Tag Archives: look for and express regularity in repeated reasoning

Blending Technology with Paper and Pencil

My geometry class is 1:1 this year; each student has her own MacBook Air. Students share responses to questions digitally in class using TI-Nspire Navigator for Networked Computers. Students explore mathematics using TI-Nspire dynamic graphs and geometry software. Students explore mathematics and share responses digitally using Demos Activity Builder. We use Canvas, an online learning management system, for assignments. We use Google Drive for sharing electronic documents with each other, and we use MathXL, online homework with built-in learning help, to practice mathematics. What place does pencil and paper have in my students’ learning and understanding of mathematics?

Even though many of the tasks that my students do for geometry take place digitally, I am convinced that pencil and paper plays an important role in how much mathematics my students not only learn but also remember. In a Wall Street Journal article, “Can Handwriting Make You Smarter?“, Robert Lee Hotz reports that students who take notes by hand usually outperform students who type notes when assessed more than one day after the class period. Students who type notes quickly type everything the professor says, but students who handwrite notes have to process the information while they are hearing it to select what is important to remember (Hotz 2016).

Hotz cites the work of Mueller and Oppenheimer published in Pyschological Science. Their research studies showed that students who took notes by hand performed better on conceptual questions than those who took notes on a laptop. Students performed about the same on factual questions. Their hypothesis for why is that students who take notes by hand choose which information is important to include in their notes, and so they are able to study “more efficiently” than those who are reviewing an entire typed lecture (Mueller and Oppenheimer 2014). Note: These studies are on college students; I have found little research on grade school students.

For several years now, my students and I have been learning how to learn mathematics using the Standards for Mathematical Practice. MP8, “look for and express regularity in repeated reasoning”, has pushed me to think about having students record what they see instead of just noticing and discussing it.

SMP8 #LL2LU Gough-Wilson

One of the ways that I’ve learned to talk about “look for and express regularity in repeated reasoning” is to ask students to notice what changes and what stays the same as we take a dynamic action on a geometric figure. Consider a recent learning episode from my classroom.

Students were told that our learning intention was “I can look for and express regularity in repeating reasoning”. The content was conceptual development of the equation of a circle in a coordinate plane using the Pythagorean Theorem. I did not share that specific content with students up front, however, because I wanted it to be revealed as the lesson progressed. I showed them a dynamic right triangle in the coordinate plane.

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What changes? What stays the same?

I could have let them simply discuss what they noticed. But instead I asked them to “Notice & Note”, using words, pictures, and numbers to write and sketch what they saw.

Then I asked them to share what they noticed with a partner and add to their own notes as desired.


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Our classroom discussion revealed that the equation of the circle formed by tracing point P was x2 + y2 = 52.

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Students continued to “Notice & Note” as they moved a circle around in the coordinate plane. What changes? What stays the same?


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As we moved P around in the coordinate plane, and then as they later moved the circle around in the coordinate plane students noted what they saw. Eventually, students generalized the center-radius form of an equation.

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Notice & Note, by Kylene Beers and Robert E. Probst, is a guide of signposts (strategies) for close reading of text. Students are taught signposts to notice while they are reading, and they are asked to stop reading and note what the signpost might imply. “Again & Again” is one signpost. Do you notice an event in the text that keeps happening again and again? Do you notice a phrase in the text that is repeated again and again? Stop reading, note it, and think about what that might mean (Beers and Probst 2013). How might we take advantage of the ways that students are learning to read text in their English Language Arts (ELA) classes to guide students in inquiry based exploration of mathematics?

In her online course, Sunni Brown, author of The Doodle Revolution, states that “Tracking content using imagery, color, word pictures and typography can change the way you understand information and also dramatically increase your level of knowledge and retention” (Brown 2016). How do we make tracking content using words, pictures, and numbers a reality in the 1:1 classroom? My experience is that it doesn’t happen without deliberate emphasis on its importance.

In Reading Nonfiction, Beers and Probst write “When students recognize that nonfiction ought to challenge us, ought to slow us down and make us think, then they’re more likely to become close readers” (Beers and Probst 2016). Our ELA counterparts are on to something. Effective classroom instruction is not just about creating learning episodes for our students to experience the mathematics using the Math Practices. Effective classroom instruction incorporates practices that will help students remember what they are learning longer than for the next test.


As I think about our district’s continued implementation of 1:1 technology, I am convinced that we need to pay attention to when we are asking, encouraging, and requiring students to use pencil and paper to create a record of what they are learning. I am interested in thinking more about how we might blend the use of dynamic graphs and geometry software with Notice & Note – using words, pictures, and numbers, along with color, so that students not only have a record of what they are learning but also have a better chance of remembering it later. And so, the journey continues …


Beers, G. Kylene, and Robert E. Probst. Notice & note: Strategies for close reading. Portsmouth: Heinemann, 2013. Print.

Beers, G. K., & Probst, R. E. (2016). Reading nonfiction: Notice & note stances, signposts, and strategies. Portsmouth: Heinemann.

Brown, S. (n.d.). Visual Note-Taking 101 / Personal Infodoodling™. Retrieved April 25, 2016, from

Hotz, Robert Lee. “Can handwriting make you smarter?” The Wall Street Journal. 04 Apr. 2016. Web. 25 Apr. 2016.

Mueller, P. A., and D. M. Oppenheimer. “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking.” Psychological Science 25.6 (2014): 1159-168. Web.


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Derivative Rules

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.


How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives.

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We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.)

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We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x).

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We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

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Posted by on September 14, 2016 in Calculus, Derivatives


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Notice and Note: Dilations

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.

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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.

SMP8 #LL2LU Gough-Wilson.png

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.

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Translate ∆DET using vector EY.

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Rotate ∆D’E’T’ about Y using angle D’YA.

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Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.

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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.

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Due to the success on the first question, I changed it up a bit with the second question.

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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 …


Posted by on November 17, 2015 in Dilations, Geometry


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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.


Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?


What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.


Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry


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Classifying Triangles

We look specifically at 45-45-90 triangles on the first day of our Right Triangles unit. I’ve already written specifically about what the 45-45-90 exploration looked like, but I wanted to note a conversation that we had before that exploration.

Jill and I had recently talked about introducing new learning by drawing on what students already know. I’ve always started 45-45-90 triangles by having students think about what they already know about these triangles (even though many have never called them 45-45-90 triangles before). After hearing about one of Jill’s classes, though, I started by asking students to make a column for triangles, right triangles, and equilateral triangles, noting what they know to always be true for each. This short exercise gave students the opportunity to attend to precision with their vocabulary.

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It occurred to me while we were talking that having students draw a Venn Diagram to organize triangles, right triangles, and equilateral triangles might be an interesting exercise. How would you draw a Venn Diagram to show the relationship between triangles, right triangles, and equilateral triangles?

In my seconds of anticipating student responses, I expected one visual but got something very different.

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What does it mean for an object to be in the intersection of two sets? Or the intersection of three sets? Or in the part of the set that doesn’t intersect with the other sets?

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Then we thought specifically about 45-45-90 triangles. What do you already know? Students practiced look for and make use of structure.

One student suggested that the legs are half the length of the hypotenuse. Instead of saying that wouldn’t work or not writing it on our list, I added it to the list and then later asked what would be the hypotenuse for a triangle with legs that are 5.


I wrote 10 on the hypotenuse and waited.

But that’s not a triangle?


5-5-10 doesn’t make a triangle.

Why not?

It would collapse (students have a visual image for a triangle collapsing from our previous work on the Triangle Inequality Theorem).

Does the Pythagorean Theorem work for 5-5-10?

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Students reflected the triangles about the legs and hypotenuse to compose the 45-45-90 triangle into squares and rectangles.

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And they constructed an altitude to the hypotenuse to decompose the 45-45-90 triangle into more 45-45-90 triangles.

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And then we focused on the relationship between the legs and the hypotenuse using the Math Nspired activity Special Right Triangles.

And so the journey continues … listening to and learning alongside my students.


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Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.


But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]

Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.


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SMP8: Look for and Express Regularity in Repeated Reasoning #LL2LU

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.


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But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4

I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3

I can look for and express regularity in repeated reasoning.

Level 2

I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1

I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.


What do you notice? What changes? What stays the same?

We use a CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning.

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What do we need to factor for the result to be (x-4)(x+4)?

What do we need to factor for the result to be (x-9)(x+9)?

What will the result be if we factor x²-121?

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What will the result be if we factor x²-a²?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1? (And then connect the result to the graph of y=x²+1.)


What happens if we factor over the set of real numbers?

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Or over the set of complex numbers?

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What about expanding the square of a binomial?

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What changes? What stays the same? What will the result be if we expand (x+5)²?

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Or (x+a)²?

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Or (x-a)²?

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What about expanding the cube of a binomial?

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Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Experiments in Learning by Doing]


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