Welcome back to our final installment of our Critical Math introduction. Critical thinking and number sense don’t just apply to younger math students, but to older ones as well! Once the beginnings of number sense are achieved (counting, grouping, fractions, proportional thinking, percents, algebra) then students are ready to build the later stages of number sense. These later stages begin at arithmetic and then build out into the core math topics such as algebra, geometry, trigonometry, and calculus.
At these later stages of math, number sense and critical thinking are paramount to success in school, as well as understanding how the math applies to the world around us. A lack of either can quickly inhibit a high school student’s ability to focus or keep interest in math. Hence it’s good to stretch those thinking muscles with practice problems that are a bit outside the normal curriculum wheelhouse. These types of problems are the same kind a student might find on the SAT or ACT, because they call into question not just whether or not a student can perform the math, but if they can know when and how to use it. Try out the example below.
A boat can travel 30 miles upstream in 2 hours, and 30 miles downstream in 1 hour. What is the speed of the river’s current, and what is the speed of the boat in still water?
Answer: This problem requires students to understand two main topics: rate-time-distance problems, and systems of equations. Since the boat has a consistent speed, which we will call X, and it is either hindered or aided by the river current, whose speed we will call R, we can set up two equations (a system) based on the information we already know.
To travel 30 miles upstream (against the river), it takes the boat 2 hours. So a distance-equals-rate-times-time equation for that scenario would look like 30 = (X-R) x 2. For the return trip, where the boat is aided by the river, the equation would be 30 = (X+R) x 1, because that trip only takes 1 hour. Between the two of these equations, using either substitution or elimination methods, we can solve for a single variable R (or X, depending on which method you employed). Using substitution, we can set (X-R) x 2 equal to (X+R) x 1, since they are both equal to 30. Using algebra, we can solve this new equation for X, giving us X = 3R. We can then substitute this new value 3R for X in one of our original equations, allowing us to solve for a numerical value of R (7.5 miles per hour).
Then, we can use that value to solve our X = 3R to solve for the numerical value of X (22.5 miles per hour). So, the boat travels at a speed of 22.5 mph in still water (where it is neither aided nor hindered by the river). The speed of the river is 7.5 mph, as we found with R.
A follow up question could ask the speed of the boat travelling up- or downstream, which we would extrapolate to be the numerical value of X-R and X+R respectively, which are 15 mph and 30 mph. Much of these questions are left to the student to understand from context clues and the bare minimum information required to solve. But, with enough practice and exposure to both number sense and critical thinking, students can begin to see through problems with ease, and find conceptual solutions quickly so they can focus on the math!
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