What is an algebra problem string? Have you heard about Number Strings or Number Talks? These are elementary school versions of what I call Problem Strings. I use them extensively in my Building Powerful Numeracy books. How can we use this cool technique in an algebra class? Why would we want to?
A problem string is a lesson format based on the idea that learning mathematics is about constructing relationships and connections. It's not the only thing that could happen in a math class, but I propose that it's an important part. See here for a brief description of problem strings in general. This post is the first in a series to introduce algebra problem strings.
Why algebra problem strings? Students answer related questions, the teacher models student thinking, students construct relationships and connections - in other words, algebra problem strings are a (relatively) short opportunity for mathematizing (structuring, schematizing). For teachers who have been looking for that balance between really cool, complicated, more open problems (think Dan Meyer's 3-act problems) and more directed, focused practice for efficiency, I invite you to consider problem strings.
The following is an example of an algebra problem string (series of algebra problems) that will take about 10-15 minutes to deliver in a typical algebra 1 or 8th grade class. Try it and tell me how it went in the comments!
Teacher: Let's do a problem string. The first problem today is "some number doubled is 20. In other words, twice a number is 20. What is that number? (The teacher writes 2n = 20 and pauses briefly.) That's easy, so I won't wait long here. Tom?
Tom: 10. (The teacher writes n = 10.)
Teacher: So, double 10 is 20. I'll just model that quickly like this.
Teacher: The next problem is 'some number doubled plus 4 is 20, or some number times 2 then add 4 is 20. What's the number?' (The teacher writes 2n + 4 = 20 and pauses, waiting for most students to signal that they have an answer.) What did you get Alyssa?
Alyssa: I got 8.
Teacher: Did anyone get anything different? No? Ok, Alyssa, how did you find 8? (The teacher doesn't cue students if the answer is correct or not.)
Alyssa: I subtracted 4 from both sides and got 2n = 16 and then divided both sides by 2 and got n is 8. (As Alyssa explains, the teachers models her thinking off to the side.)Teacher: Did anyone find that the number is 8 in a different way? Maybe by thinking about the 4 and the 20? (The teacher knows there is a different strategy at play and wants to bring it to the surface for the class to consider.)
John Michael: I just thought that something plus 4 is 20, so that has to be 16.
Teacher: So, you weren't thinking about doing anything to both sides, you were thinking about the relationship between a thing plus 4 being 20? I'll model your thinking like this. (The teacher models this thinking next to the previous strategy because the teacher wants students to consider both as viable strategies for solving this problem. As students make connections between strategies, they become more powerful problem solvers.)
John Michael: 16, and then twice the number is 16, so it's 8. (As the student talks, the teacher continues to model his thinking on the board.)Is John Michael's strategy important? Why model it? *See below for more about this.
The algebra problem string continues.
Teacher: You used different strategies, but you both found that the number is 8. Nice! The next problem in the string is 'twice some number plus 10 is 20, what is the number?' Another way to say that is 2 times a number plus 10 is 20. (The teacher writes 2n + 10 = 20 and pauses until most students signal they are ready.) Abby, what did you get?
Abby: The number is 5.
Teacher: You think the number is 5. (Writes n = 5.) Did anyone get anything different? No? Ok, Abby, how did you find the number?
Abby: I thought that 10 plus 10 is 20, so twice the number is 20.
Teacher: I hear that you knew that something plus 10 is 20 I'll model that like this. (The teacher models the thinking far off to the side so that it lines up with the similar strategy for the previous problem. One goal is to get as much of the problem string on the board at a time so that students can make connections between the problems.)
Teacher: And then since 10 plus 10 is 20, the missing value on the number line is 10.
Teacher: Abby's strategy was more like John Micheal's where he used a relationship. Did anyone use a strategy more like Alyssa's, by doing something to both sides of the equation? Zoe? (As the teacher models both strategies, students learn what their strategy can look like in mathematical notation. Also, students get the chance to consider other strategies.)
Zoe: Yeah, first subtracted 10 from sides to get 2n = 10. Then I divided both sides by 2 to get n = 5. (The teacher models this strategy as Zoe says it.)
Teacher: Why does this balance strategy work? Why are you doing things to both sides? Discuss that with your partner. (Students discuss briefly.) Why does this other strategy, on the right, work, when you use relationships? (Students discuss briefly.) We'll talk more about the connections between these strategies. First, let's do some more problems.
The string continues with the rest of the problems, one at a time, students solving, the teacher modeling student strategies, and the class discussing. Notice that some of the problems have fraction answers.
The teacher ends this string by asking students to reflect on the two main strategies that she modeled for these problems and generalize the strategies. Students partner talk and then share briefly. This is the first of many of the problem strings that the teacher will do over the next few weeks so there will be ample opportunity to flush out the ideas. You can see an example of what the board could look like at the end of the string here.
Is this how all problem strings go? Here is a basic layout for how many problem strings can be delivered:
- The teacher asks students to solve a problem.
- Students solve it any way they want to.
- The teacher asks for the solution, and if anyone got anything different.
- The teacher strategically asks student(s) to share their strategies.
- As students share their strategies, the teacher models the relationships the student used.
- This process repeats with the next question.
- During the string, there are strategic places where the teacher will ask for more or fewer strategies, choose to model certain strategies and not others, ask students to generalize a strategy that students are using or compare 2 strategies that are emerging or predict what kinds of problems are solved nicely with a particular strategy. The teacher will purposefully spend more time on some problems and less on others - creating a rhythm or cadence to the string.
Stay tuned for my next blog in this series. Each will detail a new problem string in a progression of problem strings toward solving more complex equations.
*In this problem string, the teacher models a strategy that has nothing to do with the typically taught "balance" strategy. Some students use the balance strategy and the teacher models it, but other students are using a different strategy and the teacher also models this alternative strategy, one in which students use the relationships between the numbers to solve the problem. Consider students who come to these problems with John Michael's strategy to solve 2n + 4 = 20 (he would subtract 4 from 20 to get 16) and then we tell them to subtract 4 from both sides of the equations. He isn't subtracting 4 from both sides! He is using the relationship something + 4 = 20, then something = 20 - 4.
If we don't acknowledge both strategies, we run the risk of sending the message that math is counter-intuitive, don't think in math, just memorize the teacher's method. By acknowledging both strategies while modeling the relationships, teachers give students the opportunity to construct both strategies and then we all can choose when to use the more helpful strategy. Powerful mathematizing does NOT mean using the same strategy all of the time. Powerful mathematizing means choosing the most efficient, sophisticated, even clever or elegant strategy for the particular problem. Let the problem dictate the strategy!
Hat tip to Catherine Twomey Fosnot and Martin Dolk for their Young Mathematicians at Work and Contexts for Learning work.