Week 8-alt: Non-symbolic ratios

There are researchers* (see bottom of the page), particularly in the field of neuroscience, who posit the notion that humans (and some other species, perhaps) possess an innate, non-symbolic ratio processing system (RPS). Whether or not such a thing exists, there clearly are ratio situations that we cope with successfully at an early age. Here we consider ratio tasks that lend themselves to an intuitive approach - but which might also prompt children to supplement this with an analytic method to provide certainty.

Task 08A: Here is a sketch of an everyday situation where we compare the height of objects that are different distances from the viewer. It involves two bus stops, which we can assume are the same height in reality (though not on the page), which allows us to compare the heights of the people (Heather and Tom) standing next to them.

It is pretty clear that Tom is taller than Heather. We can confirm this in various ways. For example, we can see that the height of two Toms is much closer to the height of a bus stop than the height of two Heathers:

We can express this numerically, for example by saying that a bus stop looks to be about 3 times as tall as Heather and about 2½ times as tall as Tom - or that Heather's height is about ⅓ and Tom's about ⅖ of the height of a bus stop. We can get more precise estimates by measuring the various lengths: it turns out that Heather and Tom's heights are roughly 0.31 and 0.38 of the height of a bus stop respectively.

Another approach would be to use perspective (or mathematical enlargement); the red ray touches the top of Tom's head but goes well above Heather's, so Tom is taller:

Task 08B: Here we can plainly see that Page 2 is more red (or less blue) than Page 1 and so contains more examples of the letter a.

We can confirm that Page 2 contains more as than Page 1 in various ways. For example, we could count them all! Or, more sensibly, we could compare some individual lines. Or we could analyse the individual words adaptable and adapt. The letter a occurs in 3 out of 9 or 1 out of 3 letters in adaptable, and in 2 out of 5 letters in adapt. Which fraction is larger, ⅓ or ⅖? [Similarly, we could compare the occurrence of a in adapt and in able: ⅖ and ¼ - which is larger?]

Task 08C: Here we can see that Merle has far fewer red grapes than Reece, but also fewer white grapes. It turns out that Merle has relatively more red grapes than Reece. This might not be obvious to everyone, especially as it is not easy to count the various numbers of grapes, especially for Reece. Whatever conclusion children come to, most will realise that they have to consider the numbers of white grapes as well as red. But how? - additively or multiplicatively?

It turns out that Merle has 10 red and 4 white grapes, and that Reece has 20 red and 14 white grapes. Additively, the relations are the same: they both have 6 more red grapes than white grapes. On the other hand, Reece has twice as many red grapes than Merle but over three times as many white grapes. What do these facts mean for the colour of the resulting juice?

Another approach would be to put Reece's grapes into two groups of '10 red and 4 white grapes', to match Merle's group. This would produce twice as much juice for Reece than for Merle, but the same colour - until we add the remaining 6 white grapes to Reece's juice.

Task 08D: Here it is fairly easy to see that Merle's plate is fuller than Reece's, without needing to quantify the 'fullness' or density of the sets of gooseberries.

It turns out that Reece's plate is about 1.41 (or √2) times as wide as Merle's and so has twice the area (we wouldn't necessarily expect children to determine this). However, Reece has less than twice as many gooseberries as Merle - only 18, compared to Merle's 12. So Merle's plate is indeed 'fuller'.

In the variant below, the numbers of gooseberries are unchanged as are the areas of the plates: Reece's plate again has twice the area of Merle's.

Task 08E: Here we have three rectangles which we can think of as part of a family, where the width and height of successive rectangles both change by the same amount (5 units) each time. This will lead some children to conclude that the rectangles are all the 'same shape' (in the sense of being mathematically similar) even though they might not look it!

The task can be made considerably more challenging by removing the yellow rectangle. It is quite difficult to tell, just by eye, that the red and blue rectangles are not similar. One way to try to do so is to visualise the diagonal, from bottom-left to top-right, for each rectangle. Do these diagonals overlap?

The task can be made less challenging by including a more extreme member of the family of rectangles, as in the diagram below. This might well nudge some children into seeing that the larger rectangles get progressively more square! We can quantify this in various ways. For example, the smallest of these rectangles (10 units by 5 units) is twice as wide as it is high, whereas the largest (25 by 20) is only 1¼ times as wide.

 Comments welcome! Send emails to mathsuntangle@gmail.com

 

*According to Matthews et al (below), "...the RPS may prove to be an underappreciated neurocognitive start-up tool that serves as a primitive ground for advanced mathematical concepts." (p200)