Teaching for Conceptual Understanding: Fractions, Facilitator Handbook
Teaching for Conceptual Understanding: Fractions
Professional Development
Facilitator Handbook
-SAMPLER-
Pearson School Achievement Services
Teaching for Conceptual Understanding: Fractions
Facilitator Handbook
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© 2013 Pearson, Inc.
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Printed in the United States of America.
ISBN 115502
Facilitator Agenda
Teaching for Conceptual Understanding: Fractions
Section / Time / Agenda ItemsIntroduction / 20 minutes / Slides 1–5
· Welcome and Introductions
· Agenda
· Outcomes
Activity: Simon Says
1: The Fraction Progression Holds Great Promise / 65 minutes / Slides 6–13
· Section 1 Big Questions
· What is a learning progression?
Activity: Think-Pair-Share
· Sorting through the Fractions Progression
Activity: Fraction Sort
· Organizing the Components of the Fractions Progression
· Making Connections within the Fractions Progression
Video: The Fraction Story
· Coherence through Major Clusters Connected to Supporting Clusters
Activity: Unpacking Major and Supporting Clusters
· Refining the Definition of Progression
Activity: Define/Refine
· Revisit Section 1 Big Questions
Break / 15 minutes
2: Guidance for Teaching and Learning the CCSSM Fraction Progression / 95 minutes / Slides 14–36
· Section 2 Big Questions
· Merging Content and Practice
· Representing Fractions Intuitively
· IES Recommendations
· Intuitive Meaning of Fractions at Grades 1 and 2
Activity: Intuitive Meaning of Fractions
· Major Cluster: Develop Understanding of Fractions as Numbers
Activity: Fair Sharing
Activity: Equal-Sized Parts of a Region
Activity: 1/4 of The Whole
Activity: Tangrams (Optional Extension Activity)
Section / Time / Agenda Items
Activity: Fractions on the Number Line
Activity: Justifying with Number Line Diagrams and Visual Fraction Models
Activity: Reasoning About the Size of Fractions
Activity: Sorting Cards Assessment
Activity: Share the Load
· Supporting Cluster: Reason with Shapes and Their Attributes
Lunch / 30 minutes
2: Guidance for Teaching and Learning the CCSSM Fraction Progression (continued) / 100 minutes / Slides 37–63
· Major Cluster: Extend Understanding of Fraction Equivalence and Ordering
Activity: Justifying Again with Number Line Diagrams and Visual Fraction Models
· Major Cluster: Build Fractions from Unit Fractions
· Coherence in Reasoning and Strategy of the Fractions Progression
Activity: Operations in Grade 4
· Supporting Clusters is Grade 4
· Major Cluster: Use Equivalent Fractions as a Strategy to Ann and Subtract Fractions
· Major Cluster: Apply and Extend Previous Understanding of Multiplication and Division
Activity: Create a Story
Activity: Justifying Fraction Multiplication
Activity: Justify Using Number Line Diagrams and Area Models
· Coherence in Reasoning Progresses to Grade 5
Activity: Justify Solutions to Multiplication Word Problems
· Recommendations for Teaching Fractions
Activity: Sorting the Recommendations for Teaching Fractions
· Revisit Section 2 Big Questions
Break / 15 minutes
Section / Time / Agenda Items
3: Planning So That Fractions Make Sense / 65 minutes / Slides 64–78
· Section 3 Big Question
· Anticipating Student Response
· Addressing Common Misconceptions
Video: Ally’s Common Misconceptions about Fractions
· Connecting Recommendations to the Standards for Mathematical Practice
· Planning Fractions Lessons
Activity: Planning Effective Mathematics Instruction
· Revisit Section 3 Big Question
Reflection and Closing / 15 minutes / Slides 79–86
· Outcomes
· Final Reflection: Taking Action
· Evaluation
· References
Total / 6 hours
Preparation and Background
Workshop Information
Big Ideas
· Being able to reason about the size of a fraction is key to success with fractions.
· Fractions are numbers.
· The denominator in a fraction tells the size of the fractional piece, and the numerator tells how many pieces.
· Students must understand the concept of the unit of reference and the relationship of the unit fraction.
· Students should first understand the concept, and then teachers should teach the conceptually-based algorithm.
· Misconceptions about fractions can happened because of the way fractions are talked about.
· The CCSSM provide a conceptually-based progression of fraction study.
Big Questions
· What is the Fractions progression?
· How is the Fractions progression different from a scope and sequence?
· What are the recommendations for teaching the CCSSM Fractions progression?
· How can teachers help students recognize that fractions are numbers and that they expand the number system beyond whole numbers?
· How can teachers help students understand that fractions make sense?
Assessments of Participants’ Learning during the Workshop
· Big Questions
· Standards Sort
· Fraction Definition
· Planning: Instructional Outcome Planning
· Reflection: Taking Action
Assessment Back in the School/Classroom
· Implement the instructional strategies identified as necessary for the conceptual understanding of fractions.
Outcomes
· Articulate the learning progressions necessary for students to conceptually understand fraction concepts.
· Identify strategies for helping students build their mathematical understanding of fractions.
· Use a planning template to build lessons that strategically support the conceptual development of fractions.
· Identify strategies that support simultaneous development of conceptual understanding and problem-solving skills with the intentional use of purposeful student struggle, flexible grouping, and ongoing assessments.
· Articulate common misconceptions as opportunities for students’ conceptual understanding of fractions.
Facilitator Goals
· Assist participants in understanding the progression of fraction concepts as outlined in the CCSSM.
· Deepen participants’ conceptual understanding of fractions.
· Guide participants in developing the instructional strategies needed for the conceptual understanding of fractions. The strategies include the following:
o Think-Pair-Share
o Create Vocabulary Cards
o Graphic Organizers
o Define-Refine
o Writing Reflections on Big Questions
o Mathematics Word Wall
o Student-Written Definitions with Words and Pictures
o Knowledge Rating Score
o Sorting Cards
o Story
o Using Models
Introduction (Slides 1–5)
Time: 20 minutes
Materials per Section
· Chart paper
· Markers
· Intro_Facilitator_Shapes.pdf
· Sticky tack
· Participant Workbook, pages 4–5
Topic / Presentation Points / Presentation Preview /Welcome and Introductions / · Display Slide 1.
· Begin the workshop by introducing yourself and welcoming participants to the Teaching for Conceptual Understanding: Fractions workshop.
· Go over housekeeping items such as restroom locations, lunch time, and ending time. /
Agenda / · Display Slide 2.
· Go over the workshop agenda. /
PW: Page 4
Outcomes / · Display Slide 3.
· Go over the outcomes on the slide for the day’s workshop.
· Explain that by the end of the day, participants will be able to
o articulate the learning progressions necessary for students to conceptually understand fraction concepts;
o identify strategies for helping students build their mathematical understanding of fractions;
o use a planning template to build lessons that strategically support the conceptual development of fraction concepts and operations;
o identify strategies that support simultaneous development of conceptual understanding and problem-solving skills with the intentional use of purposeful student struggle, flexible grouping, and ongoing assessments; and
o articulate common misconceptions as opportunities for students’ conceptual understanding of fractions. /
PW: Page 4
· Display Slide 4.
Activity: Simon Says
· Explain that participants are going to engage in a unique form of Simon Says.
NOTE: Direct participants to the shapes (from Intro_Factilitator_Shapes.pdf) that you placed on their tables prior to the start of the workshop. If there are more shapes than participants, set the remaining pieces to the side; however attempt to provide at least one fraction from each shape among the participants.
Option 1:
· Have each participant take one shape piece from their table. Each shape piece is a fraction of an unknown whole. In the space provided in their Participant Workbook, direct participants to build their whole based on the size of the fraction shape piece they have chosen.
· To do this, direct participants to trace their fractions in the space provided.
· After participants have traced their fractions, direct them to draw the whole that will allow them to answer the statement, “Simon says my piece is ___, and I say this is my whole.”
· An example is provided on the slide with the blue piece representing the piece the participant was given and the white piece representing what the participant drew.
· After participants have completed their drawings, have them move into one of five groups. The groups are as follows:
o Those whose pieces say:
o 1/2 A
o 1/4 A
o 1/2 B
o 1/3 B
o 1/4 B
· Within their groups, have participants introduce themselves and share their shapes. The purpose of having participants in these specific groupings is that they each have the same size fractions and the same original unit of reference (a square or a rectangle). The observation that you are trying to get participants to make is that equal shares of identical wholes need not have the same shape.
NOTE: If participants make wholes that are different from squares and rectangles, discuss the different representations that they can make from the same size and same shape shares. Then, have groups try to find one whole that will fit the scenario of each of their pieces; they can then work toward the observation that you are looking for them to make.
Option 2:
· Distribute three to five different pattern blocks per table. For example, a table of four might use the square, the triangle, the parallelogram, or the rhombus. Use the hexagon if needed.
· Use the following prompts to have participants complete three different drawings in the Participant Workbook:
o Simon Says: My piece is one half, and I say this is my whole.
o Simon Says: My piece is one third, and I say this is my whole.
o Simon Says: My piece is one fourth, and I say this is my whole.
· For the last one, have participants find other participants that had the same shape and record their observations.
· Debrief the activity with participants, and chart the observations as participants discuss them. /
PW: Page 5
Chart: Simon Says
Section 2: Guidance for Teaching and Learning the CCSSM Fraction Progression (Slides 14–63)
Time: 195 minutes
Big Questions
· What are the recommendations for teaching and learning the CCSSM Fractions progression?
· How can teachers help students recognize that fractions are numbers and that they expand the number system beyond whole numbers?
Learning Objectives
· Articulate the learning progressions necessary for students to conceptually understand fraction concepts.
· Identify strategies for helping students build their mathematical understanding of fractions.
Materials per Section
· CCSSM document
· Chart paper
· Markers
· Glue sticks
· Speakers
· Blank paper
· Rulers
· Scissors
· FacilitatorVocabCardsMasterSet.pdf (1 set total)
· CreateaStoryCards.pdf (1 set per table)
· SortingCardsTableSets.pdf (1 set per participant)
· Support documents (ConceptualFractionsTangramExt.pdf)
· WordWall.pdf (1 total)
· Participant Workbook, pages 13–36
Activity: Sorting Cards Assessment
· Point out that the sorting cards assessment that participants will now work through is another instructional strategy that they can use in their classrooms. Explain that while they sort the cards to compare fractions, they can make sets for students to use to demonstrate understanding and mastery in any area of mathematics. Prior to starting the workshop, create sets of cards for each participant using the SortingCardsTableSets.pdf file in the Additional Resources folder. Let participants know that a master set has also been provided for them in the Appendix of the Participant Workbook.
· Pass out the premade sets of sorting cards.
· Explain to participants that you want them to complete a quick assessment by sorting the cards you have given them into the categories in the Participant Workbook.
· As you discuss this activity with participants, go over the following information:
o An important concept when comparing fractions is to look at the size of the parts and the number of the parts. For example, students in Grade 3 see that for unit fractions (1/8 and 1/9), the one with the larger denominator is smaller, by reasoning, for example, that in order for more (identical) pieces to make the same whole, the pieces must be smaller. From this they reason that for fractions that have the same numerator (7/8 and 7/9), the fraction with the smaller denominator is greater. For example, 7/8 > 7/9, because 1/9 < 1/8, so 7 lengths of 1/9 is less than 7 lengths of 1/8.
o When examining fractions with common denominators, students also recognize that the wholes have been divided into the same number of parts so the fraction with the larger numerator has the larger number of parts (3/8 < 5/8). Note: Students do not learn the fundamental property of equivalent fractions until Grade 4.
o “As students move towards thinking of fractions as points on the number line, they develop an understanding of order in terms of position. Given two fractions—thus two points on the number line—the one to the left is said to be smaller and the one to right is said to be larger. This understanding of order as position will become important in Grade 6 when students start working with negative numbers” (The Common Core Standards Writing Team 2011, 4).
· To close out this activity, ask participants to answer the following prompt in the Participant Workbook:
o Examine how you were able to demonstrate proficiency in 3.NF.3d. In other words, how were you able to explain your sorting?
· Have several participants share their thoughts on this activity. /