There are different possible answers to this question, depending on the standard of proof you need and the background knowledge you bring to the question.
Mathematical consistency and patterns
Try to solve each of these problems, paying attention to the previous set of problems as you do so. Look for patterns to facilitate problem solving.
3 × 3 = ?
3 × 2 = ?
3 × 1 = ?
3 × 0 = ?
3 × -1 = ?
3 × -2 = ?
3 × -3 = ?
2 × -3 = ?
1 × -3 = ?
0 × -3 = ?
-1 × -3 = ?
-2 × -3 = ?
-3 × -3 = ?
The answers to these problems are below, but I really recommend taking the time to solve the above problems on your own first so that you get an idea of how students might think about this set of problems.
3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
At this stage, many people will notice that the answers are 3 less each time and the number being multiplied by 3 is 1 less each time, so they continue this pattern to answer the following questions.
3 × -1 = -3
3 × -2 = -6
3 × -3 = -9
Now we've decreased the first number of the pattern by 3, and you have to make some deductions about what the answer should be.
2 × -3 = -6
1 × -3 = -3
0 × -3 = 0
Now it can be seen that the answers increase by 3 every time we increase the first number, so it is reasonable to continue this pattern.
-1 × -3 = 3
-2 × -3 = 6
-3 × -3 = 9
While this pattern may seem obvious to some, when someone is still learning this concept, they have less cognitive capacity available to perform the task at hand (multiplying numbers) and perform the additional task of looking for patterns in their answers. , so this is where someone else will ask you to stop and looking for patterns in your work so far will be very helpful.
prior knowledge: You need to know what these symbols mean, what it means to find one number by another, and how negative numbers work in terms of counting down and subtracting.
Mathematical consistency and mathematical properties
Let's look at a problem that we can solve in more than one way,borrowed from Khan Academy.
5 × (3 + -3) = ?
If we first add the numbers inside the parentheses, that's 5 times 0, which is 0, since 3 + -3 = 0.
5 × (3 + -3) = 0
But what if we first distribute 5 between both terms?
5 × 3 + 5 × -3 = ?
Since distributing the 5 over the sum does not change the value of the expression, we know that it is still equal to 0.
5 × 3 + 5 × -3 = 0
But that means that 5 × 3 and 5 × -3 are opposite signs, so since 5 × 3 = 15, 5 × -3 is -15. Let's look at another example.
-5 × (3 + -3) = ?
We know that this is the same as -5 times 0, so it has a value of 0.
-5 × (3 + -3) = 0
As before, we distribute -5 between both terms.
-5 × 3 + -5 × -3 = ?
Again, the distribution of terms does not change the value of the expression on the left side of the equation, so the result is still 0.
-5 × 3 + -5 × -3 = 0
We know ahead of time that -5 × 3 is -15, so we can substitute this value for -5 × 3 on the left side of the equation.
-15 + -5 × -3 = 0
Therefore, -15 and -5 × -3 are opposites since they sum to 0, so -5 × -3 must be positive.
Nothing in what we've done for the previous two examples is specific to the value of 5 × 3, so we can repeat this argument for all the other multiplication facts we want to derive, so these two ideas can be generalized.
prior knowledge: You need to know what these symbols mean, what it means to find one number by another, how the distributive property works, and how negative numbers can be defined as the opposites of positive numbers.
Representation on a number line
Imagine that we represent multiplication as jumps on a number line.
For 3×3, we draw 3 groups of 3 moving to the right. Both the number of groups and the address of each group are on the right.
But what about 3 × -3? Now we still have 3 groups of the number, but the number is negative.
If we find -3 × 3, the size and direction of the number we've multiplied are the same, but now we're finding -3 groups of that number. One way to think about this is to think about taking 3 groups out of the number. Another is to think of -3 times a number as a reflection of 3 times the same number.
Therefore, -3 × -3 is a reflection of 3 × -3 on the number line.
However, in a sense, this visual argument is just mathematical consistency represented by a number line. If multiplication by a negative is a reflection of 0 on the number line, and we think of negative numbers as reflections of 0 on the number line, then multiplying one negative number by another negative number is a double reflection.
Context
Karen Lewtemthis analogy.
Multiplying by a negative is repeated subtraction. When we multiply a negative number by a negative number, we get minus negative.
This analogy between multiplication and addition and subtraction helps students connect the two concepts well.
Joseph Rourke sharedthis context.
A player loses $10 a day. How much more money did they have 5 days ago?
Here, the loss per day is negative and going back in time is another.
@M_Teacher_w_T compartethis analogy:
"The enemy of my enemy is my friend."
This does not point to the algebraic or arithmetic properties of numbers, but to the opposition of negative numbers.
Prior knowledge:All contexts that generate new understanding require students to understand parts of the context reasonably well, so it is especially important to investigate how students understand an idea when it is presented in context.
algebraic proof of first principles
OfDr. Alex Eustis, we have this algebraic proof that a negative multiplied by a negative is a positive.
First, it states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations. Each operation is closed, which means that using those operations (such as addition and multiplication of real numbers) leads to another number within the number system. Each operation also has an identity element, or an element that does not change another element in the system when applied to it. For example, under addition, 0 is the additive identity. In multiplication, 1 is the multiplicative identity. The full set of required axioms is given below.
| Axiom 1:a+b=b+a | (additive community) |
| axiom 2: (a+b) +C=a+ (b+C) | (Additive Associativity) |
| Axiom 3: 0 +a=a | (additive identity) |
| Axiom 4: Hay −asatisfyinga+ (-a) = 0 | (additive inverse) |
| Axiom 5: 1 ×a=a× 1 =a | (Multiplicative identity) |
| Axiom 6: (a×b) ×C=a× (b×C) | (Multiplicative Associativity) |
| Axiom 7:a× (b+C) =a×b+a×C | (Multiplicative distribution on the left) |
| Axiom 8: (b+C) ×a=b×a+C×a | (Right multiplication distribution) |
From these axioms, we can show that a negative multiplied by a negative is a positive. I will reproduce Dr. Eustis below and include reference to the axioms used. We first show thata= −(−a).
Corollary 1
| a=a+0 | (Axiom 3 and Axiom 1) |
| a=a+ (-a+ −(−a)) | (Axiom 4 applied to −a) |
| a= (a+ (-a)) + (−(−a)) | (Axiom 2 – the associative property) |
| a= 0 + (−(−a)) | (Axiom 4) |
| a= −(−a) | (Axiom 3) |
Now we know that if we enter negative numbersais equal to −(−?).
Corollary 2
| 0 =a+ (-a) | (Axiom 4) |
| 0 = (0 + 1) ×a+ (-a) | (Axiom 3 and Axiom 5) |
| 0 = 0 ×a+ 1×a+ (-a) | (Axiom 8) |
| 0 = 0 ×a+ (a+ (-a)) | (Axiom 5 and Axiom 2) |
| 0 = 0 ×a+0 | (Axiom 4) |
| 0 = 0 ×a | (Axiom 3 and Axiom 1) |
Proving that 0 = 0 ×ait's the kind of painfully obvious idea that hardly needs proof, but does establish a relationship between multiplication and additive identity in the real numbers, which is not yet included in the above axioms.
Next, we show that (−1) ×a= −a.
Corollary 3
| −a= −a+ 0 ×a | (Corollary 2 and Axiom 3) |
| −a= −a+ (1 + (−1)) ×a | (Axiom 4) |
| −a= −a+ 1×a+ (-1) ×a | (Axiom 8) |
| −a= (-a+a) + (−1) ×a | (Axiom 5 and Axiom 2) |
| −a= 0 + (−1) ×a | (Axiom 4) |
| −a= 0 + (−1) ×a | (Axiom 3) |
Now, finally, we can prove that (−a) × (-b) =ABS.
| (-a) × (-b) = (a× (−1)) × (−b) | (Corollary 3) |
| (-a) × (-b) =a× ((−1) × (−b)) | (Axiom 6) |
| (-a) × (-b) =a× (−(−b)) | (Corollary 3) |
| (-a) × (-b) =a×b | (Corollary 1) |
However, this last "test" is unlikely to justify that a negative multiplied by a negative is a positive for any student. It's the sort of thing that is a required level of justification for a mathematician interested in rigorous proofs who would probably consider other justifications "standard" and not sufficient.
However, a critical idea of proof is that the intended audience of a proof is convinced that an idea is true, and therefore I claim that the algebraic "proof" presented here is not proof at all for the majority. of people.
prior knowledge: While I added the justification for each missing proof step, I needed some fluency with the original set of axioms. I also needed to keep the overall goal in mind and be able to recognize the structure of each part of the argument and combine that structure with the axioms.
A simpler algebraic proof
This algebraic proofby Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic.
a+ (-a) = 0
a×b+ (-a) ×b= 0 ×b
ABS+ (-ABS) = 0
From this, we can show thatABSmi-ABSthey have opposite signs, and therefore a positive multiplied by a negative is a negative. Using the fact, multiplication is commutative, a negative times a positive is also negative.
Similarly, we can prove that a negative multiplied by a negative is a positive.
a+ (-a) = 0
a× (-b) + (-a) × (-b) = 0 × (−b)
−ABS+ (-a) × (-b) = 0
How do we know that -ABSis negative, and the sum of these two terms is 0, then (-a) × (-b) is positive.
prior knowledge: Background knowledge for this exam is much less than the other, but it assumes some fluency with the manipulation of algebraic structures.
Conclusion:
Since the purpose of an argument that something is true is to convince the other person of the truth of the argument, whenever someone uses any justification, representation, or proof, it is up to him to verify that his audience is convinced.