Tag Archives: Look for regularity in repeated reasoning

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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MP8: The Medians of a Triangle

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

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

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I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

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Students then interacted with dynamic geometry software.


What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.


I sent another poll.

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And then we confirmed student conjectures on the spreadsheet.

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

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Posted by on August 15, 2016 in Angles & Triangles, Geometry


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What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?


As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis


Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?



As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?


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Notice and Note: The Equation of a Circle

I wrote in detail last year about how our students practice I can look for and express regularity in repeated reasoning to make sense of the equation of a circle in the coordinate plane.

This year we took the time not only to notice what changes and what stays the same but also to note what changes and what stays the same.

Our ELA colleagues have been using Notice and Note as a strategy for close reading for a while now. How might we encourage our learners to Notice and Note across disciplines?

Students noticed and noted what stays the same and what changes as we moved point P.

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They made a conjecture about the path P follows, and then we traced point P.

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We connected their noticings about the Pythagorean Theorem to come up with the equation of the circle.

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Students moved a circle around in the coordinate plane to notice and note what happens with the location of the circle, size of the circle, and equation of the circle.

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And then most of them told me the equation of a circle with center (h,k) and radius r, along with giving us the opportunity to think about whether square of (x-h) is equivalent to the square of (h-x).

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And so the journey continues … with an emphasis on noticing and noting.




Posted by on March 19, 2016 in Circles, Geometry


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Rigor: Trig Ratios

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help our students build procedural fluency from conceptual understanding?

I wrote last year about using technology to develop conceptual understanding of Trig Ratios.

This year, we started the lesson a bit differently. I read a while back about Boat on the River, a 3-Act that Andrew Stadel had published and that Mary Bourassa had used to introduce right triangle trig, but I had never taken the time to look it up.

We watched Act 1 to begin the lesson.

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Students submitted what they noticed and wondered.

Then we thought about what information would be useful to know, along with thinking about what information would actually be reasonably attainable.

For example, many bridges are made to a certain standard height or have the clearance height painted on them. This one is no exception … the bridge height is given in the Act 2 information.

Students decided the length of the mast was attainable, too. And the angle at which the boat is leaning. Maybe there was a reading on the control panel.

So we ended up with something like this:

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Students’ experience with right triangles to this point had been the Pythagorean Theorem, similar right triangles/altitude drawn to hypotenuse, and special right triangles.

I told them we’d come back to the boat problem by the end of class.

Next, I asked students to draw a right triangle with a 40˚ angle and measure the side lengths. I collected their side lengths, again, telling them that we would use this information later in class. (I wish I had asked them to do this part the night before and submit via Google Form … maybe next year.)

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Students practiced I can look for and express regularity in repeated reasoning along with Notice and Note while first watching B move on the Geometry Nspired Trig Ratios activity and then observing what happened as I pressed the up arrow on the slider.

SMP8 #LL2LU Gough-Wilson

Eventually, we uncovered that the ratio of the opposite side to the hypotenuse of an acute angle in a right triangle is called the sine ratio. We connected that to triangle similarity as our content standard requires.


  1. Define trigonometric ratios and solve problems involving right triangles
  2. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

We checked the ratio of the opposite side to the hypotenuse for the right triangle they had drawn and measured. How close were they to 0.643? Students immediately noted that there was a problem with the ratios that were over 1 and talked about why.

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We continued practicing I can look for and express regularity in repeated reasoning along with Notice and Note to develop the cosine and tangent ratios.

And then we went back to Boat on the River. What are we trying to find? What ratio could we use? How would we know whether the boat made it?

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When I cued up the video for Act 3, the students were thrilled to find out they were actually going to get to see whether the boat made it. And at the end, they spontaneously clapped.

Someone asked me in a workshop recently how long a 3-Act takes. There are plenty on which we spend majority of a class period. Or even more than one class period. This one took less than 10 minutes of our lesson, but the payoff is worth more than our whole unit of Right Triangle Trigonometry. It gave us a way to develop the need for trig ratios that my students have just had to trust we need before. For these students, trig ratios don’t just solve right triangles; trig ratios can help with planning trips down the river. Coupled with the formative practice that students got during the next class, Boat on the River helped us balance rigor, one of the key shifts in mathematics called for by CCSS.

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And so the journey towards rigor continues … with thanks to Andrew for creating Boat on the River and Mary for blogging about her students’ experience with it and to my students for their enthusiasm about learning which made evident during this lesson through applause.

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Posted by on January 23, 2016 in Dilations, Geometry, Right Triangles


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Short Cycle Formative Assessment: 45-45-90 Triangles

I am enjoying our slow book chat on Dylan Wiliam’s Embedding Formative Assessment. (You can download the first chapter here, if you are interested.)

Chapter 1 is Why Formative Assessment Should Be a Priority for Every Teacher. Wiliam convinced me of this in Embedded Formative Assessment, but I still learned plenty from this chapter. My sentence/phrase/word reflection was actually a paragraph:

Formative assessment emphasizes decision-driven data collection instead of data-driven decision making.

As I planned our Special Right Triangles lesson for Wednesday, I decided what questions to ask based on what was essential to learn.

Level 4: I can use the Pythagorean Theorem & special right triangle relationships to solve right triangles in applied problems.

Level 3: I can solve special right triangles.

Level 2: I can use the Pythagorean Theorem.

Level 1: I can perform calculations with squaring and square rooting.

We started class with a Quick Poll.

I was surprised at how long it took students to get started. I hadn’t planned it purposefully, but the way the triangle was given forced them to make more connections than if the two legs had been marked congruent.


Eventually, everyone got a correct answer (and the opportunity to learn more about using the square root template) using the Pythagorean Theorem.

I asked them to determine the hypotenuse of a 45˚-45˚-90˚ triangle with a leg of 10 next. As soon as they got their answer, they announced “there’s a pattern”.

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They conjectured what would happen for legs of 12 and 7.

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I asked them to select a number between 20 and 100 for the leg and convince themselves that the pattern worked for that number, too.

I loved, though, that the first student whose work I saw had to convince himself that it worked for a side length of x before he tried a number between 20 and 100. I took a picture of his work and let him share it later in the class.

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Students shared their results with the whole class, and then I sent another poll.

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Which led us to reverse the question using the incorrect answer. If √6 is the hypotenuse, what is the leg?

And then a poll to determine the leg given the hypotenuse.

And another poll to determine the leg given the hypotenuse.

I set the timer for 2 minutes and asked students to Doodle what they had learned, using words, pictures, and numbers. And I was pleased that more than the majority took their doodles with them when class was over.

Wiliam says, “But the biggest impact happens with ‘short-cycle’ formative assessment, which takes place not every six to ten weeks but every six to ten minutes, or even every six to ten seconds.” (page 9)

I sent this poll first thing on Friday.

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Students gave these responses after working alone for 1-2 minutes.

I didn’t show the results, and got these responses after students collaborated with a partner for next minute or two.

When I gave a similar question a previous year, allowing collaboration, the success rate was informative but abysmal.

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And so the journey continues … focusing on decision-driven data collection, giving my students and me the opportunity to decide what do next based on “short-cycle” formative assessment.


Wiliam, Dylan, and Siobhán Leahy. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. West Palm Beach, FL: Learning Sciences, 2015. Print.


Posted by on January 9, 2016 in Geometry, Right Triangles


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Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

What do you do when the standard for the day gives away what you want students do explore and figure out on their own?

I’ve made a deal with my administrator to post the process standard for the day (Math Practice) instead of the content standard.

In many of our geometry classes, our learning goals include look for and make use of structure and look for and express regularity in repeated reasoning.

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We defined midsegment:

A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides of the triangle.

The midsegment of a trapezoid is a segment whose endpoints are the midpoints of the non-parallel sides.

Then we constructed the midsegment of a trapezoid. Students observed the trapezoid as I changed the trapezoid.

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I sent a Quick Poll: What do you think is true about a midsegment of the trapezoid?

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It creates both a triangle and a trapezoid. 1

all the midpoints form a similar triangle    1

mn is parallel to yx   2

((1)/(2))the size of the origin        1

parallel to base of triangle   1

all the midsegments make a similar triangle upside down          1

parallel to the base   1



and a trap      1

creates triangle and trapezoid        1

It will form the side of a triangle that is similar to the original.   1

The midsegment is parallel to the side not involved in making the midsegment.        1

it would be a median            1

MN is parallel to YX  1

parallel to base          1

mn parallel to yx       1

MN is parallel to the bottom line     1

XMN=XNM     1

cuts the tri into a trap and tri          1

all the segments will make a similar triangle tothe original         1

all of the midpoints connected make a similar triangle to the original one       1

creates ∆ on top + trap. on bottom 1

Triangle XMN is similar to triangle XYZ.

Line MN is parallel to line YX.          1

2/3 the largest side  1

2/3 o  1

side parallel to the midsegment is a           1

it makes a triangle and a tra            1

mn is ll to yx, mnx is congruent to triangle mon   1

In order to change things up a bit, I quickly printed the students conjectures, cut them up, and distributed a few to each team. Now you decide whether the conjectures you’ve been given are true. And if so, why?

I used Class Capture to monitor while the students talked and worked (and played/explored beyond the given conjectures).

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Then I asked what they figured out through a Quick Poll:

seg mn parallel to both seg ab and seg dc 1

parallel to both of the bases            1

it cre8 2 trap 1

mn is parallel to dc and ab  2

Trapezoid ABCD is similar to ABNM.

Lines AB and CD are parallel to the midsegment. 1

MN is parrellel to AB and DC           1

It creates a line that is parallel to the bases and forms two trapezoids. 1

nidsegment is parallel to top and bottom sides    1

((AB+DC)/(2))          2

it would be parallel to the sides above and below it        1

MN is parallel to DC and AB 1

(AB+DC)/(2)=MN     1

its parallel to DC        1

It makes two trapezoids       1

it is // 2 ab and dc    1

it forms two trapozoids        1

It is parallel to the sides above and below it.         1

makes 2 trap. 1

It makes a similar trapezoid.            1

they make 2 trapezoid, ab+dc/2     1

parellel to base of trapezoid            1

ab ll to mn ll to dc. when a parallelogram, creates 2 congruemt trapezoids      1

trapezoid ABNM is similar to trapezoid MNCD      1

all midsegments make a diala         1

And then we talked about how they knew these statements were true.

Jameria had a lot of measurements on her trapezoid. I made her the Live Presenter. What conjectures can we consider using this information?

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I made Jared the Live Presenter.

What does Jared’s auxiliary line buy us mathematically?

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I made Landon the Live Presenter.

What conjectures can we consider using this information?

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I sent a Quick Poll to formatively assess whether students could use the conjecture we made about the length of the midsegment compared to the length of the bases of the trapezoid.

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What about the midsegments of a triangle?

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And why?

We had not yet started our unit on dilations, and so there was more to the why in a later lesson.

And so the journey continues, even making deals with my administrators as needed, to create a classroom where students get to make and test and prove their own conjectures instead of being given theorems from our textbook to prove.

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Posted by on January 4, 2015 in Angles & Triangles, Geometry, Polygons


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Completing the Square on Equations of Circles

I wrote about this lesson last year. So just a few updates for this year.

Our goal – the second part of the standard:

G-GPE.A.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.

We started with a Quick Poll. I figure it’s going to be hard to complete the square if we don’t know what the square of a binomial actually is.

If someone has a counterexample, then the statement must be false. Who marked false that has a counterexample for this statement not always being true?

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One student let x=2 to show that the statement wasn’t always true.

Did anyone else use a number?

Various other numbers had been used to show the statement was false.

Did anyone show it was false a different way?

One student expanded (x+1)2 to show that it wasn’t always equal to x2+1.

We used CAS to look for regularity in repeated reasoning. What happens when you square a binomial?

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We started with the familiar, the equation for the circle with its center and radius. What happens if we expand that equation – and instead start with the expanded form? How would we go backwards to get to the center and radius form of the equation?

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More than one student couldn’t believe I made a big deal about what we needed to add to complete the square. It was so obvious to them that we needed to undo what we had done when we expanded: divide by 2 and then square.

We call this completing the square to find the center and radius of a circle.

And just in case someone needs another visual, we look at Completing the Square from Algebra 2 Nspired.

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And then we tried a few where we didn’t know the center-radius form before we started.

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And then we checked to see how well students were working on their own, finding out that we are not quite ready to move on.

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And so the formative assessment journey continues …

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Posted by on March 24, 2014 in Circles, Coordinate Geometry, Geometry


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Circles in the Coordinate Plane

CCSS-M G-GPE.A.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.

We used part of the Math Nspired activity Exploring the Equation of a Circle and part of the Mathematics Assessment Project formative assessment lesson Equations of Circles 1  for our introductory lesson on circles in the coordinate plane.

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What do you notice?

-There is a right triangle.

-The hypotenuse is always 5.

What will happen if we trace point P as we move the triangle around in the coordinate plane? What will be the locus of points that it travels?

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What can we say about the lengths of the sides of the triangle?


Would it be okay to name the lengths of the legs x and y instead of a and b, since they are horizontal and vertical lengths in the coordinate plane?

So x2+y2=52?

Yes. So for each (x,y) location of the point P, we can say x2+y2=52. That is how we describe the equation of this circle in the coordinate plane.

Now I’m going to let you play with a few more pages in your TNS document and then answer a few questions. Screen Shot 2014-03-04 at 8.32.14 AM

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Students played.

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What happens when you translate the center of the circle?

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And then students answered more formal questions about the equation of a circle to ensure that they had looked for and expressed regularity in repeated reasoning. I collected their responses (this was their bell work for the day, but I decided not to immediately show them the results. I took a glance myself to know how they were progressing).

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Next we moved to a few questions from the MAP formative assessment lesson on Equations of Circles.

I am always impressed by the progression of questioning in the lessons. I was particularly interested in how students decided whether the point (5,6) lay inside, on, or outside the circle x2+y2=36.

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One student knew that it lay outside the circle because the point (0,6) was on the circle. He reasoned that if (0,6) is on the circle, (5,6) can’t be. Another student drew the point on the grid and recognized that it could not lie on or inside the circle. Another student used the equation to show that the point (5,6) did not lie on the circle.

What if the point is too close to tell from a sketch of the graph?

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What does it mean for a point to lie on a circle? Another Quick Poll, with the idea for the question from MAP.

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I am learning to ask questions that I think are obvious.

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We looked at the coordinates that they entered, and we changed to the Graph View. I added a Teacher Equation (two, actually) to show the circle. What does it mean for a point to lie on a circle?

An aside: I was in a grade 7 classroom recently. Students were determining the x- and y-intercepts of lines given the equation. I asked the teacher to send a Quick Poll of the graph of a line and have students drop a point on its x-intercept. She was surprised to find out how many students didn’t know what the x-intercept was, and yet they’re supposedly calculating x-intercepts from equations. (Students had dropped points all over the x-axis, but only a few students had dropped a point at the intersection of the given line and the x-axis.)

Next students completed the Mathematics Assessment Project chart about equations of circles to help quell any misconceptions they might have.

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And then another Quick Poll to see how students are doing writing the equation “from scratch”.

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Which brings us a good opportunity to attend to precision. This is where, in the past, I might ask

-do you have parentheses?

-is your radius squared?

-do you have (x+5) squared and (y-1) squared?

And all of my students would have nodded.

But with Navigator, the students determine which are correct and what some need in order to be correct. They see whether their response is leveling up to the standard or not. They find out what to do for their next response to level up to the standard.

We revisited the bell work during the last few minutes of class. Students decided whether they wanted to keep their original responses or revise their response after the lesson. A few changed their responses.

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And I know who to need extra support as we continue to learn.

And so the journey continues with good evidence of what my students know and what my students still need to know …




Posted by on March 24, 2014 in Circles, Coordinate Geometry, Geometry


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Special Right Triangles

I blogged last year about providing students the opportunity to make sense of special right triangles. This year I just want to make a few observations. Over the last year, I have gone through and thrown out old binders of notes and transparencies that I used to use. I took a picture of how I used to make sure students could solve special right triangles, just as a reminder of how far we’ve come.

Special Right Triangles

In our lesson on 45°-45°-90° triangles, students use the Pythagorean Theorem to look for regularity in repeated reasoning, reason abstractly and quantitatively, and look for and make use of structure.


Students did well recognizing that a 45°-45°-90° triangle is half of a square divided by a diagonal. And they did well on the formative assessment Quick Polls that I sent to assess their progress.

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Note: These two questions were not on the same side of the page on the student handout.

But something happened when I asked them to calculate the perimeter of a non-familiar polygon.

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I monitored their progress after I sent the poll. After two minutes, I saw the following results. 4 students had a correct response, and 15 students had an incorrect response.

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After another minute, 5 students had a correct response, and 21 students had an incorrect response.

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I stopped the poll.

What would you do next?

I unchecked “Show Correct Answer” when I showed the results. And I asked a student who got 15 to explain his thinking (construct a viable argument). He counted 15 “pieces of segments” in the figure. Then I asked the class to critique his reasoning. Is every piece congruent? Another student asked to come to the board so that she could show how she used the practice look for and make use of structure.

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Of course everyone understood the mistake in measuring after the second student drew the auxiliary line. But I wonder why more students didn’t connect what we had been learning to this diagram on their own?

What would have happened if we had started class with the perimeter of the polygon? Whether I had asked them to answer it then or not, would it have made a different in what they saw later?

Not unrelated, I recently asked my 9-year old daughter to load the dishwasher. Several hours later, I opened the dishwasher and found a big surprise.


I’m not sure if you can tell or not, but among other problems, the coffee mugs still have on their lids. It’s hard to believe that AKW has ever unloaded the dishwasher, much less on many occasions.

We talked about how the dishwasher works – and why we should turn dishes towards the water. This was take 2, a week later.

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I’m still a bit flabbergasted at how difficult learning how to load the dishwasher has been without direct instruction. But I wonder what would have happened if I had asked her a different question to get her to think about how the dishwasher works. I wonder what would have happened if I had specifically asked her what she noticed when she was unloading the dishwasher, causing her to look for regularity in repeated reasoning. My daughter is still learning how to think and problem solve.

As all learners are.

And I am still learning the questions to ask, to promote thinking and problem solving, to uncover misconceptions.

As all Learners are.

And so the journey continues …

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Posted by on January 28, 2014 in Geometry, Right Triangles


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