I ended my last blog post wishing that my students had the chance to solve equations creatively. I think I have found a structure (heavily influenced by Chapter 5 of the book Thinking Mathematically) that will give students this opportunity.
I am going to call these open equations. Here’s how they work:
I project an equation, such as a + 5 = b + 8, on the board.
I ask students to spend a few minutes finding at least three sets of values for a and b that make this equation true.
Then I give the students an additional few minutes to compare their answers and look for any patterns in the responses.
Then I lead a number talk with the whole class as we discuss possible solutions and any patterns that emerge.
I like this format for several reasons:
- The right side of the equation is an expression instead of a numerical value. This is intentional. I want students to stop thinking of the equals sign as the place where the answer goes and start thinking of it as a symbol that connects two equivalent expressions. I want kids to think “Whatever I make a and b, I need to make sure that a + 5 has the same value as b + 8."
- The fact that this is a two-variable equation gets kids away from finding the answer and towards finding some answers. I think this is an important distinction because it leads kids to look for patterns within their answers, something that students rarely do when solving a problem like 2x + 3 = 15. In this problem, students might think “Wow, it looks like a is always 3 greater than b. Why is that?"
- This practice of finding pairs of values that make the solution true will be quite valuable when students begin learning about functions. Graphing a linear function just becomes another way of representing all the possible solutions for an equation. Which is exactly what the graph of a function is!
- In this format, the way students look at the expressions a + 5 and b + 8 is different than the way that students often look at the expression 2x + 3 in the problem 2x + 3 = 15. In the latter problem, students often view that expression as a series of steps that need to be undone. They go operation hunting in order to solve the equation without considering that 2x + 3 is an object in and of itself. I want kids to think of a + 5 and 2x + 3 as things. That’s how I view them and how I think other fluent algebraic thinkers see them as well. Fluent readers see sentences as things as well as seeing each component of a sentence on its own. Fluent writers know how they can manipulate the components of a sentence and still maintain its original meaning. I want my students to be fluent algebraic thinkers. I want them to see the forest and the trees.
I’ve given one simple equation above, but there are a lot of other examples that I think could help students with any number of math topics.
Are students simplifying their expressions incorrectly? Give them the open equation a - 7 + 10 = b and ask students why b is always 3 greater than a.
Are students having trouble combining like terms? Give them an open equation like a + a + a + a = b + b + b. Students will pretty quickly move from adding to multiplying a by 4 and b by 3. If you want, there is a bonus lesson about proportionality in this equation!
Want to explore the distributive property? Give students the open equation 2(a + 5) = 2b + 10 and ask them to find three pairs of answers. Why are the values of a and b always the same? Could I rewrite the equation as 2(a + 5) = 2a + 10? Can any students write another open equation where a and b are always equal?
Most importantly, I think that this format becomes more successful the more frequently it is used. I use them as warm-ups and like to think of it as a slow-motion number string, but you could work through several equations in a row if the need arises.
I have many, many more ideas for how open equations can be used as a gateway to solving equations. And I will be writing a lot more about those ideas and fleshing them out online and in my classroom.
But the purpose of open equations is not to get students to be better at solving equations. The purpose of open equations is to get students to understand equations. Once they understand how equations relate two equivalent quantities, then they can build their own strategies to solve problems like 2x + 3 = 15. And when we move from informal to formal strategies, they will have a basis for understanding those formal strategies and why they work.
Let me know what you think!