Week10: Reading Andrea Hawksley (Bridges 2015) Exploring ratios and sequences with mathematically layered beverages

 

Summary:

The given paper was a part of Bridges 2015 where the author Andrea Johanna Hawksley describes the workshop which explores the mathematical concepts of ratios, fractions, and integer sequences through the creation of layered beverages. By varying the ratios of ingredients like sugar, flavorings, and water in each layer, participants can visually and experientially understand mathematical principles. The workshop is divided into two main parts: exploring ratios with simple two-layered beverages and delving into integer sequences with more complex, multi-layered drinks.

In the first part, participants learn about ratios and fractions by creating beverages with different sweetness levels in each layer. This involves understanding how changing the relative unit volumes between layers affects the overall density and sweetness ratio of the beverage. The workshop encourages hands-on calculations and taste testing to reinforce learning.

The second part focuses on exploring integer sequences through layered beverages with many layers. Participants use sequences like arithmetic progressions or the Fibonacci sequence to determine the proportions of ingredients in each layer, creating drinks with increasing or decreasing sweetness levels. By experimenting with different sequences, participants can observe how the mathematical relationships translate into flavor experiences.

The construction of layered beverages involves following specific steps to ensure each layer is poured correctly onto the ice to maintain separation and visual appeal. The workshop provides examples, such as Fibonacci lemonade, with precise ingredient proportions for each layer. Participants can also explore other sequences like Lucas's numbers. In all such sequences, the ratio between adjacent numbers approaches the golden ratio. The workshop demonstrated how mathematical concepts are inherent in everyday activities and it can inspire others by the basic ideas of teaching math through layered drinks and try creating other mathematical foods.  

Stops:

1. When a large number of layers are used, the proportion of flavorings can increase according to various monotonic integer sequences, for example, the Fibonacci sequence. (p.519)

The above quote expands the possibility of trying different learning activities like using the Fibonacci sequence in baking by using the chocolate proportion in the proportion of the Fibonacci sequence. Using the Fibonacci sequence in this way can lead to an interesting progression of flavors, as the amount of flavoring increases more rapidly with each layer. This concept demonstrates how mathematical sequences can be applied creatively in cooking to achieve different taste experiences based on the number of layers or servings. Students can work with different recipes and not only engage in the hands-on learning experience but can learn the skill of cooking and baking too. 

2. Fractions can seem illogical and hard to conceptualize. This workshop gives a fun way for students to practice calculating fractions in an unorthodox setting using a sense seldom used in mathematics classrooms. (p. 523)

I agree with the statement that fractions sometimes feel illogical and hard to conceptualize. One interesting exercise I used to do with the students is where I used to ask students to bring their favorite flavored drinks and determine the amount of sugar in each. Students used to be tasked with determining the amount of sugar in one serving of their drink and then figuring out the total grams of sugar in the entire bottle. Initially, let's say they found that one serving contains 28 grams of sugar. To determine the total sugar content in the bottle, students must first recognize that the bottle contains 14 ounces, approximately equivalent to 414 mL, with "about 2" servings per container. Students may employ various methods such as doubling the serving amount, calculating sugar per ounce, or using ratios to scale up the sugar content from one serving to the entire bottle. By working independently or in groups and sharing their approaches, students engage in critical thinking and mathematical problem-solving skills while understanding the importance of accurately interpreting nutritional labels and serving sizes. 

in the future, I am eager to use beverage activities with my students to explore and teach integer sequences and ratios to my students.

Question: 

Could you share any hands-on activities you've employed in teaching or experienced as a student that relate to ratios and sequences?