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Milligan College, 2015 ITQ Grant Program Dr. Lyn Howell Dr. Angela Hilton-Prillhart

Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction (Grades 3 – 5). Milligan College, 2015 ITQ Grant Program Dr. Lyn Howell Dr. Angela Hilton-Prillhart Dr. Jamie Price. June 15-19, 2015. Agenda for the Week. Monday, June 15

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Milligan College, 2015 ITQ Grant Program Dr. Lyn Howell Dr. Angela Hilton-Prillhart

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  1. Putting the Pieces Together: Literacy, Modeling, and Problem-Solving for Fraction Instruction (Grades 3 – 5) Milligan College, 2015 ITQ Grant Program Dr. Lyn Howell Dr. Angela Hilton-Prillhart Dr. Jamie Price June 15-19, 2015

  2. Agenda for the Week • Monday, June 15 • 8:30 AM – 11:30 AM: Morning Session (Whole Group) • 11:30 AM – 1:00 PM: Lunch (on your own) • 1:00 PM – 4:00 PM: Accountable Talk/Lesson Planning Introduction • Tuesday, June 16 – Thursday, June 18 • 8:30 AM – 11:30 AM: Morning Sessions • 8:30 AM – 10:00 AM: Math Workshop (Group A/B) • 10:00 AM – 11:30 AM: Literary Workshop (Group A/B) • 11:30 AM – 1:00 PM: Lunch (on your own) • 1:00 PM – 4:00 PM: Lesson Planning/Afternoon math workshop

  3. Agenda for the Week • Friday, June 19 • 8:30 AM – 10:30 AM – Post test/Concept Maps • 10:30 AM – 11:30 AM – Teacher Group Presentations • 11:30 AM – 1:00 PM – Lunch (on your own) • 1:00 PM – 4:00 PM – Teacher Group Presentations

  4. Workshop Expectations Keep students at the center. Be present and engaged. Monitor air time and share your voice. Challenge with respect. Stay solutions oriented. Risk productive struggle. Balance urgency and patience. BE OPEN TO NEW IDEAS!

  5. Introduction/Ice Breaker Some Call It Art

  6. Stages for Learning Mathematics Concrete: At this stage, students are introduced to a new concept with the aid of manipulatives/hands-on work Pictures (Representational): At this stage, students are able to draw pictures to explain their reasoning used to solve a problem. Students may draw pictures to indicate what they would have done with manipulatives. Symbols (Abstract): This is the most abstract stage. Students are able to use symbols (numbers, operation signs, algorithms, etc.) to solve the problem.

  7. Stages for Learning Mathematics While the stages for learning should progress in order as students learn a concept, once students reach the symbol (abstract) stage, they should understand the relationship between the symbols and the previous two stages.

  8. Adding It Up Framework(Adding It Up: Helping Children Learn Mathematics, NRC 2001) Five Strands of Mathematical Proficiency

  9. Five Strands of Mathematical Proficiency • Conceptual Understanding—comprehension of mathematical concepts, operations, and relations • Students know more than just isolated facts and methods • Students understand why a mathematical idea is important and the kinds of contexts in which it is useful • Students have organized their knowledge into a coherent whole • Conceptual understanding supports retention; students can reconstruct facts and methods that are forgotten when needed (p. 118)

  10. Five Strands of Mathematical Proficiency • Procedural Fluency – knowledge of procedures, when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently • Students need to be efficient and accurate in performing basic mathematical computations • Students need to be able to estimate the result of a procedure • Students use a variety of mental strategies to solve various problems (p. 121)

  11. Five Strands of Mathematical Proficiency • Strategic Competence – ability to formulate mathematical problems, represent them, and solve them • Students must first understand the situation and determine the key features • Generate a mathematical representation of the problem that captures the key features and ignores irrelevant ones (drawing, equation, graph, etc.) • Students come up with multiple approaches to solving the problem and choose flexibly among various approaches (reasoning, algebraic, guess and check) (p. 124)

  12. Five Strands of Mathematical Proficiency • Adaptive Reasoning – capacity to think logically about the relationships among concepts and situations • Adaptive Reasoning is the glue that holds everything together • Includes not only informal explanation and justification of a solution, but also intuitive and inductive reasoning based on pattern, analogy, and metaphor • Ability to justify one’s work (p. 124)

  13. Five Strands of Mathematical Proficiency • Productive Disposition – refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics • If students are to develop in any of the other strands of proficiency, then they must possess a productive disposition towards mathematics • A productive disposition develops when the other strands do and helps each of them develop (p. 131)

  14. Examining Student Work • Consider the student responses to the problem on the back of the Adding It Up Handout. • What mathematical strands of proficiency are shown in each student’s response? • What mathematical strands of proficiency appear to be missing?

  15. Reflecting On Your Practice • Take a few minutes to reflect on your own practice, especially related to teaching fractions. • Within your own practice, do you feel that you help students develop all strands of mathematical proficiency related to understanding fractions? • Are there particular strands that you feel that you develop more than others? Why do you think this is so? • Are there particular strands that you feel you never really develop? Why do you think this is so? • Other comments/thoughts?

  16. Keisha’s Paycheck Keisha receives her paycheck for the month. She spends 1/6 of it on food. She then spends 3/10 of what remains on her mortgage payment. She spends 3/7 of what is now left for her other bills, and 5/8 of what now remains for entertainment. This leaves her with $300. What was her original monthly take-home pay?

  17. The rectangle represents the paycheck (our whole). One of the six parts is shaded yellow to represent the amount Keisha pays for food. 5 equal size pieces remain.

  18. food 3/10 of remaining is spent on mortgage. But, only 5 equal size pieces remain. What should I do?

  19. food Cut each piece in half to get 10 white pieces (12 pieces now in the whole). The yellow (food) section is now divided into two pieces and represents 1/6 or 2/12.

  20. Food Rent food Shade 3 of the 10 white pieces. This is rent. 7 pieces remain.

  21. Food Rent Of the 7 pieces remaining (white pieces), Keisha spends 3/7 on other bills. Shade 3 white pieces pieces to represent other. What fraction of the whole paycheck is represented by food? What fraction of the whole paycheck is represented by rent?

  22. Other Food Rent 4 white pieces remain. But, 5/8 of the remaining is spent on entertainment. What do I do?

  23. Other Food Rent Fun I need 8 pieces, so each piece is cut in half again. Five are shaded blue. There are 3 white pieces remaining. How many pieces are now in the whole? What fraction of the whole paycheck is represented by food? Rent? Other? Fun?

  24. Other Food Rent Fun $300 The 3 white pieces represent $300 left from her paycheck. How much is each white piece?

  25. Other Food Rent Fun $300 Each piece is $100. How much does Keisha make? Keisha earns $2400.

  26. Keisha’s Paycheck and Fractions • Consider the Keisha’s Paycheck problem. • What concepts related to fractions are addressed in this single problem?

  27. Keisha’s Paycheck and Mathematical Proficiency • Consider the various solutions presented to the Keisha’s paycheck problem. • What strands of mathematical proficiency are addressed in each solution?

  28. Understanding Fractions Suppose you have a fraction of the form A . B We call the value of A the numerator of the fraction and we call the value of B the denominator of the fraction. What do the numerator and denominator represent in a given fraction problem?

  29. Types of Fraction Models

  30. Area Model vs. Set Model • What are some similarities to an area model for fractions and a set model for fractions? • What are some differences between these two models? • Which model was represented by the picture solution to the Keisha’s paycheck problem? • Fraction Hexagon Task Handout

  31. Fractions as Parts of Sets • Lesson One from Lessons for Introducing Fractions by Marilyn Burns (Teaching Arithmetic Series)

  32. Homework for Tuesday

  33. For Tuesday, read selections from Chapter 7 of Adding it Up: Helping Children Learn Mathematics • Read pp. 231 – 241 (stop at Proportional Reasoning) • Read pp. 246- 247 (section titled Beyond Whole Numbers) • Be ready to have a discussion of the reading in your math session on Tuesday

  34. Task-Based Lesson Plan

  35. What is a math task? A mathematical task is a problem or set of problems that focuses students’ attention on a particular mathematical idea and/or provides an opportunity to develop or use a particular mathematical habit of mind. from http://commoncoretools.me

  36. What are the characteristics of a high-quality math task? • A high-quality math task has the following characteristics: • Aligns with relevant mathematics content standard(s) • Encourages the use of multiple representations • Provides opportunities for students to develop and demonstrate the mathematical practices • Involves students in an inquiry-oriented or exploratory approach

  37. What are the characteristics of a high-quality math task? • Allows entry to the mathematics at a low level (all students can begin the task) but also has a high ceiling (some students can extend the activity to higher-level activities) • Connects previous knowledge to new learning • Allows for multiple solution approaches and strategies • Engages students in explaining the meaning of the result • Includes a relevant and interesting context from Putting Essential Understanding of Fractions into Practice (3-5), p. 8

  38. Your Task-Based Lesson Plan Requirements/Expectations • Work with the members of your team to create a task-based lesson plan which satisfies the following: • Addresses at least one Math Common Core State Standard from your grade level related to fractions • Addresses multiple Mathematical Practices • Problem for the task must relate to a chosen piece of literature selected by your team • Problem for the task must be original work • Complete the template provided • Be prepared to present your task with the members of your team on Friday morning or afternoon

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