In our series on unpacking the 8 Mathematical Practices, we talked about what it means to reason quantitatively and abstractly. But now let’s get more practical. Here are 4 strategies to use in your classroom to develop this kind of thinking and reasoning in your students.
1. Use visual pattern tasks. When students write explicit formulas for the nth term of a visual pattern, they are naturally prompted to use algebraic symbols to represent concrete quantities in the figure. For example, consider the visual pattern below:
A student would likely use the arrangement of rows and columns to come up with the function f(n)=n(n+1)+1 to describe the number of small squares in the nth figure. *Each of the parts of this equation has meaning.* The number of squares in an array of n columns and n+1 rows is n(n+1). The extra square on the top right is then added to the total. Similarly, a student might write that f(n)=n^2 +n +1. While algebraically these two functions are equivalent, this new equation also represents the quantities in the figure. There is an n x n square. Then there is a row of n squares. Then there is the extra square on the right. Visual patterns are IDEAL for helping students connect the numbers and algebraic symbols in an equation to the quantities that they represent!
2. Start with contextual problems so students can reason about the quantities in the problem. Luckily the Math Medic curriculum is full of them. Even if you teach juniors and seniors, there’s value in pulling some Algebra 1 lessons if you want to work on this Mathematical Practice without also addressing a new content standard. Any EFFL lesson related to “representing situations with equations” is going to be great for helping students develop quantitative reasoning, like this one or this one.
This is also the approach we take when teaching students how to solve linear equations in Algebra 1 Unit 3. In Faruk’s Tutoring Business, Faruk is saving up for a pair of headphones that cost $180. He has $65 already saved up from his birthday and he gets paid $10 per hour for tutoring. Students write and solve the equation 180=65+10x where x represents the number of hours he tutors. But instead of “getting x by itself” by subtracting 65 from both sides and then dividing both sides by 10, students are asked to reason quantitatively about the situation, which leverages their intuition and sense making. Here are two scaffolded questions that help students make this jump:
How much money does Faruk still need to be able to purchase the headphones?
How many hours does Faruk need to tutor to earn that amount of money?
Once students develop this kind of reasoning, we can introduce them to a non-contextual equation like 127=70+3x and have them use the same kind of reasoning, but in a more abstract way. “In order to get to 127, the 3x has to contribute 57 to the 70 that is already there, so each individual x must contribute 19.” I think this is why reasoning quantitatively and reasoning abstractly are paired in Mathematical Practice #2!
3. Watch your language! When you’ve been doing certain problems for a long time, it can be easy to slip into some bad language habits. Consider when we teach systems of equations with a problem about adult and student tickets to a performance. A group attending the performance bought 2 adult tickets and 5 student tickets and paid $64. When we write the equation “2a+5s=64” we might read this outloud as “2 adults plus 5 students is $64” but this isn’t actually correct. The variable a isn’t just a label for adults. It needs to represent an unknown quantity, in this case the cost of an adult ticket. Ask your students to interpret “5s” and I bet the majority would say “5 student tickets” but that’s not right. “5s” represents the cost of 5 student tickets. When defining variables and when reading and interpreting parts of equations, we need to use precise language! Interpreting variables as labels rather than as quantities that vary is one of the key misconceptions students have when entering algebra–and we are partially to blame!
4. What does ____ represent in the context of this problem? This one is pretty straightforward. Math isn’t just about calculating answers, it’s about making meaning. This simple prompt can be added to almost every contextual question to provide opportunities for students to be reasoning quantitatively.
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