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E.g. what the light or sound spectrum is, what it's useful for - in very simple terms that a grandmother or a child would understand.

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I would use the idea of explaining the spectrum of light by relating it to the spectrum of flavors. Surely children and grandma must have tasted a variety of flavors in their life time. You can use different flavors, (salty, sweet, bitter, savory, sour, super sweet, super salty, super bitter, super sour, whatever you can think of, etc) to represent the different parts of the flavor spectrum.

I would then describe how different dishes of food lie in different places in the spectrum of flavors. Dessert like ice cream would fall under the spectrum of sweet. There are super sweet desserts, strawberry short cake, and there are more bland desserts such as pound cake. By establishing a point of reference in which the student can relate to and is extremely relevant in their lives, (food the most common thing.) They feel a connection. It is important to have this kind of connection before you start talking about colors. If you immediately start talking about colors, they might not have/feel immediate pleasure from it. When talking about food, it definitely brings about a state of mind that feels good.

After establishing the idea of a spectrum of flavors, you can explain to a child or grandma that they must have seen different colors of light! At this moment, the child or grandma will understand that you are trying to relate a range or "spectrum" of flavors to the range or "spectrum" of light. At this point, start describing about how you can mix colors together and how mixing colors is like mixing different flavors in a dish of food.

There are so many varieties of food, talk about their favorite food and what flavors it has in it!! By tasting the food, you are doing "flavor spectral analysis" Explain how food critics are like spectral analysis machines that are bias, LOL.

The most important thing to do is to keep it simple. This does not necessarily mean to dumb it down. Try to focus on one or two things you want them to understand, the idea that there is a range of colors/flavors and that analyzing a food dish and trying to find out its recipe is what physicists are try to do with a spectrum of light. Make sure to work off of the feedback they give you. And don't get discouraged, because you don't want them to remember you as the person who gets frustrated whenever you try to talk to them.

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I would start with talking about music - piano, a capella choir, etc. Why thirds and fifths and octaves sound good, but seconds don't.

Then maybe move on to the radio dial, eventually moving on to light and prisms.

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To a child you have to show pictures. A diamond in your ring or a wedge of glass or crystal, then the Sun light, or ..
Then the rainbow and the clouds, all the colors in the cristal, etc ...
Use the filters in the equalizer of your hi-fi (or software media player) to show that the sound is composed of many distinct sounds (freqs) that are hidden by all the other sounds. Use the back of a CD, bend it, to play with the colours. Use an aquarium, the light reflected in the skin of the fishes.

Play and record two notes at the same time in a piano (or software app) and then one at a time, and then apply FFT and show her the results.
She will understand the results as the contribution of two distinct notes (explain the harmonics later). Call the software FFT, or 'softCrystal' (to her is only the name of an active object, like 'windmill' ) and then explain that a crystal behaves like a 'softCrystal' but is made in stone (hardware) that decompose the perceived light into its components.
Now, that she have seen the decomposition of the sound and light stimulus, it's time to introduce her to the sine wave concept, amplitude and frequency, phase, and to the fact that in nature all stimulus are composed, at the lowest level, of waves.
If you have the time (a grandma always have time ;) you can explain her how the waves interfere, reflect, refract... the distinction between Light - unbounded (free) waves as the ones she obtains in the stretched long rope where she hangs the cloths to dry, and Matter - bounded waves like like the ones that persist in the surface of a drum.

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Would grandma understand FFT?? – Arnold Neumaier Mar 9 '12 at 9:28
Did I said that? Nope. But she will understand the results because she is not retarded and you are a clever guy. I will edit the answer. – Helder Velez Mar 9 '12 at 11:56

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