… or the easiest routine you’re probably already using in your classroom!
I love using the Same and Different routine with a variety of students, for a variety of reasons. It’s easy, it’s versatile, and it’s powerful learning and discussion. I think teachers do this naturally, and we can find even more success when we do it more intentionally.
First, the basics: what do I mean by Same and Different? I generally do this with the same basic structure every time:
- Post a simple prompt: two equations/graphs/tables/diagrams etc. next to each other with the questions, “What is the same?” and “What is different?”
- Give students 2-3 minutes of individual time to write down as many similarities and differences as possible.
- Team brainstorm: individuals take turns sharing with their teams and together they make a collective list.
- I vary the whole class share out: sometimes I’ll have them post their list on poster board and do a gallery walk; more often we’ll make a whole class list taking one same from every team, then one different (repeat as necessary)
I try to leave tasks open enough so I can see what students come up with, but usually I have a particular concept that I want to highlight. Here are some of my instructional goals when I use the Same and Different routine.
Intervention to correct a key misunderstanding
This is fresh on my brain because this was the purpose for a Same and Different routine I just did on Monday with my Math Lab classes (support class for struggling math students). In reviewing a recent assessment with one of our amazing Math 1 teachers, we noticed that students were using interval notation to describe the domains of both discrete and continuous functions. We designed this Same and Different task as an intervention:
We’ve been working on key features of functions and they identified both graphs as having the same intercepts, same rate of change, same max and min; all teams said that the first graph is discrete and the second is continuous. I was so proud of them using great vocabulary! We also had a little preview of their next unit: one student asked if the first graph is still linear because there’s not a line going through the points, and another student said, “I think yes because they both have a constant rate of change.” Wooooooo!
Another note: I love Same and Different because everyone can access a task. One student (who is enrolled in a below Math 1 level course) said “the first graph has just dots, but the second has a straight line connecting.” Her teammate (in Math 1, normally does not feel confident to talk in math class) said, “Oh that means the first graph is discrete and the second is continuous.” 😍
Some teams said both graphs have the same domain and range, which is exactly the conversation I wanted to have. One student said (I’m paraphrasing), “The second graph is continuous, so we have to include all of the inputs in between the whole numbers, like -3.5 and 0.2. But the first graph is just the separate points, nothing in between, so we have to write the domain and range as separate points.” Light bulbs going off everywhere, and on my follow-up exit slip, almost every student correctly identified the domain and range of one continuous and one discrete graph.
Highlight key differences to prevent common mistakes
In addition to intervening to correct mistakes, I love Same and Different for preventing mistakes by highlighting key differences between very similar-looking equations. This Same and Different task was a favorite that I did last year with my Precalculus students:
This generated a lot of discussion between students. To set the stage: we had spent a day exploring polar graphs in general, then a day exploring limaçons specifically. This was their opener on day 3. (Their ideas, I just scribed on the board; stars indicated ideas they weren’t confident with yet that they wanted to verify with Desmos)
These equations look similar but their graphs are very different, and it’s easy for students to confuse them; this task helped students to walk through those similarities and differences so they would be able to recognize and produce graphs of limaçons in the future.
Again, this task is accessible; any student can at least identify that the equations have the same numbers, same structure, the numbers are switched, and one is sine, one is cosine. These are all very important characteristics to notice, and I can guide them to see how they connect to the key features of the graph.
The task below didn’t generate as much conversation, but allowed students to see two very similar-looking conditional probability statements are very different:
Here’s another one I loved doing with my Math 2 classes:
This one was really great. Whenever we’re learning about parabolas, I want to make sure that students don’t assume that every graph they create will be a parabola. That “little 2” that every student can notice makes a big difference! We talked about key features and transformations and realized that maybe these graphs are more alike than they are different, which brings me to my next topic…
Highlight key connections between concepts/procedures
In that last task, students realized that all of the transformations we had done in Math 1 with linear and exponential graphs, still applied to quadratic (and later, absolute value) graphs! By making this connection, I did not “teach” nearly as much of graphing quadratics; we thought in terms of transformations and key features of the parent function.
How powerful is this? I gave the following task to my students, let them notice the similarities and differences, and suddenly I saved 2+ days of “teaching” operations with complex numbers that I can now use to focus on cool patterns and deeper problems.
Not all Same and Different problems have to look like this, nor do they have to revolve around whole class discussion. Here’s a homework task that I used in my Enhanced Math 2 class:
Guess what? By making the connection between these four nearly identical equations, solving equations with absolute values and square roots became just one more tool in their toolbox, rather than an entirely different idea to learn.
Activate prior knowledge
As seen in several tasks above, we can create greater coherence between topics by making connections to prior knowledge. Here’s another task I used to activate prior knowledge at the beginning of our sequences and series unit in Precalculus:
Students have so much knowledge already; we just have to figure out what it is and how we use it to build new ideas.
What’s so great about this strategy?
I love Same and Different for so many reasons. Comparing and contrasting is a key cross-curricular skill and an important life skill. More importantly IMO, the prompt is easily accessible by all students. Anyone can look at two graphs and say one is just dots while the other has a line through, or the numbers are switched, or there’s a little 2 on that one. They don’t have to have the right vocabulary or some profound observation; that’s where we as teachers come in, to build that vocabulary or guide that important connection.
Also – they’re so easy to make up! What is a mistake you want to correct? An idea you want to highlight? Try making a Same and Different task and share it on Twitter: #samediffmath. Tag me too (@kristiedonavan); I’d love to see what you come up with!