### Visualizing Maths

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But what if an object is moving in an inclined plane like a sphere. Now there is an added dimension to it, The number line by itself cannot fully represent it. It needs help. That is where the complex plane comes into picture. The real component is the component which is usually the one parallel to the x-axis. The imaginary component is the one which is usually the one perpendicular to the x-axis and parallel to the y-axis.

Irrational Numbers :- Those numbers which have a square-root attached to them. Ok, I dont know specifically of any types but i do know that i , j and k maybe used to denote the three different types in which inclination can happen. Complex numbers are used to denote inclination. So if the complex part is completely zero,only the real part remains The object now is moving in a straight line maybe even parallel to the x-axis many of the times. Now, one could also say that Complex numbers are used to denote rotation.

But then inclination and rotation are both the same. You cannot incline something without rotating it and you cannot rotate something without inclining it. The vertical component is sin and the horizontal component is cos. Complex numbers come into our lives when one force gets divided into two branches of forces. Inclining the object is just one example where the force will get divided into two.

Imagine a stick. Whenever one force gets divided to two components, its due to the force getting inclined or some other reason, the real and the imaginary parts come along. The real part is the part which is usually parallel to the horizontal or to the base surface taken as a reference. For example in the above example, the ground is the base OR reference surface. So, in this case, the real part is the part parallel to the ground surface the shadow below. The imaginary part is the part perpendicular to the ground surface the shadow on the wall.

The real part is the horizontal influence horizontal projection. Imaginary part is the vertical influence vertical projection. Whenever we thought of complex numbers, we first imagined a number line, then we imagined taking square-root of a negative number, and going still backwards at the number line. However, complex numbers are all about revolving around the number line. That is the reason why the numbers 1, i , -1 and -i keep repeating cyclically in complex numbers.

Want an example? This is very cool and a pretty big deal. Find out why. We have three colored segment in this animation. Surprisingly the length of the longest one is always the sum of the length of the two smaller ones. The theorem gives a connection between the sides and the diagonals of a cyclic quadrilateral.

In this case the length of the dashed lines is equal so the theorem can be simplified to the statement above. Take a circle and draw some points on the boundary. For every point you draw, you must also draw its antipode point on the opposite side of the circle. Draw some points in the interior wherever you want.

### As a kid, I always seemed to struggle with math.

The only stipulation is that antipodes must have opposite sign. Try it! More info and proof here. The answer is NO , you can not.

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This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented. The Gaussian curvature is an intrinsic and important property of a surface.

### The other thing about math that is so clear to me now is that it is a visual form of communication.

Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat. If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice.

However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection. The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.

## Visualizing Mathematics

The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere. So what does this flattened sphere approximated by conical strips look like? Check the image below. But this is not the only way to distribute the strips.

We could also align them by a corner, like this:. All of this is not exactly new, of course, but I never saw anyone assembling one of these. I wanted to try it out with paper, and that photo above is the result. Send me a picture if you do! Water and its circumbulative connection to the Divine breath are primordial symbols inherent to the vitality of my journey.

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Ahmed keeps a studio at Mana Contemporary. One way was found by Brooks, who was so pleased with this dissection that he made a jigsaw puzzle of it, each piece being a square. A powerful force for perception and understanding. About this free course 6 hours study. Level 2: Intermediate.

## Mathematics and Visualization

Course rewards. Free statement of participation on completion of these courses. Course content Course content. Using visualisation in maths teaching This free course is available to start right now. Free course Using visualisation in maths teaching. Activity 2: Questionnaire For each of the questions below write few very brief notes of your own. What do we mean by the term mental imagery?

What do we think is the value for the learner of working with images? Does mental imagery lend itself more easily to shape and space topics? How can we use it in number work? Can this work have a place within the School Curriculum? Do pupils improve their mental imagery as they are given more opportunities to use it? Is mental imagery part of the problem-solving process? How do you think it improves concepts and skills? Can students of different abilities cope with this approach?

Talk to at least one other person about your responses. When any form of problem is tackled using only the imagination. It helps to think more about other topics and apply the idea themselves. Sometimes it is a lot easier than doing something practically. The skill of being able to manipulate mental images improves spatial abilities. Yes, as it is easier to imagine a picture or object. I think so although algebra can be spatial How can we use it in number work? Fractions, decimals, percentages and percentage change.

## Using visualisation in maths teaching: View as single page

Four operations. Ratio; sharing and mixing. Adding — climbing up; subtraction — falling down. Fractions — imagine an object divided up. Imagine a number line.

Fractions, decimals, percentages, ratio, four rules of number, solving problems. Can this work have a place within the National Curriculum? Could be mentioned, but is more a method of delivery. A teaching method rather than a separate topic. Of course — can teaching have a place in the National Curriculum? Yes, but they find it very difficult. If they have a mental image it is easier to recall.

By visualising concepts you can generally better see the whole picture. More likely to remember having constructed your own idea of what it is. I strongly believe that complete silence must be achieved. Can pupils of different abilities cope with this approach? Yes, but the images for lower abilities must be kept simple or related to a real-life situation. Previous 2. Next 4.