The Surface Area of a Sphere

How do you help your students make sense of the formula for the surface area of a sphere?

We started with an orange that was spherical. We talked about a great circle, and we used string to draw some great circles for the sphere on a piece of paper.

If we peel the orange, and fill the circles with peeling, then how many circles will we fill? I sent a Quick Poll to get their estimates. 20% were correct, which reminds me that I must find a way to work in more estimation. (Or I’ll admit I actually wondered whether someone who taught below me could work in more estimation. I don’t know what to replace!)

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This year we didn’t actually peel the orange. I had been at #NCSM14 on the day we were going to do this in class, so after our conversation, we looked at a picture from another geometry class that their teacher had emailed me while I was out of town.


Four of the great circles were filled for this spherical grapefruit. At this point, I almost said the formula for the surface area of a sphere. My students would have nodded, and we would have moved on. But I stopped myself. Instead, I asked my students to write down on their paper what the formula was. I walked around, monitoring their work. I made myself wait on them. I thought the formula would be obvious, but it wasn’t obvious to many of my students. We talked more about the great circle, the dimensions of the great circle, and what those dimensions have to do with the sphere. A few more wrote down that the area of the surface area of the sphere was πr2*4. One student wrote down that the surface area of the sphere was πr3. I asked him to share what he had written with the rest of the class so that we could learn from this misconception. What would the units be if we calculated πr3? Cubic. What do cubic units represent? Volume. We can often use units to help us make sense of a formula.

Finally, we settled on what we would see in a textbook or on a formula chart if we looked up the surface area of a sphere: 4πr2. But it was hard won. Will my students remember it? The journey continues … with hopes that the image of the orange and 4 circles filled with peeling is imprinted in their memory.


Posted by on April 20, 2014 in Uncategorized


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Developing Conceptual Understanding of Fractions

Jill Gough and I presented a session at #NCSM14 entitled Developing Conceptual Understanding through the Progressions: Fractions, Ratios, and Proportions.

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Thanks to the work of Gail Burrill, Tom Dick, Wade Ellis, and Becky Byer, TI has just released the first lesson module in a series called Building Concepts. The first module is on Fractions, and it is available in its entirety on the TI website.

Proportional Reasoning is coming soon. Their work is based on the work of Wu: Teaching Fractions According to the Common Core Standards.

I have been saving two pictures from my daughter’s end of first grade scrapbook that I finally got to share in a PD session.


I love that fractions made Kate’s top 5 things learned in first grade. Kate also learned how to spell big words in first grade (except fractions).

On another page in her scrapbook, Kate notes that fractions are the best thing about first grade.


Look closely at Kate’s visual representation of the fraction 1/3. Kate’s picture is telling of one of the reasons we need to pay close attention to how we develop conceptual understanding of fractions for our students.

The Essential Learnings for our session:

  • I can describe a fraction a/b as a copies of 1/b.
  • I can construct questions that push and probe student thinking about fractions.
  • I can explain the role that technology plays in deepening student understanding of fractions.

While participants were gathering for the session, I sent a Quick Poll with a question from TIMSS 2011, Grade 8. Item M 02_04.

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Participants answered very closely to how US students answered.

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(And they made it easy for me to tell that 25% of them had it correct!)

US   29%
International   39%
Florida   29%
Massachusetts   47%
Korea   86%

We moved to the first activity, What is a Fraction?

tweet from Gayle about the fraction 5/3: 5 copies of 1/3. Love this CCSS language.


  • When you increase the value of D, how does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?
  • If a fraction is made up of more than five 1/5 unit fraction parts, what can you say about the value of the fraction?
  • When comparing two different unit fractions, how can you tell which fraction is greater?

I asked participants to think alone about the last question for a minute before sharing with their group. One participant noted that as the denominator of a unit fraction increases, more of them fit in a one-unit interval. Someone else noted that as the denominator of a unit fraction increases, the size of the fraction decreases. Someone else noted that the fraction farther to the right is greater.


True or False. Explain your reasoning.

  • 0 is a fraction.
  • A whole number cannot be a fraction.
  • A fraction can have many names.

We used Class Capture (taking a picture of each participants’ handheld) to discuss a few examples and counterexamples. (I wasn’t as coordinated during the PD session to take pictures of the Class Capture for this blog post as I sometimes am during class with my students!) A friend overheard one group saying that they knew whether the statements were true or false, but they were not confident that they could explain them.

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How can we increase the confidence of our teachers? How can we use visual representations of a fraction to know whether the statements are true or false? Will our students recognize the connection between what they are seeing on the number line and the truth of the given statements? Can the technology help them make their own conjectures about the validity of the statements instead of relying on their teacher for an explanation?


We moved to the second activity, Equivalent Fractions.

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What questions could you ask from looking at the first page?


As we moved to the second page of the TNS document, our focus became on how we know whether one fraction is larger than another fraction.

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Which is larger, 4/3 or 5/4? I love this question! I love that I can think through which fraction is greater using the idea that a/b is a copies of 1/b and not have to evoke any algorithmic procedure that I know for comparing fractions.


We moved to the 3rd activity, Fractions and Unit Squares.

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What is the area of the unit square? When the fraction is 1/5, what is the area of one of the rectangles?

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One the next page, we took some time to show that 4/16 of a unit square is equivalent to ¼ of a unit square.

I used Class Capture to monitor their work. Several groups showed that the shading for ¼ was the same amount for 4/16.

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I noted that none of them showed that 4/16 was equivalent to ¼ the way that I had. I saw each column as 1/4 of the whole. Each column is also 4/16.

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My students surprise me every day with seeing things differently from how I see them. What they see is beautiful. I just have to slow down and pay attention!


Our last exploration was to shade ¼ of a unit square and think about what is ½ of ¼.

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Our hour session had gone by very quickly. We revisited our essential learnings for the session.

  • I can describe a fraction a/b as a copies of 1/b.
  • I can construct questions that push and probe student thinking about fractions.
  • I can explain the role that technology plays in deepening student understanding of fractions.

At some point, I shared a picture that my geometry students had recently discussed when partitioning a segment.

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How might my high school students benefit from learning that a/b is a copies of 1/b? Will it help them when they think about partitioning the segment into equal parts?

How might learning that a/b is a copies of 1/b help my students when we are studying the unit circle and they are having to make sense of π/3, π/4, and π/6? Or comparing 2π/3 to 5π/6? Or adding π/6 to π/4?

I know I have so much more to learn about teaching fractions through conceptual understanding. The resources for these activities have been a good place to start.

And in case you are looking, another good place to learn is the Fractions Progression Module at Illustrative Mathematics.

Tonight’s IM Task Talk features two 5th grade tasks about fractions:

5.NF Connor and Makayla Discuss Multiplication

5.NF Fundraising

And so the journey continues, both as the learner and the teacher …


Posted by on April 15, 2014 in Uncategorized


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#NCSM14 Art of Questioning: Leading Learners to Level Up #LL2LU

What if we empower and embolden our learners to ask the questions they need to ask by improving the way we communicate and assess?

Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level.  Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)

On Monday, April 7, 2014, Jennifer Wilson (@jwilson828) and Jill Gough (@jgough) presented at the National Council of Supervisors of Mathematics Conference in New Orleans.

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Jill started with a personal story (you’re letting her shoot…) about actionable feedback and then gave the quick 4-minute Ignite talk on the foundational ideas supporting the Leading Learners to Level Up  philosophy.

Our hope was that many of our 130 participants would help us ideate to craft leveled learning progressions for implementing the Common Core State Standards Mathematical Practices.  Jennifer prompted participants to consider how we might building understanding and confidence with I can make sense of problems and persevere in solving them. After giving time for each participant to think, she prompted them to collaborate to describe how to coach learners to reach this target.  Jennifer shared our idea of how we might help learners grow in this practice.

Level 4:
I can find a second or third solution and describe how the pathways to these solutions relate.

Level 3:
I can make sense of problems and persevere in solving them.

Level 2:
I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.

Level 1:
I can show at least one attempt to investigate or solve the task.

 Participants then went right to work writing an essential learning – Level 3 - I can… statement and the learning progression around this essential learning. Artifacts of this work are captured on the #LL2LU Flickr page.

Here are the additional resources we shared:

How might we coach our learners into asking more questions? Not just any question – targeted questions.  What if we coach and develop the skill of questioning self-talk?

Interrogative self-talk, the researchers say, “may inspire thoughts about autonomous or intrinsically motivated reasons to purse a goal.”  As ample research has demonstrated, people are more likely to act, and to perform well, when the motivations come from intrinsic choices rather than from extrinsic pressures.  Declarative self-talk risks bypassing one’s motivations.  Questioning self-talk elicits the reasons for doing something and reminds people that many of those reasons come from within. (Pink, 103 pag.)

[Cross-posted on Experiments in Learning by Doing]


Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

Pink, Daniel H. To Sell Is Human: The Surprising Truth about Moving Others. New York: Riverhead, 2012. Print.

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Posted by on April 9, 2014 in Uncategorized


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Partitioning a Segment

CCSS-M G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

When I started writing this lesson last year, I did a Google search for ideas. What I found felt out of control to me, to say the least. Should our focus for this standard be on the formula that I found at this site?

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Or the explanation from this link?

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My search also led me to the curriculum site of the government of India, where I found some good and interesting tasks.

We started our lesson with partitioning vertical and horizontal segments.

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And we talked about the need for directed distance.

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When we were ready to move to an oblique segment, it occurred to me that maybe we should estimate where the point would be before we tried to calculate the coordinates of the point. I set up a Drop Points Quick Poll.

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75% of my students had the correct answer, although they didn’t all just estimate. (I had not marked a correct answer yet on purpose so that we could look at the graph and decide whether (3,-1) was the desired point.

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How did you get your answer? What did you notice? Some had calculated using the Pythagorean Theorem. They realized that the given segment was 3√13. So we needed to find Q such that QR was 2√13 and QS was √13. Another student was trying to come up with what he called a “modified midpoint” formula. Instead of finding the mean of the x coordinates of the endpoint, he wanted some kind of weighted mean to get the 2:1 ratio.

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I asked students what they saw in the picture.

Endpoints. A segment.

What else?

A triangle.

What kind of a triangle?
A right triangle. The segment is the hypotenuse of a right triangle.

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We found the lengths of the vertical and horizontal sides of the right triangle. Can that help us find the point that we need?

Students began to recognize several right triangles in the diagram that would be helpful. Each with the hypotenuse on the segment. Each with a vertical length of 2 (which is one-third of the whole length 6) and a horizontal length of 3 (which is one-third of the whole length 9).

We tried another problem, for which I had them enter the coordinates instead of dropping a point on the graph.

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Only 7 students have the correct response. What happened? Fractions, of course.

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Those who got it correct talked about what they did.

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And so we tried again.

This time we have 10 with the correct response until the students were dismissed to the faculty-student basketball game :-/

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At some point during class, I found what looked like a modified midpoint formula on another site.

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I included it in the students’ notes, but as I reviewed their summative assessments, I notice that no one used it. Instead, they made sense of partitioning a segment using similar triangles and proportional relationships. My hope, as the journey continues, is that they have a better chance of remembering those connections than they would a formula …


Posted by on April 5, 2014 in Uncategorized


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The Traveling Point

Locus of Points problems are not explicit in CCSS-M, but I think that they complement the G-GPE (Expressing Geometric Properties with Equations) domain nicely. I also like a question about the locus of points in a plane equidistant from two given points reminds students of the significance of the perpendicular bisector of a segment. And I am reminded by a phrase that resonated within me when I attended some Achieve the Core professional development a few months ago: The standards are the floor, not the ceiling.

The question:

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And the results – in equation form:

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And graph form. How can you tell that the points on the line are equidistant from the given points? What do you notice? How often do we ask a question like this without providing students the opportunity to connect the equation and the graph … by just asking for one or the other?

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One of the Standards for Mathematical Practice is to use appropriate tools strategically.

One of my own mathematics professors shared this problem with me after observing students work on it in a classroom in Japan. I have posed it to my students the past 8 years or so, and it works nicely with our locus of points lesson.


I find that the problem is difficult for some student to visualize. In fact, most students immediately jump to the conclusion that the point moves along the following path.

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Some students are successful when they just think about the problem (even though they some draw tentatively, in case they are wrong).

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But others need additional tools to help them “see” the math.

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Some groups were successful physically modeling the problem with an index card. One student traces the point while another student rotates the card.

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Less than half of the students were successful when I sent out the first poll. But after using appropriate tools strategically, more were successful. One group needed targeted help, which I was able to give as the others in the class moved to the next problem.

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Several years ago we began to use TI-Nspire to enhance the problem. For some students, seeing the rectangle rotate helps.

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Students use the dynamic geometry capabilities of TI-Nspire to rotate the rectangle – and then some choose to draw in the locus of points that A travels.


The physical motion that they notice as they use the rotation tool is what the rest of the students need to be successful.


This year easing the hurry syndrome meant we didn’t get to explore the quarter circle in much detail. Last year, we rotated a quarter circle in the same manner.


The first results are from asking my students the length of the path that A travels after they interacted only with pencil and paper. I didn’t mark the correct answer at this point – or even show my students the results.


At this point, I showed them the following & asked if they wanted to change their answer to the poll. (Thanks to Jeff for creating this document.)

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Students responded to this QP once they saw the object moving. You can see that more were successful. Again, I didn’t show students the results of their poll.


Next, I clicked “Show path” and let students watch.

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Do you want to keep your response? Or change it based on what you’ve now seen? Students responded to Quick Poll after seeing the trace of the path. All students didn’t need the trace – but some needed it to be able to visualize – and ultimately all students were successful.


There is another great task that we didn’t have time to explore in class last year or this year. But just posing the question is enough for a few students.

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What tools can help you build the locus of points that A travels? Or can you visualize how the string unwraps without tools?

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And so the journey continues, using tools strategically to help students visualize the results and to explore and deepen understanding of concepts …


Posted by on March 26, 2014 in Uncategorized


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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 Uncategorized


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What Is a Circle?

After reading a blog post What is a Circle that was linked to another blog post What’s in a circle, I asked my students what is a circle.

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rounded figure, no straight lines except the diameter, always 360°      1

A round 2D shape whose components are radius, diameter, and circumference. Its measure is 360 degrees, and other factors can be found with its components.         1

a regular geometric figure whos area is multiplied by pi            1

a geometric shape that is always symetrical and the ratio between its diameter and circumference is π.    1

a bunch of points that have same distance from a point which is center          1

a polygon that has no angles.          1

geometric shape that has no edges or angles and has a diameter and radius. 1

A round 2D figure that has an equidistance from any point on the circle to the center… a.k.a. constant radius       1

A round shape that curves in a consistant, unchanging motion until it reaches its starting point.     1

a round shape that has a diameter and radius that are used to find the area and circumference    1

a geometrical shape containing chords,diameters, radii, and right triangles, all composed in 360 degrees  1

a geometric figure from whose center all lines drawn to the outside are equivalent  1

a shape thats lines from the center to the edge all have equal distance           1

A closed shape with no straight edges in which a line drawn from the center to a point on the shape will always have the same distance.       1

x^(2)+y^(2)=36       1

a shape made up of points equal distant from a point    1

a shape that is round and measures to be  360°  1

a 2D round shape     1

a shape wiith no angles (corner) and an area of 360      1

area of πr^(2)           1

a geometric shape     1

a shape with no straight lines and all the lines that are drawn from its center to the outside are equal      1

a geometric shape with no sides or straight lines 1

a polgon with infinite lines of symmetry    1

a shape that all points are equal distance from the center         1

It is a geometrical shape without any corners.      1

A circle is a smooth figure that has no sides and contains a diameter,radius, and several arcs. It is always symetrical.      1

figure with rounded edges with a total of 360° and has an equal diameter all the way around       1

a shape with infinate points and no sides, just a constant curve connected.     1

a round shape that has an equal radius all around the center of that shape.  1

a 2d shape with a curved line that is 360 degrees           1


There was apparently too much time between my reading the post and my sending the question to my students. Because I completely forgot what was suggested to do with the results. I ran across the blog post again a few days later, so on our second day of circles in the coordinate plane, I gave a few of the definitions students had submitted & asked students to try to “break” the definition with a figure that met the requirements but wasn’t actually a circle.

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We are beginning to see the importance of attending to precision, not just with numerical solutions but also with our language.

I really like the idea to have students come up with a definition alone first – and then within groups before sharing with the whole class. And so I’m glad that the journey will continue next year, when I get a do over …

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Posted by on March 24, 2014 in Uncategorized


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