By thinking, no algorithms

Hi Natasha.

May I ask you which grade is your child?

The teacher wants to make children think on (the very difficult concept of ) ratios and proportional thinking.

So, let the child try to think, without urging for a result which might not even come. The exercise is about thinking not about getting a result (succeed). And the child must make drawings!

Here is my thinking. I begin always with easy numbers...

The original ratio is 5 to 3. In our example 5 people liked "this" and 3 people liked "that". Or, I can state that there were 8 people and 5 of them liked milk chocolate and 3 liked plain chocolate. (make a drawing with "dots")

5 and 3 people. So, I can see the difference: 2 people more liked milk chocolate.

Now the problem tells me that in the survey there are more than 8 people , right? Right! So how many can I add to keep the proportion?

Well, try it!

I double the people and the get 10 and 6, I double again and they get 20 and 12, and so on...(these are equivalent fractions by the way, but it's not important to know the name). The idea is that I can change the numbers of the people in the groups without changing their relative proportion.

But wait again, there is an easier way! I know that with 8 people and the proportion 5 to 3 there are 2 people more in one group!

Ok! Now the problem says that there are 46 people more in one group! How many more times is the difference of people between the groups? 23! (Why? Well, 23 times the difference of 2 makes 46!)

So now we have the key for the solution: 23!

We can change the original people by multiplying them by 23: and we get 115 (5x23) and 69 (3x23). You can also see it here:

(START) 5 and 3, difference is 2 people

(TIMES 2) 10 and 6, difference is 4 people

(TIMES 3) 15 and 9, difference is 6 people

(TIMES 4) 20 and 12, difference is 8 people

......

(TIMES 23) 115 and 69, difference is 46

So there is a total of 115+69=184 people in the survey.

The idea of the teacher is probably to make an introduction into equivalent fractions.

I know fractions are not easy (including for myself of course!) because proportional thinking is not intuitive at all. I am making a website on it (www.swiss-algebra-help.com) and the more I write the more surprises I find...

Alessandro

Switzerland