Tag Archives: Illustrative Mathematics

What I Learned Today: Scale Drawings and Maps

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered my question by asking me what I learned at school today.

It took me a while to think about what I had learned [which will make me more patient when I ask her the question again tomorrow], and then I remembered and shared with her:

We are working with some teachers who are using the Illustrative Mathematics 6–8 Math curriculum. The 7th grade teachers are in Unit 1, Scale Drawings. They are working with Scale Drawings and Maps. Today I learned to look more closely at the scale given for a map.

Look at the following for a moment. What’s the same? What’s different?





The last two are from Illustrative Mathematics, which you can download for free at

What’s different about the scales on the last two?

Attend to precision, MP6, says, “Mathematically proficient students try to communicate precisely to others. … They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.”

I’m not sure that we would have noticed a difference, except that we were trying to find some assessment items from another source and saw that many aligned to 7.G.A.1 included a scale in the form of “1 cm = 100 miles”. I’ve looked at lots of maps and I never noticed the incongruity of saying that 1 cm equals 100 miles. We don’t really mean that 1 cm equals 100 miles, right? Not in the same sense that we say 4 quarters equals $1 or 3+4=7. Is there any wonder that our students misuse the equal sign?

And so the journey continues, grateful for the authors of this curriculum who make me pay closer attention to attending to precision and grateful for my daughter who makes me think and share about what I’m learning, too …


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Using Rigid Motions for Parallel Lines Angle Proofs

CCSS-M.G-CO.C.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

After proving that vertical angles are congruent, we turned our attention towards angles formed by parallel lines cut by a transversal.

My students come to high school geometry having experience with angle measure relationships when parallel lines are cut by a transversal. But they haven’t thought about why.

We make sense of Euclid’s 5th Postulate (wording below from Cut the Knot):

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

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We use dynamic geometry software to explore Parallel Lines and Transversals:

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And then traditionally, we have allowed corresponding angles congruent when parallel lines are cut by a transversal as the postulate in our deductive system. It makes sense to students that the corresponding angles are congruent. Then once we’ve allowed those, it’s not too bad to prove that alternate interior angles are congruent when parallel lines are cut by a transversal.

But we wonder whether we have to let corresponding angles in as a postulate. Can we use rigid motions to show that the corresponding angles are congruent?

One student suggested constructing the midpoint, X, of segment BE. Then we created a parallel to lines m and n through X. That didn’t get us very far in showing that the corresponding angles are congruent. (image on the top left)

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Another student suggested translating line m using vector BE. So we really translated more than just line m. We really translated the upper half-plan formed by line m. We used took a picture of the top part of the diagram (line m and above) and translated it using vector BE. We can see in the picture on the right, that m maps to n and the transversal maps to itself, and so we conclude (bottom left image) that ∠CBA is congruent to ∠DEB: if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Once corresponding angles are congruent, then proving alternate interior (or exterior) angles congruent or consecutive interior (or exterior) angles supplementary when two parallel lines are cut by a transversal follows using a mix of congruent vertical angles, transitive and/or substitution, Congruent Supplements.

But can we prove that alternate interior angles are congruent when parallel lines are cut by a transversal using rigid motions?

Several students suggested we could do the same translation (translating the “top” parallel line onto the “bottom” parallel line). ∠2≅∠2’ because of the translation (and because they are corresponding), and we can say that ∠2’≅∠3 since we have already proved that vertical angles are congruent. ∠2≅∠3 using the Transitive Property of Congruence. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

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Another team suggested constructing the midpoint M of segment XY (top image). They rotated the given lines and transversal 180˚ about M (bottom image). ∠2 has been carried onto ∠3 and ∠3 has been carried onto ∠2. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

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Another team constructed the same midpoint as above with a line parallel to the given lines through that midpoint. They reflected the entire diagram about that line, which created the line in red. They used the base angles of an isosceles triangle to show that alternate interior angles are congruent.

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Note 1: We are still postulating that through a point not on a line there is exactly one line parallel to the given line. This is what textbooks I’ve used in the past have called the parallel postulate. And we are postulating that the distance between parallel lines is constant.

Note 2: We haven’t actually proven that the base angles of an isosceles triangle are congruent. But students definitely know it to be true from their work in middle school. The proof is coming soon.

Note 3: Many of these same ideas will show that consecutive (or same-side) interior angles are supplementary. We can use rigid motions to make the images of two consecutive interior angles form a linear pair.

After the lesson, a colleague suggested an Illustrative Mathematics task on Congruent angles made by parallel lines and a transverse, which helped me think through the validity of the arguments that my students made. As the journey continues, I find the tasks, commentary, and solutions on IM to be my own textbook – a dynamic resource for learners young and old.


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Two Wheels and a Belt

What do you do on the last day of class?

I needed to spend some time talking with students about their grades. So they worked on a task while I met with individual students. Which means I wasn’t able to orchestrate as productive of a mathematical discussion as I would have liked.

We started with the pictures, absent of any explanation or measurements. What question could we explore?

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What is the circumference of the circles?  1

What is the volume of the cylinder?           1

are circles equal?      1

are the 2 circles the same size

what are their radii  1

are the areas of the two circles equal         1

are the bases the same size 1

are the bases the same size?           1

could the one of the bases of the cylinder have the same diameter as the other?

dilation           1

does the highlighted line have any significance to the cylinder  1

does this shape qualify for a cylider           1

is it a cylinder or 2d lines and circles resembling a bike chain?  1

is it a cylinder or gears         1

is it a cylinder            1

is it a cylinder?

is it 2 similar circles with a sring around them?    1

is it a cylinder?          1

what are the wheels connected by the belt being used for?       1

what is the length of the darker line?        1

what is the length of the line around the wheels?            1

what is the radius of both circes?   1

what is the relationship between the lines and the circles          1

what is the volume of the cylinder 1

what is the volume of the figure?   1

whether its a cylinder or a dialation           1

why are the circles not touching     1


Does is help to know that these are two wheels and a belt?

Does looking at a picture of some gears help to make sense of the picture?

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I asked the question again.

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What information do you need to determine the length of the belt?

Teams worked together to make a list of measurements.

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Then they started working.

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I checked in every few minutes in the midst of meeting with students about grades. I knew some of the misconceptions that students would have because we did this task last year.

Students had the opportunity to look for and make use of structure.

Is the quadrilateral a rectangle?

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Many students were calculating as if it were.

Someone convinced us that the quadrilateral is only a trapezoid.

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Students corrected calculations and made more progress.

I checked in again.

Is the part of the belt highlighted in green a semicircle?

Many students were calculating as if it were.

Someone convinced us that the arc is greater than a semicircle.

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It wasn’t the ideal way to have class and discuss student work. But for the last day of class, it wasn’t bad.

And now the journey continues to a new school year …


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Making Time for Tasks

I co-presented a power session at the T3 International Conference Sunday morning, and I’ve posted the stories that I shared of Running around a Track I and Running around a Track II. I have also posted Running around a Track III during which I processed the feedback we received during and after the sessions.

Several participants asked about the time that rich tasks take, and so I thought it might be helpful to think about making time for tasks in this separate post. I enjoy teaching on a block schedule. I know I don’t feel as rushed as I would otherwise. But a friend asked me how often we do these types of tasks in our classes.

How often do you do this type of rich task?

There are some tasks through which the math content can be initially and naturally be learned. I think of Placing a Fire Hydrant (and last year’s notes here) and Locating a Warehouse (with last year’s notes here). There are some that are more culminating tasks for a unit, especially the modeling tasks. A typical unit for us lasts about 8 days. (See some of the Unit Student Reflections posts to know more about how we organize content: Special Right Triangles, Dilations.) We do the culminating performance type tasks 1-2 of those days towards the end of the unit. On the other days, we are learning the content using practices such as look for and make use of structure and look for and express regularity in repeated reasoning, but not always through tasks.

For example, when our learning goal is

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

I find that exploring the relationships using technology first, verifying conjectures using formative assessment questions (more skill-based practice), and then moving towards why works best for my students. If you know of a good task for learning G-C.A.2, I would love to know. But for now, at least, we end this unit with a few culminating tasks instead of beginning the unit with them. The culminating tasks are Circles in TrianglesInscribing and Circumscribing Right TrianglesTemple Geometry. I will note that while we didn’t actually do the tasks until the end of the unit, I did show students one of the diagrams from the beginning so that they could keep in mind throughout the unit where we were heading.

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So now that you know that I don’t give a rich task every day, let’s think about what steps we might take to reduce the class time for the tasks. [These questions and comments will make more sense if you have looked at how the lessons for Running around a Track I and Running around a Track II played out with my students.] One of the participants in our Sunday session wondered what would have happened if I had given the students a teaser of what was to come the day before this lesson. What if I had shown them the picture and asked what they wondered then? Or sent a link to a Google form for them to submit their question outside of class?


Or what if I actually showed them the tasks and questions (blacking out information in I that would give away II or vice versa) just so students could begin the process of thinking about the structure of the track before they did any calculations in class? I think these are great ideas – having students spend the “alone” time for processing the questions being asked could definitely make the time spent on the task take less class time.

Another question that tied into the idea of planning lessons with rich tasks was whether we use a textbook. I think of the textbook as a source of information and practice problems for students. We don’t have our units arranged like our textbook, but I do reference page numbers for each lesson on the student syllabus so that they will know where to look for extra help and practice. (Our textbook isn’t CCSS at this point, and I’m not convinced that the new ones have geometry written in the spirit of CCSS. When I’m reviewing a new textbook, I first look at the Table of Contents. When transformations is still the topic of chapter 8, I find myself skeptical.)

Someone else asked about how rich tasks complement some of the skills practice that students need. We don’t get to every practice problem that we include on our student handouts, but I have finally had the time to work through them, and so I post the worked handouts online for students to check problems they do outside of class for additional practice. We also give online practice assignments with two chances to students through Canvas so that they can get immediate feedback on what they know and don’t know. We are trying to teach our students how to use formative assessment.

[I’ve been reading Transformative Assessment and Transformative Assessment in Action, both by James Popham. According to Popham, the first level of formative assessment is when it is used by the teacher to make instructional adjustments as needed to further student learning. The second level of formative assessment is when it is used by the student to make learning adjustments as needed to further learning. The third level of formative assessment is when it is used by the class as a whole to help all students meet the learning goals of the lesson, and the fourth level of formative assessment is a transformation of the school – all teachers are learning about formative assessment in school wide PD and practicing it in their classrooms.]

My students and I have explicitly discussed in class that if you take the online practice assignment and miss every problem, you should make a learning adjustment before trying again.

So if I don’t use a textbook, how do I plan my lessons?

The top three sites that I use are Illustrative Mathematics, the Mathematics Assessment Project, and the Math Nspired lessons at TI’s site. I also follow a lot of bloggers. It helps that I’ve taught geometry for 20 years, and it helps that I’ve been making an effort for students to learn by doing for at least 17 of those years. (I owe a huge apology to all of the students I had the first 3 years.)

But it does take time. I am lucky to work with a great team of geometry teachers who are willing to help and try tasks and use formative assessment with their students. We taught our CCSS Geometry course last year for the first time, and our administrator worked it out so we could have 4 teachers and 30 students in our first block class together. We worked through the lessons together with the students and each other, and then the other 3 teachers had a planning block after that class so that they could correct everything we had done wrong the first time before they taught it in their own classrooms the rest of the day. This year, we have a team of Algebra 2 teachers doing the same thing, and next year, our Algebra 1 teachers will teach one class together. I didn’t know what my administrator would say when I proposed this idea to him, but I’ve learned it doesn’t hurt to ask. It was a total scheduling pain, and some of our other classes were more crowded, but that was worth the sacrifice for what we learned teaching the first class together.

I’ve just told you what works for us in our efforts to include more rich tasks, but it’s only one perspective. What works for you? What additional sites do you use to find rich tasks? Do you start with them more often than end with them? We’d love to learn more from you as the journey continues … easing the hurry syndrome, one task at a time.


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Running around a Track III

I co-presented a power session at the T3 International Conference Sunday morning, and I’ve posted the stories that I shared of Running around a Track I and Running around a Track II from the Illustrative Mathematics tasks Running around a Track I and Running around a Track II. In this post, I want to process the feedback that we received from the participants during and after the session. (There is not really an IM task called Running around a Track III, although there could be one from some of the suggestions participants gave for extensions!) Several participants asked about the time that it takes to do rich tasks in class. I’m going to address that conversation in my next blog post, Making Time for Tasks.

The tasks have students make sense of the lanes on an Olympic Track.

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One participant didn’t understand why I changed tasks with the two classes. I neglected to explain that during the session. I was really just doing an experiment to see how the tasks were different and how students approached them depending on the given information. Was one easier for students than the other? Did I need to scaffold the tasks for my students differently than they were written? Were students more successful with one task than with the other?

On Saturday, I went to a session by @bamentj from Darwin, Australia and learned about TodaysMeet. I wondered about using it as a backchannel during the session. A lot of participants were using Twitter throughout the conference, but we wanted a place where participants could interact with each other more than usual in a large session – and not get lost in the conference hashtag (or use two hashtags to make a subset of tweets for our session) being used by the other power sessions as well. Even though this meant that others at the conference wouldn’t find out as much as they might have otherwise about our session, we still wanted to try it. We made a room called CCSSPower. The link will only be live through March 15, 2014.

Several times we asked participants to use the protocol “I like, I wish, I wonder …” to provide feedback. So it turned out that we didn’t use TodaysMeet as effectively as we could have. In fact, one of the first posts I read was from Joe: I wonder what the purpose of today’s is. I did not see the use other than to post comments/questions that never were answered.

My reply, after the session (in 3 posts): Hi, Joe. Thank you for your comment. If we were to use TodaysMeet again, we would have a second projector to observe and use the comments. I like that the back channel can give participants a chance to communicate, whether or not the instructors are able to address the comments. But I will definitely use it differently if I use it again in a session.

I’ve also thought since my post, that since we had co-presenters, whether or not we had the second projector, one of us could have been monitoring a second computer with the backchannel while the other spoke. We learn from our mistakes, right?

As I have read through the other comments so that I could address some of the questions in this blog post that we didn’t get to address in the session, I will say that while we could have used the backchannel more effectively, it wasn’t a disaster. Those who were on the backchannel were having their own conversation. Several of the questions did come out in our whole group discussion, and some questions were answered by others in the backchannel without the presenters having to get involved.

One participant liked the pre-assessment ACT question being the same as the post-assessment.

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I am glad that one of the teachers on our geometry team had the idea to include the question in the lesson. I would not have thought of that myself. I looked back at Running around the Track I this week and noticed that one commenter suggested that the content in parts a and b of the task was on a grade 7 level. That might be true, but especially in this first year of CCSS implementation, the data I received from sending the ACT question without choices at the beginning of class (around 50% correct for both classes) indicated that our students had not thought through the mathematical content in the task before.

Several ideas for extensions came out during the session. What about having students also calculate area of the track for the ACT question? What about having student calculate the amount of paint needed for the lanes? Could you have students measure a local track with a trundle or wheel?

It occurred to me that it would be nice to take students to the track for the lesson. And then it also occurred to me that it would add at least 30 minutes to the time of the lesson for a visit to the track. I will note that one student in particular was engaged by this lesson more than any other this year. I asked him recently what his career pathway was, and he answered in all serious that is was to be a professional football player. He was an expert in class during this lesson.

Another suggestion was to use a video. I agree. If you find the right clip, please share it with me! I started by looking for a clip, but those that I found were longer than I wanted to use, and longer than I had time to search for the perfect segment to watch.

A few more comments from the back channel:

  • With regards to everyone missing the question about whether the lane lines were similar: If everyone got the quick poll wrong, I wonder how they would respond if you told them the answer is “no”, could they rethink their reasoning.
  • I wonder how you selected the student to present his “wrong” answer?
  • Perseverance is a best practice that we have to facilitate in our classrooms.
  • FYI while we were doing the 400 meter question…US women won the 4×400 meter gold at worlds in Poland.

What do you in your classroom when everyone gets a wrong answer?

If you decide to have someone explain their work anyway to correct incorrect thinking, how do you select students to present their work? Do you use a random student generator (we have one where we check roll in our PowerTeacher grade book)? Or “equity sticks” (usually tongue depressors with student names – some teachers replace and some teachers don’t replace when you call on students)? Or keep a clipboard with notes about whom you’ve chosen for whole class discussions? I’ve been trying the latter this year. It’s not perfect, but it is a start to at least paying attention to how often I call on students.

And so the journey continues … collaborating with educators from all over the world to improve our classroom practices. Thank you for the opportunity to learn from you, and thank you to Ellen from Illustrative Mathematics for sharing such great resources with us during the session and giving us a preview of what is coming soon to their website.


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Running around a Track II

I gave a different class of students Running around a Track II recently.

I started by showing them the same picture of the start of the 2012 Olympic Women’s 400 M race.


I changed the prompt, though. What do you wonder?

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I wonder why they start at different lines instead of at the same place.           1

I wonder who 1st realized that the smaller of the concentric circles would be shorter           1

I wonder whats special about this picture?           1

I wonder why if you are running on the outter circle its the same distance as running on the inner circle 1

I wonder what the distance is that they are running       1

I wonder what event they are running?    1

I wonder why tracks are ovals        1

I wonder if each one of those arcs are similar by using dialations         1

I wonder what the exact distance of all of the lanes are  1

I wonder what is happening.           1

I wonder if the circles are cocentric            1

I wonder how big the race track is as a whole.      1

I wonder why all of the starting points are in different places   1

I wonder what the ratio of the innermost ring is to the rest of the rings… or are they all congruent?           1

I wonder why the runners start at different starting positions. 1

I wonder why there are flags that are the same   1

I wonder if its better to be closer up on the outside lane than farther back on the inside lane         1

I wonder if that’s the olympics        1

I wonder which lane is the shortest distance        1

I wonder how they decided how far back to put the first runner          1

I wonder how I can figure out how to find the different measures of each of those arcs of the circle          1

[And from my cynic:] I wonder what question you’re going to ask about this image.   1

I wonder why are there so many americans         1

I wonder what shape the track is   1

I wonder why there are 3 people from us and why they are not starting from same side?   1

I wonder who the runners are        1

I wonder if the distance in the inner circle is different than the distance in the outer circle even if they start in differert places           1

I wonder if all of the starting points are marked off by distance or angle.        1

I wonder why the track runners are spaced apart rather than running all together. 1

I wonder what the arc measure of the track is?    1

Next, I sent out a Quick Poll of a question from a released ACT that students should be able answer as a result of the lesson. Who already knows how to solve this? Who knows how to solve it quickly, since the ACT is timed? I removed the choices, just to see how students answered without them. We didn’t talk about the results. I told them we would revisit the question at the end of the lesson. Just less than half of the students had it correct initially: 14/31.

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Before we started analyzing what was happening with each lane, I asked the students whether the lane lines were similar. Just for the record, I wouldn’t have thought to ask this question on my own. It was suggested in the commentary for the task on Illustrative Math, so I thought I would try it.

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I’m not sure I really had a backup plan for this formative assessment check. Not one student got this correct? How should we proceed? Understand that at this point, the students don’t know if everyone is correct or no one is correct. There is a great feature of Navigator that lets me “show correct answer” or not. While I monitor student responses with my projector screen frozen, I decide whether I will show the correct answer.

What does it mean for two figures to be similar?

Several students wanted to answer. I called on F.K., because F.K. loves dilations, and I knew that she would explain similarity in terms of transformations.

Two figures are similar if there is a sequence of transformations including a dilation that will map one figure onto the other.

Okay – so can we describe a dilation that will map one lane onto another lane? What would be the center of the dilation?

At this point, we moved to the technology. I had created the track on a Graphs page of TI-Nspire. Mainly because I wanted to prove that I could without asking my technical author friends Bryson and Jeff. It’s not beautiful, but it is functional. I didn’t even use this document in the class with Running around a Track I, but maybe it would be helpful now.

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I have a slider set up to change the radius of the circular part of the track. Are the lanes of the track similar? Does the technology help you see it?

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What happens if we dilate the straight part of the track about the center of the track by a scale factor of 1.1? And then let’s dilate the circular part of the track using the same center. (Bryson and Jeff could probably figure out how to make this happen all at once – my track is in 3 separate parts.) Is there a dilation to show that the lanes of the track are similar?

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The straight part of our track is a problem. There is no line that contains the center of dilation and the endpoint of each straight part of the track.

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One of the students wondered if each one of those arcs are similar by using dilations. Now if we are just talking the arcs, then yes, they are similar using a dilation about the center of the arc – but not using the center of the track.

The technology helped us make sense of whether the lane lines are similar.

Now back to the real task. This task is a bit different – it has less scaffolding than the first task. Students jump right in to calculating the perimeter of the track 20 cm inside lane 1 and 30 cm inside lane 2 so that they can determine how far ahead the runner in lane 2 needs to be ahead of the runner in lane 1. I sent a poll to collect the results.

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It was disastrous, and the bell was about to ring.

How in the world could I recover the lesson?

We went back to the picture, and ended class by talking in more detail about the diagrams.

We started the next class calculating the perimeter of the paint on the most inside lane.

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Even then, we didn’t have everyone with us. Don’t you love that 5 of my students just assumed the inside lane was 400 m without doing any calculations?

We went back to the questions on Running around a Track II, though and really did make progress.

To close the lesson, I sent the Quick Poll from the start of class:

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We are up to 80% correct, from just under 50% at the beginning of the lesson.

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And with choices, we have 90% correct. I finally showed the students the results and we talked about the misconception for choice C.

I think it is interesting to ask students which practices they used when working on a task.


And which would you choose if you could only choose one Math Practice?


I have shared before that my goal isn’t just to provide opportunities for my students to use the Math Practices in class – but also for them to recognize when they are using them. I ask my students to write a journal reflection each quarter on using a math practice.

N.R. writes about this task: In class today and yesterday, we worked on a problem about the track of the 400 meter dash in the Olympics. While working on this problem, we used the math practice of modelling with mathematics. We applied what we learned about circles in this unit to figuring out how far apart runners in different lanes have to start in order to run the same distance and to end at the same area in the straightway section. When solving this problem, I had to use the equations for circumference and perimeter and combine them. Once I finished working the problem, I decided that the runner in lane 1 has to start 7.037 meters behind the runner in lane 2. I also found that the runner in lane 2 has to start 7.666 meters in behind the runner in lane 3. This problem has helped me to be very attentive to detail. In this problem I had to be very careful that I worked everything correctly and completely.


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Seven Circles … Again

We tried Seven Circles I from Illustrative Mathematics a few weeks ago.

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At the end of class one day, I showed students the diagram and what question they might explore with it. I collected their responses using an Open Response Quick Poll and have shown the results below.

What does this figure have to do with geometry?            1

if we connected each top vertex of the triangles, will it make a hexagon?         1

whats the area of all the circles       1

Why are the circles in this shape?  1

what are the circles forming?          2

what is the area of all of the circles            1

why are we looking at circles?         1

are the spaces between the circles triangles?       1

what are the seven circles forming?           1

do all the circles have the same diameter?            1

do all the circles have the same diameter  1

Are the circles’ diameters the same?          1

Can the measures of the triangles that can be drawn through circles be calculated quickly  1

why are all the circles touching?     1

Why are the circles in that certain arrangement?            1

Can you find the area for that?       1

Is there a way to solve non 90° triangles? (With sin, cos, tan, or the other trig functions)      1

Can the circles be mapped onto each other with a rigid motion?           1

when you look at the image what do you see?      1

are the 6 figures that look like triangles in the gaps of the circles considered triangles since their sides arent straight    1

what are the circles for        1

are all of the circles congruent to each other?       1

What is the significance of the circular pattern?   1

How can you find the measurement of each circle           1

What are the triangular looking spaces in between the triangles called?          1

Some students were interested in the space between the circles. Other students wondered whether the circles were congruent. The task is given below.

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My students felt like it was pretty obvious that this could work with 7 congruent circles. I gave them different sized coins so that they could play. What if the circle in the middle is not congruent to the others? Will this work for 6 congruent circles? Or 8 congruent circles?

After students played for a few minutes, I sent them a TNS document that a friend made to explore this task. I used Class Capture to watch while students used the technology to make sense of the necessary and sufficient conditions for 6 circles and 7 circles in the given arrangement. Who had something interesting to discuss with the whole class?

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Many students saw the regular pentagon or regular hexagon with vertices at the centers of the outside circles and used that to make sense of the mathematics. While I was watching them, I was trying to figure out how we should proceed as a class. We started with Claire’s work. What do we know?

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We saw a dilation. We saw central angles of a regular pentagon. We saw isosceles triangles, which we bisected to make right triangles. We saw an opportunity to use right triangle trigonometry. We looked for and made use of structure. We reasoned abstractly and quantitatively.

And before the bell rang, we looked back at the picture with 7 circles and recognized that the 30-60-90 triangles require that the radius of the center circle equal the radius of the outer circles.

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We only touched the surface of what we can learn from this task. Last year, we didn’t even do that. Last year, I shared the task with students during their performance assessment lesson, but we spent all of our time on Hopewell Triangles. This year, we got to it, but I know that our exploration could have been better. We began to answer what are the necessary and sufficient conditions for 6 circles. And in the process, we came across an argument for why the 7 circles must be congruent. But we didn’t really solve the conditions for 6 circles.

I wanted to write about this as a reminder that we are all learning. In this journey, I am finding good tasks out there to try with my students. And I am more confident about some than about others. Even though I don’t know exactly how the tasks should play out in the classroom, I am going to keep trying them. And I’m not going to throw out the task just because we didn’t get as deep into the mathematics as I wish we had. I will try again next year as the journey continues …

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Posted by on February 24, 2014 in Circles, Geometry, Right Triangles


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Placing a Fire Hydrant

This task is from Illustrative Mathematics. Students are asked to place a fire hydrant equidistant from three locations.

My students worked on paper first. I was impressed with their work this year. More so than last year, which could have to do with us swapping Units 1 and 2 from last year to this year. They used paper, rulers, folding, and compasses.

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Several of them realized that if they could find the circle that contained all three locations, the center would be equidistant (and thus the location of the fire hydrant). However, their methods for finding a circle to contain all three points were not very precise (which meant they didn’t already know everything they needed to know about triangle centers).

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After students worked on paper & a few students presented their ideas (I deliberately sequenced them, after reading Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions), we tried to solve a simpler problem.

I asked two students in the room to stand up front. I asked another student to stand equidistant from the two students. The student stood at the midpoint of the segment containing student 1 and student 2 as endpoints. Is student 3 the only person who can be equidistant from students 1 and 2? A student in the back of the room said that she could be equidistant as well. She stood in the back. Another student in the middle of the room said that he could be equidistant as well. He stood in the middle. What is true about students 3, 4, and 5? They are all the same distance from student 1 and student 2. What else is true about students 3, 4, and 5? They are in a line. Yes, they are collinear. What else is true about students 3, 4, and 5? What would happen if we drew the line containing those students? What relationship would that line have to the segment containing students 1 and 2 as endpoints? Somehow, we got perpendicular bisector out of the conversation. Note: I’m pretty sure I read about this idea in a Mathematics Teacher years ago, but I do not have a reference for the article.

What happens if we add a 3rd person to our original problem? Are students 3, 4, and 5 equidistant from students 1, 2 and our 3rd person? No. Can anyone be equidistant to all three of our people? The students recognized the significance of taking the points two at a time – and then eventually determined that the perpendicular bisectors would be concurrent at our point of interest.

We moved to our dynamic geometry technology. Students constructed the perpendicular bisectors of the given triangle and moved the triangle. We created the circumscribed circle. We paid attention to what happened to the circumcenter. Is the circumcenter always a good location for the fire hydrant? I used Class Capture to monitor student progress.

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Can the location of the circumenter give us information about the type of triangle? I made a student the Live Presenter to show us what she found about the type of triangle and the location of the circumcenter.

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Finally, we ended with the straightedge and compass construction for the perpendicular bisector, focusing on the question “What segments, angles, arcs, and triangles are always congruent in the construction”.

For whatever reason, I made the list on the non-interactive whiteboard, but I marked the student findings on the interactive whiteboard. We got into isosceles triangles, congruent base angles, rhombi, and more.

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And so the journey continues…


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Reflected Triangles

We used a task from Illustrative Mathematics as part of a performance assessment on Rigid Motions. 

On the next page, △ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.


Students used TI-Nspire to construct the line of reflection, and we asked them to explain their construction on paper. I monitored their work using Class Capture to see what approaches the students were using. This year, almost everyone created segments BB’ and CC’. Some also created segment AA’. Some then constructed the midpoint of the segments then created the line through those midpoints. Some used the perpendicular bisector tool. Our class discussion focused on what were the fewest number of objects we could construct to get the line of reflection.



And then I asked H.K. if she would share her work. She had the same idea as most of the others, but instead of drawing a segment with a vertex and its image as the endpoints (AA’, BB’, or CC’), she constructed the midpoint of segment BC and the corresponding midpoint of segment B’C’. Then she joined those image and pre-image points with the segment tool (actually vector tool) and constructed the perpendicular bisector of the segment. That’s a big deal. H.K. reminded us of an important property of rigid motions. Most of the segments and angles that we found congruent as we were exploring focused on the vertices of the triangle and their images. But the same is true from every point on the pre-image to its corresponding point on the image – not just from the vertices. Thanks for the reminder, H.K.


And so the journey continues ….




Posted by on September 2, 2013 in Geometry, Rigid Motions


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Two Wheels and a Belt

Two Wheels and a Belt (and Why I am Convinced That the Standards for Mathematical Practice Must Be How We Do Math)

One of the last tasks that we gave our geometry students this year was Two Wheels and a Belt from Illustrative Mathematics.

A certain machine is to contain two wheels, one of radius 3 centimeters and one of radius 5 centimeters, whose centers are attached to points 14 centimeters apart. The manufacturer of this machine needs to produce a belt that will fit snugly around the two wheels, as shown in the diagram below. How long should the belt be?


The correct answer is 53.42 cm, which several students got, in more than one way.

Some used correct mathematical reasoning to get the correct answer.


Others used incorrect mathematical reasoning to get the correct answer.


It is unfortunate that the incorrect reasoning produced the correct answer. What is more unfortunate is that this incorrect reasoning goes unobserved when our focus is only on answers. When our focus is on how we do the math and not just on what we get as an answer, students and teachers can learn more about mathematics.

As soon as the students began to construct viable arguments and critique the reasoning of others, the misconceptions in the incorrect solution became evident.

And so the journey to help students know and understand mathematics continues …


Posted by on June 18, 2013 in Circles, Geometry, Right Triangles


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