The Commutative Properties of Addition and Multiplication
state that the order of two addends (e.g., (4+2 or 2+4) or two factors (e.g., 4 x 2 or 2 x 4) does not affect the sum or product, respectively. The root word of commutative is commute, which means to interchange. Therefore, we can reverse the order of two addends or two factors without changing the result. Sometimes, the commutative property of multiplication can be especially confusing to students. For example, in an addition operation, the addends represent subgroups consisting of the same things: If 3 + 4 is expressed as 4 + 3, only the order of the subgroups changes. In the grouping interpretation of multiplication, 4 x 5 and 5 x 4 represent different groupings- four groups of 5 does not look the same as five groups of fours. Students need to model many multiplication equations in order to see that products are identical. The commutative property of multiplication is also confusing because switching the factors in some multiplication word problems switches the relationships. Sometimes the switched relationships are similar and still make sense; other times they change the problem completely. A good device for justifying the commutative law is the array model for multiplication. The following word problems below can be used to illustrate the commutative property of addition and multiplication.
Sally collected aluminum for two days. On Friday morning she collected 20 cans and Friday night she collected 25 cans. On Saturday morning Sally collected 25 cans but on Saturday night only collected 20. Did she collect more on Friday than Saturday?
This problem demonstrates the commutative property of addition. Changing the order in which Sally collected the cans do not change the result.
Friday Morning/Evening 25 cans + 20 cans
Saturday Morning/Evening 20 cans + 25 cans
2. Royale went to a 20% off everything sale at Sports Authority. She brought some running shoes marked at $89.99 and a jersey marked at $39.99.
a) With 6% sales tax, what will be the total cost?
b) Will it be better to pay the sales tax first, and get the discount on the total bill, or to
get the discount first, and only pay sales tax on the discounted price?
In order to find the total cost, first find the price with discount first, then taxes ( P is the original price)
= p x (.8) x (1.06)
=$129.98 x .8 x (1.06) = 110.21
The sale cost is $110.21
The price with tax first then discount on everything
= p x (1.06) x (.8).
=129.98 x (1.06) x .8
Answer: No, there is no difference in the total cost. The prices are the same because (p x .8) x 1.06 = (p x 1.06) x .8. This is an excellent instance of the commutativity of multiplication because the order of operation does not affect the product.
The addition and multiplication properties can also be demonstrated by the following example: Consider that 2 + 2 give the same answer as 2 x 2. Although 2 is the only number with this property, there are many pairs of different numbers a and b which can be substituted in the equations above. They may be fractions, but they must have a product which is exactly equal to their sum. For example, there are infinitely many pairs of numbers which have the same sum and product. If one number is a, the other number can always be found simply by dividing a by a-1 because a + b = ab then a = ab-b = (a-1)b. Dividing both side by a-1 assuming this is not zero gives a/a-1 =b. For example: 3 times 1 ½ equals 3 + 1 1/2 (Inverse Properties of Addition and Subtraction).
Source: Sam Loyd,
Mathematical Puzzles of Sam Loyd Selected and Edited by Martin Gardner
, Dover Publications, Inc., New York, 1959, Puzzles 51, p. 53.