It is week three of #MTBoS Blogging Initiative and the focus is on questioning.
Last week I visited a 5th grade classroom in my district and had the opportunity to sit down with the children and talk to them about their work. As a former Algebra I & II educator, I was very much at home, up to my elbows in visual patterns, input-output tables, and graphs using Desmos.
In grade 5, the learners interact with multiplicative and additive patterns (y=ax and y=x+a).
Though I appreciate teachable moments to support learners as they inquire about the math beyond the scope of their current class, I am always hesitant for them to jump to the abstract before they have a grasp of the concrete, pictorial, tabular, and graphical representations. This is where the questioning came in. At this table, shown in this picture below, you can see the pattern growing (with a hexagon in the middle and triangles added each stage).
In the picture you can also see the input-output table. When I first sat down with the group, their x-column and y-column was complete (indicating the stage number and total number of shapes used to create the image). The children were attempting to use guess-and-check to complete the process column. Though eventually they arrived at the correct expression for the relationship, this method is not efficient nor transferable. In an attempt to direct them back to the visual pattern, I asked:
How is the 6 from your expression shown in the pattern of shapes?
After a long silence, the response was not what I expected.
It was just random luck.
So, I directed them to use words to describe the pattern in the shapes. Stage 1 has a hexagon and 6 triangles. Stage 2 has a hexagon the 6 triangles from before and 6 more. Stage 3 has a hexagon, 6 triangles, 6 triangles, and 6 more triangles. Then came the ah, ha! And one child said it:
The repeated addition of the 6 is multiplication. 6x means adding 6 each time!
What a response! But, what about the 1? Remember, this is beyond the scope of this class. But, since the students created their own visual patterns, there was not safety net to keep them from creating a pattern of the form y=ax+b.
Next, I asked:
If this is Stage 3, this is Stage 2, and this is Stage 1 (pointing to the visual pattern), what would Stage 0 look like?
There would be 1 hexagon. 1! That’s where the one came from!
So what about the graph? They entered the table and equation into Desmos and again, without hesitation, announced proudly, the 1 is where the graph starts! Yes – the initial condition, the y-intercept, yay! And if 1 is where the graph starts, then that must be where the pattern started. In no time we were talking about the coordinate points from the table on the graph.
And that is why I love Algebra. Oh, and that these children in elementary school were using Desmos and airdropping images on their iPads of the visual patterns they created.