Additive and Multiplicative Relationships - Math Rocks!!
PDF | The present study explores relationships between additive and multiplicative structures in the context of proportional reasoning. Spanish primary school children were given a test which involved twelve .. Examples of problems considering the number structure (versions I and N) and the. Try to give me a simple definition of exponentiation, which is The real question is "How do we introduce multiplication to children?" Professor. See Using Arrays to Show Multiplication Concepts. will learn how to use arrays to show the relationship between multiplication and division. For example.
If 50 divided by five is 10, then 10 times five is And you could do it the other way around. What is 50 divided by 10 going to be?
How do I know that? Well, five times 10, five times 10, five times 10, is equal to Is equal to So let's keep thinking about this. If someone walked up to you in the street again, and said blank, blank, divided by, divided by two, blank divided by two is equal to, is equal to nine.
How would you figure out what blank is? Something divided by two is equal to nine. Each 4 you subtract is a group or basket. A farmer has 24 apples. How many baskets of 4 can she fill? Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to illustrate the multiplication process. Arrays can also illustrate division. Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related.
If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.
In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find the number of counters in each group. I always quote "God created the integers, all the rest is the work of man" and cite the grman author Google it and that I am not violating separation of Church and State.
Of course I am not teaching Category Theory to year-olds. But, as you have shown, Mark, it is good for us to know, to get clear on the basics of Math. Now, I have had an on again, off again, tentative relationship with Category Theory since I was a teenager. There is a gradual and accelerating takeover of parts of Math, including Foundational, by means of Category Theory.
This has a bright side and a dark side. Category Theorists have beliefs in things even sillier than the infinities of Set Theory, about which I need not believe in completed infinity, can use the Cantor stuff, have seen the morass deeper in theory, and don't matter for Science anyway except as to whether or not space or time are continuous or discrete, which don't use either sets or categories anyway.
Category Theory is more "gestalt" and less "analytical" in terms of which half of your brain is engaged. This is a paradigm shift, socially, I can say without accepting or denying the claims on either side of the battle.
In any case, Engineering and Science can pretty much watch with detached amusement, or ignore the fight, until the winners start new invasions.
Grade 3: Relating Multiplication and Division: Overview
The Categorists tend to see Biology and Sociology and Economics as ripe for conquest, due to "networks" and their uses. Decategorification and recategorification are not merely clumsy big words.
They code for an agenda. First, that the real world has structures which set theory throws away. So how do we put the real world structure back into Math? I have tried making lesson plans based about "Stuff, Structure, and Properties", i.
There is "stuff" in the world. All models are wrong, but some models are useful. Stars are real, but they do things we don't understand. People are real, ditto. Part of the problem is the education of the teachers themselves, and partly a systematic problem that teachers are required to teacher almost all subjects especially in elementary levels.
Multiplication Lesson Plans
I think you have to introduce the idea of multiplication as repeated addition as I think that's how the concept was invented and used by our ancestors. Yes the concept gets more abstract as more math was developed through the ages but I think the idea of introduce ideas as they were developed through history make sense. As long as the concepts get corrected and refined as the kids develop. I think this is called the "genetic approach".
Relating division to multiplication (video) | Khan Academy
That's where the problem with the teachers gets in the way: Thinking of multiplication as scaling rather than repeated addition is indeed a great way to look at things and it also emphasizes geometrically Devlin's point that numbers come equipped with two operations.
Viewing addition as shifting and multiplication as scaling is an elegant, concise, and geometrically intuitive way to understand how the two operations are distinct.
The question is, however, can this concept be effectively taught to little kids, kids who don't yet know how to multiply two whole numbers together? Do third graders have intuition regarding the real line developed enough to understand the shift vs.
- Relating division to multiplication
- Multiplicative relationships
- Maths KS1 / KS2: The relationship between multiplication and division
I don't remember how I thought about numbers when I was in the third grade. If kids this age do, in fact, understand the number line model then I think its a great idea to emphasize the differences between addition and multiplication in this way. Log in to post comments By jc not verified on 25 Jul permalink I really don't see the problem with saying that multiplication is repeated addition, because that is what it is.
That is that is how it was originally defined. It's true, of course, that the concept has been generalized to more advanced mathematical structures.
But does it make sense to teach children these deeper abstractions, without first explaining their original meaning? Also, if we are worried about advanced definitions, it should be noted that scaling is not the same as multiplication. Row count would be: Row 3 of 5 in above diagram gives correct answer. Log in to post comments By Tony Jeremiah not verified on 25 Jul permalink I think some of the earlier comments get a little to tied to the abstract mathematics of multiplication.
We are dealing with 2nd and 3rd graders here. A big part of it is memorizing the multiplication tables, so they can do basic math in their head. These kids don't even know fractions, so the repeated addition works perfectly. After they learn fractions you can knock out some of addititive intuition because they won't need it any more. This seems like a pointless discussion because I don't think anyone is struggling because they are too attached to the repeated addition they learn in elementary school.
Log in to post comments By Jim RL not verified on 25 Jul permalink For what little it is worth, my intuitive model of arithmetical multiplication never had to get beyond repeated addition of integers. For example, to multiply the square root of two by pi, I first aproximate them as rational numbers, say 1. Then I consider that to be the same as addingcopies of , and then shifting the decimal place back to just after the first digit of the result. For multiplications involving negative numbers, I first get the result for all positive numbers, and then apply the laws of signs to the result.
So for me, any arithmetical result I could get from multiplication of two complex numbers can be considered as mainly due to repeated additions. I realize there are more abstract forms of math where multiplication is a defined operation than has nothing to do with arithmetic, but when we are teaching arithmetic I think defining multiplication as repeated addition is quite practical.
I wonder if it's because I was way more interested in algebra then counting as a child.