Julia wrote on mathematics24x7
My students really struggle with Geometric Mean, and I am finding that many math teachers don't even know what this is. Does anybody have good suggestions for teaching this concept?
Replies:
This is how I teach the concept of GM.
Before teaching GM, students learn the meaning of Arithmetic Mean. Suppose we have a rectangle with sides x and y. Arithmetic Mean of x and y is (x+y)/2 , which geometrically is side of a square whose perimeter is same as the given rectangle. Isn't it? (Perimeter of given rectangle is 2x+2y and perimeter of square of side (x+y)/2 is also 2x+2y.
Once this idea is clear, then the concept of Geometric Mean is introduced.
Take a rectangle of sides x and y , then geometric mean is side of a square whose area is same as the given rectangle.
In both cases it is square root of xy.
Beth said...
I usually remind my students what they know about the arithmetic mean and show them how two numbers have an arithmetic mean that is the same distance away from both of them on a number line. They see that the two numbers are the same distance in an "addition/subtraction sense". Then I use a nice integer geometric mean and help them discover that the two numbers are now the same "distance" away from each other in a "multiplication/division sense." For example, the GM of 3 and 12 is 6; 3x2 = 6 and 6x2=12. Whereas the AM of 3 and 12 is 7.5; 7.5-3 = 4.5 and 12-7.5 = 4.5.
Michael said...
in the right - angled triangle the height dropped onto hypotenuse is the geometric mean between the segments of the hypotenuse which are created . The students must reveal this fact during learning the similarity in triangles - without any struggle, but by investigation, maybe guided by their teacher.
And, if we speak about another mean, arithmetic mean, so the midline in trapezoid is the arithmetic mean between it`s bases. And also , this fact we reveal and don`t put upon students in the final form.
Ryan said...
To make it relevant, Geometric Mean does have an application in artistic viewing. Using the width (and possibly height?) of a piece of an art (or nowadays flat-screen LCD TV's), the Geometric Mean actually is the exact location you should stand away from the artwork/TV to get maximum enjoyment. Research has proven this if you meander through the internet. Also, store like Best Buy are really starting to get into this with the increase in LCD TVs and home surround sound theaters.
All these are fantastic ideas. Try them out in your classroom. Make Math enriched and enjoyable.
My students really struggle with Geometric Mean, and I am finding that many math teachers don't even know what this is. Does anybody have good suggestions for teaching this concept?
Replies:
This is how I teach the concept of GM.
Before teaching GM, students learn the meaning of Arithmetic Mean. Suppose we have a rectangle with sides x and y. Arithmetic Mean of x and y is (x+y)/2 , which geometrically is side of a square whose perimeter is same as the given rectangle. Isn't it? (Perimeter of given rectangle is 2x+2y and perimeter of square of side (x+y)/2 is also 2x+2y.
Once this idea is clear, then the concept of Geometric Mean is introduced.
Take a rectangle of sides x and y , then geometric mean is side of a square whose area is same as the given rectangle.
In both cases it is square root of xy.
Beth said...
I usually remind my students what they know about the arithmetic mean and show them how two numbers have an arithmetic mean that is the same distance away from both of them on a number line. They see that the two numbers are the same distance in an "addition/subtraction sense". Then I use a nice integer geometric mean and help them discover that the two numbers are now the same "distance" away from each other in a "multiplication/division sense." For example, the GM of 3 and 12 is 6; 3x2 = 6 and 6x2=12. Whereas the AM of 3 and 12 is 7.5; 7.5-3 = 4.5 and 12-7.5 = 4.5.
Michael said...
in the right - angled triangle the height dropped onto hypotenuse is the geometric mean between the segments of the hypotenuse which are created . The students must reveal this fact during learning the similarity in triangles - without any struggle, but by investigation, maybe guided by their teacher.
And, if we speak about another mean, arithmetic mean, so the midline in trapezoid is the arithmetic mean between it`s bases. And also , this fact we reveal and don`t put upon students in the final form.
Ryan said...
To make it relevant, Geometric Mean does have an application in artistic viewing. Using the width (and possibly height?) of a piece of an art (or nowadays flat-screen LCD TV's), the Geometric Mean actually is the exact location you should stand away from the artwork/TV to get maximum enjoyment. Research has proven this if you meander through the internet. Also, store like Best Buy are really starting to get into this with the increase in LCD TVs and home surround sound theaters.
All these are fantastic ideas. Try them out in your classroom. Make Math enriched and enjoyable.
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