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SMP6: Attend to Precision #LL2LU

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

I can attend to precision.

CCSS.MATH.PRACTICE.MP6

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But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?

 

Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

 

How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written

y+4=3(x-2)

And then he wrote:

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He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.

 

Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

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Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]

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Systems of Equations – Take 3

What do you do with Systems of Equations in a high school Algebra 1 class?

Our students have some experience with systems from grade 8, but not as much as they eventually will with full implementation of our new standards.

NCTM’s Principles to Actions lists Establish mathematics goals to focus learning as one of the Mathematics Teaching Practices. We can tell that having a common language to talk about what we are doing is helping our students communicate to us about what they can and can’t (yet) do.

We started our unit on Creating Equations & Inequalities with the following leveled learning progression and questions:

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 3: I can solve a system of linear equations.

Level 4: I can create a system of equations to solve a problem.

 

NCTM’s Principles to Actions lists Elicit and use evidence of student thinking as one of the Mathematics Teaching Practices. We need to know what students are thinking so that we can move their thinking forward. We created a leveled learning formative assessment so that we could see where students are.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

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We have this y= Question configured to generate a graph preview. The equations students enter graph as they are typed, so the students are able to check their thinking as they go. (Note that we don’t have to configure the question to graph the equation as it is entered; we are choosing to do so while students are learning.)

The following are the results from one of our Algebra 1 classes. This is my favorite question.

How can we use the equation to know whether it contains the point (2,3)?

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How can we use the graph to know whether it contains the point (2,3)?Screen Shot 2014-11-05 at 2.36.56 PM

What happens when we give students a point & ask them to create two equations that contain the point?

Do they know that they are creating a system?

Does it help them to know that eventually we will give them the equations and ask them for the point?

 

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

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And results from the second question:

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Level 3: I can solve a system of linear equations.

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What’s significant about the green point?

What does it have to do with the given equations?

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At least half of our students don’t yet understand what the solution is.

 

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And only a very few are able to solve the system from the question on 3.3, which is okay. This is the first day of the unit. We have the information we need to know how to proceed with the lesson.

 

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I wrote about some of the tasks we used on the first two days of the unit here.

On the third day, we tried Dueling Discounts from Dan Meyer’s 101 Questions, which went better than Dan’s Internet Plans Makeover.

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And more:

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Maybe it’s time to generalize our results?

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On the last day we tried Candy & Chips from 101 Questions, which also went well.

NCTM’s Principles to Actions lists Build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. We know we aren’t there yet, but we are definitely making progress as 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. SMP7_Number SMP7_Algebra   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]

 
 

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SMP7: Look For and Make Use of Structure #LL2LU

SMP 7

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.

 
 

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