Informal to Formal: Building Confidence

I teach a math support class (called Math Studio in my school) which is a second period of math for a small group of students.  For me, there is a magic one can create in this class.  With just the right amount of preview and scaffolding in math support, students arrive in regular math class later in the day ready to shine.  And many have never felt that shine in math class before, so it’s incredibly rewarding for me to watch and for them to experience.

Next week we begin our 8th grade unit on transformational geometry and today in Math Studio we developed an informal understanding of various transformations, going especially deeper on translations.

First, students did Shifting Shapes, a relatively new Desmos activity which provides a visual and conceptual look into rigid and non rigid motion.  They are given a square with draggable vertices and asked to make their own shape.  In subsequent slides their shape undergoes various transformations which they are asked to describe in words.

Here is an example of one student’s shape (original in grey).

I so love both the informal and formal language they used to describe it:

“It went woop and to the side”

“It did a backflip”

“My shape rotated”

“It was turned sideways.”

Here is a second transformation (from a subsequent slide–again grey is original and red is after the transformations)

Students descriptions are again quite elegant:

“It went smaller then to the left”

“It got small then ran away”

“The shape got smaller then moved to the side”

I have to admit, I LOVED the idea that a translation is when a shape runs away from itself.  I ended up naming this idea after the student who said it and for the rest of the period, when we talked about translations, we talked about it being a slide, but also it running away from itself.  I’m smiling thinking about this idea as I type this.

While Shifting Shapes was a great, quick way to get some informal language out about transformations, I then went deeper into the mechanics of translations.  Again, I wanted to spend time looking at them informally first.

Using GeoGebra, I drew a polygon and a vector. Screen Shot 2019-08-29 at 10.55.17 AM Students then talked about what they saw and made predictions of what the arrow on the vector meant.  I explained that a translation was the same as the shape ‘running away” and the vector showed the direction and distance that it would ‘run’ (or slide).

Screen Shot 2019-08-29 at 3.51.29 PMWith this drawing on my LCD, I gave students 2-3 minutes to go to a vertical white board, draw the original shape, then draw their prediction of where the translated image would be.

They then came back to their seats where I showed in GeoGebra what the translated figure landed, and we discussed their drawings.  At first, their drawings had the translated image in the right general area, but they were considering only the direction of the vector, not distance.  I had them predict where the end of the vector would be if I moved the beginning of it to A.  I modeled this on several vertices as we discussed, like this:

We did this back and forth between vertical white boards and watching GeoGebra about 5 times with different figures and vectors.  By the end, maybe 15 minutes later, nearly every student could accurately draw the translation, label its vertices with A’, B’, etc, and explain whether the translated figure would overlap the original one or not.

I am hoping this deep conceptual dive, rich with visual models and lots of time to both practice and refine ideas will provide the scaffolding my Math Studio students need to feel confident and skilled when we cover this concepts in 8th Grade math beginning on Tuesday.  By no means will I eliminate the visual models in my regular math classes, but the reality is that we’ll move to the informal a bit more quickly and with a few less examples.  This blend of GeoGebra and Desmos within a single lesson felt seamless to me as the 8th grade Common Core transitional geometry work blends so well with the strengths of each of these tech tools.

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