Visual Representations Of Quadratic Expressions For .

Visual Representations of QuadraticExpressions for Students with LDTricia K. Strickland, Ph.D.NCTM Regional Conference – HartfordOctober 26, 2012Strickland, 20121

Advance Organizer Overview of my study Instructional Unit – Exploring QuadraticExpressions through Multiple Representations Data Implications for practiceStrickland, 20122

Rationale for my Dissertation Study Through my 12 years of teaching algebra to high schoolstudents with and without disabilities, I struggled to find away to make factoring quadratics accessible and fun. Common Core Standards -– Interpret the structure of expressions Interpret expressions that represent a quantity in terms of its context.– Interpret parts of an expression, such as terms, factors, and coefficients.– Interpret complicated expressions by viewing one or more of their parts as asingle entity. Use the structure of an expression to identify ways to rewrite it. Choose and produce an equivalent form of an expression to reveal andexplain properties of the quantity represented by the expression.– Factor a quadratic expression to reveal the zeros of the function it defines.– Perform arithmetic operations on polynomialsStrickland, 20123

Independent tical Features ofInstruction DesignCRA –IStrategyNCTM ProcessStandardsGraphicOrganizerStrickland, 2012Variety of TasksContextualizedTasks withTabular DataSymbolicalManipulation4

Concrete-Representational-AbstractIntegration StrategyStrickland, 20125

Transitioning from concretemanipulatives to abstract notationStrickland, 20126

Abstract Phase - Graphic Organizerx132x2x²26x1010x130Important to use with numbers to large for blocks orsketches.Strickland, 20127

Abstract Phase - Marcia’s Graphic OrganizerStrickland, 20128

Contextualized Problemswith Tabular DataStrickland, 20129

Instructional Unit on Multiplying LinearExpressions and FactoringQuadratic Trinomial Expressions Lesson 1 – Multiplying linear expressions to form aquadratic expression (positive terms only) Lesson 2 - Multiplying linear expressions to form aquadratic expression (positive and negative terms) Lesson 3 – Transitioning from algebra blocks toabstract notation with support from a graphicorganizer Lesson 4 – Multiplying linear expressionsembedded in area problems using the box method.Strickland, 201210

Instructional Unit Lesson 5 – Exploring factoring a quadratictrinomial through qualitative data Lesson 6 – Factoring using Algebra Lab Gear Lesson 7 – Discovering rules for factoringquadratics trinomials Lesson 8 – Factoring using the box method Lesson 9 – Transforming a quadraticexpression in standard-form to factored formvia area problems.Strickland, 201211

Lesson 3Launch:Our neighborhood swimming pool has a lengththat is 2-meters longer than the width.Use algebra blocks to determine the area of the pool.Sketch the blocksWrite the area equation: length · width area.Strickland, 201212

Lesson 3Launch:Our neighborhood swimming pool has a lengththat is 2-meters longer than the width.Use algebra blocks to determine the area of the pool.Sketch the blocksWrite the area equation: length · width area.(x 2) x x ² 2xStrickland, 201213

On Saturday mornings, a section of the swimming pool isroped off for swimming laps.That section makes the pool 1-meter shorter in width.Strickland, 201214

On Saturday mornings, a section of the swimming pool isroped off for swimming laps.That section makes the pool 1-meter shorter in width.Use algebra blocks to determine the area of the pool.Sketch the blocksWrite the area equation: length · width area.(x 2) (x – 1) x² x -2Strickland, 201215

Discussion Questions for Launch What happens to the shape of the pool? What happens to the size of pool? Which block representation is geometricallycorrect? Explain. How do we interpret the visual representationfor the incorrect geometric representation?Strickland, 201216

Lesson 3 Objective:Since we will not always be able to carry blocks aroundwith us, we need to learn a new way to multiplybinomials. Today we will use the box. Investigation:– Look at your rectangle that you just made with thealgebra blocks for the expression x2 x - 2.– Can you separate that rectangle into 4 sectionsthat would look something like this?– Physically move the blocks into the four sectionsand explain to us why you chose to place theblocks where you did.Strickland, 201217

We will be using a box that looks likethis:Physically move the blocks into the box and write the symbols.Strickland, 201218

Lesson 5Launch: Find the area of the rectanglebelow.x 2x 1Students should be using the box method and abstractnotation only.Strickland, 201219

Lesson Objective: We have been solving areaproblems by multiplying two linear binomialexpressions to create quadratic expression.Today we are going to start with the quadraticexpression and find the dimensions.Strickland, 201220

Investigation: The area of a rectangular fencedin backyard is x2 3x 2. What are thedimensions of the backyard?Each of us has a backyard with different xvalues. Let’s figure out the area of each of ourbackyards by completing the table below.Students use a graphing calculator to completetable. Students will be given a Cue Card.Strickland, 201221

Namex-value in yardsArea of backyardusing x2 3x 2Malka430Chana542Tricia656Strickland, 201222

Teacher modeling with “Think Aloud” What shapes of backyards can we make if weknow that the area of the rectangularbackyard is 30 square yards? On graph paper,draw all the different rectangles you can makewith an area of 30 square yards.Strickland, 201223

Student Activity – work with a partner Student 1: On graph paper, draw all thedifferent rectangles you can make with anarea of 42 square yards? Student 2: On graph paper, draw all thedifferent rectangles you can make with anarea of 56 square yards? Do you notice any similarities in thedimensions across these sets?Strickland, 201224

Discussion Questions How can we use symbols to show the patternwe just discovered? What are the dimensions of the anyrectangular backyard that has an area of x2 3x 2? Would this work for all values of x? Would thiswork when x is 20? (See next slide.) Explainhow you know it would work for any x-value?Strickland, 201225

Students’ worksamplesStrickland, 201226

Lesson 7 – Discovering rules forfactoring quadratics trinomials From Algebra Lab Gear (Piciotto, 1995) Discussion Questions:– What did you notice about the shape of therectangle when we changed the constant?– What did you notice about the shape of therectangle when we changed the linear coefficient?– Which term (constant or linear) had a greaterimpact on the shape of the rectangle?– Is there a greater increase in overall value / quantityof the rectangle when we increase the constant orlinear term?Strickland, 201227

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Lesson 8Factoring using the Box MethodX²45Area (Length) (width)x² 14x 45 Strickland, 201229

Results from Multiple Probe DesignStrickland, 201230

10080Cheryl Pretest60Cheryl Posttest40Cheryl Transfer20Cheryl Mainten010080Cindy Pretest60Cindy Posttest40Cindy Transfer20Cindy Mainten010080Sasha Pretest60Sasha Posttest40Sasha Transfer20Sasha Mainten010080Anna Pretest60Anna Posttest40Anna Transfer20Anna Mainten010080Marcia Pretest60Marcia Posttest40Marcia Transfer20012345678Strickland, 2012910Marcia Mainten11121314151631

Social Validity Intervention was beneficialWould recommend to other studentsLab Gear and Box Method most beneficialMixed results regarding– word problems– tables of data– discussing the task with teacher and/orclassmatesStrickland, 201232

Implications for Practice Access to general education curriculum Developed a rich understanding of quadraticsembedded within the area context Blending of instructional practices Possible Tier 2 intervention in RtI Affordable and feasibleStrickland, 201233

Questions / Commentsstrickland@hood.eduStrickland, 201234

Lesson 5 – Exploring factoring a quadratic trinomial through qualitative data Lesson 6 – Factoring using Algebra Lab Gear Lesson 7 – Discovering rules for factoring quadratics trinomials Lesson 8 – Factoring using the box method Lesson 9 – Transforming a quadratic expression in standard-form to