Coherence – Fractions Across the Grades February 2013 Network Team Institute Coherence – Fractions Across the Grades NOTE THAT THIS SESSION IS DESIGNED TO BE 75 MINUTES IN LENGTH; 5 MINUTES ARE ALLOTTED FOR THE SESSION OPENING (SLIDES 1-3). Turnkey Materials Provided in Addition to PowerPoint: Instructional Shifts Progression of Standards Related to Fractions Number and Operations—Fractions progressions document Measurement and Data (measurement part) progressions document Counting and Cardinality / Operations and Algebraic Thinking progressions document Additional Suggested Resources: Electronic access to elementary progressions documents We have just completed a walkthrough of the new modules. Those of you who attended the Grade 3—Module 5 session are already aware that its content is the introduction of fractions. We will now begin a study of how fractions are developed across the grades that includes considering foundational understandings from PK through 2nd grade.

Coherence – Fractions Across the Grades February 2013 Network Team Institute Session Objectives Explore the development of fractions across the grade levels. Examine and study the PK–5 standards and related modules foundational to fractions. Examine the K–5 progressions documents with particular emphasis on Number and Operations—Fractions. Explore how content knowledge directed by the standards and progressions documents supports the implementation of coherence in the classroom. Consider how G3—M5 supports the fraction progression. The objectives of this session are to: Explore the development of fractions across the grade levels. Examine and study the PK–5 standards and related modules foundational to fractions. Examine the K–5 progressions documents with particular emphasis on Number and Operations—Fractions. Explore how content knowledge directed by the standards and progressions documents supports the implementation of coherence in the classroom. Consider how G3—M5 supports the fraction progression.

Coherence – Fractions Across the Grades February 2013 Network Team Institute AGENDA FRACTIONS ACROSS THE GRADES: A Progression of Standards A Progression of Concrete and Visual Models The Progressions Documents: Number and Operations—Fractions 20 MINUTES ARE ALLOTTED TO WALK THROUGH AND ANALYZE STANDARDS FOUNDATIONAL TO FRACTIONS (SLIDES 4-11). Let’s begin with a look at the progression of standards that lay a foundation for, and then develop, an understanding of fractions.

Standards of Grade 3—Module 5 Coherence – Fractions Across the Grades February 2013 Network Team Institute Standards of Grade 3—Module 5 Develop understanding of fractions as numbers. 3.NF.1, 3.NF.2, 3.NF.3 Reason with shapes and their attributes. 3.G.2 Most of you have just attended a full session examining Grade 3—Module 5 and the standards that begin to formalize students understanding of fractions as numbers. Those of you who were in the Grade K session were asked to review these Grade 3 standards after getting resituated in our whole group.

Foundational Understanding and Skills Coherence – Fractions Across the Grades February 2013 Network Team Institute Foundational Understanding and Skills Review the standards of G3—M5. What understanding and skills are needed before learning these standards? Spend one minute writing down your answer to the following: What understandings and skills are needed before embarking on learning these standards? (Select volunteers to share their thoughts, noting their thoughts on a flip chart. If not brought up, suggest the following.) Understanding the notion of one half and one fourth. Connecting the words one-half and one-fourth to the representation of a regularly shaped area broken into 2 or 4 equal sized parts.

Foundational Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute Foundational Standards What Grade 2 standards are foundational to the G3—M5 standards? Examine the Grade 2 standards and identify those foundational to the Grade 3—Module 5 standards. Using the handout, “Fraction Standards Across the Grades,” write the standards you find to be foundational in the light yellow boxes to the left of the Grade 3 standards shown in light green. (Allow participants to work independently for 3-4 minutes.)

A Progression of Foundational Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute How does your list compare with the standards shown here? (Call on volunteers to speak to differences and facilitate a discussion of how strongly the standards support the G3—M5 standards. Do not emphasize standards as being either, ‘yes – foundational’ or ‘no - not foundational’. Recognize that, in theory, one could draw connections between almost all content. Rather allow participants to share their thinking with each other and compare it to what is presented.) A Progression of Foundational Standards

Foundational Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute Foundational Standards Continue to map backwards noting the understanding, skills and related standards from Grades 1, K, and PK. Continue to move backwards through the grades, examining the standards of grades 1, K, and PK and identify standards foundational to the grade 2 standards we just chose. (Allow participants to work independently for 3-4 minutes.)

A Progression of Foundational Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute How does your list compare with the standards shown here? (Again, call on volunteers to speak to differences and facilitate a discussion of how strongly the standards support the G3—M5 standards.) A Progression of Foundational Standards

A Progression of Fraction Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute A Progression of Fraction Standards The G3—M5 standards are foundational to what standards in grades 4 and 5? Take 2 minutes now to consolidate your thoughts on the reflection question, and then share your thoughts with a partner at your table.

A Progression of Fraction Standards Coherence – Fractions Across the Grades February 2013 Network Team Institute How does your list compare with the standards shown here? (Again, call on volunteers to speak to differences and facilitate a discussion.) A Progression of Fraction Standards

Coherence – Fractions Across the Grades February 2013 Network Team Institute AGENDA FRACTIONS ACROSS THE GRADES: A Progression of Standards A Progression of Concrete and Visual Models The Progressions Documents: Number and Operations—Fractions 15 MINUTES ARE ALLOTTED TO WALK THROUGH THE MODELS USED IN THE DEVELOPMENT TO FRACTIONS (SLIDES 12-13). Now that we have explored the progression of standards relating to fractions, let’s have a look at the concrete and visual models used to support understanding of these topics.

Demonstration – A Progression of Concrete and Visual Models Coherence – Fractions Across the Grades February 2013 Network Team Institute Demonstration – A Progression of Concrete and Visual Models << For turnkey purposes, a video of the NTI presentation of this portion of the session will be available. >>

Coherence – Fractions Across the Grades February 2013 Network Team Institute AGENDA FRACTIONS ACROSS THE GRADES: A Progression of Standards A Progression of Concrete and Visual Models The Progressions Documents: Number and Operations—Fractions 30 MINUTES ARE ALLOTTED FOR A STUDY OF THE K-5 PROGRESSIONS DOCUMENTS (SLIDES 14-19). For this final portion, you will need to have a printed copy of the Number and Operations—Fractions progression document and of the Measurement and Data (measurement part) progressions document

Techniques for a Study of Progressions: Coherence – Fractions Across the Grades February 2013 Network Team Institute Techniques for a Study of Progressions: Denote your reactions to the text with a personalized coding system. For example: Light bulb moment ? Questions about this Important – revisit this portion To facilitate this study we will employ three techniques. The first technique is to create a quick and easy set of 3 or 4 symbols with which to mark the text, denoting your reactions as you are reading it. Let’s use this technique now as we read pages the first three pages, which are actually pages 2-4, of the Number and Operations—Fraction progression. Start by giving some thought to the symbols you will use, and then begin reading, using the symbols as you go. (Allow participants time to read, resuming when it appears most have finished.)

Collaborative Study – Number and Operations— Fractions Coherence – Fractions Across the Grades February 2013 Network Team Institute Collaborative Study – Number and Operations— Fractions Who is willing to share some of their notes on the text they marked? (Allow 3-5 participants to share. Document their thoughts on a flip chart note.)

Techniques for a Study of Progressions: Coherence – Fractions Across the Grades February 2013 Network Team Institute Techniques for a Study of Progressions: Create a visual link (brackets, arrows, or color coded highlights) between standards in the margin and related text. The second technique we will use is creating a visual link of some sort between the standards shown in the right-hand margin and the portions of the text that relate to that standard. For example, you can highlight the number of the standard on the right, and then underline in the same color highlight each line of text related to that standard. Or, you can simply bracket text related to a standard and create an arrow connecting the bracket to the standard number. Go ahead and do this now. (Allow for 1-2 minutes for participants to mark the text accordingly.) These notations are very helpful when going back through and looking for clarifications on a given standard.

Selected Excerpts from the Progressions: Coherence – Fractions Across the Grades February 2013 Network Team Institute Selected Excerpts from the Progressions: Words / sentences to use when defining or describing content: For example, we “partition” the shape, not “divide.” Connection to mathematical practices Models and representations High level goals for this content in a given grade Areas of possible student misconceptions << For turnkey purposes, a video of the NTI presentation of this portion of the session will be available. >> Let’s review some of the findings of our study of the document. The progressions documents help answer several questions for us about presentation of the material: (CLICK TO ADVANCE 1st BULLET.) Words and sentences to use (and not to use) when defining or describing – for example, they use the word partition is the word we want to use in lieu of break up, separate, divide. Who can find another example where they help us with the proper terms to use for students? Additional words and sentences – a number line is like an infinite ruler, ‘equal shares’ in grades 1 and 2, ‘equal parts’ in grade 3, 3 fourths is what you get when you put three of the one fourths together, do not use proper or improper, equivalent fractions are fractions that are the same point on a number line. (CLICK TO ADVANCE 2nd BULLET.) Connections to Mathematical Practices – the progression gave two examples of how the study of fractions needs to be connected with attention to precision – the first was in requiring of ourselves and of students giving an explanation to specify the whole. They gave a lovely example on the bottom right corner of the first page (actually ‘page 2’) where if you haven’t specified what one whole is, you can’t express the shaded region as a fraction of that whole. What is the second way that a study of fractions in 3rd grade should promote student practice of attention to precision? (Describing what equal parts is, that, in reference to the area model, it does not have to be the same shape, only the same area.) (CLICK TO ADVANCE 3rd BULLET.) As we’ve already reviewed, they lay out the models and representations that can be used. (CLICK TO ADVANCE 4th BULLET.) High level goals of the study at a given grade level – grade 3’s goal is to see unit fractions as building blocks of fractions in the way that the number 1 is the building block of the whole numbers. (CLICK TO ADVANCE 5th BULLET.) Areas of possible student misconceptions – see if you can identify where they spoke of an area of possible misconception. (page 3 – when asked to show ¾ on a number line they may point to the whole number 3, thinking of it as ¾ of the way from 0 to 4.)

Techniques for a Study of Progressions: Coherence – Fractions Across the Grades February 2013 Network Team Institute Techniques for a Study of Progressions: Read the first five paragraphs of page 5 in the Number and Operations—Fractions progression is a line by line fashion. Check for comprehension at every component of each sentence. Mark anything that ties back to Grade 3 with a “3,” and anything that is new content with a “4.” Now let’s do a deep read – read the first five paragraphs of page 5 in the progression document. Read it line-by-line pausing to ensure comprehension of each phrase. As you read also make note of the grade level that each sentence is referring to – knowledge carried over from Grade 3 or knowledge new to Grade 4. If you are responsible for grades 4 or 5, it would be very helpful to continue this exercise reading through grade 5. It will support your work in the next 3 sessions. Any questions you have on your reading can be asked during tomorrow’s – Q&A session / office hours.

Studying Progressions of Topics Coherence – Fractions Across the Grades February 2013 Network Team Institute Studying Progressions of Topics Reflections: How does what is gained from a study of the progressions differ from what is gained from a study of the standards? What do these two essential documents offer to our profession? 5 MINUTES ARE ALLOTTED FOR THE SESSION CLOSING (SLIDES 20-21). Take one moment now to collect your thoughts on the reflection questions. (Allow 1 minute for reflection.) Who is willing to share their thoughts? (Facilitate a discussion.)

Coherence – Fractions Across the Grades February 2013 Network Team Institute Next Steps How can you increase coherence in your school’s delivery of topics? How can you initiate a study of progressions with your colleagues? Here are some questions to think about as you leave this session. How can you increase coherence in your school’s delivery of topics? How can you initiate a study of progressions with your colleagues?