Proportional Reasoning

Proportional reasoning is one of the biggest math ideas students will develop over the junior years. It applies to almost all areas of the curriculum, and is considered a critical concept for success in secondary math.
Despite this, the curriculum has a few expectations dealing with rates and ratios under a Proportional Relationships heading, and then leaves other proportional problems as unconnected ideas in number sense and other strands.
Proportional reasoning is, most simply, the ability to compare two things using multiplicative thinking, and then apply this to a new situation.  It is also frequently defined as just the ability to compare quantities multiplicatively.

Here’s a good example.

In 30 minutes, Bob reads 60 pages of a book. How long should it take him to read 150 pages?





You can compare pages to minutes (pages are twice as much as minutes), and then apply this understanding to the new situation of 150 pages.



prop2You could also compare the 60 pages to 150 pages (2 and a half times more), and then apply this relationship to the amount of time.

There are several other strategies students will naturally use for problems like these – I’ll try to get a set of student samples posted.

The research for developing proportional reasoning skills indicates several strategies, most of which are the opposite to approaches in most textbooks.

1. Proportional reasoning is best developed in investigative problem solving lessons.
In the junior grades most texts show students ratios, then rates, and then maybe has them solve unit rate problems. Larger proportional reasoning problems of the type above might appear in this chapter but usually are in the multiplication / division section with no reference to proportional reasoning skills.

2. Student understand best when multiple strategies are shared and discussed.

3. Problems should start with high context, hands on situations.


Elastic Video

This is a great video from Japan where they try to shoot the world’s largest elastic band.  It makes a great proportional reasoning problem solving lesson.

The link to the right is from youtube, but I have an edited version I can send out to AMDSB teachers (too large to post here).  The hard copy is also useful if you don’t have internet access in your room.

Giant Handprint Lesson

This lesson is from AIMS.  Basically a giant handprint is left on the wall over night, and students need to figure out how tall the giant was that left the handprint.

This is a good book for looking at proportional reasoning.  The link to the left is a few of the situations in the book converted to metric, which makes the problems more approachable for grades 4 to 6.
Ratio cards are suggested as one way to approach proportional reasoning informally.  Students compare various visual representations of ratios intuitively.



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  37. I found my way easier.

    If it takes him 30mins to read 60 pages,
    So to read 30 pages half 30 which is 15
    Therefore it’s gonna take her 15 minutes to read 30 pages
    It takes 1h 15 minutes to read 150 pages

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