Coherence – Fractions Across the Grades
Sequence of Sessions
Overarching Objectives of this February 2013 Network Team Institute
- Participants will explore GK—M5 and G3—M5 and be prepared to train others to teach these modules.
- Participants will examine and study the K–5 progressions documents with particular emphasis on Number and Operations—Fractionsand be prepared to teach others about the development of fractions across the grades.
- Participants will explore how content knowledge directed by the progressions documents supports the implementation of coherence in the classroom.
- Participants will deepen their study of rigor by examining its relationship to coherence using examples from GK—M5 and G3—M5 and be prepared to promote coherent and balanced instruction.
High-level Purpose of this Session
- Explore the development of fractions across the grade levels.
- Examine and study the K–5 progressions documents and PK–5 standards 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.
Related Learning Experiences
- Participants have just finished studying either GK—M5 or G3—M5. They now begin to explore how content from grades K–2 supports G3—M5 as well as how G3—M5 is foundational to 4th and 5th grade content.
- The focus of this session – coherence – is carried over to the next three sessions on the components of rigor. In those sessions, participants will examine and reflect upon how the cross-grade content knowledge developed during this session can be manifested in the implementation of the three components of rigor.
Key Points
- When studied in conjunction with the Standards, progression documents offer guidance on terminology, models, connections to mathematical practices, grade level goals, and common student misperceptions.
- Coherent content knowledge facilitates teaching with cross-grade coherence and bridging gaps in prerequisite knowledge.
Session Outcomes
What do we want participants to be able to do as a result of this session?
- Teach lessons related to fractions with clear cross-grade coherence.
- Guide other teachers through a study of the progressions documents as they relate to development of fractions across the grades.
- Understand and convey to their colleagues that content knowledge directed by the progressions supports teaching with greater coherence.
- Understand and convey to their colleagues how the GK—M5and G3—M5 modules align with the progressions documents.
How will we know that they are able to do this?
- Participants will report ease and confidence in redelivery of this information.
- Participants will report techniques for studying progressions documents independently and with colleagues.
Session Overview
Section / Time / Overview / Prepared Resources / Facilitator PreparationOpening / 0:00-0:05
(5 min) /
- Link to previous sessions; frame the session, referencing the agenda.
- Introduce objectives and sequence for this session.
- Session PowerPoint
- Instructional Shifts
- Review session notes and PowerPoint presentation.
Walk Through and Analyze Standards Foundational to Fractions / 0:05-0:25
(20 min) /
- Examine standards foundational to understanding fractions.
- Progression of Standards Related to Fractions
Walk Through of the Models Used in the Development of Fractions / 0:25-0:40
(15 min) /
- Examine the models used in development of fractions.
Study of the K–5 Progressions Documents / 0:40-1:10
(30 min) /
- Study of the K–5 progressions documents with emphasis on Number and Operations—Fractions.
- Number and Operations – Fractions progression document
- Measurement and Data (measurement part) progression document
- Counting and Cardinality / Operations and Algebraic Thinking progression document
Closing / 1:10-1:15
(5 min) /
- Reflect on the benefit of studying the coherence of concepts through a study of the Standards and progression documents.
TOTAL TIME / 75 min
Session Roadmap
Opening
Time: 0:00-0:05
- Link to previous sessions; frame the session, referencing the agenda.
- Introduce objectives and sequence for this session.
Materials used include:
- Session PowerPoint
- Instructional Shifts
(SLIDE 1) 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.
(SLIDE 2) 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.
Walk Through and Analyze Standards Foundational to Fractions
Time: 0:05-0:25
- Examine standards foundational to understanding fractions.
Materials used include:
- Progression of Standards Related to Fractions
(SLIDE 4) 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.
(SLIDE 5) 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.
(SLIDE 7) 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.)
(SLIDE 8) 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.)
(SLIDE 9) 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.)
(SLIDE 10) This time let’s move forward, identifying the grade 4 standards that are supported by the G3-M5 standards. (Allow participants to work independently for 3-4 minutes.)
(SLIDE 11) How does your list compare with the standards shown here? (Again, call on volunteers to speak to differences and facilitate a discussion.)
Walk Through of the Models Used in the Development of Fractions
Time: 0:25-0:40
- Examine the models used in development of fractions.
Materials used include:
(SLIDE12) 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.
(SLIDE 13) < For turnkey purposes, a video of the NTI presentation of this portion of the session will be available. >
Study of the K–5 Progressions Documents
Time: 0:40-1:10
- Study of the K–5 progressions documents, Number and Operations—Fractions and Measurement and Data.
Materials used include:
- Number and Operations—Fractions progressions document
- Measurement and Data (measurement part) progressions document
- Counting and Cardinality / Operations and Algebraic Thinking progressions document
(SLIDE 14) 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.
(SLIDE 15) 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.)
(SLIDE 16) 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.)
(SLIDE 17) 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.
(SLIDE 18) < For turnkey purposes, a video of the NTI presentation of this portion of the session is 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.)
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.
Closing
Time: 1:10-1:15
- Reflect on the benefit of studying the coherence of concepts through a study of the Standards and progression documents.
Materials used include:
(SLIDE 20) Take one moment now to collect your thoughts on the reflection questions.
- 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?
Who is willing to share their thoughts? (Facilitate a discussion.)
(SLIDE 21) 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?
Turnkey Materials Provided
- PowerPoint Presentation
- Instructional Shifts
- Progression of Standards Related to Fractions
- Number and Operations—Fractions progressions document
- Counting and Cardinality / Operations and Algebraic Thinking progressions document
- Measurement and Data (measurement part) progressions document
Additional Suggested Resources
- Electronic access to all K–5 progressions documents