Tag Archives: construct a viable argument and critique the reasoning of others

The Circumference of a Cylinder

We talked about pi earlier this week in geometry, and we used Andrew Stadel’s water bottle question to start.

I’m not one to pull of the wager that Andrew used (unfortunately, my students will agree that I am a bit too serious for that), but we still had an interesting conversation.

Compare the circumference and height of the water bottle.

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Here’s what they estimated by themselves.

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Then they faced left if they thought height > circumference, straight if =, and right if height < circumference. (I saw Andrew lead this at CMC-South year before last … I certainly didn’t think of it myself.) They found someone who agreed with their answer, and practiced I can construct a viable argument and critique the reasoning of others.

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Next they found a second person who agreed, and practiced I can construct a viable argument and critique the reasoning of others again. (By this time, we decided it was easier to raise 1, 2, or 3 fingers based on answer choice rather than turn a certain direction as it was a challenge for some to see someone turned the same direction.) Finally, they found someone who disagreed, and practiced I can construct a viable argument and critique the reasoning of others.

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I sent the poll again.

It didn’t change much.

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So without discussion, I sent a poll with a bit more context … a cylindrical can holding 3 tennis balls. Would the can of tennis balls help them reason abstractly and quantitatively?

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Apparently not.

Here’s what they thought by themselves.

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And here’s what they thought after talking with someone else.

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The clock was ticking. I still wanted us to talk about pi. I asked someone who correctly answered to share her thinking with the rest of the class to convince them. And we used string to show that the water bottle circumference was, in fact, longer than its height.

I intended to follow up with this Quick Poll. But I was in a hurry and forgot. Maybe next year.

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You can find more number sense ideas from Andrew here.

I’ll look forward to hearing about how they play out in your classroom, as the journey continues …


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Popcorn Picker + #ShowYourWork

Towards the end of our geometry course last year, we focused on students being able to say:

I can show my work.

How often do our students understand what we mean when we say, “show your work”?

Jill Gough’s Show Your Work learning progression has been an important addition to our classroom.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem.

A correct solution isn’t enough … we want the reader (and sometimes grader) to understand our solution without having to ask any questions.

We continue to use Dan Meyer’s Popcorn Picker 3-Act, even though I keep thinking we shouldn’t need to do this in high school. The Quick Poll results, however, provide evidence that we aren’t wasting our time.

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You can read more about how the 3-Act lesson played out in last year’s post.

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My purpose for posting about the lesson again is to consider the value added when we ask students to practice “show your work” and provide students the opportunity to give other students feedback on what they’ve shown … when we provide students an opportunity to practice SMP3, “I can construct a viable argument and critique the reasoning of others”.

We used the “I like …, I wish …, What if (or I wonder) …” protocol for providing feedback.

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Kato Nims recently posted Illuminating Success and Growth, where she shares her students’ first attempts at giving feedback to each other this year. She writes, “It is clear to me that as our work together continues that it will be important for me to model how to give effective feedback so that we can all benefit from the unique perspectives that are represented in our classroom.”

Some of the feedback that my students gave each other is helpful, but more of it is not. How do we teach students to give productive feedback to each other? Would my giving feedback on the feedback have been helpful? At what point have we spent too much time on this activity and need to call it “done”? But then how often am I so focused on teaching content that I neglect to provide students the opportunity to grow as learners? Isn’t learning how to give feedback important for all of us? The tasks we choose are so important … which ones will further both our content and our practice learning goals?

And so the journey continues … with many more questions than answers.


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#AskDontTell: Pythagorean Relationships

I have been invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom. You can read my second post here. While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

My post starts with a quote by one of my students: A few weeks ago, I overheard one student telling another, “Will you help me figure this out? Don’t just tell me how to do it.” How many of the students in our care are thinking the same thing? How often do we tell them how to do mathematics? How often do we provide them with “Ask, Don’t Tell” opportunities to learn mathematics?

After reading the post, John Golden tweeted the following:

John had no idea that I happen to be reading Creating Cultures of Thinking by Ron Ritchart (you can preview the first chapter at the link), and so the language that he chose to use was timely. I’m deep in the midst of thinking about how we teach our students to learn … about the cultures that we are creating with our students.

Ritchart quotes Lev Vygotsky: “Children grow into the intellectual life of those around them.” And then says himself, “… learning to learn is an apprenticeship in which we don’t so much learn from others as we learn with others in the midst of authentic activities.” [p. 20]

Ritchart later asks, “What difference does it make if a teachers asks, ‘Is your work done?’ or ‘Where are you in your learning?’” [p. 44]

I wonder what you think. Does it matter whether we ask our students whether they are finished with their work or where they are in their learning? I think it might. Focusing on the learning instead of the work creates a culture of thinking. Focusing on the learning instead of the work causes students to say, “Will you help me figure this out? Don’t just tell me how to do it.”

And so the journey of creating cultures of thinking continues …


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Which One Doesn’t Belong?

You’ve seen “which one is different” before.

(I first remember seeing this particular question from John Bament at a T3 session in 2014, although he might have gotten it from somewhere else. He sent it to the participants as a Quick Poll and showed us our quite varied results.)

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You’ve seen “Odd One Out” before.

These two images come from the Mathematics Assessment Project formative assessment lesson on Comparing Investments.

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I observed this lesson in a classroom a few weeks ago. It didn’t bother students that more than one answer can be correct, and they naturally explained why they chose what they did without the teacher even having to prompt them with “How did you get that?” or “Why?”

My coworker and I introduced Christopher Danielson’s Which One Doesn’t Belong to our beginning K-2 teachers recently. They began to think immediately about how they could do something similar with language as well as math. (And they were thrilled to learn something in PD that they could immediately take back to their classrooms.)

When I recently learned about Mary’s Which One Doesn’t Belong site, I decided to spend some time on it during our recent Math PLC meeting.

We started with a page from Christopher’s shape book. Our assistant principal (former history teacher) was thrilled to be able to immediately participate in our discussion. (How many of our students feel the same when we offer them low-floor, high-ceiling tasks?)

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We did a number WODB (one teacher fist-pumped another assistant principal when they figured out that 9 didn’t belong since the sum of its digits isn’t 7). Thanks, Pam!

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Then we moved to Rachel Fruin’s geometry Which One Doesn’t Belong. Our history teacher-turned assistant principal was still able to participate. She didn’t have the same vocabulary that the rest of the math teachers in our department had when stating why one doesn’t belong, but she learned some math vocabulary and we learned to see the images through different eyes during our shared experience.

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We ended our PLC with Hunter Patton’s Graphs & Equations 7.

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I recently heard that one measure of the success of professional development is whether the teacher’s practice changes as a result of what was learned. (Another part to this would of course be how long the teacher’s practice changes … one lesson? A few lessons? Or permanent change in lessons?) So I was thrilled to notice that the teacher with whom I share a room gave her precalculus students a WODB to try at the end of their opener later that day.

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They were studying rational functions. Which one doesn’t belong?

Before I knew it, students were in different corners of the room based on their initial responses.

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They shared thoughts with each other before sharing with the whole class.

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I tried the geometry WODB with my geometry students yesterday. I asked them to send me their response so that I could decide whether moving to one of the four corners of the room would be worthwhile. I asked bottom left to gather, bottom right to gather, and then top left & top right to gather. Why doesn’t your choice belong?

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Now work on your mathematical flexibility. Instead of being satisfied with one way to answer, find multiple responses.

Find a reason that each one doesn’t belong, and let me know when you do by selecting that choice on the new Quick Poll (now multiple response).

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Now sorted by individual responses so I can see which students need support:

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I’ve offered problem solving points for students who create their own WODB, and I look forward to seeing the results. Thank you, Mary, for creating a place for us to share and learn together … for creating a site that our teachers were able to immediately incorporate into their own learning and their students’ learning.


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Half of a Square

Sketch a smaller square inside the given square so that the area of the smaller square is half the area of the larger square. Write an explanation to convince another that your new, smaller square is truly half the area of the original square.

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I’m sure that I didn’t write this question, but I don’t remember from where it came.

What would your students get right?

What misconceptions might they have?

I was surprised at some of my students’ misconceptions (and miscalculations) and explanations.

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Measuring errors.

Calculation errors. Lots of “simple” calculation errors: Half of 36 is 16. Half of 25 is 13.25. Half of 5 is 3.5.
More measuring errors.

Square vs. rectangle errors.

And someone getting at the immeasurability of irrational numbers.

The test was summative. But even though the test was summative in my gradebook, I can still use the responses to inform the learning experiences that I give my students in the future. And so I will, as the journey continues …


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Conditional Statements & Instructional Adjustments

I wrote recently about A-S-N-T-F.

I also wrote recently about what happens when students don’t recognize that a statement and its converse are different. We explicitly worked on conditional statements after that lesson.

So we defined converse/inverse/contrapositive/biconditional using symbolic logic, and then students decided which was which for a given conditional statement: If a shape has four sides, then the shape is a rectangle.

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What happens when you change which statement is the conditional statement?

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Again, students figured out which was which using the “new” conditional.

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We formatively assessed their progress on conditional statements:

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And then we were ready to start thinking about the truth value of the statements.

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Or maybe we weren’t ready actually ready, but we started thinking about the truth value anyway.

I never click on the student results for a Quick Poll in front of the students for the first time. My projector remote is an extension of my hand (except when I’ve carelessly laid it down and can’t find it), and so while students are working, I freeze the screen to look at the results and decide whether to make an instructional adjustment and if so, what instructional adjustment to make before I show the results to the students. Sometimes (as is the case in the contrapositive QP above), I have time to go talk to students who have answered incorrectly to clear up misconceptions while other students are still working. Sometimes (as is the case with this question), I deselect “Show Correct Answer” before displaying the results to students.

A student usually asks, “So who is correct?”

To which I reply, “So who is correct?”

I asked students to find someone in the room at a different table who answered differently. Convince them you are correct. Then let them convince you they are correct. (Practice construct a viable argument and critique the reasoning of others.) Then send in your answer again.

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Sometimes the responses change to 100% correct. Sometimes they don’t. And so I have to decide my next instructional adjustment. Do we have time to try this again? Or is the clock ticking more quickly?

My decision this time was to have a student come draw her counterexample.

What do we mean when we say two angles are supplementary?

Does that counterexample convince those who say the statement is true?

What if we are evaluating whether this statement is A-S-N?

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And so the journey continues, grateful for technology that gives every student in my classroom a voice – from the quietest to the loudest – so that I can make more informed decisions for when to make instructional adjustments.

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Posted by on November 29, 2014 in Angles & Triangles, Geometry


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What’s My Rule

Instead of having a whole lesson of What’s My Rule explorations, we are adding one exploration to each bellringer during our unit on Rigid Motions. From yesterday:

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Students move point Z and observe how W follows. Z is mapped to W according to some rule that the students are trying to determine

I’ve written about this exploration before, so I want to focus on what was different this year.

Students constructed viable arguments and critiqued the reasoning of others. We are learning how to attend to precision, so we were lenient in giving credit to responses for which the oral explanation helped us make sense of the written explanation.

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One team wrote that if you added something to Z and subtracted something from W, then the points would map onto one another. I wouldn’t have worded what they were trying to say like they did. But they were getting at some important mathematics. Ultimately, they were trying to convey that Z and W are the same distance from the origin. We constructed a circle with the origin as the center and Z as one of the points on the circle and noticed that both Z and W always lie on the circle.

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A few other students said that the rule was to reflect Z over the line y=x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=x?

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Others thought that the rule was to reflect Z over the line y=-x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=-x?

Other students noticed that we could describe the rule using a rotation of Z 180˚ about the origin.

No one noticed that we could reflect Z about the x-axis and then about the y-axis. So what happens when no one notices something we want them to notice? I could have moved on. It wouldn’t have been detrimental to my students learning of mathematics if they didn’t know that. But I didn’t. Instead I asked whether there was a reflection that we could use to map Z onto W. I gave students just seconds to think alone and then time to talk with their teams. I monitored their team talk. 5 teams said that we could reflect Z about the perpendicular bisector of segment ZW to map Z onto W. Yes. Not what I was expecting … but absolutely true.

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One team said that we could reflect Z about the line y=x and then about the line y=-x to map Z onto W. Oh…we can reflect Z about y=x and then y=-x? How can you show that?

What happens when you reflect (x,y) about y=x? (y,x)

What happens when you reflect (y,x) about y=-x? (-x,-y)

Is there another sequence of reflections that will map Z onto W?

Reflecting about y=-x and then y=x.

Is there another sequence of reflections that will map Z onto W?

Teams worked together – and after another few minutes, they figure out that reflecting about y=0 and then x=0 would work. Or reflecting about x=0 and then y=0.

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And then we were called to the cafeteria for school pictures.

And then a student came up to me in the line for school pictures and asked whether there would be an infinite number of pairs of lines about which we could reflect Z onto W.

Are there an infinite number of pairs of lines that will work?

What relationship do the pairs of lines have that we found?

y=x and y=-x; y=0 and x=0

What is significant about the pairs of lines?

After a few more questions, the students around us in line for pictures noted that the lines are perpendicular.

So if we reflected Z about y=2x, then about what other line would we need to reflect Z’ to get W?

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I will be the first to admit both that of course all of this makes sense mathematically, and also that I’ve never thought about it before. And so the journey continues … ever grateful for the students with whom I learn.


Thanks to Michael Pershan for sharing Transformation Rules.


Posted by on August 21, 2014 in Coordinate Geometry, Geometry, Rigid Motions


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A Tank of Water

Towards the end of last semester, we worked on a task that Tom Reardon shares as The Great Applied Problem.

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What is your question? (Note that this lesson was before I read Michael Pershan’s post considering how we ask students about what we can explore mathematically.)

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why are we always looking from the top and never the side      1

other than circles and rectangles what are the cross sections of shapes?         1

are the 2 bases the same size          1

what are the sizes of the different cross sections?            1

i wonder how to find the volume of a partially filled cylinder in which the base of the liquid is a sector     1

how many cubic units of water are in the shape currenty?        1

is the water filling up or flowing out. what is the length of each cord as it fills 1

I wonder if there is a relationship between the diameter of the base and the change in dimensions of the rectangle…    1

the areas of the bases created by the water         1

What is the total volume of the tank          1

what is the smallest rectangle possible to be a cross section?     1

at what rate does the volume change        1

is it filling up or draining     1

How much water is in the cylinder?           1

how much water will the cylinder hold      1

how much water would it take to fill the tank       1

I wonder what the volume is of the water in the tube.    1

how much water will it hold            1

will the water flow through the straw        1

what is the volme ofthe water         1

what is the volume of the water in the cylinder?  1

does the circumference change      1

how would the shape change as the cylinder shift           1

whats the ratio of the volume of water to the cylinder    1

how much more water do you need to fill it up    1

How much water can the tank hold?          1

how will the volume change if the water increases          1

What are the measures of its radius and horizontal height, or of either if the volume is given?        1


We settled on how much water is in the tank.

What is the least amount of information you need to answer the question?

Teams worked together to make a list of the measurements they wanted to use for their calculations. Very few teams wanted the same information. Some differences were minute, such as one wanting the radius and another wanting the diameter. Or one wanting the depth of the water and another wanting the distance between the center and the chord. Or one wanting the radius and another the length of the chord. Some differences were bigger, such as one wanting the ratio of water to total volume.

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What information was I willing to give them?

Thanks to a spreadsheet included in Tom’s problem, I had plenty of measurements from which to choose to give students. But I hadn’t thought through whether I was willing to give the ratio of water to total volume instead of the length of the radius. I ended up not giving the ratio. But I did give some teams the length of the chord instead of the radius.

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As I watched students work, I noticed that they had the opportunity to look for and make use of structure. Dylan Wiliam talks about asking students questions that push their thinking forward and probe their understanding. This task did just that.

What misconceptions would students have about this figure?

We talked about what’s there that’s not there. What do you see that isn’t pictured?

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A semicircle

What else do you see that isn’t pictured?

A diameter

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What else do you see that isn’t pictured?

A radius that forms a right triangle with half of the chord.

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What else do you see that isn’t pictured?

An isosceles triangle formed by two radii and the chord.

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What else do you see that isn’t pictured?

A sector.

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What else do you see that isn’t pictured?

The region formed by decomposing the sector into a triangle and a segment of a circle.

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Is that same region a semicircle?
It’s not. But that is what a lot of students thought when they first looked.


We are learning to look deeper to make sense of the structure before we jump into calculating.

One student reflected on working through this task in class. She also talks about using the practice make sense of problems and persevere in solving them: “trying to find what we needed to know”, “tried different ways to find the area”, “drew diagrams”, “made a plan”, “discussed different approaches”. She also talks about the math practice construct viable arguments and critique the reasoning of others: “We all decided what we wanted to know to figure out how much water was in the tank. And then we tried to explain our reasoning to the class. We all discussed what we wanted to know then decided together what we really needed to know.”

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I’ve used this task for several years, but I’ve never introduced it like this before. Previously, I’ve asked my students only to calculate.

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And so the journey to be less helpful continues …


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Evaluating Statements about Enlargements

We recently used the Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Enlargements.

I had just returned from NCSM where I heard Tim Kanold’s session “Beyond Teaching for Understanding: The elements of an authentic formative assessment process”. In the session, he suggested that no more than 35% of class should be the teacher leading from the front of the classroom. I was determined to figure out how this played out in my classroom when I got back to school. I also found a blog post where he talks about leaving the front of the classroom behind.

We started with Candy Rings.

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When I got the results from the first poll, I knew I was in trouble. Why did you choose “correct”, Amber? Two of the small rings has the same total circumference as one of the large rings. Why did you choose “incorrect”, Ryan? Some of the pieces on the larger ring look broken. Those on the smaller ring are closer together.

Against my better judgment, I asked the next question. After all, construct a viable argument and critique the reasoning of others is how we are learning math, right?

If the price of the small ring of candy is 40 cents, what is a fair price for a large one?

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About half of the class used proportional reasoning, deciding that 80 cents was fair. The other half used business reasoning. Some included tax. Some decided that the larger portion should have a bit of a discount. All of them had an argument for why they chose what they did.


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No one thought that Jasmina reasoned correctly about the amount of pizzas.

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And only a few insisted on using business sense to come up with a fair price for a large pizza.

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Students then worked with their teams to determine whether their cards with statements about enlargements were true or false. I gave each team 6 cards to evaluate. They reasoned abstractly and quantitatively. They constructed viable arguments and critiqued the reasoning of others. They even had a good time.

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This particular class clocked in at exactly 65% peer-to-peer discussion. Even then, the 35% whole group discourse wasn’t just one raised student hand at a time. I used the Quick Poll results to selectively call on students who don’t always raise their hand. They presented their argument and the class decided whether to buy it or not. I was there to facilitate the conversation and move it forward. I’m not sure whether that only counts as “leading from the front of the classroom”.

Either way, the journey to leave the front of the classroom behind continues …


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Length & Area – Sliding Triangles

We started our unit on Geometric Measure & Dimension with a Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Length and Area.

I chose the card that each group would explore ahead of time. I only gave each group one card, and I had another task ready for them if they finished early. They didn’t. Students chose whether they wanted to explore on paper or using technology.

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Each card had a hint, which I held until I felt like the group might need it. While students were working, I monitored their progress.

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The second group had Sliding Triangles.

If you slide the top corner of a triangle from left to right, its area stays the same.

If you slide the top corner of a triangle from left to right, its perimeter changes.

They read through their card and drew a few diagrams, but they decided to spend most of their time building the scenario using our dynamic geometry software. This group did not need the hint card.

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When it was time for this group to present their work, we sent the Quick Polls so that we would know what students instinctually thought, even though they had not all had time to explore the statements in depth. I am learning to make use of the TI-Nspire Navigator allowing me to send more than one Quick Poll at a time.

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I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

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I used the Live Presenter feature of Navigator to make one of the student’s calculators live on the projector at the front of the room. They grabbed and moved the point that changed the top vertex of the triangle, noting with the square they had constructed off the base that the height of the triangle remained constant.

The area of the triangle will always be the same.

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What about the perimeter? The perimeter changes.

Does the perimeter always change? Sometimes the perimeter is the same.

When is the perimeter the same? When one triangle is a reflection of the other about the perpendicular bisector of the base.

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And so the journey continues …



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