I have been selected by my college to teach integration to kids in the age group of 8-12. I am an engineering major who has finished Calculus 1 and 2 but I have no idea how to teach integration from scratch to kids that small and at the same time make it fun for them. I am asked to create a lesson plan, worksheets, manipulatives etc.
Any help is appreciated. Parents or teachers, your ideas will really come handy.
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The furthest you can teach them before having to explain derivatives is Riemann Sums. This gives them the geometric intuition behind the integral, but does not give a useful method of evaluating integrals analytically. You could just tell them that $$\int_a^bx^n\,dx=\frac{1}{n+1}\left(b^{n+1}-a^{n+1}\right)$$ and have them check by using Riemann sums to see if the formula gives the correct area. But I'm not a fan of telling kids to use certain formulas without proving that they work, because then they'll feel that mathematics is just memorizing formulas.
As for making the learning process fun, you could frame your integration problems by using physics. Give them polynomials for the velocity of a race car as a function of time and ask them to determine who would win a race after a given amount of time. If they don't understand the relationship between velocity and displacement, you could explain that to them.
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First, your task is impossible to teach rigorously. Since these students probably don't have the algebra skills necessary to set the groundwork to success in calculus. But this is what I would do as my lesson plan.
First define what area even means in terms of area of a rectangle. The area of a rectangle is $length*width$ or $a*b$.
Next show areas are additive and show other areas.
So the area of a triangle is $\frac{1}{2}a\cdot b$.
But WHY? Well explain to them area is additive and show them this is what the area of a triangle has to be, to be consistent with the area of a rectangle.
Then provide as many proofs as you want to show that Area of A + Area of B=Area of A & B.
I would also stick to geometric ways of showing most of my ideas.
Also show how rectangle A, $1\:\cdot \:2$, has the same area as rectange B, $\sqrt{2}\cdot \sqrt{2}$.
And as a geometric way to help them understand so you can cut one rectangle to achieve the other rectangle, reinforcing areas can be subtracted and added.
Anyway, once you have them interested in the idea that area that if two things have the same area you can geometrically chop one thing into pieces and form the other thing; then introduces curves. This idea of area being able to cut pieces and rearrange it to form another thing with the same area is totally destroyed with objects with curved boundaries.
For example what does it mean to have an area of $\pi $.Well then teach them that they can find area in terms of rectangles which they already know.
Show them they can approximate by making small enough rectangles. Obviously we don't care to much about computations as much as giving them a general understanding, so I wouldn't go overboard with it. I wouldn't introduce limits directly, but make sure they understand the smaller the rectangles the better approximation they get.
Now at this point I finally start talking about curves and introduce some notation. I start with a simple curve and partition it into rectangles.
I say $dx$ represents the width of the tiny rectangle, $f(x)$ is the height, $\int $ is sum of, and $b$ and $a$ is my interval. I am not getting technical whatsoever to what these actually represent! Not defining limits, not going into summations, anti-derivatives or the fundamental theorem.
Show them something like $\int _0^1x^2dx=\frac{1}{3}$ and ask them what $\int _0^13x^2dx$
Finally teach them about how they can approximate areas of some small curves.
I would use desmos online graphing calculator to show them the how to use rectangular area approximations.
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To be honest, when I first was introduced to calculus, I was introduced of it in terms of physics. I didn't even know what a limit was. I think instead of over complementing them you should make it as graphical as you possibly can. Teach them basic physics, so they can relate the math to the world around them. Come up with good word problems that would make sense to the students. I don't have a concrete lesson plan for you like some of the other answers. BUT, here is a video series that I thought was pretty good. You should obviously go much slower and simplify the videos. Also turn it into word problems. Don't focus on computations, focus on insight in how these ideas explain physics.
https://www.youtube.com/watch?v=dXGjJSMZGDA
Note: I didn't post this video onto youtube. So if this answer is useful, let the credit go to the poster of the video.
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I don't have any specific advice, but I remember hearing about a book that was really good; Calculus for Young People. Maybe you could get in contact with the author as well?
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This may not answer your question. But it will probably be a good response. DON'T TEACH CALCULUS to kids! I suggest you teach them something more enriching and realistic like paw88789 said. There isn't a need for these kids to learn calculus at such a young age. Calculus can be very confusing and complex, they should wait till they mature a little bit.
I have been selected by my college to teach integration to kids in the age group of 8-12.
Go to who ever selected you to teach calculus and explain to them how unrealistic his or her goal is to accomplish. Is there a reason why they want you to teach calculus to these kids?
I am an engineering major who has finished Calculus 1 and 2 but I have no idea how to teach integration from scratch to kids that small
You shouldn't have an idea on how to teach it. Some of these kids won't know the multiplication table, how are you going to teach them calculus? The answer is you wont.
and at the same time make it fun for them.
Teach them about something they can understand. And that applies to them in nature. Something they can talk about with their friends. Or just be an easy going teacher. Maybe make some math jokes. But stay away from calculus. Not that calculus can't be fun, but they can't understand it or apply it.
I am asked to create a lesson plan, worksheets, manipulatives etc. Any help is appreciated. Parents/Teachers, your ideas will really come handy.
If you really can't avoid teaching them something more useful or passing on the offer. Then teach them about slopes instead. Then show them the bare bones concept of derivatives. But stay away from integration and anti-differentiation. Anti-differentiation is like anti-multiplying or factoring, it requires a lot more skill. Derivatives itself are a huge stretch, but is a whole lot more reasonable of a task than integration. Ideally stay away from calculus altogether.
Most of these kids are not going to be math majors and have no purpose for learning calculus especially at that young of an age.
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I don't have an exact answer because I honestly don't know what I would do if someone told me I had to teach little kids calc, however I have a book that may lead you in the right direction. It is called "The Cartoon Guide to Calculus" by Larry Gonick.
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Like many comments and answers here indicate it is impossible to teach integration (and in fact any calculus concept) to students of age 8-12 years. Any efforts to do this will be wasted because the students at this age don't have the basic knowledge of alagebra/trigonometry which is essential to study calculus.
My own encounter with calculus started at age of 13 with some crappy textbook (most calculus textbooks fall in this category) and I did know trigonometry at that time so was able to learn mechanical differentiation and integration. But then I had no idea whatsoever about what these concepts meant and I could not do any problem which was based on application of these concepts. And interestingly I could not handle evaluation of limits and derivatives using limit definition.
In my opinion it is better to teach students of age 11-12 something which is at the very heart of "foundations of calculus". And that is the appreciation of order relations between rational numbers and more importantly density of rationals. This is something that can be grasped easily by a student who knows how to do $+, -, \times, /$ with fractions. But unfortunately this aspect of density is never emphasized when dealing with fractions. When we were studying rational numbers at the age of 12 years we did get some exercises like "find 10 rationals between $1/2$ and $2/3$" but it was based on a general algebraic procedure to find $n$ equally spaced numbers between $a, b$. Even more interesting was the technique of expressing the numbers in decimals and then find the intermediate numbers by changing some decimal digits.
It is also better to let the students appreciate the simple fact that there is no least positive rational number. And such facts form the backbone of typical arguments used in analysis proofs. This also helps the students to deal with the notions of infinitely small and infinity. And if students are taught this part well, they can be presented with the Dedekind's theory of real numbers at age 16-17 just before they start to learn calculus.
I was lucky enough to follow this route to calculus with Hardy's A Course of Pure Mathematics. However proceeding in this fashion has a side effect of wiping out almost 30-40 percent of ungraduate real analysis and that is probably one reason this particular approach to calculus is not followed.
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For children of this age, you're probably better off introducing integration as a method to compute averages. The problem with the traditional approach is that this builds on a lot of math that the children have not yet been exposed to. Any attempt to go down the traditional part would end up being very boring to most of the children. We can circumvent that problem by exploiting the fact that today's children are fluent in using their smartphones and computers. In their free time, they take pictures, record videos which they share with each other on social media.
One can hook into their experiences to introduce integration, by giving them assignments like taking pictures and analyzing the pictures using programs like ImageJ. A picture magnified sufficiently becomes a set of gray values. The children can be taught about noise, e.g. they can be let to stumble into that when given the assignment to enhance contrast allowing them to peek into a dark areas. Stretching the gray values will look like a magical method to see things that were invisible in the original picture, but we now get a lot more noise. So, what is noise and how can we get rid of that? Thing is that the children will be very motivated to understand all of this.
Simple methods to reduce noise involve taking local averages can be taught. When the children do this they also see that such manipulations come with unwanted side effects. One can then ask how a picture viewed on a a large scale where it has smoothly changing gray value, changes when we take such averages. So, integration pops up quite naturally within this topic.
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I am writting from Hungary, sorry for the language mistakes. I think also, we can not teach any concepts of calculus for Young children, but we can speak with them about infinity small, infinity big numbers, about change, area, approximate calculation, estimation. After playing and understanding mathematics idea, children can make some problems of calculus, and also they can understand what are the differences beetwen knowing calculus and knowing just $f'(x)$ in the case of $f(x)= x^n$.
Define integration as area under a curve.
Use examples to approximate well-known areas (start with line, e.g. area of trapezoid, and make it more complex like semi-circle or half-ellipse or parabola) with rectangles and see the numbers getting closer to what the area is.
You cannot teach analytic integration because they won't be able to find the anti-derivative, so stay in geometric approximations. Not sure you can do much else at that age.