Multiplication and Addition Relationships - Essay Example

There is a clear relationship between the mathematical processes of multiplication and addition. Multiplication, in terms of addition, is the repeated addition of the same number to itself a certain number of times. For every multiplication problem, there is a way to write the equation with the sole use of addition. An illustration of this point can be displayed by showing that the mathematical equation, "5 x 3," is the same as "5 + 5 + 5." Five multiplied by three means that three groups of five are being added together. Understanding the relationship between multiplication and addition can have significant benefits on a student's ability to utilize the skill of multiplying numbers. The relationship between multiplication and addition can also be seen in various mathematical properties. The commutative property is one that applies to both multiplication and addition problems. The implications of this property are that in a multiplication equation, one can multiply the numbers in any order to get the same product, and in an addition equation, one can add the numbers in any order to get the same sum. An example of the commutative property being used in addition is the equation, "10 + 2 = 12." If the numbers 10 and 2 were to be switched (2 + 10), the sum would still be 12. The equation, "2 x 5," can be utilized to demonstrate the commutative property in multiplication. Two times 5 equals 10, and when the numbers switch places, 5 times 2 still yields a product of 10. The commutative property is connected to the thinking strategy of thinking about multiplication in terms of adding groups of numbers. When students see 5 x 2 they may first think that means 5 groups of 2, which is 2 + 2 + 2 + 2 + 2. The commutative property lets them know that it can also mean 2 groups of 5, which is a much simpler 5 + 5. The associative property is similar to the commutative property, but applies to equations that have more than two numbers and have at least two of the numbers grouped. Like the principle of the commutative property, the associative property dictates that the numbers can be multiplied or added in any order, regardless of the groupings. An example of the associative property in addition is the equation, "3 + (2 + 8)." The grouping of 2 + 8 implies that these two numbers must be added first, but the associative property allows the numbers to be added in any order without the sum being changed. Whether you add 2 + 8 + 3 or 3 + 8 + 2 or 8 + 3 + 2, the sum is always 13. The equation, "(4 x 2) x 3," can be utilized to exemplify the associative property as it applies to multiplication. The grouping of 4 x 2 implies that these two numbers must be multiplied first, but the associative property allows the numbers to be added in any order without the product being altered. Whether you multiply 4 x 2 x 3 or 2 x 4 x 3 or 3 x 2 x 4, the product is always 24. Similar to the commutative property, the associative property helps students use the thinking strategy of adding together groups of numbers, because it allows them to solve the problem in the order that is easiest. When students multiply (2 x 6) x 5, which is 12 + 12 + 12 + 12 + 12, it may be easier for them to multiply (5 x 2) x 6, which is 10 + 10 + 10 + 10 + 10 + 10. It is easier to count by 10's than 12's. The distributive property involves breaking down multiplication problems, and it uses addition as a crucial tool. While many students are able to memorize the products of multiplying numbers 1 through 10, numbers greater than 10 that aren't multiples of 10 may begin to pose difficulties. The distributive property makes this sort of multiplication easier, and can be exemplified by the equation, "4 x 56." The distributive property holds that (4 x 50) + (4 x 6) = 4 x 56. To make 56 an easier number to multiply by, 56 can be separated into two numbers that add up to 56, and then each can be multiplied by 4 and added together. Four times 50 is an easy product for students to multiply because it is 50 + 50 + 50 + 50, or 200. Four times 6 is also easy for students because they learn and memorize the products of multiplying the numbers 1 through 10; the product is 24. By adding together 200 and 24, the answer 224 is found, which is the same product found by multiplying 4 x 56 on a calculator. While many students would have trouble counting by 56's, fewer students have trouble counting by 6's and 50's. One common conceptual error of multiplication is that students have trouble understanding the rule that any number multiplied by 0 equals 0. An instructional strategy could be used to help students visualize why this rule is the case, rather than telling them merely that the rule exists. Use an overhead projector to display five chips or markers for the class to see. First, explain that multiplying two numbers, like 5 x 1, means one group of five. Ask them to count how many chips there are and tell you what 5 x 1 equals. Add five more chips, but keep them grouped separately from the first five, and show them that this represents 5 x 2, because it is two groups of five. Ask them to count how many chips there are and tell you what 5 x 2 equals. Now ask them what 5 x 0 equals, considering that this means ZERO groups of five. After a couple of guesses, remove all of the chips and tell them that now there are NO groups of five, so it equals 0. Repeat this process with different numbers and at the end, reiterate the rule that they now understand - that any number times 0 equals 0. Another common conceptual error of multiplication involves students memorizing answers to numeric equations like 3 x 4 without knowing what it really means. A useful strategy in fixing this problem is the use of word problems. Instead of always asking simple number problems, the use of word problems lets students see how multiplication is applied and why it works the way it does. An example of a word problem for 3 x 4 could be the following: Johnny just got allowance and wants to buy candy for his four best friends. If Johnny wants to give three pieces of candy to each of his four friends, how many pieces of candy does Johnny need to buy The students could be asked not only for the answer, but also for the numeric multiplication problem that leads to the answer. This will let students see what multiplication is used for. Read More