Remove any red-green pairs (zero) from your remaining collection.After you remove the counters (which are shown by what you circled in part a), show what is left in your collection below. Then, circle 3 more subsets of 3 green counters so that four separate subsets are circled. To show it on paper, circle a subset of 3 green counters in the collection above in part a. You have just removed 4 subsets of 3 green counters from zero. Then, remove 3 more subsets of 3 green counters. From your collection, remove a subset of 3 green counters.Write down what your collection looks like here: For this example, make a collection of 14 red and 14 green counters. We first need to form a collection of counters that represents zero so that it will be possible to remove 4 subsets of 3 green counters. Convince yourself it wouldn't matter here, either.) We did this kind of an exercise in exercises 1 - 3. (We could use a more complicated collection and still arrive at the same answer. The simplest collection to represent 3 is 3 positives, or 3 green counters. For this problem, we need to remove 4 subsets of a collection of counters that represents 3 from a collection of counters representing zero. This exercise shows you how to use the above definition to multiply \(-4 \times 3\). So, here is the comprehensive definition for multiplying any two integers, using positive and negative counters. The trick to doing this is to remove the subsets from a collection of counters representing zero. If m is negative, this definition doesn't make sense since you certainly can't combine a negative number of subsets! The way we'll revise this definition to include the possibility that m may be negative is to agree that if m is negative, we REMOVE \(m\) subsets of a collection of counters representing \(n\). The number that the resulting collection represents is the answer to the problem \(m \times n\). If m is a whole number and n is any integer, \(m \times n\) is obtained by combining m subsets of a collection of counters representing n. Let's look once more at the definition for multiplying \(m \times n\), when m is a whole number.ĭefinition: Multiplication of a Whole Number times an Integer using the Repeated-Addition Approach, using positive and negative counters Okay, now that you've mastered how to multiply a whole number by an integer, let's work on how we can use the counters to multiply a negative integer by an integer. \(2 \times -6\) = RRRRRR + RRRRRR = RRRRRRRRRRRR = -12 The number that the resulting collection represents is the product (answer). Multiplying \(2 \times -6\) means to combine 2 subsets of 6 red counters. For exercise 3, did you choose 4 negatives as your representation? If so, did you notice you didn't have to remove any red-green pairs to answer part c? From now on, let's do it the easy way, using the simplest collection possible. You'll always end up with a collection that represents -12. To compute \(3 \times -4\), you could combine 3 subsets of a collection of 8 reds and 4 greens, or you could combine 3 subsets of a collection of 7 reds and 3 greens, etc. Well, I hope you got the answer of -12 for exercises 1, 2 and 3, since \(3 \times -4 = -12\)! This illustrates that it doesn't matter exactly which collection of counters you use to represent -4, as long as the collection really is -4. What number does the collection in part b represent? _.Combine the counters together and show what the large collection looks like below: Now, form two more collections (for a total of 3 subsets) of counters that you had for part a.Show what your collection looks like below: Do it the easy, natural way, using the least number of counters possible. Form a collection of counters to represent -4.Okay, let's do \(3 \times -4\) one more time, choosing the easiest way to represent -4.
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