Day 3
Math & Science Collaborative at the Allegheny
Intermediate Unit
Sharing
Share the mental math activity
and game you did with your
class.
What successes did your students
have?
What challenges did they face?
Math & Science Collaborative at the Allegheny
Intermediate Unit
Read and Discuss
 Discuss: “Why Children Have Difficulties
Mastering the Basic Number Combinations
and how to Help Them.” from TCM, August 2006
 What are some reasons why students have
difficulty mastering basic combinations?
 What new ideas did you discuss in your
group about the relationship between
number sense and fluency after reading
the article?
Math & Science Collaborative at the Allegheny
Intermediate Unit
Modules are built on the strands
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foundational development occurs PreK- Grade 225%
K-2=49%
16%
30%
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Mental Math
 Encourages students to build on number
relationships to solve problems instead of
memorized procedures
 Using number relationships helps students
develop efficient, flexible strategies with
accuracy
 Causes students to be efficient to avoid
holding numerous quantities in their heads
 Strengthens students’ understanding of place
value
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Video: 16 + 15
 How does the teacher use the hundred chart to help
students think about efficiency with addition
strategies?
 What examples of composing and decomposing
numbers do you notice on the students’ strategies?
 Madeline uses a doubles +1 strategy to solve the
problem. What math understandings must be in place
for her to approach the problem in this way?
 What evidence indicates that the students understand
and can apply place value?
Math & Science Collaborative at the Allegheny
Intermediate Unit
Video: 70 - 54
 What structures are present in the classroom
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community that allow:
Mary to be comfortable thinking about her errors with
the class;
Incorrect answers to be posted; or
Students to change their answers when presented with
other students’ reasoning?
How is the relationship between addition and
subtraction used to build efficient strategies?
Why did Grant choose to increase both numbers by 6
Are there other ways he could have altered the problem
to make friendly numbers?
Math & Science Collaborative at the Allegheny
Intermediate Unit
Place Value Standards
 PA Core Standards talk about “strategies
based on place-value, properties of
operations, and/or relationships between
operations.”
 Look at the standards. What strategies are
they focusing on? What are “strategies
based on place-value, properties of
operations, and/or relationships between
operations.?
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Intermediate Unit
2.OA.2
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Intermediate Unit
1.OA.6
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Intermediate Unit
PA Core
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Intermediate Unit
Precursors to Fluency
 The sequence of number names, both starting at 1 and
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not starting at 1
How to count a set, keeping track of the items they
counted
Understanding relationships of more, less , and same
Skip counting starting from 1 and from other numbers
Cardinality
Conservation
1-to-1 correspondence
Making tens – link to understanding place value
Subitizing
Decomposing and composing numbers
Understanding part-part-whole
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Number sense at the
Allegheny Intermediate Unit
Fluency demands…
 An understanding of the meaning of
operations and their relationship to
each other.
 The knowledge of a large repertoire of
number relationships. (The “patterns”
of our number system.)
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Fluency demands…
 A thorough understanding of the base-ten number
system, how numbers are structured in this
system, and how this system behaves in different
operations.
 Knowing how a number can be composed and
decomposed and using that information to be
flexible and efficient with solving problems.
Involves three components.
Math & Science Collaborative
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Fluency
 Efficiency implies that the student does not get bogged down in
many steps or lose track of the logic of the strategy. An efficient
strategy is one that the student can carry out easily, keeping
track of sub-problems and making use of intermediate results to
solve the problem.
 Accuracy depends on several aspects of the problem-solving
process, among them, careful recording, the knowledge of basic
number combinations and other important number
relationships, and concern for double-checking results.
 Flexibility requires the knowledge of more than one approach to
solving a particular kind of problem. Students need to be flexible
to be able to choose an appropriate strategy for the problem at
hand and also to use one method to solve a problem and another
method to double-check the results.
Math & Science Collaborative
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How Can We Help Students with
Facts?
 Ongoing practice and engagement with
math facts tasks
 Hands-on activities and thoughtful
discussions
 Conceptual understanding of operations
 Strategic thinking
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Conceptual Understanding
 Understanding operations
 Symbolic representations
 Relationship between parts and whole
 Investigating the meaning of facts through hands-
on activities and thoughtful discussions
 Understanding is gained through:
 Problem posing
 Hands-on exploration
 Classroom discussions
 Real-world examples
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Meaningful Practice
Builds on understanding of operations and
using strategic reasoning to explore math facts
 Practice 5 – 10 minutes daily throughout the
school year
 Vary the practice activities - ensures that
students are motivated and engaged
 Automaticity is achieved through brief,
frequent, interactive activities
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Understanding Addition and
Subtraction
 Address the big ideas
 Guide the types of questions that are posed
 Explore symbolic representations
 Use models to represent addition and
subtraction
 Number lines, manipulatives, ten frames,
number charts
 Explore concepts through problems and
literature
Math & Science Collaborative
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Understanding Addition and
Subtraction
 What does it mean to understand addition?
 Look at the standards for the operations.
Read the standards dealing with addition
an dsubtraction.
 Then talk with your group. Develop a chart,
graphic, or model showing what students
need to know to show an understanding of
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addition. Math
at the Allegheny Intermediate Unit
Building Understanding While
Focusing on Fluency
 Use models to represent addition and
subtraction
 Ten frames, manipulatives, number lines
 Use problem contexts/real-life situations
 Make sure all four categories of problems are
addressed
 Productive talk
 Classroom
environment
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Big Ideas Central to Understanding
Addition and Subtraction
 Our number system is a system of patterns.
 The order of factors does not change the
sum.
 Addition and subtraction are inverse
operations.
 Numbers are flexible. They can be broken
apart to more easily perform an operation.
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Questions Posed during Class
Discussion
 What does each number represent?
 What patterns do you notice in the sums (or
differences)?
 Does the order of the addends affect the sum? Give
examples to justify your thinking.
 Do you notice a connection between this addition
equation and this subtraction equation?
 Can you break apart one o the addends to more
easily find the sum?
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Intermediate Unit
Classroom Environment
 Discussion
 Partner Work
 Interactive bulletin boards
 Word walls
 Centers
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Introducing Concepts of Addition
and Subtraction
 Compare methods for solving problems.
 Provide multiple opportunities for students to
represent and solve addition problems.
 Number lines
 Manipulatives
 Ten frames
 Number charts
 Explore subtraction
Math & Science Collaborative at the Allegheny
Intermediate Unit
Possible Teaching Sequence
Mastering the Basic Math Facts in Addition and Subtraction
 +1/+2
 +0
 +10
 Doubles
 Making ten
 Using tens
 Using doubles
Math & Science Collaborative at the Allegheny Intermediate Unit
Looking at One Example: Plus 1 & Plus 2
What are the big ideas around adding with 1 & 2?
 The sum when 1 is added to a quantity is the next
counting number.
 Our number system is a system of patterns.
 Addition is a joining or combining process.
 Subtraction is a separation or comparison process.
 The order of addends does not change the sum.
 Addition & subtraction are inverse operations.
Math & Science Collaborative at the Allegheny Intermediate Unit
Possible Questions to Support the Big Ideas of
+1, +2
 When you add 1 (2), how do you know what the
sum will be? Explain your thinking.
 Does the order of the addends change the sum?
Give examples to justify your thinking.
 How might a number line (or number chart) help
with +1 and +2 facts?
 How do you know when your answer makes sense?
 What is 30 + 1? Explain how you know.
 What is a way to find a -1/-2 fact?
Math & Science Collaborative at the Allegheny Intermediate Unit
Mouse Co
https://my.
Literature Connection
 Mouse Count – read and discuss. Add cubes to a jar
while reading story.
 Mice in a Jar activity. For each bag, count the number
of mice. Then add 1 & record the result in a number
sentence. Then add 2 mice & record. Find a way to
organize your data.
 Debrief activity
 What questions might you ask
about the data collected?
Math & Science Collaborative at the Allegheny Intermediate Unit
Word Problems and Number Strips/Charts
 Students need to visualize the facts using a
concrete model and move from
concrete/visual experiences to symbolic
representations. (C-R-A)
Math & Science Collaborative at the Allegheny Intermediate Unit
Word Problems and Number Strips/Charts
 Using the 20’s Chart, represent each problem:
 Katie took 5 bites of an apple. Then she took 1
more. How many bites did she take?
 Kevin put 7 rocks in his wagon. Then he put 2
more rocks in his wagon. How many rocks did
he put in his wagon?
 If Kevin took two rocks out of his wagon, how
many rocks would be in his wagon?
Math & Science Collaborative at the Allegheny Intermediate Unit
Word Problems and Number Strips/Charts
 Pose problems such as the following:
 Allison had 5 lollipops and 1 peppermint. How
many pieces of candy did she have?
 Brendan had 1 lollipop and 5 peppermints. How
many pieces of candy did he have?
 Allison won 3 tickets at the arcade. Then she won 2
more. How many tickets did she win?
 Brendan won 2 tickets at the arcade. Then he won 3
more. How many tickets did Brendan win at the
arcade?
What is this series of tasks allowing students
to develop?
Math & Science Collaborative at the Allegheny Intermediate Unit
Check for Understanding
 Jimmy says that 2 + 3 has a different sum than 3 +
2. Do you agree with him? Why or why not?
 Why might you use a prompt such as this one?
Math & Science Collaborative at the Allegheny Intermediate Unit
+1/+2
 +1/+2 Spin
 Challenge students to summarize their thinking
about +1/+2
 Did the number strip help you add 1 or 2?
How?
 Do you need the number strip to add 1 or 2?
 What if you did not have a number strip, how
could you add 1 or 2?
Math & Science Collaborative at the Allegheny Intermediate Unit
Building Automaticity
 Short practice – daily routine
 Fact card practice
 Fact card jumps
 Games
 I Spy 1 More
 Hop the Line
 Dot Addition
 Plus One Two Bingo
Math & Science Collaborative at the Allegheny Intermediate Unit
Connect to Subtraction
 Revisit Mouse Count
 Create 1 Less/2Less Story Problems
 Hop the Line Subtraction
 Less Than the Ten-frame
 Minus One, Minus Two Bingo
 What’s in the Bag
 Dot Card Subtraction
 Spinning More or Less
Math & Science Collaborative at the Allegheny Intermediate Unit
Other Groups of Facts
For the group of facts assigned to your group, read the pages
concerning your group of facts.
 Adding 10- Ch. 4
 Doubles- Ch. 5
 Making ten- Ch.6
 Using tens- Ch.7
 Using doubles – Ch. 8
Be ready to share out on the following questions:
1. Math big ideas?
2. Key Questions related to big ideas?
3. Strategies to explore those facts?
4. Connection to Subtraction?
Math & Science Collaborative at the Allegheny Intermediate Unit
Article Discussion
 Read and discuss: “Fluency: Simply Fast
and Accurate? I Think Not!” by NCTM
Past President Linda M. Gojak
 How do the ideas in the article resonate with
your ideas?
 What is the same? What is different?
 What stood out for you?
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Fluency
Look back at your ideas about
fluency.
What ideas, if any, do you want
to add to your ideas?
Math & Science Collaborative
at the Allegheny Intermediate Unit