Tag Archives: ask don’t tell

Hinge Questions: Dilations

Students noticed and noted.

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I wanted to be sure that they could answer a dilations question based on their observations. I had two questions premade in my set of Quick Polls. Which question would you ask?

In the past, I would have asked both questions without thinking.

I am learning, though, to think more about which questions I ask. If we only have time to ask a few questions, which questions are worth asking?

From slide 34 in Dylan Wiliam’s presentation at the SSAT 18th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

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I decided to send the second poll. I decided that if they get that one right, they can both dilate a point about the origin and pay attention to whether they are given the image or pre-image. If I had sent the second poll, I wouldn’t know whether they could both do and undo a dilation.

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Next we looked at this question.

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Students worked on paper first.

Then some explored with technology.

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What do you want your students to know about the relationships in the diagram?

What question would you ask to see whether they did?

I asked this question to see what my students were thinking.

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And so the journey to write and ask and share and revise hinge questions continues …


Posted by on December 20, 2016 in Coordinate Geometry, Dilations, Geometry


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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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The Magic Octagon – Dan’s, Andrew’s, and mine

I had saved Andrew’s post in my folder for a recent lesson, which was about Dan’s video.

We paused halfway in, and students decided where it would be. They answered a Quick Poll to let me know, and by the time they had all answered, some had changed their minds.


We quickly looked at the responses, and they decided using time would be easier to decipher than some of the other descriptions.

I sent a second poll. I waited for everyone to answer, even the ones who wanted to take their time thinking about it.


And then we continued to watch.

We paused for the last question, they discussed with their team, and then we finished watching.

Good conversation. But we didn’t get to the sequel proposed by one of Andrew’s students: If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too? Why or why not?

So I emailed that question to my students.

  • Yes, the two points move like opposite hands on a clock moving closer to each other and overlapping at 5:00. At about 11:00 they would overlap again. Otherwise, there is no overlap.
  • They would be at 5:00. This is because when he flips the magic octagon, the back arrow also flips, causing the new time to be 3:00 instead of 9:00. This means that if you were to find a line of reflection, you could flip the octagon on that line and the arrow would always land right where the previous one did. If this was on transparent paper, you can see that if one arrow points to 5:00, then the other one would be pointing at 7:00. But if you were to flip the octagon on the reflection line which intersects 12:00 and 6:00, then you would continuously get 5:00 because of the reflection.

As I got the responses from students, I realized that I wished I had asked a different question. While I did include why or why not, and it was obvious from the responses that students didn’t just answer yes or no, I wish I had asked “At what time(s), if any, are the front side and back side arrows at the same time?”

I am reminded of something I can no longer find that I read in a book. A group of teachers observed a “master” teacher for a lesson and then went back to their own classrooms to teach the lesson. The teachers asked the same questions that the master teacher asked; however, the lessons didn’t go as hoped. The teachers were not asking questions based on what was happening in their own classrooms; they were asking questions based on what had happened in the other classroom.

I love reading blog posts and learning from so many mathematics educators. They give me ideas that I wouldn’t have on my own. In fact, as my classroom moved toward more asking and less telling, I used to say that my most important work happened before the lesson, collaborating with other teachers and deciding what questions to ask. I’ve decided otherwise, though. My most important work happens in the moment, not just asking, but also listening. And then, if needed, adjusting what I planned to ask next based on the responses of the students in my care. And so the journey will always continue …


Posted by on November 15, 2016 in Geometry, Rigid Motions


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MP5: The Traveling Point

How do you give students the opportunity to practice “I can use appropriate tools strategically”?

When we have a new type of problem to think about, I am learning to have students give their best guess of the solution first. I’ve written about The Traveling Point before.

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Students sketched the path of point A. How far does A travel?

Students used paper and polydrons, their hands and string.

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I sent a poll to find out what they were thinking about the distance traveled.

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Students then interacted with dynamic geometry software. Does seeing the figure dynamically move help you better see the path?

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Does seeing the path help you calculate how far A travels?

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And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

And the journey for my own learning continues. Thanks to Howard for correcting me. The second two moves do not travel a distance of 6, but the length of the circumference of the quarter circle.

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One student figured that out by the time the bell rang.

I look forward to redeeming this lesson this year, as the journey continues …


Posted by on August 23, 2016 in Geometric Measure & Dimension, Geometry


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Properties of Definite Integrals

I love the day in calculus that students figure out properties of a definite integral using the Calculus Nspired activity Definite Integral. When I first started using Math Nspired activities, I often copied the student handout. I’ve used them long enough now that there are only a few that I still copy. Instead, I will show some of the questions from the student handout at the front of the room as a guide for student exploration, or I will just use the questions myself to guide our whole class discussion.

We practice look for and express regularity in repeated reasoning. What changes and you move a and b? What stays the same?

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Students explore individually first and write what they notice. While they share with a partner, I monitor their conversation, select what they notice that needs to be brought to the attention of the whole class, and sequence their findings in our whole class discussion.

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The functions in the activity are carefully chosen – one is odd, and one is even – so that students can begin to generalize what will be true for definite integrals of odd and even functions as well as all functions.


I used to tell my students properties of definite integrals.

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Now they tell me.

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(At least those who notice the word NOT.)

And so the journey continues, providing students opportunities to figure out the mathematics we want them to know without just telling it to them.




Posted by on February 10, 2015 in Calculus


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