The Circumference of a Circle

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

Create an argument for comparing the height and circumference of the bottle.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

And sent another Quick Poll.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …

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Visual: SMP-7 Look for and Make Use of Structure #LL2LU

How do our learners determine an equivalent expression to 4(x+3)-2(x+3)? How would they determine the zeros of y=x²-4? How might we provide opportunities for them to successfully look for and make use of structure?   We want every learner in our care to be able to say I can make look for and make use of structure.  (CCSS.MATH.PRACTICE.MP7) But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?   One of the CCSS domains in the Algebra category is Seeing Structure in Expressions. Content-wise, we want learners to

• “use the structure of an expression to identify ways to rewrite it. For example, see x⁴–y⁴ as (x²)²–(y²)², thus recognizing it as a difference of squares that can be factored as (x²–y²)(x²+y²)”
• “factor a quadratic expression to reveal the zeros of the function it defines”
• “complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines”
• “use the properties of exponents to transform expressions for exponential functions”.

  How might we offer a pathway for success? What if we provide cues to guide learners and inspire noticing?   Level 4 I can integrate geometric and algebraic representations to confirm structure and patterning. Level 3 I can look for and make use of structure. Level 2 I can rewrite an expression into an equivalent form, draw an auxiliary line, or identify a pattern to make what isn’t pictured visible. Level 1 I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.   Illustrative Mathematics has several tasks to allow students to look for and make use of structure. We look forward to trying these, along with a leveled learning progression, with our students. 3.OA Patterns in the Multiplication Table 4.OA Multiples of 3, 6, and 7 5.OA Comparing Products 6.G Same Base and Height, Variation 1 A-SSE Seeing Structure in Expressions Tasks

Animal Populations

Delivery Trucks

Seeing Dots

Equivalent Expressions

  Leveled learning progression posters [Cross posted on Experiments in Learning by Doing]

 

SMP7: Look For and Make Use of Structure #LL2LU

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

I can look for and make use of structure.
(CCSS.MATH.PRACTICE.MP7)

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

 

Level 4
I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

 

Are observing, associating, questioning, and experimenting the foundations of this Standard for Mathematical Practice? It is about seeing things that aren’t readily visible.  It is about remix, composing and decomposing what is visible to understand in different ways.

How might we uncover and investigate patterns and symmetries? What if we teach the art of observation coupled with the art of inquiry?

In The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators, Dryer, Gregersen, and Christensen describe what stops us from asking questions.

So what stops you from asking questions? The two great inhibitors to questions are: (1) not wanting to look stupid, and (2) not willing to be viewed as uncooperative or disagreeable.  The first problem starts when we’re in elementary school; we don’t want to be seen as stupid by our friends or the teacher, and it is far safer to stay quiet.  So we learn not to ask disruptive questions. Unfortunately, for most of us, this pattern follows us into adulthood.

What if we facilitate art of questioning sessions where all questions are considered? In his post, Fear of Bad Ideas, Seth Godin writes:

But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones.  Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed.

How might we create safe harbors for ideas, questions, and observations? What if we encourage generating “bad ideas” so that we might uncover good ones? How might we shift perspectives to observe patterns and structure? What if we embrace the tactics for asking disruptive questions found in The Innovator’s DNA?

Tactic #1: Ask “what is” questions

Tactic #2: Ask “what caused” questions

Tactic #3: Ask “why and why not” questions

Tactic #4: Ask “what if” questions

 

What are barriers to finding structure? How else will we help learners look for and make use of structure?

 

[Cross posted on Experiments in Learning by Doing]

 

Dyer, Jeff, Hal B. Gregersen, and Clayton M. Christensen. The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators. Boston, MA: Harvard Business, 2011. Print.