Addition and multiplication are often referred to as binary operations. We can operate on only two numbers at a time. For example, (2+ 3, 9 x 8). If a computation involves three addends, we first add two of the numbers and then add the third to the previous sum. For example, (2+3) +4 = 2+ (3+4) or (a + b) + c = a + (b + c) for all numbers a, b, and c.
The associative property states that the way in which three or more addends or factors are grouped before being added or multiplied does not affect the sum or product. The property is usually used to simplify calculations when adding numbers. The associative property is also used to regroup compatible numbers in order to simplify the calculations. Compatible numbers refer to friendly numbers whose sums or products are easily to calculate mentally. For example 25 x 4 = 100, and 35 and 65 are compatible because 35 + 65 = 100. In general, numbers that can be combined to form multiples of 10, 30, 100, 200, and 1000 are compatible.
The associative and commutative rules can be combined to show that, if you have a collection of numbers to be added the ways you pair up the numbers to do the addition, and the order in which you do the addition have no effect on the outcome. This is sometimes called the Any Which Way Rule. It is the practical outcome of the Associative and Commutative Rules, and gets used in almost every calculation. Because the Associative and Commutative Rules hold also for multiplication, so does the Any Which Way Rule.
The following word problem demonstrates the associative property of addition:
1a.There is a prize for selling the most tickets to the school play. Alphonz, Bela and Chalfont are the leaders. Alphonz sold 42 the first week, 59 the second week and 78 the third week. Bela sold 59 the first week, and 78 the second week but has a disappointing third week, with only 42 sold. The first week, Chalfont sold 78, the second week he sold 59, and the third week he also sold 42. Who wins the prize?
All three won the prize because all three sold 179 tickets during the three weeks.
Alphonz sold 42 + 59 + 78 = 179
Bela sold 59 + 78 + 42 = 179
Chalfont 78 + 59 + 42 = 179
1b. There is a prize for selling the most tickets to the school play. Alphonz, Bela and Chalfont are the leaders. Alphonz sold 42 the first week, 59 the second week and 78 the third week. Bela sold 60 the first week, and 79 the second week but had a disappointing third week, with only 40 sold. The first week, Chalfont sold 77, the second week he sold 58, and the third week he sold 44. Who wins the prize?
Alphonz sold 42 + 59 + 78 = 179
Bela sold 60 + 79 + 40 = 179
Chalfont 77 + 58 + 44 = 179
This is a more subtle use of the Any Which Way Rule. One does not have to do each calculation separately and compare. One can just see that 60 + 79 + 40 = (59 + 1) + (78 + 1) + (42-2) and recombine.
2. Darcy, Egmont and Finian each have a box. Darcy’s box is 12 inches long, 8 inches wide and 5 inches high. Egmont’s box is 8 inches wide, but it is a foot deep, and it is also 5 inches high. Finian’s box is only 5 inches wide, but it is eight inches wide, and it is the tallest box-a full 12 inches high (these are all inside measurements.) They are arguing about which box will hold the most. Which box has the largest volume?
This problem represents both the commutative and associative properties of multiplication. In this problem we have three boxes. First, we know that Darcy’s box is 12 inches long times 8 inches wide x 5 inches high. We can represent this by multiplying the length times the width times the height. We can represent this by V= lwh and substitute numbers for literal expressions to solve. For example V= 12 x 8 x 5 =480 square inches. In Egmont’s box we need to change a foot to 12 inches and then solve V= wlh and substitute numbers for literal expressions, V= 8 x 12 x 5 = 480 square inches and Finian’s box V = 5 x 8 x12 = 480 square inches. Each box has the same volume because the way in which three or more factors are grouped before being multiplied does not affect the product.
3. On a business trip in Upstate New York, Mr. Floyd stopped several times to buy gas. His car held 12.4 liters when he filled up the first time. At his next gas stop, his car held 22.8 liters. The last time he stopped for gas, the car held 18.6 liters. How many liters of gas did his car use on the trip? Answer: 12.4 + 22.8 + 18.6 = 53.81liters of gas
This can be done mentally if you use the Any Which Way Rule and compute it as (12, 4 + 18.6) + 22.8 = 31 + 27.8 = 53.8
4. The senior class was selling tickets to their play. They had two prices, $5 and $8. The tickets were on sale during the week before the play. On Monday, they sold 23 tickets at $5 and 14 at $8. Tuesday, they sold 31 tickets at $5 and 22 at $8. Wednesday sales were 46 at $5 and 28 at $ 8. Thursday sales were 39 at $5 and 32 $8. Friday, including sales at the door, they sold 22 at $5 and 56 at $8. How much money did they take in ticket sales? You can solve this problem by multiplying the number of tickets sold each day using the two different prices and adding up the total amount of tickets sold for the week.
Ticket Sold Ticket Sold Price $5.00 Ticket Sold Price $8.00
Monday 23 $115 14 $112
Tuesday 31 $155 22 $176
Wednesday 46 $230 28 $224
Thursday 39 $195 32 $256
Friday 22 $110 56 $448
Total Amount 161 $805 152 $1216
This problem is about the Distributive Rule (as well as the Any Which Way Rule). Total sales
= (23 + 31 + 46 + 39 + 22) x 5(70 + 45 + 46) x 5 + (14 +22 + 28 + 32 + 56) x 8 + (50 + 70 + 32) x 8
= 161 x 5 + 152 x 8
= 805 + 1216