supporting rigorous mathematics teaching and learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of EducationHigh School MathematicsAlgebra 1

Engaging In and Analyzing Teaching and Learning

RationaleAsking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

By engaging in a task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

Common Core State Standards for Mathematics, 2010


Session Goals

Participants will:

• develop a shared understanding of teaching and learning; and

• deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.


Overview of Activities

Participants will:

• engage in a lesson; and

• reflect on learning in relationship to the CCSS.


Looking Over the Standards• Look over the focus cluster standards.

• Briefly Turn and Talk with a partner about the meaning of the standards.

• We will return to the standards at the end of the lesson and consider:

What focus cluster standards were addressed in the lesson?

What gets “counted” as learning?


Bike and Truck Task

A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.D

ista

nce

from

sta

rt of

road

(in

feet

)

Time (in seconds)


Bike and Truck Task

1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.

2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.

3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words.

4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack.

The Structures and Routines of a Lesson

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

MONITOR: Teacher selects examples for the Share,

Discuss, and Analyze Phase based on:

• Different solution paths to the

same task

• Different representations

• Errors

• Misconceptions

SHARE: Students explain their methods, repeat others’

ideas, put ideas into their own words, add on to ideas

and ask for clarification.

REPEAT THE CYCLE FOR EACH

SOLUTION PATH

COMPARE: Students discuss similarities and

difference between solution paths.

FOCUS: Discuss the meaning of mathematical ideas in

each representation

REFLECT: by engaging students in a quick write or a

discussion of the process.

Set Up of the Task

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on

Key Mathematical Ideas

4. Engage in a Quick Write


Solve the Task(Private Think Time and Small Group Time)

• Work privately on the Bike and Truck Task.

• Work with a partner and then others at your table.

• Consider the information that can be determined about the two vehicles.


Expectations for Group Discussion

• Solution paths will be shared.

• Listen with the goals of:– putting the ideas into your own words;– adding on to the ideas of others;– making connections between solution paths;

and– asking questions about the ideas shared.

• The goal is to understand the mathematics and to make connections among the various solution paths.


Bike and Truck Task

A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.D

ista

nce

from

sta

rt of

road

(in

feet

)

Time (in seconds)


Bike and Truck Task

1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision.

2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description.

3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words.

4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack and why.


Discuss the Task(Whole Group Discussion)

• How did you describe the movement of the truck, as opposed to that of the bike? What information from the graph did you use to make those decisions?

• In what ways did you use the information you determined about the two vehicles to determine which vehicle was first to reach 300 feet from the start of the road?

• When, if ever, is the average rate of change the same for the two vehicles?


Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

• Which CCSS for Mathematical Content did we discuss?

• Which CCSS for Mathematical Practice did you use when solving the task?

Linking to Research/LiteratureConnections between Representations

Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987

Five Different Representations of a Function

Language

TableContext

Graph Equation

Van De Walle, 2004, p. 440

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Creating Equations* (A–CED)

Create equations that describe numbers or relationships.A-CED.A.1 Create equations and inequalities in one variable and use them to

solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

*Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Reasoning with Equations and Inequalities (A–REI)

Solve equations and inequalities in one variable.

A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A-REI.B.4 Solve quadratic equations in one variable.

A-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.


The CCSS for Mathematical ContentCCSS Conceptual Category – AlgebraReasoning with Equations and Inequalities (A–REI)Represent and solve equations and inequalities graphically.A-REI.D.10 Understand that the graph of an equation in two variables is the set of

all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(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. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.


The CCSS for Mathematical ContentCCSS Conceptual Category – FunctionsInterpreting Functions (F–IF)Interpret functions that arise in applications in terms of the context.F-IF.B.4 For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( )★ . Where an entire domain is marked with a star, each standard in that domain is a modeling standard.


What standards for mathematical practice made it possible for us to learn?

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.Common Core State Standards for Mathematics, 2010

Research Connection: Findings by Tharp and Gallimore

• For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.”

• They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support.

• For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.

Tharp & Gallimore, 1991


Underlying Mathematical Ideas Related to the Lesson (Essential Understandings)

• The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values.

• A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables (NCTM, EU 2b).

• The average rate of change is the change in the dependent variable over a specified interval in the domain. Linear functions are the only family of functions for which the average rate of change is the same on every interval in the domain.

Essential Understandings EU #1a: Functions are single-valued mappings from one set—the domain of the

function—to another—its range.

EU #1b: Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions.

EU #1c: The domain and range of functions do not have to be numbers. For example, 2-by-2 matrices can be viewed as representing functions whose domain and range are a two-dimensional vector space.

EU #2a: For functions that map real numbers to real numbers, certain patterns of covariation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has.

EU #2b: A rate of change describes how one variable quantity changes with respect to another—in other words, a rate of change describes the covariation between variables.

EU #2c: A function’s rate of change is one of the main characteristics that determine what kinds of real-world phenomena the function can model.

Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics.

Essential UnderstandingsEU #3a: Members of a family of functions share the same type of rate of change.

This characteristic rate of change determines the kinds of real-world phenomena that the function can model.

EU #3c: Quadratic functions are characterized by a linear rate of change, so the rate of change of the rate of change (the second derivative) of a quadratic function is constant. Reasoning about the vertex form of a quadratic allows deducing that the quadratic has a maximum or minimum value and that if the zeroes of the quadratic are real, they are symmetric about the x-coordinate of the maximum or minimum point.

EU #5a: Functions can be represented in various ways, including through algebraic means (e.g., equations), graphs, word descriptions, and tables.

EU #5b: Changing the way that a function is represented (e.g., algebraically, with a graph, in words or with a table) does not change the function, although different representations highlight different characteristics, and some may only show part of the function.

EU #5c: Some representations of a function may be more useful than others, depending on the context.

EU #5d: Links between algebraic and graphical representations of functions are especially important in studying relationships and change.

Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics.

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