Rigor Breakdown:
Procedural Skill and Fluency
Grades 3-5
Sequence of Sessions
Overarching Objectives of thisNovember 2012 Network Team Institute
- Participants will understand the features of A Story of Units and how the curriculum is aligned with the CCLS, Learning Progressions, Instructional Shifts, Publishers’ Criteria, PARCC, Major Content Emphases, and the Pre-Post Math Standards.
- Participants will understand how to use the How to Implement A Story of Units and A Story of Units: A Curriculum Overview for Grades P-5 documents and be prepared to share these documents with others.
- Participants will explore G2—M3 and G5—M3 and be prepared to train others to teach these modules.
- Participants will extend understanding of the structure of these modules to other modules in A Story of Units.
- Participants will analyze assessment tasks associated with G2—M3 and G5—M3.
- Participants will examine evidence of all three components of rigor in A Story of Units and be prepared to promote balanced instruction.
High-Level Purpose of this Session
- Understand the fluency component of rigor called for in the Standards, as defined by the Instructional Shifts and the Publishers’ Criteria
- Examine various activities that promote fluency in A Story of Units
- Compare and contrast fluency activities and analyze the impact and advantages of each
- Highlight Standards for Mathematical Practice in fluency activities in A Story of Units
- Recognize the balance and intensity of all three components of rigor in A Story of Units
Related Learning Experiences
- This session was preceded by an overview of rigor and a breakdown session focusing on one component of rigor – conceptual understanding. This session focuses on the fluency component of rigor. The next session will focus on application.
Key Points
- In each grade, students are expected to master certain fluencies over the course of the year.
- Fluency can be promoted in a variety of ways.
- Fluency work is usually done as a warm-up at the beginning of a lesson for approximately 10-20 minutes.
- Fluency represents a major part of the instructional vision that shapes A Story of Units. Frequent, rich opportunities to help students achieve fluency are woven throughout A Story of Units.
- Fluency work also provides opportunities to nurture the Standards of Mathematical Practice. The Standards for Mathematical Practice are embedded into all components of rigor and are not separate from the content instruction.
Session Outcomes
What do we want participants to be able to do as a result of this session?
- Identify key fluencies per grade as outlined by the CCSSM
- Recognize and understand the different types of fluency activities in A Story of Units
How will we know that they are able to do this?
- Participants will share their understanding of fluency requirements and fluency activities with colleagues.
- Participants and their colleagues will successfully implement fluency activities as described in A Story of Units.
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
- Examine fluency as defined in the Shifts and Publishers’ Criteria
- Examine how A Story of Units implements fluency
- Review the flow and objectives of this session
- Session PowerPoint
- Review session notes and PowerPoint presentation
- Download CCLS for easy reference to Standards of Mathematical Practice as needed
- Download Crosswalk of Instructional Shifts for easy reference as needed
- Download Publishers’ Criteria for easy reference as needed
Fluency Work –
Counting Exercises / 0:05-0:25
(20 min) /
- Engage in examples of fluency as counting exercise
- Watch a video clip of fluency as counting exercises
- Analyze the impact and advantages of counting exercises
- Video Clip: Skip-counting by Fractions
- Get a stopwatch
- Review and download video file as a back-up in case there is a problem with the embedded video
Fluency Work–
Choral and White Board Exchanges / 0:25-0:35
(10 min) /
- Learn about choral and white board exchange fluency activities
- Engage in an example of choral and white board exchange fluency work
- Watch a video clip of choral and white board exchange fluency work
- Reflect upon how fluency can also support conceptual understanding
- Video Clip: White Board Exchange
- Arrange for personal dry erase boards and markers per participant
- Review and download video file as a back-up in case there is a problem with the embedded video
Fluency Work –
Sprints / 0:35-0:55
(20 min) /
- Engage in an example of Sprints
- Learn about Sprints
- Watch video clip of Sprints
- Look for evidence of mathematical practices in Sprints
- G5—M3—B—L4 Sprints A and B: Equivalent Fractions
- Fluency Progression
- Video Clip: Sprints – Fluency in Action
- Get a stopwatch
- Review and download video file as a back-up in case there is a problem with the embedded Video
Closing / 0:55-1:00
(5 min) /
- Learn about banks of fluency activities provided with A Story of Units
- Summarize key points
- Reflect on next steps
- Reflect on the role of fluency in balanced instruction
- How to Implement A Story of Units
- Fluency Progression
TOTAL TIME / 60 min
Session Roadmap
Opening
Time: 0:00-0:05
- Link to previous sessions; frame the session, referencing the agenda
- Examine fluency as defined in the Shifts and Publishers’ Criteria
- Examine how A Story of Units implements fluency
- Review the flow and objectives of this session
Materials used include:
- Session PowerPoint
(SLIDE 1) In the last session we looked at three approaches to developing conceptual understanding. Conceptual understanding should be developed before related fluency work begins. In this session we will explore three different forms of fluency work.
(SLIDE2) The objectives of this session are to:
- Understand the fluency component of rigor called for in the Standards, as defined by the Instructional Shifts and the Publishers’ Criteria
- Examine various activities that promote fluency in A Story of Units
- Compare and contrast fluency activities and analyze the impact and advantages of each
- Highlight Standards for Mathematical Practice in the fluency activities in A Story of Units
- Recognize the balance and intensity of all three components of rigor in A Story of Units
(SLIDE4) The Publishers’ Criteria specifies that, “the word ‘fluently’ is used [in the standards] to set expectations for fluency.” (page 5) The document goes on to demand that, “Materials in grades K-6 help students make steady progress throughout the year toward fluent (accurate and reasonably fast) computation…. Progress toward these goals is interwoven with students’ developing conceptual understanding of the operations in question.” (page 10)
(SLIDE5) This is an inventory of all the K-5 standards that call for fluency:
K K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2 2.OA.2 Add/subtract within 20 (know single-digit sums from memory)
2.NBT.5 Add/subtract within 100
3 3.OA.7 Multiply/divide within 100 (know single-digit products from memory)
3.NBT.2 Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT.5 Multi-digit multiplication
(SLIDE6) Fluency promotes automaticity,acritical capacity that allows students to reserve theircognitive resources for higher-level thinking.
(CLICK TO ADVANCE SECOND BULLET ON SLIDE 6) Fluency represents a major part of the instructional vision that shapes A Story of Units. In this curriculum, fluency is a daily, substantial, and sustained activity supported by the lesson structure.
(CLICK TO ADVANCE THIRD BULLET ON SLIDE 6) A Story of Units includes 10 to 20 minutes of daily fluency work. A variety of suggestions for fluency activitiesare offered. They are strategically designed for the teacher to easily administer and assess. Note that the time spent each day will vary depending on the lesson and your students’ current skill level.
(CLICK TO ADVANCE FOURTH BULLET ON SLIDE 6) The fluency activities in A Story of Units are generally high-paced and energetic, getting students’ adrenaline flowing, and creating daily opportunities to celebrate improvement. From the beginning of the year, students see their accuracy and speed measurably increase both as individuals and as a class. Like opening a basketball practice with team drills and exercises, both personal and group improvements are exciting and prepare the players for the application in the game setting.
(CLICK TO ADVANCE FIFTH BULLET ON SLIDE 6) By encouraging students to recognize patterns and make connections within the lessons, the fluency exercises in A Story of Units support the other two components of rigor as well as the Standards for Mathematical Practice.
(SLIDE7) Fluency can be worked on in a variety of ways. In this session we will watch video clips and experience for ourselves three general categories of the fluency activities that can be found in A Story of Units: counting exercises, choral and white board exchanges, and Sprints.
Our goals for the session are to become familiar with the key fluency requirements per grade and to become familiar with several examples of fluency exercises that can be used in your classrooms to promote fluency.
Fluency Work – Counting Exercises
Time: 0:05-0:25
- Engage in examples of fluency as counting exercises
- Watch a video clip of fluency as counting exercises
- Analyze the impact and advantages of counting exercises
Materials used include:
- Stopwatch
- Video Clip: Skip-counting by Fractions
(CLICK TO ADVANCE ANIMATION ON SLIDE 7) We will begin with counting exercises.
(SLIDE8) This is one category of fluency work that is appropriate for all grade levels. You may have heard of exercises like ‘skip-counting’ or ‘happy counting.’ In a counting exercise, everyone might count chorally or individual students might count taking turns one after the next competing in a whole-class timed challenge. A counting exercise might also be individual timed challenges. In some cases, the students or the teacher may use a manipulative as the students count.
(CLICK TO ADVANCE SECOND BULLET ON SLIDE 8) Counting exercises typically require counting both forward and backwards. The fluency of being able to count backwards is often neglected, which is unfortunate, as it plays a vital role in students’ ability to subtract.
(CLICK TO ADVANCE ANIMATION ON SLIDE 8) Thus, counting exercises that encourage going in both directions apply directly to the addition and subtraction fluency standards of grades K-4.
(CLICK TO ADVANCE THIRD BULLET ON SLIDE 8) Skip-counting is a precursor to multiplication and thus also applies to the 3rd grade standard 3.OA.7 for multiplying and dividing within 100. In both 4th and 5th grade, skip-counting can occur in fractional amounts, counting by 1/3’s for example. In this case, the fluency activity also serves to solidify the conceptual understanding of the number line.
(SLIDE9) Let’s do a counting exercise together. While participating, reflect on these questions. We will discuss your thoughts in response to these questions afterwards.
- How could counting exercises have an impact or a benefit in your classes, school or district?
- How might counting exercises address trouble areas that your students experience during concept development and application work?
If you coordinate with other teachers of your grade, you can get a bit of a class competition going to see which class can count by 3’s up to 30 and back down to zero the fastest. We will simulate that competition here amongst ourselves. Now that you have been divided into classes, stand up and form a circle with your class. You’ll be Class A, (point throughout the room and continue naming the classes as letters.)
Class A will demonstrate with me. I’ll start counting with ‘3’, then the next person says ‘6,’ (point around the circle to indicate that the students take turns as the counting goes up by 3’s to 30), we only say ‘30’ once, now we are headed back down, so that’s ’27’ (pointing to the next person), ‘24’, … and we go all the way to 0, before I will click the timer to stop. OK, that took us a xx seconds. I will give each group 1 min to practice before we hold the official competition.
I need silence as we begin. Class A will start. I will choose where it starts and I will start the timer when that person says ‘3’. (Indicate who will start in Class A and time them as they count by 3’s up to 30 and back down to 0.)
Continue to let each “class” have a turn and record the times for each class on the flip chart.
This took us several minutes because we let several classes have a try, but normally you could hold the competition once each day for one week and then once a week you could hold it again, allowing your class to try to improve their time. It is great fun to chart their progress, showing how their speed improves from the beginning of the year. You can have similar charts and competitions for counting by other numbers. Regardless of what number you are counting by, you will start with that number, go up to a number 10 times that amount, and then backwards down to zero.
(SLIDE10) Let’s watch a video clip of a skip-counting exercise practiced in grade 5. While watching the video, continue to reflect on the questions. Later you will have an opportunity to discuss your thoughts.
This video showsAnita Meadows, a teacher at Saint Edward’s School in Vero Beach, Florida leading students in skip-counting by eighths. You will notice that the pace in this video is a bit slow as students are just beginning to gain fluency with this particular exercise.
(SLIDE 11 – Play Video Clip: Skip Counting by Fractions)
(SLIDE 12) Before discussing your thoughts on the video, let’s try one more counting exercise. This time we will count in unison by sixths using mixed numbers as we go beyond 1.
Let’s start by skip-countingby 1/6’s with mixed numbers up to 2 wholes. (Participants will skip count out lound as a whole group.) I’llnow draw out the number line to show the sixths.
(SWITCH TO DOCUMENT CAMERA and draw a number line from 0 to 2 with 1/6th intervals as shown.)
(Ask the following, filling in the equivalent fractions above or below the existing number line as students give them.)
T: Raise your hand if you can say 2/6 as the largest unit? (1/3)
T: Raise your hand if you can say 3/6 as the largest unit? (1/2)
T: Raise your hand if you can say 4/6 as the largest unit? (2/3)
T: Raise your hand if you can say 1 2/6 as the largest unit? (1 1/3)
T: Raise your hand if you can say 1 3/6 as the largest unit? (1 1/2)
T: Raise your hand if you can say 1 4/6 as the largest unit? (1 2/3)
Now we will try skip-counting by sixths again, this time using the largest unit. I will give you a minuteto practice first; you may use the number line as you practice with a your partner at your table.
Let’s begin.
I will point up as we increase, then use this symbol (make a fist or a flat hand) to pause at the current number; and I will point down when I want us to count backwards. Try not to look at the number line as we count.
We will start with 0:
(Pointing upward and making a rhythmic motion to indicate the speed of the counting:) 0, 1/6, 1/3, ½, 2/3, 5/6, 1
(Make a fist or flat hand, then point downward with rhythmic motioning:) 5/6, 2/3, ½
(Make a fist or flat hand, then point upward with rhythmic motioning:) 2/3, 5/6, 1, 1 and 1/6, 1 and 1/3, 1 and ½, 1 and 2/3, 1 and 5/6, 2, 2 and 1/6, 2 and 1/3, 2 and 1/2
(Make a fist or flat hand, then point downward with rhythmic motioning) 2 and 1/3, 2 and 1/6, 2, 1 and 5/6. STOP. Well done.
(SWITCH BACK TO POWER POINT)
(SLIDE13) Turn and talk with a partner at your table about your response to these questions.
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
(SLIDE14) Let’s summarize some key points about counting exercises:
- They are appropriate for all grade levels, PK-5.
- (CLICK TO ADVANCE SECOND BULLET ON SLIDE 14) Counting forward and backwards supports addition and subtraction.
- (CLICK TO ADVANCE THIRD BULLET ON SLIDE 14) Skip-counting supports multiplication.
- (CLICK TO ADVANCE FOURTH BULLET ON SLIDE 14) Counting exercises can be untimed choral counting, or timed competitions, either as a class moving from student to student allowing classes to compete against each other by posting each class’ best time, or as individuals competing with both for their best personal time and with one another.
Fluency Work – Choral and White Board Exchanges
Time: 0:25-0:35
- Learn about choral and white board exchange fluency activities
- Engage in an example of choral and white board exchange fluency work
- Watch a video clip of choral and white board exchange fluency work
- Reflect upon how fluency can also support conceptual understanding
Materials used include:
- Personal Dry Erase Boards and Markers
- Video Clip: White Board Exchange
(SLIDE15) In this next set of fluency examples, we’ll look at activities involving choral and white board exchanges.
(SLIDE16) These are quick, efficient interactive drills and games. The teacher may use verbal call outs, flash-card style manipulatives, or an overhead projector. Students respond either chorally or in writing on personal white boards or use manipulatives displayed at their desks.
The personal white board has a number of benefits. During written fluency exercises the personal white board gives the teacher instant feedback about student understanding and the boards are motivating to students. While students do have to commit to writing something down, they can also erase quickly so that they do not have of suffer the evidence of mistakes.
(CLICK TO ADVANCE SECOND BULLET AND SUB-BULLETS ON SLIDE 16) This type of choral or white board exchange can be applied to a wide range of skills from how many to make 10 games to computing one less than, one more than (e.g., what is one less than 6, what is 2 more than 7), to writing expressions to represent words (e.g., show me an expression for 2 less than the product of 4 and 5.) As we mentioned with skip-counting by fractional amounts, sometimes fluency exercises can support and develop conceptual understanding as well.
(SLIDE17) While you watch the video clips, notice how conceptual understanding is being reinforced. After the video, you will have a chance to share your thoughts with your table.
This video shows Anita Meadows and her 5th grade students at Saint Edward’s School in Vero Beach, Florida using a white board exchange.
(SLIDE 18 – Play Video Clip: White Board Exchange)
(SLIDE 19) Now let’s engage in a white board exchange. Get your white boards and markers ready. Let’s begin.
Show me an expression for the product of 6 and 2. (As participants raise and show you a correct answer of 12 acknowledge as many of them as is feasible saying, “Very Good,” “Yes,” or “Nice”, if a participant writes an incorrect answer say to them, “Listen again, I want the product of 6 and 2.”)
T: Show me an expression for the product of 6 and 2.
T: Show me an expression for 1 more than the product of 6 and 2.
T: Show me an expression for the difference of 6 and 2.
T: Show me an expresssion for 1 more than the difference of 6 and 2.
T: Show me an expression for the quotient of 6 and 2.
T: Show me an expression for 1 more than the quotient of 6 and 2.
(SLIDE20) Turn and talk with a partner at your table about how these activities reinforce conceptual understanding.
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
(SLIDE21) Let’s review some key points specific to choral and white board exchanges:
- (CLICK TO ADVANCE FIRST BULLET ON SLIDE 21) Exchanges can be used as a quick form of assessment. Note that white boards exchanges provide a better opportunity to assess students individually than do choral responses, and that any choral exchange can be done as a white board exchange.
- (CLICK TO ADVANCE TO SECOND BULLET ON SLIDE 21) Exchanges allow for rapid completion of multiple practice problems.
- (CLICK TO ADVANCE TO THIRD BULLET ON SLIDE 21) Exchanges create intensity in student practice.
- (CLICK TO ADVANCE TO FOURTH BULLET ON SLIDE 21) Exchanges give studentsimmediate feedback.
Fluency Work – Sprints
Time: 0:35-0:55