One might now notice that the answers are going up by 3 each time as we increase the first number, and so it is reasonable to continue this pattern. While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they have less cognitive capacity available to accomplish the task at hand multiplying numbers together and accomplish the additional task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful.
Prerequisite knowledge : One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction. Since distributing the 5 across the addition does not change the value of the expression, we know this is still equal to 0.
Again, the distribution of terms does not change the value of the expression on the left-hand side of the equation, so the result is still 0. Prerequisite knowledge : One has to know what these symbols mean, what is meant by finding one number times another, how the distributive property works, and how negative numbers can be defined as the opposites of positive numbers.
Both the number of groups and the direction of each group are to the right. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number. In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.
Karen Lew has this analogy. Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative.
This analogy between multiplication and addition and subtraction helps students nicely connect the two concepts. Joseph Rourke shared this context.
How much more money did they have 5 days ago? Here, the loss per day is one negative and going backwards in time is another.
This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers. Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually.
From Dr. Alex Eustis , we have this algebraic proof that a negative times a negative is a positive. First, he 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 these operations such as addition and multiplication on the 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. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below. So what mathematics could guide us in our thinking here? As the area model is just a representation our belief in expanding brackets, the area model should hold for negative numbers too! Even though geometrically it makes no sense to have a negative side length in a geometric figure, we see that the mathematics each diagram represents is still correct mathematics.
People try to give concrete meaning to all this with models of soldiers marching on number lines turning different directions, systems of profit and debt, working with temperatures above and below freezing, and so on. Each model is good for illustrating SOME aspect of the arithmetic of negative numbers, but not all. We must start instead with a discussion on what we think should be true about multiplication in general and how it behaves. And for students ready for it, the axiomatic approach clinches it.
I just know it is algebraically consistent. Your support is so much appreciated and enables the continued creation of great course content.
Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks! Video transcript Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers But now you are faced with a conundrum What happens when you multiply negative numbers?
Either when you multiply a positive number times a negative number Or when you multiply two negative numbers So, for example You aren't quite sure what should happen if you were to multiply and im just picking two numbers where one is positive and one is negative What would happen if you were to multiply 5 times negative 3 You're not quite sure about this just yet You're also not quite sure what would happen if you multiply two negative numbers.
So lets say negative two times negative 6 This is also unclear to you What you do know, because you are a mathematician, is however you define this or whatever this should be It should hopefully be consistant with all of the other properties of mathematics that you already know And preferably all of the other properties of multiplication That would make you feel comfortable that you are getting this right.
That's also consistent with the intuition of adding negative three repeatedly five times, now look above above us slightly higher so you can see ideas of multiplying two negatives, but we can do the exact same product experiment. We want whatever this answer to be consistent with the rest of mathematics that we know so we can do the same product experiment.
What would negative two times six plus negative six to be equal to. Well, six plus negative six is going to be zero. I'll leave you there and I'll see if I can make a few other videos that can also give you a conceptual understanding of why these are true.
0コメント