The Distributive Property Using Tiles © Math As A Second Language All Rights Reserved next #4 Taking the Fear out of Math

next In this and the following several discussions, our underlying theme is… Our Fundamental Principle of Counting The number of objects in a set does not depend on the order in which the objects are counted nor in the form in which they are arranged. For example, in each of the six arrangements shown below, there are 3 tiles. © Math As A Second Language All Rights Reserved

next © Math As A Second Language All Rights Reserved In our previous three discussions we used the above principle to demonstrate the properties of closure, commutativity and associativity. next Notice that in each of these discussions we limited ourselves to the situations in which either addition was the only operation or multiplication was the only operation. That is, we talked about such things as and 2 x 3 x 5. However, we did not talk about a computation in which both addition and multiplication were involved.

© Math As A Second Language All Rights Reserved The property that links addition and multiplication is known as the distributive property or, more formally, as “the distributive property of multiplication over addition”. Let’s introduce this new property by discussing a “real life” situation.

next © Math As A Second Language All Rights Reserved next You are selling an item at a price of $6. You want to determine how much money you made by selling that item during those two days. On the first day you sell 4 of these items and on the second day you sell 5 more of these items.

© Math As A Second Language All Rights Reserved One way to determine how much money you made each day is to add the two daily results. In other words, on the first day you made (4 × 6) dollars. On the second day you made (5 × 6) dollars. Thus, altogether for the two days, you made (4 × 6) + (5 × 6) dollars. next

© Math As A Second Language All Rights Reserved Another way to determine how much money you made is to find the total number of items you sold during the two- day sale and then multiply this number by the price of each item that you sold. In other words, since you sold 4 items the first day and 5 items the second day, the total number of items you sold during these two days was (4 + 5), and at $6 per item, you made (4 + 5) × 6 dollars. next

© Math As A Second Language All Rights Reserved (4 × 6) + (5 × 6) = (4 + 5) × 6 And since the two answers are the same (namely, $54) it follows that… Since the numbers 4, 5, and 6 were used only for illustrative purposes we can generalize the above result by, for example, replacing 4 by b, 5 by c, and 6 by a to obtain the more general result… (b × a) + (c × a) = (b + c) × a

next © Math As A Second Language All Rights Reserved (a × b) + (a × c) = a × (b + c) A more common form of the distributive property is obtained by using the commutative property 3 times: Replace b × a by a × b, replace c × a by a × c and replace (b + c) × a by a × (b + c) to obtain… (b × a) + (c × a) = (b + c) × a

next © Math As A Second Language All Rights Reserved The Distributive Property If a, b, and c are any whole numbers, then a × (b + c) = (a × b) + (a × c). Finally, the above equality is usually written by interchanging the right and left sides of the equation to obtain the most general form of the distributive property.

next © Math As A Second Language All Rights Reserved Because the times symbol looks like the letter x it is conventional to replace the notation a × b by either ab, a b, (ab), or a(b). next a(b + c) = (ab) + (ac) Thus, the above equality is usually written as… Notes

next © Math As A Second Language All Rights Reserved Another common agreement is that if there is ambiguity when grouping symbols are omitted, we perform all multiplications before we perform any additions. next a(b + c) = ab + ac Thus, the above equality is often further abbreviated as… Notes

next © Math As A Second Language All Rights Reserved Notice the importance of the grouping symbols. If we omitted them and wrote ab + c, the agreement that we performed multiplications before we performed additions would have led to the equality ab + c = (ab) + c and (ab) + c ≠ a(b + c). Special Note

next © Math As A Second Language All Rights Reserved To show that a(b + c) ≠ (ab) + c, all we have to do is show one example in which the equality is false. If we let a = 6, b = 4 and c = 5, we see that 6 ×(4 + 5) = 6 × 9 or 54, but (6 × 4) + 5 = or 29. Special Note

next © Math As A Second Language All Rights Reserved Later in our course we will discuss whole number multiplication in greater detail. For now, however, we wish to show how our relatively simple introduction can be a segue to more computationally complicated applications of the distributive property.

next © Math As A Second Language All Rights Reserved However, it would be extremely cumbersome to use “skip counting” to compute, say 4 × 63. This is where are adjective/noun theme becomes very useful. By way of illustration, it is relatively easy to “skip count” to conclude that 4 × 6 = 24. That is, we count 6, 12, 18, 24, 30, 36, note 1 Skip counting is a less technical way of listing the multiples of a number. For example, when we are skip counting by 6's we are listing the multiples of 6. In the language of multiples, the fact that 4 × 6 = 24 is stated as 24 is the 4th multiple of 6 (and by the commutative properly it is also the 6th multiple of 4). next

© Math As A Second Language All Rights Reserved 4 × 6 apples = 24 apples Once we know that 4 × 6 = 24, we also know that… next 4 × 6 dollars = 24 dollars 4 × 6 tens = 24 tens

© Math As A Second Language All Rights Reserved 4 × 63 = With this in mind we may think of 63 as being 6 tens + 3 (ones) and then by the distributive property we see that… next 4 × (6 tens + 3 ones) = 4 × 6 tens + 4 × 3 ones = 24 tens + 12 ones = =252.

© Math As A Second Language All Rights Reserved In terms of a real-life example, suppose you sell 63 pens and make a profit of $4 on each pen you sell. To find the total profit, you know that if you sell 60 pens the profit is $240 and the 3 additional pens yield a $12 profit; and hence your total profit is $240 + $12 or $252.

next © Math As A Second Language All Rights Reserved Tiles and the Distributive Property While we do not have to use tiles to demonstrate the distributive property, it might be an easier way for younger students to internalize the concept.

next © Math As A Second Language All Rights Reserved Tiles and the Distributive Property For example, to show that 2 × (3 + 4) = (2 × 3) + (2 × 4), we could have the students look at 2 rows, each with 7 tiles, as shown below. In the above arrangement, we have 2 rows each with (3 + 4) tiles; so the total number of tiles is 2 × (3 + 4). next

© Math As A Second Language All Rights Reserved Tiles and the Distributive Property The number of tiles will remain the same if we rearrange them as shown below. In the above arrangement, we have one group of tiles that consists of 2 rows, each with 3 tiles (2 × 3 tiles) and another group that consists of 2 rows, each with 4 tiles (2 × 4 tiles). So altogether there are (2 × 3) + (2 × 4) tiles. next

© Math As A Second Language All Rights Reserved Tiles and the Distributive Property And since the number of tiles is the same in each case… 2 × (3 + 4) next (2 × 3) + (2 × 4) …we see that 2 × (3 + 4) = (2 × 3) + (2 × 4).

Hopefully, this snippet helps you understand the power of the distributive property and our adjective/noun theme. In any event this concludes our discussion of the properties of whole number arithmetic. introduce them. © Math As A Second Language All Rights Reserved 5(3 + 4)5(3) + 5(4) addition multiplication Associative We will return to this topic a bit later in our course when there will be a better motivation to introduce them. next