Are you familiar with Notice & Note: Strategies for Close Reading? Here’s a link to Heinemann’s Notice & Note learning community, and here’s a sample PDF. I wonder whether our Standards for Mathematical Practice are similar to the Notice and Note literary signposts.
It’s not enough to just read a text. We want students to read for understanding and comprehension. The literary signposts help students with close reading of a literary text.
Similarly, it’s not enough to just explore math with dynamic graphs and geometry. We want students to explore for understanding and comprehension. The math practices help students learn how to interact with a mathematical problem or concept … and what to notice.
Last week, we explored dilations.
What do you need for a dilation?
A figure, a point (which we’ll call the center of dilation), and a number (which we’ll call the scale factor)
We used our dynamic geometry software to perform a dilation.
About what things might you be curious as you explore dilations?
(I thought of Kristin when I used the word curious.)
What happens when the center of dilation is inside the pre-image?
What happens when the center of dilation is on the pre-image? (on a side, on a vertex)
What happens when the scale factor is between 0 and 1?
What happens when the scale factor is negative?
How do the corresponding side lengths in the pre-image and image relate to each other?
I asked students to practice look for and express regularity in repeated reasoning as they explored the dilation. Do you know what it means to look for and express regularity in repeated reasoning?
Find a pattern.
Yes. Figure out what changes and what stays the same as you take a dynamic action on the dilation. Begin to make some generalizations about what you notice.
And don’t just notice, but actually note what you’re thinking.
The room got quiet as students noticed and noted their observations about dilations. I monitored student work both using Class Capture and walking around to see what students were noting.
(I promise I’ve tried to make it clear to students that dilation has 3 syllables and not 4 … but we do live in the South.)
Eventually, they shared some of their findings with their team, and then I selected a few to note their observations for the whole class.
BB showed us what happened when he perfomed a dilation with a scale factor of -1. He had noted that it was the same as rotating the pre-image 180˚ about the center of dilation.
SA talked with us about when the dilation would be a reduction. She had decided it wasn’t enough to say a scale factor less than 1 or a fractional scale factor but that we needed to say a scale factor between 0 and 1 or between -1 and 0.
FK showed us that when she drew a line connecting a pre-image point and its image, the line also contained the center of dilation.
PS noted that when the scale factor was 2, the length of the segment from the center of dilation to a pre-image point equaled the length of the segment from the pre-image point to its image.
When the scale factor was 3, the length of the segment from the center of dilation to a pre-image point equaled one-half the length of the segment from the pre-image point to its image.
We next determined a dilation and set of rigid motions would show that the two figures are similar.
Translate ∆DET using vector EY.
Rotate ∆D’E’T’ about Y using angle D’YA.
Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.
Then we looked at dilations in the coordinate plane. I knew that my students had some experience with this from middle school, and so I sent a Quick Poll to see what they remembered.
Due to the success on the first question, I changed it up a bit with the second question.
But I wonder now whether I should have started with the second question. If they could do the second question, doesn’t that tell me they can also do the first?
I’ve rearranged the polls to try that the next time I teach dilations.
We ended the lesson with a triangle that had been dilated. Where is the center of dilation?
And so the journey continues, with hope that noticing & noting will make a difference in what students learn and remember …