Efficient Polynomial Multiplication, Division, Factoring, and Completing the Square

Transcription

1 Efficient Polynomial Multiplication, Division, Factoring, and Completing the Square 2011 NCTM Regional Conference & Eposition Albuquerque Raymond C. Johnson School of Education University of Colorado at Boulder Freudenthal Institute US November 3, 2011 This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

2 Disclaimer Two things before we begin: Don t give me credit. Teachers are as teachers do. Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

3 How important are polynomials? Very early in our mathematical education in fact in junior high school or early in high school itself we are introduced to polynomials. For a seemingly endless amount of time we are drilled, to the point of utter boredom, in factoring them, multiplying them, dividing them, simplifying them. Facility in factoring a quadratic becomes confused with genuine mathematical talent. I. N. Herstein, Topics in Algebra (1975, p. 153) Image (GNU FDL) Wikimedia Commons

4 Approaches to algebra Traditional Manipulating variables Solving equations Simplifying epressions Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

5 Approaches to algebra Traditional Manipulating variables Solving equations Simplifying epressions Civilized Functions Multiple representations Contet Table Graph Equation Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

6 The makings of a learning progression Assuming students can: multiply with both constants and variables; find the area of a rectangle; use the distributive property; and model the distributive property and partial products with a subdivided rectangle; Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

7 The makings of a learning progression Assuming students can: multiply with both constants and variables; find the area of a rectangle; use the distributive property; and model the distributive property and partial products with a subdivided rectangle; we want to build a learning progression that includes: multiplying polynomials; factoring quadratics; completing the square; and polynomial division; Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

8 The makings of a learning progression Assuming students can: multiply with both constants and variables; find the area of a rectangle; use the distributive property; and model the distributive property and partial products with a subdivided rectangle; we want to build a learning progression that includes: multiplying polynomials; factoring quadratics; completing the square; and polynomial division; so that students have the formal algebra skills to simplify polynomial epressions and solve polynomial equations. Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

9 Theory: Realisitic Mathematics Education (RME) Founded by Dutch mathematician Hans Freudenthal Realistic properly translated is to imagine Progressive formalization / iceberg metaphor Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

10 Theory: Realisitic Mathematics Education (RME) Founded by Dutch mathematician Hans Freudenthal Realistic properly translated is to imagine Progressive formalization / iceberg metaphor What is a contet for quadratics? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

11 The strengths and weaknesses of projectile motion Contet Table Graph Equation Image (CC) Flickr

12 Visualizing the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) (Slower Formalization) (Faster Formalization) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

13 The reality of the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

14 What this learning progression isn t This progression is likely to start midway through an Algebra 1 course and end in the second half of Algebra 2. It is not all the mathematics in between. Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

15 What this learning progression isn t This progression is likely to start midway through an Algebra 1 course and end in the second half of Algebra 2. It is not all the mathematics in between. This progression is not a complete guide to learning about polynomials. Other contets and other models would be needed in parallel learning progressions. Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

16 What this learning progression isn t This progression is likely to start midway through an Algebra 1 course and end in the second half of Algebra 2. It is not all the mathematics in between. This progression is not a complete guide to learning about polynomials. Other contets and other models would be needed in parallel learning progressions. The intersection of this learning progression with parallel learning progressions is left as an eercise to the reader. Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

17 What this learning progression isn t This progression is likely to start midway through an Algebra 1 course and end in the second half of Algebra 2. It is not all the mathematics in between. This progression is not a complete guide to learning about polynomials. Other contets and other models would be needed in parallel learning progressions. The intersection of this learning progression with parallel learning progressions is left as an eercise to the reader. If learning progressions have dimensions, think of this one as long and thin, not short and wide. (Or macro instead of micro.) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

18 Contet Image (CC) Flickr House Garden

19 Building on partial products Given the dimensions of Peter s house and patios, find the area of the house, each patio, and garden. 40 ft 8 ft 30 ft House 5 ft Garden Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

20 Building on partial products Given the dimensions of Peter s house and patios, find the area of the house, each patio, and garden. 40 ft 8 ft 40 ft 8 ft 30 ft House 30 ft House 1200 sq ft 240 sq ft 5 ft Garden 5 ft 200 sq ft Garden 40 sq ft What is the total area of the house, patios, and garden? Can you find it more than one way? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

21 Building on partial products Given the dimensions of Peter s house and patios, find the area of the house, each patio, and garden. 40 ft 8 ft 40 ft 8 ft 30 ft House 30 ft House 1200 sq ft 240 sq ft 5 ft Garden 5 ft 200 sq ft Garden 40 sq ft What is the total area of the house, patios, and garden? Can you find it more than one way? As a sum: = 1680 sq ft As a product: (40 + 8)(30 + 5) = (48)(35) = 1680 sq ft Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

22 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? House 5 ft Garden Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

23 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? = 2 sq ft House How can you write the area of each patio? 5 ft Garden Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

24 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? = 2 sq ft House How can you write the area of each patio? 8 = 8 sq ft and 5 = 5 sq ft What is the area of the garden? 5 ft Garden Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

25 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? = 2 sq ft House How can you write the area of each patio? 8 = 8 sq ft and 5 = 5 sq ft 5 ft Garden What is the area of the garden? 5 8 = 40 sq ft Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

26 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? = 2 sq ft House How can you write the area of each patio? 8 = 8 sq ft and 5 = 5 sq ft 5 ft Garden What is the area of the garden? 5 8 = 40 sq ft How can you write the total area of the house, patios, and garden? Is there more than one way? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

27 Peter s friend Lisa wants to have patios and a garden, too. Peter knows Lisa s house is square, but doesn t know how big, so he just labels the length and width of Lisa s house. 8 ft How can you write the area of the house? = 2 sq ft House How can you write the area of each patio? 8 = 8 sq ft and 5 = 5 sq ft 5 ft Garden What is the area of the garden? 5 8 = 40 sq ft How can you write the total area of the house, patios, and garden? Is there more than one way? As a sum: sq ft As a product: ( + 8)( + 5) sq ft Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

28 Models of to models for Sometimes we want to transition away from less-formal contets and models. Other times we have to. Consider the multiplication of ( + 2)( 3): In contet: -3 ft With Algebra Tiles: 3 House 2 sq ft 3 sq ft 2 ft 2 sq ft Garden 6 sq ft +2 (Not realistic ) (Still useful) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

29 Models of to models for Now consider the multiplication of ( )(3 + 4) With Algebra Tiles: Dimensions? With Bo/Table: What role should the FOIL strategy play in this progression? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

30 The reality of the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

31 Content progression: What are the dimensions? Robin wants patios and a garden net to his house, arranged in a rectangle the same way Peter and Lisa have theirs arranged. Robin has sq ft of space. Use Algebra Tiles to model Robin s house, patios, and gardens. House Garden Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

32 Content progression: What are the dimensions? Robin wants patios and a garden net to his house, arranged in a rectangle the same way Peter and Lisa have theirs arranged. Robin has sq ft of space. Use Algebra Tiles to model Robin s house, patios, and gardens. House Garden Which arrangement makes a rectangle? What are its dimensions? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

33 Models for grouping : a ab b a + b y b y b a ya ab 8 7 What is the pattern here? Where in the model progression do students have to be for it to work? How do you guide this re-invention? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

34 Factoring problem string Problem #4 with bo/table: If students are still dependent on Algebra Tiles, where do you epect them to struggle? Why? What happened with problem #6? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

35 The reality of the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

36 Content progression: What if we can t make a rectangle? If you re trying to solve = 0, which of these equivalencies offers a way forward? ( + 5)( + 1) ( + 4)( + 2) ( + 3) 2 5 Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

37 Completing the square of ?? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

38 Completing the square of ? 2 3? +3 3 Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

39 Completing the square of ? ? ( + 3) 2 5 = 0 Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

40 Completing the square of ? ? ( + 3) 2 5 = 0 ( + 3) 2 = 5 ( + 3) 2 = ± = ± 5 = 3 ± 5 What role should the quadratic formula play in this progression? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

41 Completing the square problem string Problem #3 with bo/table: 2 If students are still dependent on Algebra Tiles, where do you epect them to struggle? Why? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

42 The reality of the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

43 Content progression: Dividing polynomials The last problem in multiplying polynomials was ( )(3 + 4): Knowing the patterns of like terms, can you fill in what s missing if the product is ? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

44 Content progression: Dividing polynomials The last problem in multiplying polynomials was ( )(3 + 4): Knowing the patterns of like terms, can you fill in what s missing if the product is ? What are the factors (dimensions) of the second bo? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

45 Knowing patterns of like terms, can you fill in what s missing if the product is ? 3 +5 Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

46 Knowing patterns of like terms, can you fill in what s missing if the product is ? 3 +5 How is this different than asking students to divide ? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

47 Dividing polynomials problem string What role should long and synthetic division play in this progression? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

48 The reality of the learning progression Multiplying Polynomials Factoring Quadratics Completing the Square Polynomial Division Area Contet Algebra Tiles Bo/Table Formal ( Above Water ) Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

49 Discussion How much do we need to teach quadratics and polynomials? Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

50 Discussion How much do we need to teach quadratics and polynomials? Does it help to classify learning progressions? content vs. models vs. student learning macro vs. micro parallel and intersections continuous vs. discrete Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29

51 Discussion How much do we need to teach quadratics and polynomials? Does it help to classify learning progressions? content vs. models vs. student learning macro vs. micro parallel and intersections continuous vs. discrete Resources for this progression College Prepatory Mathematics (CPM) - Pamela Weber Harris Building Powerful Numeracy for Middle + High School Students NCTM s Focus in High School Mathematics: Reasoning and Sense Making Contact information raymond.johnson@colorado.edu Raymond C. Johnson (NCTM 2011) Efficient Polynomial Progression November 3, / 29