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Document Sample
Conceptual Teaching of
Whole Number
Operations
Day 2
Welcome Back!
Russell Larson
Elementary Math Coordinator
Pflugerville ISD
Russell.larson@pflugervilleisd.net
512-594-0123
Norms
Collaborate with an open mind!
Silence your cell phone.
Take care of your needs as they arise.
Enjoy Yourself…Suffering is Optional!
What is Learned Here…Leaves Here!
Objectives
Understand that efficient computation comes
from conceptual understanding of numbers.
Understand that efficient computation can be
accomplished in many ways. Including, but not
limited to, an algorithm.
Understanding of number is based on place
value and the ability to compose and decompose
numbers.
Building understanding of whole number
operations is accomplished through the use of
concrete models and pictorial representations,
before moving into the abstract.
Developing Computational
Fluency with Whole Numbers
(Homework)
• As a group, share your V.I.P.’s and discuss
the relevance of each point.
• Choose two points to share. Write them
on sentence strips.
• Post in room. Be ready to share your
group’s thoughts
Relationship of Base 10 Blocks to
Whole Numbers and Decimals
Boy… we can confuse our students…..
Closest to 1
1. 2 – 3 players
2. Each player begins with a Flat ( worth the value of 1 whole)
3. On you turn, you roll 2 number cubes
1. Decide how to arrange the digits as a decimal amount less than 1
2. Use Base 10 Blocks to show how to add/subtract that decimal amount
3. Record your action on the recording sheet
4. After each turn, all players check the trading and recording
5. Play continues for a total of 10 rounds
6. At any given time during the game, a player may decide to
skip 1 turn (and ONLY 1 turn) even after rolling the number
cubes
7. Winner is the player closest to 1 at the end of 10 rounds
Other Decimal Activities
What’s 1?
Tenths or Hundredths?
Making and Writing Decimals
Decimal Mirrors
BREAK TIME
See you in 15 minutes
TAKS Problem Study!
3.3 (A B) 3.4 (B C)
4.3 (A B) 4.4 (A B D E)
5. 3 (A B C D E) 5.4
Multiplication/Division
Concrete
Pictorial
Abstract
Flexible Strategies for
Multiplication
Complete-Number Strategy (repeated addition)
Students do not break numbers apart.
Partitioning Strategies for Multiplication
By Decades
Partitioning the Multiplier
By Tens and Ones
Other Partitions
Compensation Strategies for Multiplication
Area Model
Each table pick and present
Division
Zach has ____ pencils. They are packed
____ pencils to a box. How many boxes of
pencils does he have?
(12, 3) (28, 4) (34, 8) (110, 10)
Bart has ____ boxes of pencils with the
same number of pencils in each box. All
together he has _____ pencils. How many
pencils are in each box?
(5, 15) (6, 24) (8, 42) (9, 108)
Types of Division
Name the two types of division.
Measurement Division
Partitive Division
What is the difference?
Measurement Division:
Number of groups is unknown
Partitive Division:
Number of items in each group is
unknown.
Flexible Strategies for Division
Jigsaw
1’s: Read pgs. 121-123
2’s: Read pgs. 124-125 (Develop written record)
3’s: Read pg. 125 (Record explicit trades) -126
4’s: Read pgs. 127-128
Lunch Time
Multiplication & Division
Games Rotation
The Product Game
Target Multiplication
It’s In the Bag
Multiplication… Arrgggg
X 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
Multiplication… Doubles
X 0 1 2 3 4 5 6 7 8 9
0 0
1 2
2 0 2 4 6 8 10 12 14 16 18
3 6
4 8
5 10
6 12
7 14
8 16
Multiplication… Fives
X 0 1 2 3 4 5 6 7 8 9
0 0 0
1 2 5
2 0 2 4 6 8 10 12 14 16 18
3 6 15
4 8 20
5 0 5 10 15 20 25 30 35 40 45
6 12 30
7 14 35
8 16 40
Multiplication… Zeros & Ones
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 15
4 0 4 8 20
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 30
7 0 7 14 35
8 0 8 16 40
Multiplication… Nines
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 15 27
4 0 4 8 20 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 30 54
7 0 7 14 35 63
8 0 8 16 40 72
Multiplication… Helping Facts
Doubles and Doubles Again
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
Multiplication… Helping Facts
Doubles and One More
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
Multiplication… Helping Facts
Just 6 more to learn
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
Multiplication… DONE……
Add the 10’s, 11’s, and 12’s
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 6 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
6 0 6 12 18 24 30 36 42 48 54
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
Math Problem Solving:
The primary goal
of mathematic is
to enable students
to develop their
ability in
“Mathematical
Problem Solving”.
Word Problems….
ARRGH !
Word Problems…
Overcoming the Fear!
Put your pencil down. Read the problem.
After everyone in your group has finished
reading the problem, have each person
tell one thing that they KNOW about the
problem. (state a fact)
Next, have each person ASK a question
about the problem.
Now SOLVE the problem with your group.
Now Try It!
In building the road through the
subdivision, a low section in the land was
filled in with dirt that was hauled in by
trucks. The complete fill required 638
truckloads of dirt. The average truck
carried 6 ¼ cubic yards of dirt, which
weighed 17.3 tons. How many tons of dirt
were used in the fill?
Word Problems…
Overcoming the Fear!
Building the Context w/ Students
What is happening in the problem?
What will the answer tell us?
Will that be a small number of tons or a
large number of tons?
About how many do you think it will be?
Now Try It!
In building the road through the
subdivision, a low section in the land was
filled in with dirt that was hauled in by
trucks. The complete fill required 638
truckloads of dirt. The average truck
carried 6 ¼ cubic yards of dirt, which
weighed 17.3 tons. How many tons of dirt
were used in the fill?
Two-Step Problems
Step 1
It took 3 hours for the Joneses to drive
the 195 miles to Washington. What
was their average speed?
Write a second problem that uses the
answer to the first problem.
The Jones children remember crossing the river
2 hours after they left home. About how far from
home is the river?
Step 2
Combine the previous 2 questions to
create a new problem. Leave out the
question from the first problem.
What is the new problem?
It took 3 hours for the Joneses to drive the 195 miles to
Washington. The Jones children remember crossing the
river 2 hours after they left home. About how far from
home is the river?
Practice
Given problem: Tony bought 3 dozen eggs
for $0.89 per dozen. How much was the
bill?
Write a second problem that uses the
answer to the first problem.
Combine the two problems to create a two
step Hidden Question Problem:
Given problem: Tony bought 3 dozen eggs
for $0.89 per dozen. How much was the
bill?
Second Problem: How much change did
Tony receive from $5.00?
Two step problem with hidden question:
Tony bought 3 dozen eggs for $0.89
per dozen. How much change did
Tony receive from $5.00?
Step 3
Pose standard multi-step problems and
have students identify and answer the
hidden question.
Sample:
Willard’s Sales decides to add widgets to
its line of sale items. To begin with Willard
bought 275 widgets wholesale for $3.00
each. In the first month the company sold
205 widgets at $5.00 each. How much did
Willard make or lose on the widgets?
Problem Solving Detectives
Approach
Read, Explain, Reflective
Reasoning
2 – 3 students in a group
Witness, Detective, and By
Stander
Witness – Sees everything
Detective – Solves the crime
By Stander – Observes all
Let’s Practice!
Get in groups of 3
One will be the witness, one will be the
detective, one the by-stander (silent
partner)
Each get a the correct card for your part
Listen for directions
Witness and By-Stander will be the only
ones to read the next slide.
Witness: Read this information and be ready to
answer questions from the detective.
Maria and her family drove 1,236
miles on vacation. Monica and her
family drove 376 miles on their
vacation. What was the total number
of miles both families spent driving
on their vacations?
Detective: Ask all the questions you need.
Write down the responses from the Witness.
By-stander can take it all in.
Research says you have to read
the problem multiple times in
order to get all the details and
information.
Allow the witness to review the
problem several times, and the
detective needs to write down the
facts they learn.
Extension to Word Problems to
Provide More Meaning
Oral Explanations
Journal / Reflection
Multiplication and Division
Activities
Work through the series of activities
provided in the packet.
Talk with your table:
What are the differences in these types of
activities and that of a typical worksheet?
Which is better to build conceptual
understanding?
Book Walk
A-Z
Reflection
Please take a few minutes to
reflect on your day!
Think about what we’ve done and how it
will effect the kinds of experiences you will
want to have with your students.
