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Presented by: Rebecca Gaddie
Charles Rutledge Renee’ Yates
Day 3 of 3
On a computer or device other than the one on which you are viewing the webcast, please join us at
www.todaysmeet.com/fluency
Who?
Location (school or district)?
How many are in your group (if more than one)?
• Develop understanding of the Standards for
Mathematical Practice and their connection to conceptual teaching and learning
• Explore progressions of the Kentucky Core Academic Standards that lead to mathematical fluency
• Examine how conceptual understanding, application, and procedural skill provide essential components in creating fluency
Component 1A – Knowledge of Content and Pedagogy
Component 2B – Establishing a Culture for Learning
Component 3B – Questioning and Discussion Techniques
Tools and Strategies
CRA and Multiplication
Number Talks
Effective Drill
Aligning Tasks to Standards
Mathematical fluency means having a deep understanding of mathematical concepts, which results in the facility to efficiently and accurately access, compare, and apply strategies, knowledge, and skills in a variety of contexts.
As Defined by Committee for Mathematics Achievement (CMA)
Efficiency
Accuracy
Flexibility
Being able to pass a timed fact test
Automaticity
Memorizing facts for quick recall
“How many do you see?”
“How do you see it?”
“Does anyone see it a different way?”
“What if we…”
“How many more…”
“How many less…”
Double Ten Frames
Double Bead Strings
Rekenrek
20 Bead Strings
10 Frame Addition Go Fish/Matching 10 Frame Flash Cards Dice Go Fish/Matching
100 Rekenrek
100 String
Bundling Straws
Bean Cups
Mini Ten Frames
Bean Sticks
Base Ten Blocks
BUILD IT!
+
+
Combine
Share
“How many do you see?”
“How do you see it?”
“Does anyone see it a different way?”
“What if we…”
“How many more…”
“How many less…”
Double Ten Frames
Double Bead Strings
Rekenrek
20 Bead Strings
10 Frame Addition Go Fish/Matching 10 Frame Flash Cards Dice Go Fish/Matching
100 Rekenrek
100 String
Bundling Straws
Bean Cups
Mini Ten Frames
Bean Sticks
Base Ten Blocks
BUILD IT!
+
+
Combine
Share
DRAW IT !
+
+
Combine
Share
SCREEN IT!
+
+
Combine
Share
Write It!
+
Write It
+ 50 50 = 100
Combine
Share
BUILD IT!
+
+
Combine
Share
DRAW IT !
+
+
Combine
Share
SCREEN IT!
+
+
+
+
Combine
Share
Write It!
+
+
+
+ 19 70 = 89
Combine
Share
Write It
BUILD IT!
+
+
Combine
Share
DRAW IT !
+
+
Combine
Share
SCREEN IT!
+
+
+
+ Combine
Share
Write It!
+
+
+
+
Combine
Share
Write It
56 36 = 92
Multiplication and Division Reasoning
CCSS.Math.Content.3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
2 + 2 + 2 =
REPEATED ADDITION
OR EQUAL GROUPING
So, is skip counting the same as repeated addition?
3 x 2
REPEATED SUBTRACTION
OR EQUAL SHARING
15 ÷ 5 = 3
Multiplication
◦ When the number and size of groups are known.
Division
◦ When either the number of sets or the size of the set is unknown.
RECTANGULAR ARRAY
AREA MODEL
Concrete Manipulatives
Representational Drawings, mental
images
Abstract Symbolic/verbal
Concrete
Build It with Manipulatives
Representational
Use drawings or illustrations
Abstract
Write using symbols
How many different ways can you name the number of cookies in this pan of cookies?
2.OA.3
Write a number sentence to represent the total number of fish in these tanks. Students may begin with 3 + 3 + 3 + 3 and we can scaffold this to 4 groups of 3 or 4 x 3
Suppose there are 4 tanks and 3 fish in each tank. The total number of fish in this situation can be expressed as 4 x 3 = 12. a. Describe what is meant in this situation by 12 ÷ 3 = 4 b. Describe what is meant in this situation by 12 ÷ 4 = 3
Source: illustrativemathematics.org
2 types of division problems
Partition - When you know the number of groups “I have 21 caramels. If I share them with myself and 2 of my friends, how many do we each get?” Quotition - When you know the number in each group. “I have 21 caramels. I’m allowed to eat 3 per day. How many days will they last me?”
What is one take-away or question you have about Multiplication and Division?
Let’s Develop Fluency through
Instructional Practices
Opportunities for a class to come together and share their mathematical thinking.
Problems are designed to elicit specific strategies that focus on number relationships.
Fosters and encourages mental math strategies.
Incorrect answers are used as opportunities to investigate misconceptions and learn from mistakes.
Students analyze their own thinking – metacognition- high level thinking
How are students using number relationships to solve the problem?
How would you describe the classroom community and environment?
Which strategies demonstrate accuracy, efficiency, and flexibility?
What is the teacher’s role?
How are students using number relationships to solve the problem?
How would you describe the classroom community and environment?
Which strategies demonstrate accuracy, efficiency, and flexibility?
What is the teachers role?
Author Sherry Parrish
contains video examples
for grades K, 2, 3, and 5
as well as explanations
of strategies shown
on the video clips.
http://mathsolutions.com/educator-tools/
Grade Required Fluency
K Add/subtract within 5
1 Add/subtract within 10
2 Add/subtract within 20 (mental strategies)
Add/subtract within 100 (strategies)
3 Multiply/divide within 100 (strategies)
Add/subtract within 1,000 (strategies)
4 Add/Subtract multidigit whole numbers (standard algorithm)
5 Multidigit multiplication (standard algorithm)
“…strategy development and general number sense…are the best contributors to fact mastery.”
“Do not subject any student to fact drills unless the student has developed an efficient strategy for the facts included in the drill.”
John Van de Walle
Do students have an efficient strategy for all the facts included in the drill?
Is the drill anchored to a strategy?
Are all facts within the number range of the specific grade level for fluency?
On a worksheet, have students circle the facts that belong to a strategy they have been working on and answer only those facts. The same approach can be used with two or three strategies on one sheet.
Mix ordinary flash cards from two or more strategies into a single packet. Prepare simple pictures of labels for the strategies in the packet. Students first match a card with a strategy and then uses the strategy to answer that fact.
Doubles plus 1
Doubles Doubles plus 1 Neither
3 + 3=
7 + 7 =
6 + 6 =
5 + 6 =
6 + 7 = 9 + 1 =
3 + 4= 7 + 3 =
Since a few sums go beyond 10, this would be a fluency drill for Grade 2. Number range is important when selecting equations for students.
“Timed test of basic facts may be used for diagnostic purposes – to determine which combinations are mastered and which remain to be learned.” Van de Walle
Marilyn Burns “Children who perform well under time pressure display their skills.”
Students should drill on addition and subtraction facts to 10, since that is their fluency benchmark.
Anytime students are working equations and problems beyond the sum of 10, they need tools and resources for support.
This next section is for illustrative purposes only. These are not mandated timeframes.
It is designed for teachers to deepen their understanding and importance addition and subtraction plays in their grade levels.
K CC 1 K CC 2
K CC 3 K CC 4
K CC 4a K CC 4b
K CC 4c K CC 5
K CC 6
K CC 7
K G 1
K G 2
K G 3
K G 4
K G 5
K G 6
K OA 1
K OA 2
K OA 3 K OA 4
K OA 5
K NBT 1
K MD 1
K MD 2
K MD 3 25 Standards How many deal with addition and subtraction of whole numbers?
1 MD 1 1 MD 2 1 MD 3 1 MD 4 1 G 1 1 G 2 1 G 3
1 OA 1 1 OA 2 1 OA 3 1 OA 4
1 OA 5 1 OA 6 1 OA 7 1 OA 8
1 NBT 1
1 NBT 2a 1 NBT 2b 1 NBT 2c 1 NBT 3
1 NBT 4 1 NBT 5 1 NBT 6
2 MD 5
2 MD 6 2 MD 7 2 MD 8 2 MD 9 2 MD 10 2 G 1 2 G 2 2 G 3
2 OA 1 2 OA 2 2 OA 3 2 OA 4 2 NBT 1a 2 NBT 1b 2 NBT 2 2 NBT 3 2 NBT 4
2 NBT 5 2 NBT 6 2 NBT 7 2 NBT 8 2 NBT 9 2 MD 1 2 MD 2 2 MD 3
2 MD 4
3 MD 1 3 MD 2 3 MD 3 3 MD 4 3 MD 5 3 MD 6 3 MD 7 3 MD 7a 3 MD 7b
3 MD 7c 3 MD 7d 3 MD 8 3 G 1 3 G 2
3 OA 1 3 OA 2 3 OA 3 3 OA 4 3 OA 5 3 OA 6 3 OA 7 3 OA 8 3 OA 9
3 NBT 1 3 NBT 2 3 NBT 3 3 NF 1 3 NF 2 3 NF 2a 3 NF 2b 3 NF 3 3 NF 3a 3 NF 3b 3 NF 3c 3 NF 3d
In grades K-3 there are about 110 content standards
When completing this activity, I typically get between 37-47 responses for standards dealing with addition and subtraction
(discrepancies usually occur depending on whether or not individuals divide place value into a separate topic)
What does this all mean to fluency?
This means that 34%-43% of the standards in K-3 deal with addition and subtraction.
With all of the field trips, benchmark testing, holidays, etc.; let’s assume you get 170 instructional days in per year.
This means over the course of these four years a student will receive about 680 instructional days in math.
If we were to spend 34%-43% of our days teaching addition and subtraction that would be a total of 231-292 days.
Kindergarten: App. 50% of standards About 85 days of instruction
First Grade: App. 45-70% of standards About 77-119 days of instruction
Second Grade: App. 40-50% of standards About 68 to 85 days of instruction
Third Grade: App. 10% of standards About 17 days of instruction
Rigor ◦ Conceptual Understanding
◦ Application
◦ Fluency/Procedural Skill
What is the content? ◦ Topical Alignment vs. Congruency
3.NF.3a 3.NF.3b 3.NF.3d
3.NBT.2
Thank you for joining us for the past 3 sessions!
Please share your feedback on Today’s Meet with us.
[email protected] ◦ www.rebeccagaddie.com
[email protected] ◦ www.charlesrutledge.weebly.com
[email protected] ◦ www.reneeyates2math.com
Mathematical discussions are a key part of
current visions of effective mathematics
teaching
• To encourage student construction of
mathematical ideas
• To make student’s thinking public so it can be
guided in mathematically sound directions
• To learn mathematical discourse practices
To make student-centered instruction more manageable by moderating the degree of improvisation required by the teachers and during a discussion.
1. Anticipating
2. Monitoring
3. Selecting
4. Sequencing
5. Connecting
likely student responses to mathematical problems
• It involves considering:
• The array of strategies that students might use to
approach or solve a challenging mathematical task
• How to respond to what students produce
• Which strategies will be most useful in addressing the
mathematics to be learned
likely student responses to mathematical problems
• It is supported by:
• Doing the problem in as many ways as possible
• Discussing the problem with other teachers
• Drawing on relevant research
• Documenting student responses year to year
students’ actual responses during independent work
• It involves:
• Circulating while students work on the problem and
watching and listening
• Recording interpretations, strategies, and points of
confusion
• Asking questions to get students back “on track” or to
advance their understanding
students’ actual responses during independent work
• It is supported by:
• Anticipating student responses beforehand
• Carefully listening and asking probing questions
• Using recording tools
student responses to feature during discussion
• It involves:
• Choosing particular students to present because of
the mathematics available in their responses
• Making sure that over time all students are seen as
authors of mathematical ideas and have the
opportunity to demonstrate competence
• Gaining some control over the content of the
discussion (no more “who wants to present next?”)
student responses to feature during discussion
• It is supported by:
• Anticipating and monitoring
• Planning in advance which types of responses to
select
student responses during the discussion
• It involves:
• Purposefully ordering presentations so as to make
the mathematics accessible to all students
• Building a mathematically coherent story line
• It is supported by:
• Anticipating, monitoring, and selecting
• During anticipation work, considering how possible
student responses are mathematically related
student responses during the discussion
• It involves:
• Encouraging students to make mathematical
connections between different student responses
• Making the key mathematical ideas that are the
focus of the lesson salient
student responses during the discussion
• It is supported by:
• Anticipating, monitoring, selecting, and sequencing
• During planning, considering how students might be
prompted to recognize mathematical relationships
between responses
Thank you for joining us for the past 3 sessions!
Please share your feedback on Today’s Meet with us.
[email protected] ◦ www.rebeccagaddie.com
[email protected] ◦ www.charlesrutledge.weebly.com
[email protected] ◦ www.reneeyates2math.com
