Category Archives: Algebra

SMP-6: attend to precision #LL2LU

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

I can attend to precision.
(CCSS.MATH.PRACTICE.MP6)

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 Easing the Hurry Syndrome]

Visual: SMP-8: look for and express regularity in repeated reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

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

I can look for and express regularity in repeated reasoning.
(CCSS.MATH.PRACTICE.MP8)

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But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Easing the Hurry Syndrome]


Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.

SMP-8: look for and express regularity in repeated reasoning #LL2LU

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

I can look for and express regularity in repeated reasoning. (CCSS.MATH.PRACTICE.MP8)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning?

What do we need to factor for the result to be (x-4)(x+4)?
What do we need to factor for the result to be (x-9)(x+9)?
What will the result be if we factor x²-121?
What will the result be if we factor x²-a2?

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We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1. (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers?

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Or over the set of complex numbers? 

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What about expanding the square of a binomial? 

What changes? What stays the same? What will the result be if we expand (x+5)²?  Or (x+a)²?  Or (x-a)²? 

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What about expanding the cube of a binomial?  Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Easing the Hurry Syndrome]

 

 

How do we use the December exam as formative assessment? (TBT Remix)

“For assessment to function formatively, the results have to be used to adjust teaching and learning; thus a significant aspect of any program will be the ways in which teachers make these adjustments.”
Inside the Black Box: Raising Standards Through Classroom Assessment Black and Wiliam

How do students reflect on their work?  What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?

I’m interested and curious about different strategies and methods used to help learners process and reflect on their exam experience and the accumulation of what they know.  Since each learner will have different bright spots and strengths, what strategies are used to differentiate for intervention and enrichment?

We aim to get “in the weeds” about reflection and intervention.  We want every child to reflect on what they could demonstrate well and where they need additional help.  We do not want them to move to the next year with any doubt or weakness if we can help now.  But, how do we know who needs help?  We collect data, and we let our learners gather data.  We need to be informed; they need to be informed.  We are a team working toward the goal of mastery or proficiency for all learners.

Our process:

  1. Return the exam to the learner on the first day back.
  2. Have each learner complete the exam analysis and reflection form (shown below) to identify strengths and areas of need.
        1. Circle the number of any missed problem.
        2. Begin, and possibly complete, correcting missed problems to review the material and determine if any error was a simple mistake or if more help is needed.
        3. Write a reflection about strengths, struggles, and goals.
        4. Report results on our team’s Google doc. (This is a copy; feel free to explore and “report” data to see how it feels. You can view the results here.)
  3. Meet in team to review all results and analyze for groups to design and provide necessary intervention and additional learning experiences.
  4. All assessments 2nd semester will have questions from first semester essential learnings to offer learners the opportunity to show growth and to help with retention.

We would love it if others would share methods and strategies for helping learners grow from an exam experience.  How do students reflect on their work?

What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?


How do we use the December exam as formative assessment? was originally posted on January 4, 2012.

SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 2)

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

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?
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A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?
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In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?

Level 4:
I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations.

Level 3:
I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.   

Beginning:
I can identify and connect the units involved using an equation, graph, or table.

Middle:
I can attend to and document the meaning of quantities throughout the problem-solving process.

End:
I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion.

Level 2:
I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:
I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

What evidence of contextualizing and decontextualizing do you see in the work below?

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[Cross-posted on Easing the Hurry Syndrome]

SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

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

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively.  (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)

Level 4:
I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:
I can reason abstractly and quantitatively.

Level 2:
I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:
I can define variables and constants in a problem situation and connect the appropriate units to each.

You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively.  Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway.  How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.

[Cross-posted on Easing the Hurry Syndrome]

 

Visual: SMP-5 Use Appropriate Tools Strategically #LL2LU

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

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

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Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display  confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

Suppose you are solving an equation.

Are you practicing use appropriate tools strategically if you use the numerical solve command on your graphing calculator?

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Or what about using your calculator to substitute values of x until you find a value that makes a true statement?

Screen Shot 2014-09-14 at 4.07.28 PM Are you practicing use appropriate tools strategically if you use a computer algebra system to explain your steps? 

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Or what if you use the graphing capability of your handheld?

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Consider each of the following learning goals:

I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and I can construct a viable argument to justify a solution method.  CCSS-M A-REI.A.1.

I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS-M A-REI.B.3.

I can explain why the x-coordinates of the points where the graphs of the equations y=(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. CCSS-M A-REI.D.11.

Does use appropriate tools strategically depend on the learner? Or the learning goal? Or the teacher? Or the availability of tools?

[Cross posted on Easing the Hurry Syndrome]