Balanced Math K-8.ppt - mnpsmath by xiaopangnv

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```									Balanced Math
K-8 Framework
October 2010

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Norms

• Be prompt
 Return from breaks on time
• Be respectful
 Put cell phones on silent.
• Be a polite and positive participant
 Speak in a normal tone of voice, and listen
attentively.
• Be a problem solver

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Who are these guys?

This is who I am
by the numbers…..
1985, 25,
155551

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By the numbers…….

• A significant year in my
life was 1985
• I’ve spent 25 wonderful
years with MNPS.
• My favorite palindrome
is 155551.
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numbers…
• Select 5 numbers that are meaningful to
you that will help someone understand
who you are.
• Then write a sentence or question for
each number, leaving a blank line where
the number should go.
• Share you numbers and sentences with
your neighbor. See if he or she can
match the correct number to the line.

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Activity: Group and Label

• Write each of your numbers on a post it.
One number per post it.
in the middle and eliminate any duplicates.
• Then group your numbers and label them
according to some common
characteristics. Then turn you labels over.
• Visit another table and try to figure out
there groupings.
• Discuss how you can use this activity in
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Why a Balanced Approach?

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Why a Balanced Approach?

States…

“The mutually reinforcing benefits of
conceptual understanding, procedural
fluency, and automatic, i.e. quick recall of
facts.”

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Conceptual
Understanding

Computational                        Problem
Fluency                            Solving

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Why a Balanced Approach?

Math
• Large group instruction
• All students work on the same level
• Primarily instruction and practice from
text book
• Emphasis on paper and pencil work
• Individual work
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Why a Balanced Approach?
Approach
• Successful for some students
• Even less successful for struggling
students
• Encourages emphasis on computation
skills
• Little opportunity for communication
• More emphasis on evaluation, rather
than assessment for learning

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Why a Balanced Approach?
The Statistics
• Fifth graders spend more than 90
percent of their time in their seats
listening to the teacher or working alone
and only about 7 percent of their time
working in groups.

times as much instruction in basic skills
as instruction focused on problem
solving or reasoning. The ratio was 10:1
The Global Achievement Gap by Tony Wagner.
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Why a Balanced Approach?
To teach math more effectively,
teachers must…
• reach students at all levels of
achievement

• provide diverse methods of learning

• allow more opportunities for
observation and communication by
students

• encourage active engagement by
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Teaching Procedures
What the Research and Experts Say
 According to the National Mathematics
 understand key concepts
 achieve automaticity as appropriate (e.g.,
 develop flexible, accurate, and automatic
execution of standard algorithms
 Computational Proficiency is dependent on
 automatic recall of math facts
 a solid understanding of core concepts

National Mathematics Advisory Panel. Foundations for Success: The Final
Report of the National Mathematics Advisory Panel, U.S. Department of
Education: Washington, DC, 2008.
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Conceptual vs. Procedural

Striking the Balance

 Vocabulary                                 Four
development                                 operations
 Manipulatives                              Standard-
algorithms
 Multiple
representations                            Step-by-step
methods
 Pictures
 Math facts
 Real-life
contexts                                   Release of
responsibility
 Practice

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Day            15 min                              40 minutes                         5 minutes
1    Math         Mental    Concept Lesson                                           Closure/
Review       Math      Problem solving, manipulatives, small groups, centers    math
journals
2    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

3    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, small groups , centers

4    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

5    Math Facts Practice/   Problem-based activities, centers, games, small groups
Math Review Quiz
6    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

7    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

8    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

9    Math         Mental    Concept Lesson                                           Closure
Review       Math      Problem solving, manipulatives, small groups, centers

10     Assessment/Math                                       Assessment
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Review Quiz
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•One more/one less, before/after, a given number

•Counting by twos, fives, tens

•Doubles

•Fact families

•Measurement (time, money, calendar, feet, etc.)

•Math Vocabulary/Math Word Wall

•Estimation

•Math Around the Room
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Math Review and Mental
Math- Rally Coach
Partners Take turns, one solving a problem while the other coaches.
Each pair needs one set of problems and one pencil.

Person A                                    Person B

• Partner A solves the first • Partner B watches and
problem. Talking out         listens, checks, coaches
their thinking.              if necessary and praises.
• Partner A watches and      • Partner B solves the
listens, checks, coaches     next problem. Talking
if necessary, and praises.   out their thinking.

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Balanced Math
What is Conceptual Understanding?
 It is the underlying knowledge behind the concept
 Teaching techniques include: concrete models, vocabulary
connections, problem solving, real-life applications, etc.
 Conceptual understanding is important for two main reasons:
 In order to apply knowledge to new situations
 Subsequent math concepts rely on students’ ability to
understand the current concept

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Conceptual Understanding

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Vocabulary Development

Content vocabulary words are used within the subject
matter you are teaching (e.g., fractions, decimals).
Academic vocabulary is the higher-level language
needed to understand the content (e.g., analyze,
identify).
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Vocabulary Development
Word Wall
 Bulletin board display of
key vocabulary or concept
words
 Students can be involved
in their creation
 Include illustrations,
photos, examples
 Refer to the words often
during instruction

http://www.kealakehe.k12.hi.us/amilwordsmedia/FWWordWall/FWWordW
all-Images/21.jpg

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Vocabulary Development
Developing Math Vocabulary to
Make Concepts Accessible

Identify      Decide on            Engage Your
Vocabulary      Teaching             Students!
Words        Strategies

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Group and Label Activity

Purpose:
Group and Label asks students to conceptualize
their way to deep understanding by organizing
mathematical data into meaningful categories.
Students analyze a collection of mathematical
information, group the items into categories,
and label each category in a way that explains
why the items go together. Finally, students
use their labeled groups to generate a set of
hypotheses or generalizations, which they
revisit periodically and refine in light of new
Learning.

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Problem Solving

A Starting Point
for Problem Solving

Many children think of mathematics as a
subject that relies on their memorizing
facts and practicing skills. But, the true
test of children’s success in mathematics is
when they can’t remember a fact or have
forgotten a skill, they are able to think,
reason, and solve problems and make
sense of mathematical ideas.
Marilyn Burns

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Thought is
STRATEGIC…
Therefore, we need a
classroom that models
instructional practices and
strategies that enhance
students’ ability to think.

We can't solve problems by using the
same kind of thinking we used when we
created them.
-- Albert Einstein
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Teaching through Problem
Solving

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How Much Does Matt Weigh?

What We Know…….
• Matt’s head weighs 15 lbs
• His torso weighs as much as
• His legs weigh as much as his

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Using
Manipulatives

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Using Manipulatives

What Are Manipulatives?
Manipulatives are colorful, intriguing
materials constructed to illustrate and
model mathematical ideas and
relationships and are designed to be
used by students in all grades (Burns
and Sibley 2008).

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Using Manipulatives

Why Use Manipulatives?

Using Manipulatives, “helps
students understand the
mathematical concepts and
processes, increases thinking
flexibility, provides tools for
problem-solving, and can
reduce math anxiety for some
students. (The Education
Alliance 2006).

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Using Manipulatives
Classroom Management Tips for Manipulatives
• Organize manipulatives.   • Set expectations.

• Give students time to
• Model examples.                  practice.

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Laura went shopping

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Using Manipulatives
Phases of Instruction and Learning
C-R-A

-Moving with Math

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Using Manipulatives

Concrete
Hands-on teaching method using
manipulatives such as:
• Algebra Tiles
• Toothpicks
• Counting blocks
• Unifix Cubes
• Cuisenaire Rods
• Food
• Algeblocks
• Balance scales
• Hands-On Equations
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Using Manipulatives
Representational
Uses:
• Pictures
• Tally marks
• Diagrams
• Drawings
• Maps
• Graphs
• Charts

Relates directly to the manipulatives
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Using Manipulatives
Abstract
A teaching method using written words
and symbols.
• Graphs (meaning)
• Matrices
• Estimation
• Predictions
• Tables (ex. slope)
• Oral explanations
• Systems of equations

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Differentiating to Meet Students Needs
– What do you want students to learn during the activity?
– All students should work toward the same objective.

Objective: Students will demonstrate understanding of the
relationship between factor, product, and multiple.

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Differentiating to Meet Students Needs
students to complete.
1) On graph paper, draw an array to represent the multiplication problem 3 x 4.
What is the product? How do you know?
2) Make an array with a different length and width, but with the same product as
3 x 4.
3) Write a multiplication number sentence to represent your array.
4) For both arrays, complete the sentences below by writing the number in the
blank.

This array has ____ rows by _____ columns.
This array has _____groups of _____.
5) List the factors shown in both arrays.
6) The product is also called a multiple. How is the
multiple related to its factors?
7) List some other multiples and their factors.
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Differentiating to Meet Students Needs
• Revise the activity for your English language learners.

4
1) An array is putting objects in equal rows and
2
columns to make a rectangle. On graph paper, draw
an array to show the multiplication problem 3 x 4.            array
2) The product is the answer to a multiplication
problem. What is the product of 3 x 4?
How do you know?                                       3x4=
3) Draw an array with a different length and width. Use                     product
the same number of squares, so the array has the
same product as 3 x 4.
length

width
4) An example of a multiplication number sentence is 2
x 4 = 8. Write a multiplication number sentence for

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Differentiating to Meet Students Needs
Revise the activity for your English language learners.
column
5) For both arrays, complete the sentences by writing
the number in the blank.                              Row
This array has ___ rows by ___ columns.
This array has      groups of ____.
6) The factors are the number of objects in a row and a
column of the array. Write the factors shown in the
row of columns of both arrays.
7) The product is also called a multiple. How is the
multiple related to its factors. You can show your
8) Draw arrays showing other multiples and their
factors.

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Differentiating to Meet Students Needs

1) Write a number sentence with repeated addition to represent this array.
2) Write a multiplication number sentence to
represent the array.
3) What is the product of the number sentence? How do you know?
4) For the array, complete the sentences by writing the number in the blank.
This array has ____ rows by _____ columns.
This array has _____groups of _____.
5) What are the factors of the product shown in this array?
6) The product is also called a multiple. How is the
multiple related to its factors
7) Using manipulatives, make an array with a different factors,
but with the same product as 3 x 4. Draw this array.
Name the factors and the multiple.
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Differentiating to Meet Students Needs

1) On graph paper, draw an array to represent the multiplication problem 6 x 4.
What is the product? How do you know?
2) Make at least two arrays with different factors, but with the same product as
6 x 4.
3) Write multiplication number sentences to represent your arrays.
4) List the factors shown in all of your arrays. Does the product or multiple have
any other factors, not shown in your arrays?
5) How is the multiple related to its factors?
6) Write a real-life scenario to represent one of your arrays. What are the
factors, product, and multiple in your scenario?

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Game Time!

Concept-Based Games

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• Math Journals
• Reflect individually and with
whole-group
• Record representation of key
concepts
• Opportunity to use math
vocabulary/word wall in context
• Pose questions
• Making their thinking visible!

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Three-Way-Tie

1. Along each side of the triangle, the student
writes a sentence that clearly relates the two
terms.
2. Have students use their three sentences to
develop a brief summary of the concept.
3. Allow students time to share and explain what
they wrote on their organizers.

PRODUCT, FACTORS,
MULIPLES
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FORMATIVE—checking on learning
as students progress

SUMMATIVE—checking on learning
at the end of the learning
experience

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Assessment
“When the cook tastes the
soup, that’s formative; when
the guests taste the soup,
that’s summative.”

(Stake, 2005)

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Bringing It
Altogether

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Bringing It Altogether
Mix-Freeze-Share
• What resources do you use
to teach mathematics?
• How do you determine what
to teach?
• What is the process you use

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Bringing It Altogether
Things to Consider
• Amount of instructional time for each phase
• Meeting students’ needs
• Grouping of students
• Strategies for engaging students
• Differentiation strategies to be used
• How are students’ assessed on the TCAP?

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Balanced Math Lesson

• In a grade level group, develop a Balanced
Math Lesson that you could use in your class
this year.
• We will share with the group in a gallery walk at
the end of the day.
• On chart paper, include your math
review/mental math, strategies and methods for
teaching the concept, and the closure.

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