Proportional reasoning can be difficult for middle school students. Sometimes they are able to use an algorithm to solve a problem, but have no concept of the meaning behind the algorithm.
Most math problems involving shapes with proportional reasoning involve empty shapes. Why not use a picture to demonstrate proportionality?
In most computer programs, when you lock the aspect ratio of a picture both the height and the width change at the same time (lower photo). If the aspect ratio is not locked the width and the height do not change in a proportional way (upper photo).
After using photographs to demonstrate proportionality there are a few exercises that you can do in order to move to deeper understanding.
- Have the students create their own examples of proportional and non-proportional pictures. After printing them have them measure the sides to create math problems.
- After creating proportional and non-proportional photographs they can demonstrate their knowledge by using open rectangles. To draw rectangles in Word choose View-Toolbars-Drawing. Use the rectangle tool on the toolbar to create either a congruent (proportional) rectangle or a non-proportional rectangle. The secret is to hold down the shift key while dragging the shape, which also works to keep a line straight as you extend it. Have students first create a proportional rectangle using the shift key, and then a non-proportional rectangle by not using the shift key.
- Create a page of different rectangles and have the students choose which are congruent (proportional) or in the same family.
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