FALL 2011 MATHEMATICS SOL INSTITUTES Grade Band Team Members Vickie Inge, University of Virginia Patricia Robertson, Arlington Public Schools, retired Beth Williams, Bedford County Public Schools Vandi Hodges, Hanover County Public Schools, retired GRADE BAND: 3-5

Today’s Institute Objective To improve mathematics instruction by providing district-level trainers with professional development resources focused on facilitating students' mathematical understanding through mathematical problem solving, communication, reasoning, connections, and representation. 2

Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 Work alone for 5 minutes to put these fractions in order from least to greatest. Do not use a common denominator or convert to a decimal. 3

Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 Work with a shoulder partner to justify that the fractions are in the correct order. Which ones were easier to determine? How did you think about completing this task? 4

Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 What are the big ideas about fractions brought forward by this task? Which fractions were the easiest to compare? Which fractions were the more challenging? 5

Mathematics Standards of Learning for Virginia Public Schools, 2009 Handout 1.Goals Introductory Paragraph 2.Mathematical Problem Solving Goal 3.Mathematical Communication Goal 4.Mathematical Reasoning Goal 5.Mathematical Connections Goal 6.Mathematical Representations Goal 6 Virginia Process Goals

7

Mathematical Process Skills Student Look-fors Read your assigned part of the handout. Underline the key ideas. Share findings between table partners. 8

Fraction Track Activity You need: Fraction Cards Fraction Track Game Board 20 beans /counters Play with 1 or 2 players or with 2 pairs. Playing to 1 Use only the fraction cards that are equal to 1 whole or less than 1. Mix the cards, and place the deck of cards facedown. Use one Completed Fraction Track Game Board that goes to 1 for each pair of players. Place seven beans on the game board, one on each track, at zero. Players take turns drawing the top card and moving a bean (or beans) to total the amount on the card. The goal is to use a series of moves to have beans land on exactly the number 1. When you land on 1 you win the bean. When a bean is won, place a new bean at 0 on the same track so that the next play has a bean on every track for the next player’s turn. If you cannot move the total amount of your Fraction Card you lose that turn. 9 Adapted from NCTM Illuminations

Fraction Track Activity Closing Reflection Questions: What were some of the strategies you used to play the game? What mathematical concepts are connected to the strategies? 10

Preparing to View the Video Use the Mathematical Process Skills -“Student Look-fors” Recording Form Handout while viewing the video. Review the indicators under the Skill Area your table was assigned. In the note’s section record specific evidence to support the indicator(s) observed in the assigned Skill Area. 11

Mathematics Process Skills “Student Look-fors” Table Group Debrief At the top of the paper, write the Skill Area the table group addressed. As a group select one or two the indicators in your assigned Skill Area that seemed to be most visible. On the chart paper record specific evidence for one or two indicators identified in the Skill Area. Post your chart on the wall. Return to your table and discuss charts for other Skill Areas. 12

13 4.2The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction. UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) ESSENTIAL UNDERSTANDINGSESSENTIAL KNOWLEDGE AND SKILLS A fraction is a way of representing part of a whole (as in a region/area model or a measurement model) or part of a group (as in a set model). A fraction is used to name a part of one thing or a part of a collection of things. In the area/region and length/measurement fraction models, the parts must be equal. In the set model, the elements of the set do not have to be equal (i.e., “What fraction of the class is wearing the color red?”). The denominator tells how many equal parts are in the whole or set. The numerator tells how many of those parts are being counted or described. When fractions have the same denominator, they are said to have “common denominators” or “like denominators.” Comparing fractions with like denominators involves comparing only the numerators. Strategies for comparing fractions having unlike denominators may include comparing fractions to familiar benchmarks (e.g., 0,, 1); finding equivalent fractions, using manipulative models such as fraction strips, number lines, fraction circles, rods, pattern blocks, cubes, Base-10 blocks, tangrams, graph paper, or a multiplication chart and patterns; and finding a common denominator by finding the least common multiple (LCM) of both denominators and then rewriting each fraction as an equivalent fraction, using the LCM as the denominator. A variety of fraction models should be used to expand students’ understanding of fractions and mixed numbers: Region/area models: a surface or area is subdivided into smaller equal parts, and each part is compared with the whole (e.g., fraction circles, pattern blocks, geoboards, grid paper, color tiles). Set models: the whole is understood to be a set of objects, and subsets of the whole make up fractional parts (e.g., counters, chips). Measurement models: similar to area models but lengths instead of areas are compared (e.g., fraction strips, rods, cubes, number lines, rulers).  A mixed number has two parts: a whole number and a fraction. Equivalent fractions name the same amount. Students should use a variety of models to identify different names for equivalent fractions. Students should focus on finding equivalent fractions of familiar fractions such as halves, thirds, fourths, sixths, eighths, tenths, and twelfths. Decimals and fractions represent the same relationships; however, they are presented in two different formats. The decimal 0.25 is written as. When presented with the fraction, the division expression representing a fraction is written as 3 divided by 5. All students should  Develop an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on a number line.  Understand that a mixed number is a fraction that has two parts: a whole number and a proper fraction. The mixed number is the sum of these two parts. Use models, benchmarks, and equivalent forms to judge the size of fractions. Recognize that a whole divided into nine equal parts has smaller parts than if the whole had been divided into five equal parts. Recognize and generate equivalent forms of commonly used fractions and decimals. Understand the division statement that represents a fraction. Understand that the more parts the whole is divided into, the smaller the parts (e.g., < ). The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Compare and order fractions having denominators of 12 or less, using manipulative models and drawings, such as region/area models. Compare and order fractions with like denominators by comparing number of parts (numerators) (e.g., < ). Compare and order fractions with like numerators and unlike denominators by comparing the size of the parts (e.g., < ). Compare and order fractions having unlike denominators of 12 or less by comparing the fractions to benchmarks (e.g., 0, or 1) to determine their relationships to the benchmarks or by finding a common denominator. Compare and order mixed numbers having denominators of 12 or less. Use the symbols >, <, and = to compare the numerical value of fractions and mixed numbers having denominators of 12 or less. Represent equivalent fractions through twelfths, using region/area models, set models, and measurement models. Identify the division statement that represents a fraction (e.g., means the same as 3 divided by 5). VADOE Curriculum Framework Grade 4

14

Analyze the unpacked standard. Identify the Mathematical Process Skills and specific indicators that are implicit and/or explicit in the Standard. Record ideas in the Classroom Instruction column. 15 analyze the unpacked standard and identify the Process Goals and where possible specific indicators that are implicit and/or explicit in the Standard. Record ideas in the Classroom Instruction column.

Encourage Mathematical Communication with Talk Moves 1.Teacher revoices a student’s reasoning for the purpose of clarification and advancing student thinking. 2.Student revoices another student’s reasoning to make sense themselves and advance the other students’ reasoning and understanding. 3.Asking students to justify or prove someone else’s reasoning. Using justification or proof to allow for respectful discussion of ideas. 4.Asking student to build on the group’s reasoning by connecting and extending another student’s idea. 5.Wait time (means to make the other things happen). 16

Task Sort Divide up the tasks so that at least 2 people will complete each task in the set. Write on the task whether you consider the task low or high cognitively demanding. Work independently before discussing with your partner whether you thought it was a low or high cognitively demanding task. Discuss the tasks that you and your partner solved. Read the remainder of the tasks and without working these tasks indicate if you think the task is a low or high cognitively demanding tasks. 17

Characteristics of Rich Mathematical Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content (Heibert et. al, 1997) Require Justification or explanation (Boaler & Staples, in press) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi &Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003) 18

Task Analysis Guide – Lower-level Demands Involve recall or memory of facts, rules, formulae, or definitions Involve exact reproduction of previously seen-material No connection of facts, rules, formulae, or definitions to concepts or underlying understandings Require limited cognitive demand Focused on producing correct answers rather than developing mathematical understandings Require no explanations or explanations that focus only on describing the procedure used to solve 19 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Task Analysis Guide – Higher-level Demands Focus on use of procedures for developing deeper levels of understanding of concepts and ideas Suggest broad general procedures with connections to conceptual ideas (not narrow algorithms) Provide multiple representations to develop understanding and connections 20 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Task Analysis Guide – Higher-level Demands Require complex, non-algorithmic thinking and considerable cognitive effort Require exploration and understanding of concepts, processes, or relationships Require accessing and applying prior knowledge and relevant experiences to facilitate connections Require task analysis and identification of limits to solutions 21 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press DOING Mathematics

Factors Associated with Impeding Higher-level Demands Shifting emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient or too much time to wrestle with the mathematical task Letting classroom management problems interfere with engagement in mathematical tasks Providing inappropriate tasks to a given group of students Failing to hold students accountable for high-level products or processes 22 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Consolidating Today’s Work Today we have looked at: Using rich task to engage students Using the Cognitive Demand Framework to analyze mathematical tasks Identifying the Mathematical Process Goals called for in the standards in the Virginia SOL Curriculum Framework Using the Process Goals and “Student Look-fors” in the classroom setting Encouraging mathematical conversation using Talk Moves in the classroom setting 23

Consolidating Today’s Work Exit Card: As you reflect on our work together today respond to the following question: When a classroom teacher pays attention to each of these areas how is student engagement and ultimately student learning impacted? 24