This week I read Exploring Ratios and Sequences with Mathematically Layered Beverages, written by Andrea Hawksley and published in Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture in 2015.
The article describes a workshop that was offered at the
2015 Bridges conference, where ratios and integer sequences are explored by creating
mathematical beverages via layering liquids of different sweetness levels (and therefore
different densities). The rationale and
the mathematics behind the activity are discussed, but the materials/ingredients
and procedure for running the workshop are also included, making it relatively
easy to borrow and adapt this activity for classroom use. Below is a photograph taken from the article.
The article explains that liquids of varying sweetness
levels will have different densities and therefore can be layered in a drinking
glass, so long as they have different colors and are poured into the glass in
order from most dense to least dense. The
layers will mix together and ruin the layering effect if errors are made.
STOP – Math problems that mark themselves!
I think this is a really great activity for a number of reasons, but one of them is because of the nature of this activity, the math problems mark themselves, giving the learner immediate feedback! If an error is made calculating the density of each layer prior to pouring, it will be noticed because they won’t layer properly. Each layer can also be tasted before pouring, and their sweetness will correspond to their relative density.
The article provides a number of examples of math topics that can be experienced by creating layered beverages. Fractions and ratios can be explored when creating the liquids of differing density, as well as when mixing them together in various concentrations and volumes. Sequences such as the Fibonacci numbers can also be explored (see picture from the article below), so long as the numbers do not grow in size too quickly, as this results in the first layers being too watery and the later ones being much too sweet. The author also claims that the best tasting lemonade actually results from mixing together amounts of lemon juice and sugar in the golden ratio! (1 part lemon juice to 1.618 parts simple syrup, which of course could be approximated by 3 parts lemon to 5 parts simple syrup, or 5 parts lemon to 8 parts syrup, etc.) I think I’m going to have to try this – I’m really excited to taste the golden ratio! The author also suggests that adults could also have fun with this activity by adding different amounts of alcohol to different layers – this also sounds like it’s worth a try if you ask me!
STOP - Tasting Mathematics
I think it would be fun to explore what other math concepts taste like - for example constants like e and pi, or perhaps the trig ratios from special triangles (45-45-90 and 30-60-90). What math concepts would you like to taste?
Activity: Mathematically Interesting Ways of Lacing Your
Shoes
I chose to do this activity because I remember watching a
TED talk about a decade ago that claimed that most people are tying their shoes
wrong – the classic shoelace knot is essentially a modified version of either a
square knot (a ‘good’ knot) or a granny knot (a ‘crappy’ knot). If you use the granny knot variant, your
laces are much more likely to come undone, especially with slippery and/or
round laces. Well, I checked which way I
tie my shoes and to my surprise I’d been doing it ‘wrong’ since childhood! Since then I’ve changed the way I tie my shoes,
which was very strange at first since the motions of tying shoes is such an
engrained routine.
STOP – You're Doing it Wrong!
Watch the TED Talk on YouTube that I’ve linked below (it’s only 3 minutes) and see if you’ve also been tying your shoes non-optimally for decades like I was!
Hi Reed,
ReplyDeleteThis summary was super interesting to me! I love the idea of 'math that checks itself'. If students did calculations with density wrong, their drink would not later correctly and this, be visibly evident whether or not their calculations were correct. I wonder how this could be expanded to other concepts. If students were given a liquid with an unknown density, they could work out the density using other liquids as referents too... Interesting things to think about here.
In your comments on tasting the golden ratio- I am wondering if there are other drinks that taste the best if made according to the golden ratio. Hot chocolate, for example. There could be some interesting ratio math activities here as well.
Hi Reed,
ReplyDeleteLike Megan, I also loved your idea of "math that checks itself." This is a concept that I think could be embedded into a lot of math teaching, since so much math can be verified using other methods such as the inverse operations (using the answer value to find the question values) like in arithmetic, algebra, geometry, trigonometry, etc. Adding a step for checking the math could be a form of math-assisted self-evaluation, that applies additional mathematical thinking. In the case of the mathematical beverages, though, there's an added layer (pun intended) of being able to visually SEE the math and whether it's been done correctly. When errors have occurred, students may be able to visually decipher where or how the ratios may have gone wrong. This makes it more accessible and engaging for a wider range of learners.
I'm curious to try the golden ratio lemonade and how it might achieve a status of 'best'. Personally I prefer lemonade on the less sweet side, but I'm open to the wonders of mathematical perfection that the golden ratio can bring. When the author says, '3 parts lemon to 5 parts simple syrup', I wonder if that ratio would work out for both volume (ml) and weight (g). I generally prefer working with weights and scales since I find it to be easier and more accurate to measure out than measuring cups and spoons.
Your reflections on the mathematically correct laces are interesting. In his video, John Halton defines the 'best' lacings as the shortest lacings, and the strongest lacings. To be strong, they have to be 'tight' lacings, which means that each eyelet needs to connect to the opposite side. Does Terry Moore's method achieve one or both these two conditions?