# Engaging students: Ratios and rates of change

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Kelly Bui. Her topic, from Algebra: ratios and rates of change.

How could you as a teacher create an activity or project that involves your topic?

The activity I created would involve having the entire class make Rice Krispies treats as either groups or partners. The recipe I linked below calls for 6 cups of Rice Krispies, but for the sake of the activity, each table will receive 1 ½ cups of the cereal. Every table will receive the original recipe and determine how many large marshmallows they will need and how many tablespoons of butter they will need for the recipe to be modified to using only 1 ½ cups of cereal. This activity will allow students to use the ratios to convert measurements, such as 40 marshmallows / 6 cups of Rice Krispies. After a group finishes their calculations and finds the ratio of each ingredient in respect to the amount of cereal, they can begin making their Rice Krispies treats. To extend this to a project, the Rice Krispies Treats activity can be done in class and students will be assigned to find a recipe which involves either using the ratio to create a smaller or larger serving of the recipe.

How can this topic be used in your students’ future courses in mathematics or sciences?

Ratios are introduced in middle school when we compare a part to another other part or part to whole. Students also begin to grasp that 12 inches / 1 foot is a relationship between two quantities because there are 12 inches per foot. We also see the use of ratios in high school chemistry when converting units. A simple ratio we first learn is that density is the ratio of mass to volume. This can then be extended, for example, when students begin to solve for the number of moles of an element given its mass in grams. Before teaching a chemistry class that 1 mole = 6.022×10^23, instructors could begin with simple conversions of the length of a state in miles and converting that length into inches. Once students understand the process and the concept that we are taking one unit and converting it to another unit, it will be easier to apply it to more complex situations in chemistry.

As a class, to get into the process of using ratios to convert units, the students can make their own conversion ratios with different objects to model this relationship. For instance, 4 fire extinguishers are the length 1 lab table and 8 lab tables are the length 1 school bus, and based on these ratios, students must find the length of a school bus in terms of fire extinguishers. This activity will allow the students to use objects they see every day and create a relationship among them.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Shark Tank is a show which involves 5 or 6 sharks (investors) and entrepreneurs that go into the “shark tank” to pitch their ideas seeking one or more partners who will invest in their business. Most entrepreneurs seek a money amount for an amount of stake in their company or business. If the entrepreneurs are lucky, they will get a deal with one or more of the sharks. In the video below, Aaron Krause pitches his product, the “Scrub Daddy” in which he asks for a \$100,000 investment for a 10% equity in his company. We see the topic of ratios appear in this business-related show because 10% equity of \$100,000 means he values his company at \$1 million, in other words our ratio is 10% / \$100,000. This ratio can be used to find the value of the company at 100%. In addition, the sharks also like to know the breakdown of the cost per unit. In this video, Mr. Krause states that it takes \$1.00 to create a scrub daddy and he sells it for \$2.80 wholesale. This gives the sharks the knowledge of how much they would earn for 1 Scrub Daddy. Given the sharks are willing to negotiate, like in the video, Lori gets 20% equity of the company. For each \$2.80 / 1 Scrub Daddy, she will earn \$0.56.

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