Tag Archives: Hannah’s Rectangle Problem

Notice & Note: Dilations

How do you give your students the opportunity to practice MP8: I can look for and express regularity in repeated reasoning?

SMP8 #LL2LU Gough-Wilson

We started our dilations unit practicing MP8, noticing and noting.


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What would you want students to notice and note?

How do students learn what is important to notice and note?

An important consideration when learning with self-explanation is to look at the quality of the explanation itself. What are the students saying or writing? Are they just regurgitating bits of text or making connections to underlying principles? Do the explanations contain predictions about what is going to happen, try to go beyond the given instruction or do they just superficially gloss over what is already there? Students who make principle-based, anticipative, or inference-containing explanations benefit the most from self-explaining. If students seem to be failing to make good explanations, one can try to give prompts with more assistance. In practice, this will likely take iteration by the instructor to figure out what combination of content, activity and prompt provides the most benefit to students. (Chiu & Chi, 2014, p. 99)

We had a brief discussion about what might be important to notice and note. We’ve also been working on predictions, thinking about what you expect to happen before trying it with technology:

What happens when the center of dilation is on the figure, outside the figure, and inside the figure?

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What happens when the scale factor is greater than 1? Equal to 1? Between 0 and 1? Less than 0?

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I observed, walking around the room and using Class Capture, selecting conversations for our whole class discussion.

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Here’s what NA noticed and noted.


We looked at Hannah’s Rectangle, from NCSM’s Congruence and Similarity PD Module. Students had a straightedge and piece of tracing paper.

Which rectangles are similar to rectangle a? Explain the method you used to decide.Hannahs Rectangle.png

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What would you do next? Would you show the correct responses? Or not?

Would you start with an incorrect answer? or a correct answer?

Would you regroup students based on their responses?

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I started with a student who didn’t select G and then one who did. Then I asked a student who selected C to share why he chose C and didn’t choose F. We ended by watching Randy’s explanation on the module video.

And so the journey continues, always wondering what comes next (and sometimes wondering what should have come first) …

Chiu, J.L, & Chi, M.T.H. (2014). Supporting self-explanation in the classroom. In V. A. Benassi, C. E. Overson, & C. M. Hakala (Eds.). Applying science of learning in education: Infusing psychological science into the curriculum. Retrieved from the Society for the Teaching of Psychology web site:


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Posted by on December 19, 2016 in Dilations, Geometry


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Are you aware of all that NCSM has to offer? Many resources are free on their website, and there are additional resources with a paid membership when you login to My NCSM.

I learned about their Illustrating the Standards for Mathematical Practice modules at the NCSM conference a few years ago. While I haven’t reviewed all of them, I’ve been using ideas from their Congruence & Similarity module ever since both with my students and in professional development with teachers.

One of my favorite tasks from the module is Hannah’s Rectangle Problem.

Students determine which rectangles are similar to rectangle a and explain how they know.


Tracing paper, scissors, and straightedges were available for students to use. (Although my straightedges really are rulers, I did ask my students to refrain from measuring.)

I set the class mode to individual and sent a Quick Poll to collect their results.

I think CS shared with the whole class first. (I’m writing the post two months after the lesson & can’t remember for sure.) CS had traced rectangle a & transformed the rectangle so that it shared a center with each rectangle.

I know that c, b, e, and g work because the space between the dilated rectangles has the same “width” all the way around. CS showed us what he meant, talking about the “frame” that forms.

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AC added that she did that, too, but she eyeballed the length between corresponding vertices to check for similarity.

Another student noted that h works because the two rectangles are congruent, which of course made us think about why similarity does not imply congruence but why congruence does imply similarity.

What about one that doesn’t work?

CS shared what he did with rectangle d, and showed how the horizontal width of the “frame” isn’t the same as its vertical width.

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Can we check whether one rectangle is a dilation of rectangle a using a different center?

At this point, we watched the video that accompanies the NCSM learning module, so that students could see Randy’s method. Had one of my students come up with Randy’s method, of course we wouldn’t have watched the video, but they didn’t, and so I was glad for a different source of learning other than me.

(Randy draws the diagonals of the rectangles from a shared vertex to show whether or not the rectangles are similar.)

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I love that the rectangles not only gave my students the opportunity to remember how we define similarity from grade 8:

CCSS-M 8.G.A.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

But also gave them the opportunity to practice one of our high school geometry standards as well:

CCSS-M G-SRT.A.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.

Students then dilated a figure using our dynamic geometry software and played. I watched using Class Capture.

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What happens when you move the center of dilation?

What happens when the scale factor is 3?

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What changes when the scale factor is –3?

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What is significant about a scale factor of –1?

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And students transferred what they had learned to paper:

Dilate ΔXYZ about point C by a scale factor of 2.


Dilate points A and B about point O by a scale factor of 3.


And so the journey continues, with many thanks to NCSM for all of the work they do to provide worthwhile resources to the mathematics education community.

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Posted by on January 10, 2015 in Dilations, Geometry


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