The Mathematics of Risk
Redwood Coast Montessori: Eureka, CA | Designing Teachers: Michelle Dobrowolski & Hart Ford

We make choices that balance risks and rewards everyday. Some of these choices are small. In a soccer game, you might choose how to wrestle the ball away from your opponent. Other choices are larger and deal with our health, relationships, the environment, and finances. When making these decisions, we weigh the potential risks and rewards and also consider the probability of different outcomes.

In The Mathematics of Risk students examined risk though a mathematical lens. First, they considered how slope affects the level of risk in real-life situations, such as on a slide, rollercoaster, or skate ramp. Then, throughout a Socratic Seminar, research, and iterative brainstorming, students identified a real-world decisions that force people to balance risks and rewards and that can be evaluated with probability and other statistical data. The questions at issue ran the gamut from “How do social factors and access to birth control affect teen pregnancy?” to “What should I do after high school?” to “What type of protein should I eat?” Students used the research to design a board game that used probability as well as research about their question at issue. The project culminated in a game day so that students could play their games with their families and students from another campus.

Academic Skills & Content:

CCSS.MATH.CONTENT.8.F.A.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

CCSS.MATH.CONTENT.7.SP.C.8: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

CCSS.MATH.CONTENT.7.SP.C.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

CCSS.MATH.CONTENT.7.SP.C.6:  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Suggested Duration: 9 weeks

Final Products: What the Students Made
Final Products: What the Students Made

Students created games that utilized risk-taking and probability in order to work. Here, you can see two examples of “Project in Progress” posters in which students provided an update on the game they were in the process of developing.

Core Practice 1: Differentiation
Core Practice 1: Differentiation

In differentiation, students are given a task that every student can access and that gives them freedom to push their thinking. This approach, which Jo Boaler refers to as low floor / high ceiling, finds a sweet spot between scaffolded instruction and open-ended inquiry. This is effective for a class of students who have varying levels of understanding.

The paper roller coaster is well-suited for differentiation. Every student, regardless of math skill, is able to build a linear slope and describe it as rise over run. For some students, this was a challenge that pushed their mathematical thinking. Some students incorporated features that were not required, like loops and curves. Others took it to a higher level of thinking by including exponential and quadratic slopes. Others constructed mega coasters during their extra work time. During the process, all students engaged in mathematical thinking at the level that met their needs.

Core Practice 2: Student Choice
Core Practice 2: Student Choice

Having a discussion about risk and the thought processes that go into life decisions is an important step toward the final product. Students participated in a socratic seminar prompted by a case study. In the case study, an adolescent boy asks his parents if he can go on a ski trip, but is vague on key details. Students discussed how each family member in the case study had experiences that formed different opinions that, ultimately, affected their comfort level of risk. Students asked questions that would help the family better assess the risks of the trip, such as the character of the chaperone, the probability of an avalanche, and the boy’s skill level.

After the seminar, students choose an example of a risk based on interest or self-discovery. They begin to analyze the risk by weighing risks and rewards and by collecting probability and other statistical data. Because students choose topics of personal, direct interest, they were motivated to collect as much information as possible. Max, a 10th grade math student, used this project to answer the question: “What should I do after high school?” Obviously, he was highly motivated to research the costs of different education options, salary ranges for different professions, and the effect of education on total lifetime earnings. Other students explored topics that looked at the risk vs. reward with with drugs or alcohol experimentation or driving a car. Ashton, an 8th grade student, wanted to explore head injuries associated with sports like soccer. Supporting choice gives students permission to pursue questions that are relevant to their passions and futures.

Core Practice 3: Critique of Exemplars
Core Practice 3: Critique of Exemplars

Critiquing examples was a crucial step before students began to design and build their own games. Students and staff brought in a variety of games based on probability. Some were more sophisticated strategy games, such as Settlers of Catan, Risk, Forbidden Island, Above & Below, and Pandemic. Others were very simple, straightforward games, such as Life and Exploding Kittens. Through the critique, students identified elements of mechanics and design that made each game effective and also identified how the games used probability. In a whole group discussion, we generated a list of common elements and also highlighted ideas for making probability more complex. In Catan, for example, outcomes are based on the sum of two dice, making some values more probable than others. In Above & Below and Dungeons & Dragons, outcomes are based on the dice rolls as well as character bonuses.

Students borrowed and adapted exemplary strategies, mechanics, and designs for their own games. Their own games became more complex and interesting because they evaluated their work against the work of professionals.

As icing on the cake, the critique of exemplars fosters equity. As in any classroom, some students had a great deal of prior experience with the content, while others had none. Together, students shared a common experience and built background knowledge.


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