6.2: Multiplication
- Page ID
- 70321
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You will need: Positive and Negative Counters (Material Cards 18A and 18B)
Now, we'll explore how to multiply positive and negative numbers using the counters. Let's look again at a definition of multiplication for whole numbers.
If \(m\) and \(n\) are whole numbers, then \[m \times n = n + n + n + n + ... + n, \nonumber \] where there are \(m\) add ends of \(n\) in this sum.
This definition makes sense as long as \(m\) and \(n\) are positive numbers. In fact, we can even make sense of and modify this definition using counters as long as m is positive by rewording the definition slightly, as stated below:
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 \(n\). The number that the resulting collection represents is the answer to the problem \( m \times n\).
This exercise shows you one way to use the above definition to multiply \(3 \times -4\). In this problem, the definition can be used because 3 is a whole number and -4 is an integer. We need to combine 3 subsets of a collection of counters that represent -4.
- Use your positive and negative counters to represent a collection of counters that represent -4. For this exercise, choose a collection of 5 negatives and 1 positive. Show what your collection looks like below:
- Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
- After removing any red-green pairs (zero) from your collection in part b, show the collection that remains below. What number does this represent? _____
Do \(3 \times -4\) again, using a different collection of counters to represent -4.
- Use your positive and negative counters to represent a collection of counters that represent -4. This time, choose a collection of 6 negatives and 2 positives. Show what your collection looks like below:
- Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
- After removing any red-green pairs (zero) from your collection in part b, show the collection that remains below. What number does this represent? _____
Okay, let's do \(3 \times -4\) one more time, choosing the easiest way to represent -4.
- Form a collection of counters to represent -4. Do it the easy, natural way, using the least number of counters possible. Show what your collection looks like below:
- Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
- What number does the collection in part b represent? _____
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. 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. You'll always end up with a collection that represents -12. 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.
Use your counters to do each of the following multiplication problems using the definition of multiplying a whole number by an integer. Then, explain what the multiplication problem given means in terms of the counters, and explain and show the individual steps. Use the following example as a model.
Multiply \(\bf 2 \times -6\).
Solution
Multiplying \(2 \times -6\) means to combine 2 subsets of 6 red counters. The number that the resulting collection represents is the product (answer). \(2 \times -6\) = RRRRRR + RRRRRR = RRRRRRRRRRRR = -12
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a. Multiply \(4 \times -2\) This means to |
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b. Multiply \(3 \times 5\) This means to |
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c. Multiply \(5 \times -3\) This means to |
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d. Multiply \(7 \times 2\) This means to |
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e. Multiply \(0 \times -3\) This means to |
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. Let's look once more at the definition for multiplying \(m \times n\), when m is a whole number.
Definition: Multiplication of a Whole Number times an Integer using the Repeated-Addition Approach, using positive and negative counters
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. The number that the resulting collection represents is the answer to the problem \(m \times n\).
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 trick to doing this is to remove the subsets from a collection of counters representing zero. So, here is the comprehensive definition for multiplying any two integers, using positive and negative counters.
- Case 1: 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 \(n\). The product of m and \(n\), \(m \times n\), is the number that the resulting collection represents.
- Case 2: If \(m\) is negative and \(n\) is any integer, \(m \times n\) is obtained by removing |m| subsets of a collection of counters representing \(n\) from a collection of counters representing zero. The product of \(m\) and \(n\), \(m \times n\), is the number that the resulting collection represents.
This exercise shows you how to use the above definition to multiply \(-4 \times 3\). For this problem, we need to remove 4 subsets of a collection of counters that represents 3 from a collection of counters representing zero. The simplest collection to represent 3 is 3 positives, or 3 green counters. (We could use a more complicated collection and still arrive at the same answer. We did this kind of an exercise in exercises 1 - 3. Convince yourself it wouldn't matter here, either.)
- 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. For this example, make a collection of 14 red and 14 green counters. Write down what your collection looks like here:
- From your collection, remove a subset of 3 green counters. Then, remove 3 more subsets of 3 green counters. You have just removed 4 subsets of 3 green counters from zero. To show it on paper, circle a subset of 3 green counters in the collection above in part a. Then, circle 3 more subsets of 3 green counters so that four separate subsets are circled. After you remove the counters (which are shown by what you circled in part a), show what is left in your collection below.
- Remove any red-green pairs (zero) from your remaining collection. Show this on paper by crossing off or circling any red-green pairs (zero) from your collection shown in part b. Show the collection that remains below. What number does this collection of counters represent?
Use the definition again to multiply \(-4 \times 3\). Remember, you need to remove 4 subsets of a collection of counters that represents 3.
- 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. This time, put in the smallest collection of counters possible so that you'll be able to remove 4 subsets of 3 green counters. Write down what your collection looks like here:
- From your collection, remove four subsets of 3 green counters (take out one subset of 3 green counters at a time). You have just removed 4 subsets of 3 green counters from zero. To show it on paper, circle 4 different subsets of 3 green counters in the collection shown in part a. After you remove the counters (which are shown by what you circled in part a), show what is left in your collection below.
- What number does the above collection represent?
Did you notice that if you start out with a minimal collection to represent zero, you don't have to remove any red-green pairs when you get to part c? That is the easiest way to do it because you know how to calculate what to take out before you start the problem. However, if you were teaching this to someone who couldn't figure that out ahead of time, you might always start out with zero being 20 (or some other agreed upon number) of each counter.
On the next page, the steps are shown for the example below– how to use your counters to do a multiplication problem when the first number is a negative integer. It's necessary to explain what the multiplication problem given means in terms of the counters, and then explain and show the individual steps. Use the example as a model for the exercises that follow.
Multiply \(-4 \times -3\) by using the definition of multiplying integers.
Solution
Multiplying \(-4 \times -3\) means to remove 4 subsets of 3 red counters from a representation of zero. The number that the resulting collection represents is the answer.
Since there are 12 greens remaining, the answer is +12. Didn't you always wonder why a negative number times a negative number equaled a positive number? Using the definition of multiplying integers with counters, you can really see why it's true. After you do a few more problems, feel free to go out and show all of your friends who still don't understand it or just memorized the rules because someone told them that's the way it is and to just accept it. Okay, enough of that unbridled enthusiasm from me for now.
It's time for you to work a few problems. Some have negative numbers before the multiplication sign (start with a representation of zero and remove subsets as in the last example) and some have whole numbers before the multiplication sign (so just combine subsets together like you did in the earlier exercises of this exercise set). It's the number before the multiplication sign that will indicate which case of the definition you will use.
Use your counters to do each of the following multiplication problems using the definition of multiplying two integers with positive and negative counters. Then, explain what the multiplication problem given means in terms of the counters, and explain and show each of the individual steps. Use the example above as a model when the first number is negative.
a. \(-5 \times 3\) ____ This means to ____________________________________________________
Show work and all steps below. Then, state the answer to the problem.
b. \(-3 \times 2\) ____ This means to ____________________________________________________
Show work and all steps below. Then, state the answer to the problem.
c. \(2 \times -3\) ____ This means to ____________________________________________________
Show work and all steps below. Then, state the answer to the problem.
NOTE: Although the answer to part b is the same as part c due to the commutative property of multiplication, the problems mean different things, the steps are not alike and the problems are done differently.
d. \(-2 \times 3\) ____ This means to _______________________________________________________
Show work and all steps below. Then, state the answer to the problem.
e. \(3 \times 2\) ____ This means to _______________________________________________________
Show work and all steps below. Then, state the answer to the problem.
f. \(0 \times -4\) ____ This means to _______________________________________________________
Show work and all steps below. Then, state the answer to the problem.
g. \(-4 \times 0\) (this means something different than \(0 \times -4\)) ____ This means to _____________________________
Show work and all steps below. Then, state the answer to the problem.
Let's do some more problems when the first integer is negative, using a chart to keep track of the counters. The first column will be the multiplication problem, the second will show what representation of zero is being used, the third will explain the meaning of the multiplication problem (what has to be done in all of these cases, subsets are being removed from the representation of zero), the fourth will show how many of each of the counters are left after the removal of the subsets, and the last column will be the answer, obtained from the representation shown in the fourth column.
Fill in all of the blanks. FILL IN THE ENTIRE TABLE ALL THE WAY TO THE BOTTOM!! You should be using your real manipulatives (red and green counters) as you do (most of) these problems. I've done the first one for you to use as a model.
| problem | Counters for Zero | The meaning of the problem | counters remaining | answer |
|---|---|---|---|---|
| \(-3 \times 6\) | 20G 20R | remove 3 sets of 6 G from zero | 2G 20R | -18 |
| \(-3 \times 6\) | 23G 23R | |||
| \(-3 \times 6\) | 18G 18R | |||
| \(-4 \times 4\) | 18G 18R | |||
| \(-4 \times -4\) | 18G 18R | |||
| \(-5 \times 2\) | 14G 14R | |||
| \(-2 \times 5\) | 13G 13R | |||
| \(-5 \times -2\) | 11G 11R | |||
| \(-2 \times -5\) | ||||
| \(-3 \times 3\) | 10G 10R | |||
| \(-3 \times -3\) | 12G 12R | |||
| \(-7 \times 2\) | ||||
| \(-2 \times 8\) | 18G 18R | |||
| \(-2 \times -8\) | 18G 18R |
After doing all of the exercises in this Exercise Set, the rule for multiplication of integers should make sense.
Rule for Multiplying Two Integers Together:
- To multiply two integers, first multiply the absolute values of the integers. To determine the sign of the product: it is positive if both integers have the same sign (both positive or both negative); otherwise it is negative (if one of the integers is positive and the other integer is negative). If one of the integers is zero, the answer is zero.
If you are multiplying more than two integers together, the sign of the product can be determinined by how many negative numbers are being multiplied. For every two negative numbers being multiplied together, the answer is positive. Therefore, if there are an even number of negative numbers being multiplied, the sign of the product is positive. If there are an odd number of negative numbers being multiplied, the sign of the product is negative. If one of the integers is zero, the answer is zero.
Although we haven't covered division in this exercise set, the rule for determining the sign of a quotient is the same as the rule for determining the sign of a product.
Determine if each product is negative (–), positive (+) or zero (0). Do not compute
Remember that for a set to be closed under multiplication, the product of any two elements in the set must be in the set. To prove a set is not closed under multiplication, you need to provide a counterexample.
For each of the following sets, determine if the set is closed under multiplication. Provide a counterexample if it is not closed.
- Integers
- Positive Integers
- Negative Integers
- {-1, 0}
- {1, -1}
- {-1, 0, 1}
Less than (<) and Greater than (>) signs are used to order numbers. ](a < b\) if \(a\) is to the left of \(b\) on the number line. \(a < b\) (read "a is less than b") can also be written as \(b > a\) (read "b is greater than a.)
Decide which of the following are true if a, b and c are any integers, p is a positive integer and n is a negative integer. Provide a counterexample if it is false.
- If a < b and b < c, then a < c
- If a < b, then a + c < b + c
- If a < b, then ap < bp
- If a < b, then ap > bp
- If a < b, then an > bn
- If a < b, then an < bn
Exercise 11a, 11b, 11c, and 11e are the four properties of Ordering for Integers. 11a is called the Transitive Property for Less Than. 11b is called the Property of Less Than and Addition. 11c is the Property of Less Than and Multiplication by a Positive 11e. is the Property of Less Than and Multiplication by a Negative. If the less than symbols in 11a, 11b and 11c were replaced with greater than symbols, you would have the corresponding properties of greater than. For part 11e, if both signs were switched, you would have the Property of Greater Than and Multiplication by a Negative.
Fill in the following properties, if a, b and c are any integers, p is a positive integer and n is a negative integer.
- Transitive Property for Greater Than:
- Property of Greater Than and Addition:
- Property of Greater Than and Multiplication by a Positive:
- Property of Greater Than and Multiplication by a Negative:
11.a,b and c are similar to properties of equality. The difference between equalities and inequalities (whether there is a less than or greater than symbol) if that when both sides of an inequality are multiplied by a negative number, the inequality sign changes directions.