Calculus at a Fifth Grade Level - Printable Version

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Calculus at a Fifth Grade Level - Kyng - 12-28-2023


"Calculus"... I'm sure that word sends shivers down the spines of many people here. The mere sight of a '∫' or a 'd/dx' in their maths book is enough to send them running for cover. But, does it have to be this way? The above video aims to explain calculus at a level that a fifth-grader can understand - and, I'll summarise the first half of the video below:

• People find calculus hard because it is different. It introduces concepts like 'limits', 'derivatives' and 'integrals', that they haven't encountered in any of the mathematics that they've studied elsewhere. So, it should become much easier if we establish a firm understanding of the underlying concepts first!
  
  
• Infinity (∞): People often describe 'infinity' as "the biggest number". But that's not really true: 'infinity' is not a number. Instead, 'infinity' is just a convenient way of saying "We can't put a number to it because it goes on forever". For example, imagine someone asks, "How many numbers are there between 1 and 2?". Well, there's 1.1, and there's 1.11, and there's 1.111, and we can just go on like this forever, adding as many 1s as we like. We can't answer this question with a number, because no matter how many we've found, we can always find more. So, the answer to the question is 'infinitely many' (or, for brevity, 'infinity').
  
  
• Limits: Now, let's go in the other direction, and imagine what 1/∞ is . We know how fractions work: as the number on the bottom gets bigger, the fraction gets smaller. 1/1 = 1; 1/10 = 0.1; 1/100 = 0.001; and so, it might be tempting to say "1/∞ = 0". But again, this is technically not right - because, just as ∞ isn't a number, "1/∞" isn't a number either! We can't say "1/∞ = 0"... but what we can say is "1/∞ leads to 0", or "as x goes to ∞, 1/x goes to 0". This is what's known as a 'limit': the limit of 1/x (as x goes to infinity) is 0. Or, the way to express that in mathematical symbols is as follows:
  
  
  
  
  
• Limits of areas - Now that we know what a 'limit' is, let's apply the same concept to the area of a shape. For example, let's imagine we want to find the area of a triangle, which is 5cm tall and 5cm long. Now, some of you might remember the formula "½ base x height", and come up with the answer "½ x 5 x 5 = 12.5cm²" . But, if you don't remember that, we can approximate the area by covering the shape in rectangles:
  
  
  
  In our first example, we have 5 rectangles, arranged in a way that looks a bit like a triangle. Each rectangle has width 1, so the area of the shape is just 1 + 2 + 3 + 4 + 5 = 15cm². That's not a bad estimate, but we can improve it by making the rectangles thinner... and thinner, and thinner. And as we do, the shape looks more and more triangle-like - and the area gets closer and closer to the 'correct' value of 12.5cm². Indeed, as the width of the rectangle goes to zero, the area of the shape goes to 12.5cm²!
  
  But it's not just triangles: if we can apply this technique to any shape, then we can find the area of anything! This is what that '∫' symbol means: the expression "∫₀⁵ x² dx" looks very frightening, but it really just means "the area under the graph y=x² between x=0 and x=5", and we can find that area by using this "infinitely thin rectangles" technique .

Like I said, that's just the first half of the video. You can go ahead and watch the rest if you want - but, if you need a breather, that's fine too. I don't want this post to get overwhelming, so I'll stop here before I continue with the second half of the video (and, if you have any questions, go right ahead and ask them !)

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RE: Calculus at a Fifth Grade Level - Kyng - 01-07-2024

Now, to cover the second half .

• Slope (or gradient): Let's imagine we have a graph of a straight line that goes from (0,0) to (5,10). It's quite easy to calculate the gradient of this line: it goes up 10 and across 5, so the gradient is just 10/5 = 2 . Or, if it was going from (0,10) to (5,0), then the line would go down 10 and across 5, so the gradient would be -10/5 = -2:
  
  
  
  
• Slopes of curves: But what about the gradient of a curved line? That's more complicated - because the gradient constantly changes . But can we find the gradient of the curve at any given point? As it happens, we can - by using an approximation technique similar to the one we used for areas.
  
  For example, let's look at the graph y = x²:
  
  
  
  We can't measure the gradient directly: it's a curve, not a straight line . But we do know some points that the graph goes through, because we know what the first few square numbers are! The first five square numbers are 1, 4, 9, 16, and 25 - so, in addition to (0,0), the graph goes through (1,1), (2,4), (3,9), (4,16), and (5,25).
  
  So, what we can do is mark these points on the graph, and draw straight lines between these points, so we end up with a bunch of straight line segments that looks like our curve:
  
  
  
  We can then calculate the gradient of each line segment, in the usual way. The gradient of the line from (0,0) to (1,1) is 1; the gradient of the line from (1,1) to (2,4) is 3; the gradient of the line from (2,4) to (3,9) is 5; the gradient of the line from (3,9) to (4,16) is 7; and the gradient of the line from (4,16) to (5,25) is 9.
  
  You might have noticed something here: every time we go up by 1, the gradient increases by 2. Indeed, the gradient is always twice the x-value of the midpoint of the line segment. For example, the first line segment has midpoint 1/2 and gradient 1; the second line segment has midpoint 3/2 and gradient 3; and so on. So... at the point x, perhaps the gradient of the graph is 2x?
  
  To confirm that hypothesis, we could do what we did before for the areas - and take smaller and smaller line segments; work out the gradient of each one; and figure out what would happen as the line segments get infinitely small. I won't do that here - but, if we did, we would find that our hypothesis is correct: at the point x, the gradient of this graph is indeed 2x . Or, to use the notation that mathematicians use:
  
  
  if y = x², then y' = 2x.

Anyway, I hope this thread has at least given you some idea of what calculus is about. If you have any questions, then please let me know !

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