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how to teach area of a circle
Having trouble teaching area to 6th grade students any suggestions?
(1) Get some of the graph paper that has big squares in dark green lines and little squares one tenth of an inch or whatever in lighter lines. Draw some circles of radius one, two, three, and four inches on this paper, with center at the center of the paper. Pass these around the class randomly mixed, and have each student count up the area of one circle. (Note: if they do not aleady know the concept of area, that area is the number of square units covered on a surface, or if they don’t know that the squares one-tenth by one-tenth are hundredths, or if they do not already know how to combine the whole squares and the fraction-decimal squares into a single sum, you absolutely *must* teach these basic concepts first, especially the concept of area.)
Then collect all the circles of the same size and note the areas found on the board. Throw out any that seem totally out of range, of course. Take an average for each radius.
Try to look for patterns in the areas. Do they seem to double when radius doubles and triple when radius triples? No, doesn’t work. Have kids actually try this — is there a pattern to measured area divided by radius? OK, well, area covers a surface of two dimensions — how about radius times radius? Divide measured area by (radius times radius) — do we see a pattern? (I hope). You should get a decent approximation of pi.
(2) Is there any other way to explain this?
Yes! Here is a way I found in the textbook that I was taught out of, back in the dark ages, an excellent series:
First, actually experiment and check out the circumference rule. Take a whole bunch of round things – cans, bottles, plates, can lids, etc. etc. Give one to each student. Have them measure the circumference with a piece of paper, marking wher it meets and then flattening out and measuring accurately (if at all possible use metric or tenths of an inch — fractions will lead to some heavy calculation here). Measure the diameter accurately using cardboard calipers — two pieces of light cardboard with a right angle cutout, put the edges together and slide them in until they will just slip over the round object, tape at this size, then measure. Make a table of circumference and diameter, and llook for a pattern. If you divide circumference over diameter, you should get a pretty good estimate of pi.
Then, cut a large circle out of light cardboard. Colour one half red and leave one white. Cut the halves apart. Fold in half carefully and unfold (folding over and over will get inaccurate) and refold each half and continue until each half is divided into sixteen little pie slices. Very carefuly cut from the *center* towards the edge until there is just a little strip holding your pieces together, but don’t cut right apart (or if you miss, tape back at the edge! — actually, best to reinforce the edge with tape anyway.) Now you have the perfect circle area demonstration tool. Put it together in the original form, and you have a circle. Pull the points of the pie slices gently apart until the long edge is nearly straight and fit the two halves together alternating red-white-red-white slices (nearly triangles) and you have an almost-rectangle. Its width is clearly the radius of the circle, and its length is clearly half the diameter which is one half pi D which is pi times r (work this out with the class) area of your almost-rectangle is then pi r times r, or pi r squared. But of course you can put it back together to make a circle, and voila.
***************************
This all will take two or three class sessions to teach, but is very worth doing — students will understand and remember, as opposed to messing up formulas for years. Also good to hand out *after* the experiment some papers reminding how it works, with *pictures*. Then do practices, if possible allied to real life — area of a decorative flower bed, area of a stavge to paint, area illumiminated by a spotlight, etc..
(1) Get some of the graph paper that has big squares in dark green lines and little squares one tenth of an inch or whatever in lighter lines. Draw some circles of radius one, two, three, and four inches on this paper, with center at the center of the paper. Pass these around the class randomly mixed, and have each student count up the area of one circle. (Note: if they do not aleady know the concept of area, that area is the number of square units covered on a surface, or if they don’t know that the squares one-tenth by one-tenth are hundredths, or if they do not already know how to combine the whole squares and the fraction-decimal squares into a single sum, you absolutely *must* teach these basic concepts first, especially the concept of area.)
Then collect all the circles of the same size and note the areas found on the board. Throw out any that seem totally out of range, of course. Take an average for each radius.
Try to look for patterns in the areas. Do they seem to double when radius doubles and triple when radius triples? No, doesn’t work. Have kids actually try this — is there a pattern to measured area divided by radius? OK, well, area covers a surface of two dimensions — how about radius times radius? Divide measured area by (radius times radius) — do we see a pattern? (I hope). You should get a decent approximation of pi.
(2) Is there any other way to explain this?
Yes! Here is a way I found in the textbook that I was taught out of, back in the dark ages, an excellent series:
First, actually experiment and check out the circumference rule. Take a whole bunch of round things – cans, bottles, plates, can lids, etc. etc. Give one to each student. Have them measure the circumference with a piece of paper, marking wher it meets and then flattening out and measuring accurately (if at all possible use metric or tenths of an inch — fractions will lead to some heavy calculation here). Measure the diameter accurately using cardboard calipers — two pieces of light cardboard with a right angle cutout, put the edges together and slide them in until they will just slip over the round object, tape at this size, then measure. Make a table of circumference and diameter, and llook for a pattern. If you divide circumference over diameter, you should get a pretty good estimate of pi.
Then, cut a large circle out of light cardboard. Colour one half red and leave one white. Cut the halves apart. Fold in half carefully and unfold (folding over and over will get inaccurate) and refold each half and continue until each half is divided into sixteen little pie slices. Very carefuly cut from the *center* towards the edge until there is just a little strip holding your pieces together, but don’t cut right apart (or if you miss, tape back at the edge! — actually, best to reinforce the edge with tape anyway.) Now you have the perfect circle area demonstration tool. Put it together in the original form, and you have a circle. Pull the points of the pie slices gently apart until the long edge is nearly straight and fit the two halves together alternating red-white-red-white slices (nearly triangles) and you have an almost-rectangle. Its width is clearly the radius of the circle, and its length is clearly half the diameter which is one half pi D which is pi times r (work this out with the class) area of your almost-rectangle is then pi r times r, or pi r squared. But of course you can put it back together to make a circle, and voila.
***************************
This all will take two or three class sessions to teach, but is very worth doing — students will understand and remember, as opposed to messing up formulas for years. Also good to hand out *after* the experiment some papers reminding how it works, with *pictures*. Then do practices, if possible allied to real life — area of a decorative flower bed, area of a stavge to paint, area illumiminated by a spotlight, etc..