6-8 Interventionists Training Session 2 September 25, 2013

Today’s Learning Targets We are learning to … Understand the meaning and importance of “explicit and systematic instruction.” Deepen our understanding of the Number and Operations: Fractions Domain

Success Criteria We will be successful when we can… Identify what “systematic and explicit instruction” looks like during PIE time. Clearly explain the mathematical content in selected Grade 3-5 CCSSM standards and be able to provide examples of the mathematics.

A Research Based Approach to Intervention

Emerging Key Research Themes Increased instructional time in addition to core mathematics taught in Tier 1. Small-group instruction utilized in all tiers Explicit methods of instruction (e.g., C  R  A, Talk Moves) Use of concrete and pictorial representations to facilitate conceptual understanding Strategy instruction for problem solving (e.g., Think Aloud) Focus on problem solving skills (not just computation) Careful alignment of instruction and content in Tier 1 and Tier 2 Screening and progress monitoring to target deficit areas Source: Adapted from Newman-Gonchar, R., Clarke, B., & Gersten r. (2009). A summary of nine key studies: Multitier intervention and response to interventions for students struggling in mathematics.Retrieved from

IES Practice Guide: Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle School Students US Department of Education Research-based education practices Committee Chair: Russell Gersten Published by: What Works Clearinghouse (April 2009)

Explicit and Systematic Instruction What is it? How might it look with my PIE group?

Explicit and Systematic Instruction Read and highlight Recommendation 3 With your group come to consensus on 3-4 important points you would like to share with the group.

Explicit & Systematic Instruction Summarize key aspects of “explicit and systematic instruction” as defined by this reading. What do we need to know as teachers to do this well in our intervention groups?

A structure for explicit and systematic instruction Symbols Give a context: tell a story Explain orally and/or in writing Make a picture Use concrete models: manipulatives

Laying the Foundation Initial Fraction Concepts Supporting the Transition From the Whole Number World

Read the grade level focus for 3 rd through 5 th Grade 3rd Grade: Read 3 rd Grade Critical Area #2 p. 21 Review 3 rd grade Cluster Statements p th Grade: Read 4 th Grade Critical Area #2 p Review 4 th Grade Cluster Statements p.48 5 th Grade: Read 5 th Grade Critical Area #1 p. 33 Review 5 th Grade Cluster Statements p Be prepared to summarize the Standards’ progression of Number and Operations: Fractions. 3 rd through 5 th Progression

Where does the difficulty with fractions begin? Research says… Premature experience with formal procedures may lead to symbolic knowledge that is not based on understanding impeding students’ number and operation sense. Some students “have a continuing interference from their knowledge of whole numbers.” Difficulty reasoning with fraction symbols as quantities. Knowledge that is dependent primarily upon memory, rather than anchored with a deeper understanding of foundational concepts, contributes to incorrect use and misunderstanding of formal algorithms. ---A Focus on Fractions

Learning Target: Deepen our understanding of the Number and Operations: Fractions domain Examine fractions as numbers using models Deepen understanding of partitioning Understand and use unit fraction reasoning Read and interpret the cluster of CCSS standards related to fraction concept development.

So let’s step back into the elementary world for a minute….

Fractions as Numbers 3 4 What are ways we want students to “see” and “think about” fractions?

CCSSM 1.G.3 and 2.G.3 How does fraction work begin in Grade 1 and Grade 2?

Examining Partitioning....“early experiences with physically partitioning objects or sets of objects may be as important to a child’s development of fraction concepts as counting is to their development of whole number concepts” (Behr and Post, 1992)

Stages of Partitioning Read pgs. 71 – 75 (through stages of partitioning) Highlight key phrases from the reading * Star the important ideas ? Question mark the confusing thoughts Table group discussion summarizing and clarifying thoughts Work through problem number 1 on pages 77 & 78

Whiteboard Work Take a slate and divide it in thirds On each 1/3 draw a model; a area model, a set model, and a number line. Discuss the features of each model.

Features of Models What is the whole? How are equal parts defined? What does the fraction indicate? Area Model The whole is determined by the area of a defined region Equal area The part covered of whole unit area Set Model The whole is determined by definition (of what is in the set) Equal number of objects The count of objects in the subset of the defined set of objects. Number Line Unit of distance or length (continuous) Equal distance The location of a point in relation to the distance from zero with regard to the defined unit.

Fractions Composed of Unit Fractions Fold your fraction strip to show ¾ How do you see this fraction as ‘unit fractions’?

Making Fraction Strips White: whole Green:halves, fourths, eighths Yellow:thirds, sixths, ninths ?:twelfths Note relationships among the fractions as you fold. Remember – no labels.

Looking at a Whole Arrange the open fraction strips in front of you. Look at the thirds strip. How do you see the number 1 on this strip using unit fractions? In pairs, practice stating the relationship between the whole and the number of unit fractions in that whole (e.g., 3/3 is three parts of size 1/3).

CCSSM 3.NF.1 Understand a fraction1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. How do you make sense of the language in this standard connected to the previous activity?

Extension of Unit Fraction Reasoning Jason hiked 3/7 of the way around Devil’s Lake. Jenny hiked 3/5 of the way around the lake. Who hiked the farthest?  Use fraction strips and reasoning to explain your answer to this question.

Extension 2 Jim and Sarah each have a garden. The gardens are the same size. 5/6 of Jim’s garden is planted with corn. 7/8 of Sarah’s garden is planted with corn. Who has planted more corn in their garden?  Use fraction strips and reasoning to explain your answer to this question.

Whiteboard Work write on your whiteboard the reasoning that you used to explain your answer. Be sure your reasoning is connected to unit fractions and fraction strips.

Reflect Think of the work you have done with your PIE group up to this point… How will this knowledge help you be more explicit with your instruction? Where might you step in with explicit instruction?

A structure for explicit and systematic instruction Symbols Give a context: tell a story Explain orally and/or in writing Make a picture Use concrete models: manipulatives

Planning for Instruction

What is instructional supports are avaialble? MTSD RtI Math Resources Conceptua Fractions Targeted Problem Solving Tasks – Howard County Math Wikis

MTSD RtI Math Resources cfm General Documents Grades 3-7 Intervention Guides Organized by Standard Conceptua “Homework” – Paper and Pencil Tasks

Conceptua Fractions Opener Guided Practice & Skills Check Closer Remediation Lessons Tool Investigations Teacher & Student Dashboard LOVE the visual models! LOVE the opportunity for translational teaching! NEED context for Big Ideas 1-7

Resources to Support Context Web Resource #1: Learn Zillion A good website to support using visual models to add and subtract fractions: addition-of-fractions-with-unlike-denominators Lessons 3 through 8 offer 3-4 minute “mini-lessons” showing how to solve addition and subtraction of fraction story problems using different models including number lines and tape diagrams. The links are found on the left side of the screen. Web Resource #2: Thinking Blocks – will give you story problems and modeling tools Accessing Thinking Blocks: Click on either “Adding Fractions with Like Denominators

Illustrative Mathematics

Howard County Math Wikis One resource for problem solving tasks ss.org/Grade+1+Home ss.org/Grade+1+Home