiplication and not addition will enable them to deal with this problem.
Another conceptual error arises from confusion of the associative rules with Distributive property rules. Students forget these rules and in many cases, they forget to interpret the signs ending up solving the problems wrongly.
For example if a problem of 4+ (3+5) is provided to the student, they sometime confuse it with 4*(3+5) and instead of solving it as 4+(8)= 12 they end up with (12)+(20)= 32. Here, explaining to the student how the sign before the parenthesis affect the numbers within the parenthesis will enable them when it is needed to distribute and when not. Also this calls for intensive practice and reiteration to develop student’s habit.
Understanding that math is a confusing subject to many students, it is essential to parents, teachers and tutors. This is because mathematics contains many rules as well as formulas to memorize and recall. One way of assisting students to memorize these rules and formulas is correlating them with examples in real life.
This will create reasons behind the formulas enabling students to identify problems within them. For example, in multiplication of length and width to obtain an area, this can be correlated to real life examples such amount of paint to apply on a classroom wall.
On the other hand, when adding length twice and width twice to get the perimeter of a rectangle, this can be correlated with measurement of a fence around a piece of land. Thus math will become less threatening to the student as they will view it as real life phenomenon. Moreover, integrating the examples with multiplication and addition will assist the students in reinforcement of these concepts.