Mathematics: An informal approach to solving proportions
An informal approach to solving proportions
Traditional textbooks show students how to set up an equation of two ratios involving an unknown, "cross-multiply", and solve for the unknown. This can be a very mechanical approach and will almost certainly lead to confusion and error. Although you may wish eventually to cover the cross-product algorithm, it is well worth the time for students to find ways to solve proportions using their own ideas first.
If you have been exploring proportions informally, students will have a good foundation on which to build their own approaches.
To illustrate some intuitive approaches for solving proportions, consider the following tasks:
In the first situation, it is perhaps easiest to determine the cost of one widget - the unit rate or unit price. This can be found by dividing the price of three widgets by 3.
Multiplying this unit rate of $0.80 per widget by 10 will produce the answer. This approach is called a unit-rate method of solving proportions. Notice that the unit rate is within a ratio.
In the second problem, a unit-rate approach could be used, but the division does not appear to be easy. Since 12 is a multiple of 4, it is easier to notice that the cost of a dozen is 3 times the cost of 4. This is called a factor-of-change method. It could have been used on the first problem but would have been awkward.
The factor of change between 3 and 10 is
Multiplying $2.40 by
will produce the correct answer.(When you multiply entries in a ration table, you are using a factor of change.)
Although the factor-of-change method is a useful way to think about proportions, it is most frequently used when the numbers are compatible. Students should be given problems in which the numbers lend themselves to both approaches so that they will explore both methods.
The factor of change is a between ratio.
Resources:
http://www.cehd.umn.edu/ci/rationalnumberproject/93_4.html
http://cargalmathbooks.com/The%20EOQ%20Formula.pdf
http://slideplayer.com/slide/7642490/
http://elementary-math-resources.wiki.inghamisd.org/file/view/chapter%20from%20Van%20de%20Walle.pdf/461802390/chapter%20from%20Van%20de%20Walle.pdf
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I really love all these steemiteducation articles, they are all of great quality, useful and enjoyable to read. Congratulations for this great article! It can sound as easy stuff, but many adults can't even do simple math like this. Thank you for sharing! @OriginalWorks
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I'm for anything that makes mathematics easier for students to learn. The US is already way behind the power curve compared to the rest of the world.
I believe math concepts are easier to learn within a 'real world' situation. The answer to "why should I learn this" should be self evident.
(I am sure you know this, I just wanted to reply to your post)