3 Tips for Creating a Powerful Same/Different Prompt
I have been all-in on the Same/Different discourse structure since it was shared by Brian Bushart of Round Rock ISD. Since observing and conceptualizing how it streamlines the conversation in the vein of Which One Doesn't Belong? and Estimation 180, I sought to jump in early on the creation process even as colleagues would destructively critique them in earshot: "I REALLY don't like the ones that just have methods and no visuals."
There's still a lot to analyze because the prompt continues to grow in application and purpose, especially at the middle school level and up. However, the engagement potential remains just as high as I demonstrated on Do The Math a few weeks ago.
These are 4th-6th graders deconstructing the process of "dividing across" with fractions using common denominators, and the capacity to see this using Same/Different was impressive.
After presenting the topic at the MidSchoolMath National Conference in Santa Fe, I wanted to put forward a triad of tips for the design process to ensure that the prompt you present is dynamic and accessible.
1. Know What You Want Your Students to Connect
In the Dividing Fractions prompt from the show, I wanted students to see that the process becomes simpler when the same expression is converted to common denominators. I did not impose an equal sign onto the expression because I initially wanted students to recognize that both expressions were equivalent, since the fractions in both had the same value. Students would then associate that since the fractions were equivalent, then logically their quotients must be equivalent. If that is the case, what do I notice about how I solve both of these? When students identified that the first set "Keep-Changed-Flipped" to 12/10, they noticed that those same values appeared as the numerators in the second set, then connected that 12/10 is a fractional representation of "12 divided by 10" and can divide straight across. The realization that when dividing common denominators across will always lead to a denominator of 1 then becomes a beautiful "light bulb moment."
2. The Two Don't Have to Be "The Same"
I created this graphing prompt using the Desmos graphing calculator to highlight some minor differences when graphing absolute value functions. The one on the left is the function y = |x-3|, while the right represents y = |x|-3.
While both functions have similar components, the functions themselves are fundamentally different, which is a key discovery to arrive at for students. Impressively, as is the case with many Same/Different prompts, students also identify their own observations, like the 8th grader who identified that the two overlap when x ≥ 3.
3. The Connection Should NOT Be Immediately Apparent
I get that onions are a nice metaphor for things that have layers, but the thing with onions is that the deeper layers don't necessarily trigger a positive reaction! I will say that a great Same/Different prompt is like a 3D Magic Eye picture, in that it's confusing to see at first (and may cause a minor strain!), but as you keep examining it things begin to pop out at you!
In this example, it may be clear that the two will have the same solution since they involve the same operation and values. However, the way in which the expanded form is presented and misaligned from place value requires the student to re-evaluate what they know to be true about subtraction algorithms. I have heard several students claim that because place values are not lined up vertically, the first version is by nature "WRONG." It is only after revisiting it and breaking down each smaller piece do they recognize that the calculations involved are much more straightforward and efficient in this case, as opposed to the standard algorithm.
-DR