Building Polynomial Functions
Algebra Tasks That Promote Reasoning and Problem Solving Ben Sinwell NCTM Annual Meeting, San Antonio, TX April 7, 2017 [email protected] We will meet in Reflection Cove Number #4 after this session
Mathematics Teaching Practices 1. Establish Mathematics Goals to Focus Learning
2. Implement Tasks That Promote Reasoning and Problem Solving 3. Use and Connect Mathematical Representations 4. Facilitate Meaningful Mathematical Discourse 5. Pose Purposeful Questions
6. Build Procedural Fluency from Conceptual Understanding 7. Support Productive Struggle in Learning Mathematics 8. Elicit and Use Evidence of Student Thinking -- From NCTM’s Principles to Actions
Which One Doesn’t Belong (www.wodb.ca)
Choose one of the ‘boxes’ above and write a sentence or two that explains why it does not belong. Repeat this for another ‘box.’ I will ask you to read your explanation(s) to your partner. You may be asked to share with the group (class).
Mathematics Teaching Practices 1. Establish Mathematics Goals to Focus Learning
2. Implement Tasks That Promote Reasoning and Problem Solving 3. Use and Connect Mathematical Representations 4. Facilitate Meaningful Mathematical Discourse 5. Pose Purposeful Questions
6. Build Procedural Fluency from Conceptual Understanding 7. Support Productive Struggle in Learning Mathematics 8. Elicit and Use Evidence of Student Thinking -- From NCTM’s Principles to Actions
Adapting Tasks • Textbooks relabeling old content as “Common Core” without much actual revision • Lots of good problems in textbooks and online, but often too helpful & do too much problem formulation for students • Still use other people’s resources, but adapt to incorporate SMP • Adaptations must be consistent with and advance your objective. We will meet in Reflection Cove Number #4 after this session.
Worthwhile Tasks • Focused on important mathematics; clear mathematical goal • Provide opportunities for discussion • Provoke thinking and reasoning about the mathematics; high level of cognitive demand • Engage students in the mathematical practices and/or process standards • Create a space in which students “wonder, notice, are curious” SSTP, 2013
Objectives Look at and employ strategies for adapting tasks to one’s current practice: • Reversibility
• Flexibility • Generalization
Source: “Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach” (Dougherty et al., 2014) Adapted in collaboration with Alyssa Hosler and Jason Slowbe.
Questioning Strategies for Adapting Tasks Reversibility Instead of: What is the vertex of: f(x) = (x – 3)2 + 5 ?
Do this: Write a function where the vertex is (3, 5). “Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach” (Dougherty et al., 2014)
Reversibility for Algebra • Instead of finding the x-intercept of a linear function…
• Find coefficients for y = ax + b such that the xcoordinate of the x-intercept is positive.
Questioning Strategies for Adapting Tasks Reversibility Instead of: What is the asymptote of: f(x) =
Do this: Write a function where the asymptote is x = 2
Reversibility for Algebra • Think about the mathematics you are teaching (or just taught) and write a reversibility question. • Feel free to work on one with your shoulder partner.
Reversibility is like working backward from the answer to a problem.
Questioning Strategies for Adapting Tasks Generalization Instead of: Solve: Do this: “Write a quadratic equation whose solutions are integers.” “Is it possible to predict if the solutions are integers just by looking at the equation?”
Questioning Strategies for Adapting Tasks Generalization Instead of: Solve: Multiply xm by xn Do this: “Pick a value for m and n and give an equivalent expression. Choose two more values. Repeat. What do you notice mathematically.”
Generalization for Algebra 1 or 2 • The x-coordinate of the vertex of y = x2 – 4x – 5 is at x = 2. – Write another quadratic function whose vertex is at x = 2. – Write another such function… – …and another such function… – Now write an easy rule describing ALL of the quadratic functions whose vertex is at x = 2.
Generalization for Algebra 2 • For FACTOR BY GROUPING: The binomial (x + 3) is a factor of the cubic y = x3 + 3x2 – 6x – 18. –Change two of the four coefficients to obtain a different cubic for which (x + 3) is still a factor. –Find another… and another…
Generalization for Algebra 2 • For FACTOR BY GROUPING: The binomial (x + 3) is a factor of the cubic y = x3 + 3x2 – 6x – 18. –Now change the other two coefficients you have not already changed to obtain a different cubic with (x + 3) as a factor. –Write a rule describing all of the cubic functions for which (x + 3) can be factored by grouping. –Does this rule work for all cubic functions with (x + 3) as a factor?
Generalization for Algebra • Think about the mathematics you are teaching (or just taught) and write a generalization question. • Feel free to work on one with your shoulder partner.
Generalization questions get at the structure of the mathematics or repeated reasoning.
Questioning Strategies for Adapting Tasks Flexibility • Solve 2x – 8 = 3x + 4 Now solve this equation a different way or • Solve 2x – 8 = 12 2(x+2) – 8 = 12 2(2x+2) – 8 = 12
Flexibility for Algebra Part 1: • For 3(x – 5) – 7 = 26. • Now solve for x using a different set of algebraic steps. • Which method do you prefer? Why?
Flexibility for Algebra Part 2: • For 3(x – 5) – 7 = 25. • Solve for x using the same two methods from Part 1. • Which method do you prefer? Explain.
Flexibility for Algebra Part 3: What kinds of values for c in 3(x – 5) – 7 = c would you definitely prefer one method over the other? Explain.
Flexibility for Algebra 2 Part 1: • For 2∙3x = 162, solve for x. • Now solve for x using a different set of algebraic steps. • Which method do you prefer? Why?
Flexibility for Algebra 2 Part 2: • For 2∙3x = 163, solve for x using the same two methods from Part 1. • Which method do you prefer? Explain.
Flexibility for Algebra 2 Part 3: What kinds of values for c in 2∙3x = c would you definitely prefer one method over the other? Explain.
Flexibility for Algebra • Think about the mathematics you are teaching (or just taught) and write a flexibility question. • Feel free to work on one with your shoulder partner.
Flexibility questions get at doing problems in different ways or with different methods (and can include their pros and cons).
Strategies for Adapting Tasks Writing prompt Ask students to make and defend a decision Aiden: “When the degree of a polynomial increases, its number of changes in direction also increases.” Do you agree with Aiden? Why or why not?
Strategies for Adapting Tasks Writing prompt Ask students to make and defend a decision Delisha: “When the exponent in an exponential function increases, the out put increases.” Do you agree with Delisha? Why or why not?
Strategies for Adapting Tasks Uniqueness Is there more than one solution? How do you know?
Strategies for Adapting Tasks Opening Them Up
Making it not so teacher-centered. Give the students some freedom. Non-prescriptive. Learning how to Learn
Having students explain.
Reflect What did you find interesting or surprising about adapting tasks today? What is something that you learned or thought about different?
Worthwhile Tasks • Focused on important mathematics; clear mathematical goal • Provide opportunities for discussion • Provoke thinking and reasoning about the mathematics; high level of cognitive demand • Engage students in the mathematical practices and/or process standards • Create a space in which students “wonder, notice, are curious” SSTP, 2013
Which One Doesn’t Belong (www.wodb.ca)
Write a sentence explaining why one of the ‘boxes’ above does not belong? Be prepared to share your reasoning.
Which One Doesn’t Belong (www.wodb.ca)
Using a graph, explain which one doesn’t belong. Using an equation, explain… Create your own which one doesn’t belong. Replace the ‘box’ you chose with a different table…
Which One Doesn’t Belong (www.wodb.ca)
• Choose new values for y that don’t change the problem. • Choose new values for x … • Write a general rule that explains how to replace the values.
Reflect Mathematics Teaching Practices 1.Establish Mathematics Goals to Focus Learning 2.Implement Tasks That Promote Reasoning and Problem Solving 3.Use and Connect Mathematical Representations 4.Facilitate Meaningful Mathematical Discourse 5.Pose Purposeful Questions
6.Build Procedural Fluency from Conceptual Understanding 7.Support Productive Struggle in Learning Mathematics 8.Elicit and Use Evidence of Student Thinking 37
Thank You [email protected] We will meet in Reflection Cove Number #4 after this session.
Mathematics Teaching Practices 1. Establish Mathematics Goals to Focus Learning
2. Implement Tasks That Promote Reasoning and Problem Solving 3. Use and Connect Mathematical Representations 4. Facilitate Meaningful Mathematical Discourse 5. Pose Purposeful Questions
6. Build Procedural Fluency from Conceptual Understanding 7. Support Productive Struggle in Learning Mathematics 8. Elicit and Use Evidence of Student Thinking -- From NCTM’s Principles to Actions
Which One Doesn’t Belong (www.wodb.ca)
Choose one of the ‘boxes’ above and write a sentence or two that explains why it does not belong. Repeat this for another ‘box.’ I will ask you to read your explanation(s) to your partner. You may be asked to share with the group (class).
Mathematics Teaching Practices 1. Establish Mathematics Goals to Focus Learning
2. Implement Tasks That Promote Reasoning and Problem Solving 3. Use and Connect Mathematical Representations 4. Facilitate Meaningful Mathematical Discourse 5. Pose Purposeful Questions
6. Build Procedural Fluency from Conceptual Understanding 7. Support Productive Struggle in Learning Mathematics 8. Elicit and Use Evidence of Student Thinking -- From NCTM’s Principles to Actions
Adapting Tasks • Textbooks relabeling old content as “Common Core” without much actual revision • Lots of good problems in textbooks and online, but often too helpful & do too much problem formulation for students • Still use other people’s resources, but adapt to incorporate SMP • Adaptations must be consistent with and advance your objective. We will meet in Reflection Cove Number #4 after this session.
Worthwhile Tasks • Focused on important mathematics; clear mathematical goal • Provide opportunities for discussion • Provoke thinking and reasoning about the mathematics; high level of cognitive demand • Engage students in the mathematical practices and/or process standards • Create a space in which students “wonder, notice, are curious” SSTP, 2013
Objectives Look at and employ strategies for adapting tasks to one’s current practice: • Reversibility
• Flexibility • Generalization
Source: “Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach” (Dougherty et al., 2014) Adapted in collaboration with Alyssa Hosler and Jason Slowbe.
Questioning Strategies for Adapting Tasks Reversibility Instead of: What is the vertex of: f(x) = (x – 3)2 + 5 ?
Do this: Write a function where the vertex is (3, 5). “Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach” (Dougherty et al., 2014)
Reversibility for Algebra • Instead of finding the x-intercept of a linear function…
• Find coefficients for y = ax + b such that the xcoordinate of the x-intercept is positive.
Questioning Strategies for Adapting Tasks Reversibility Instead of: What is the asymptote of: f(x) =
Do this: Write a function where the asymptote is x = 2
Reversibility for Algebra • Think about the mathematics you are teaching (or just taught) and write a reversibility question. • Feel free to work on one with your shoulder partner.
Reversibility is like working backward from the answer to a problem.
Questioning Strategies for Adapting Tasks Generalization Instead of: Solve: Do this: “Write a quadratic equation whose solutions are integers.” “Is it possible to predict if the solutions are integers just by looking at the equation?”
Questioning Strategies for Adapting Tasks Generalization Instead of: Solve: Multiply xm by xn Do this: “Pick a value for m and n and give an equivalent expression. Choose two more values. Repeat. What do you notice mathematically.”
Generalization for Algebra 1 or 2 • The x-coordinate of the vertex of y = x2 – 4x – 5 is at x = 2. – Write another quadratic function whose vertex is at x = 2. – Write another such function… – …and another such function… – Now write an easy rule describing ALL of the quadratic functions whose vertex is at x = 2.
Generalization for Algebra 2 • For FACTOR BY GROUPING: The binomial (x + 3) is a factor of the cubic y = x3 + 3x2 – 6x – 18. –Change two of the four coefficients to obtain a different cubic for which (x + 3) is still a factor. –Find another… and another…
Generalization for Algebra 2 • For FACTOR BY GROUPING: The binomial (x + 3) is a factor of the cubic y = x3 + 3x2 – 6x – 18. –Now change the other two coefficients you have not already changed to obtain a different cubic with (x + 3) as a factor. –Write a rule describing all of the cubic functions for which (x + 3) can be factored by grouping. –Does this rule work for all cubic functions with (x + 3) as a factor?
Generalization for Algebra • Think about the mathematics you are teaching (or just taught) and write a generalization question. • Feel free to work on one with your shoulder partner.
Generalization questions get at the structure of the mathematics or repeated reasoning.
Questioning Strategies for Adapting Tasks Flexibility • Solve 2x – 8 = 3x + 4 Now solve this equation a different way or • Solve 2x – 8 = 12 2(x+2) – 8 = 12 2(2x+2) – 8 = 12
Flexibility for Algebra Part 1: • For 3(x – 5) – 7 = 26. • Now solve for x using a different set of algebraic steps. • Which method do you prefer? Why?
Flexibility for Algebra Part 2: • For 3(x – 5) – 7 = 25. • Solve for x using the same two methods from Part 1. • Which method do you prefer? Explain.
Flexibility for Algebra Part 3: What kinds of values for c in 3(x – 5) – 7 = c would you definitely prefer one method over the other? Explain.
Flexibility for Algebra 2 Part 1: • For 2∙3x = 162, solve for x. • Now solve for x using a different set of algebraic steps. • Which method do you prefer? Why?
Flexibility for Algebra 2 Part 2: • For 2∙3x = 163, solve for x using the same two methods from Part 1. • Which method do you prefer? Explain.
Flexibility for Algebra 2 Part 3: What kinds of values for c in 2∙3x = c would you definitely prefer one method over the other? Explain.
Flexibility for Algebra • Think about the mathematics you are teaching (or just taught) and write a flexibility question. • Feel free to work on one with your shoulder partner.
Flexibility questions get at doing problems in different ways or with different methods (and can include their pros and cons).
Strategies for Adapting Tasks Writing prompt Ask students to make and defend a decision Aiden: “When the degree of a polynomial increases, its number of changes in direction also increases.” Do you agree with Aiden? Why or why not?
Strategies for Adapting Tasks Writing prompt Ask students to make and defend a decision Delisha: “When the exponent in an exponential function increases, the out put increases.” Do you agree with Delisha? Why or why not?
Strategies for Adapting Tasks Uniqueness Is there more than one solution? How do you know?
Strategies for Adapting Tasks Opening Them Up
Making it not so teacher-centered. Give the students some freedom. Non-prescriptive. Learning how to Learn
Having students explain.
Reflect What did you find interesting or surprising about adapting tasks today? What is something that you learned or thought about different?
Worthwhile Tasks • Focused on important mathematics; clear mathematical goal • Provide opportunities for discussion • Provoke thinking and reasoning about the mathematics; high level of cognitive demand • Engage students in the mathematical practices and/or process standards • Create a space in which students “wonder, notice, are curious” SSTP, 2013
Which One Doesn’t Belong (www.wodb.ca)
Write a sentence explaining why one of the ‘boxes’ above does not belong? Be prepared to share your reasoning.
Which One Doesn’t Belong (www.wodb.ca)
Using a graph, explain which one doesn’t belong. Using an equation, explain… Create your own which one doesn’t belong. Replace the ‘box’ you chose with a different table…
Which One Doesn’t Belong (www.wodb.ca)
• Choose new values for y that don’t change the problem. • Choose new values for x … • Write a general rule that explains how to replace the values.
Reflect Mathematics Teaching Practices 1.Establish Mathematics Goals to Focus Learning 2.Implement Tasks That Promote Reasoning and Problem Solving 3.Use and Connect Mathematical Representations 4.Facilitate Meaningful Mathematical Discourse 5.Pose Purposeful Questions
6.Build Procedural Fluency from Conceptual Understanding 7.Support Productive Struggle in Learning Mathematics 8.Elicit and Use Evidence of Student Thinking 37
Thank You [email protected] We will meet in Reflection Cove Number #4 after this session.
