Transcript Document

Components of Number Sense
Overview
Unit 4
What’s the “Big Idea”?
• Multiple authors recommend that teachers utilize the “Big
Ideas” in their instruction.
• Generally authors present the “Big Ideas” as lesson specific.
• Often lessons are disconnected from each other and / or
focus on one “Big Idea” for that lesson.
Big Ideas Revisited
• We use the “Components of Number Sense” to connect
every part of the curriculum in almost every lesson, thus,
making our lessons more powerful and more efficient.
• Outcome equals stronger teachers and stronger math
students.
How does this fit with the
NCTM?
• The NCTM utilizes a strands curriculum
• The Components of Number Sense are not in
conflict with the NCTM strands.
• The Components enable teachers to build
true understanding of the mathematical
strands.
• This helps maintain and build connections for
the way students think about mathematics.
NCTM Strands Without
Number Sense
NCTM Strands With Number Sense
Components of Number Sense
• What are the Components of Number
Sense?
• Language Connections
• Classroom Implementation of the
Components of Number Sense
Quantity and
Magnitude
Numeration
Equality
Base Ten
Hundreds
Tens
Units
One
tenth
One
hundredth
Weights
102
101
100
10-1
10-2
Digits
6
5
4
5
2
Weighted
Value
600
50
4
0.5
0.02
* Table taken from Binary Number Systems, 2007
Total
654.52
Form of a
Number
Proportional
Reasoning
Geometric and
Algebraic
Thinking
“Arithmetic in American
Mathematics Education”
1. What are the steps for solving a mathematical
problem in China?
2. What is the American equivalent to Lie Shi?
3. Using 10 sticks model a figure and then talk
about the mathematical expressions that exist
within the visual.
4. What are the key points that Ma makes in this
article?
Reflection
• Write down individually:
– Reinforced
– Learned
– Want to Know
• Share with a partner or a small group
• Complete this task in five minutes.
Algebraic and Quantity/
Geometric
Magnitude
Thinking
Proportional
Reasoning
Numeration
Language
Form of a
Number
Equality
Base Ten
The Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
Analyze the Lesson!
How is that each component of the
Components of Number Sense gets
expressed in the teaching of the lesson as
we connect the language and build true
mathematical understanding?
Story One
• You have a dime (1/10 of a dollar) and
seven nickels (7/20 of a dollar).
• In fractional form show how much money
you have all together.
Story Two
• A team has two team members
– 3-point shots 1/10 and 7/20.
• How did the two of them perform as a
team ?
Just Do It!
Language and Math Language
• Is math-language language?
• How can we use some of what we do in
reading instruction to improve our
mathematics instruction?
Algebraic
and
Geometric
Thinking
Quantity/
Magnitude
Proportional
Reasoning
Numeration
Language
Form of
Equality
a
Number Base Ten
Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
We have to teach math like we
teach reading!
• How is math exactly like a mystery novel?
• Do we really try to see the BIG picture?
• How does our chapter in the novel fit the
whole story?
Two stories
• You have a dime (1/10 of a dollar) and
seven nickels (7/20 of a dollar).
– In fractional form show how much money
you have all together.
• Two team members record their 3-point
shots as 1/10 and 7/20.
– How did they do as a team?
Set 2 WCPSS Middle School Remedial Warm-ups
Developed with Wake County Public School System/Valerie Faulkner
© 2006 Valerie N. Faulkner/Wake County Public School System
Trashketball
The Power to Understanding Percents,
Fractions and Decimals
Let’s Try It!
• Agreements
– How far will the foul line be from the
basket?
– Are backboards allowable?
– What else?
• Demo with two players
– How hot is their shot as a team?
– Representing their stats
Trashketball Procedures
•
•
•
•
Team members rotate shots.
After you shoot, you record your result.
Return to end of line for your next shot
Team Members stay positive with team members:
“Nice shot!” “Nice try.” “You’re getting better.”
“We’re doing great.”
• Stay humble with other teams: “Great
percentage!”, “Thanks”.
• In teams of four, play Trashketball.
Trashketball Stats
Group Name _____________
Group Members: 1._________,(2)________,(3) _________(4) __________
Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Total made/Total taken
_______/20 Decimal _____ Percentage_____
What was our percent?
How do we determine this ratio?
___ / ___
How Do We Represent the Money
Problem Story One?
How do we determine this ratio?
1 Dime
7 Nickels
_45_ /_100__
Trashketball Stats for Story Two
Group Name _____________
Group Members: 1._________,(2)________,(3) _________(4) __________
1. Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10
Total made/Total taken
1/10
Decimal ___.10__
Percentage__10%_
2. Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Total made/Total taken
7/20
Decimal __.35___
Percentage_35%___
100s Grids for Story Two
How can our ratios be represented?
___ / 10
___ / 20
How Did the Team Do Altogether?
Story Two
How do we determine this ratio?
__8_ /_30_
“Move it over two”
or
Percentage and Decimal relationship using
the Components of Number Sense
45% and .45
out of 100
out of 1
How can the
Quantity
and its connections to the
Math Structure
provide understanding for the
Symbolic?
What Do We Ask?
•
•
•
•
•
•
•
Quantity/Magnitude
Numeration
Equality
Base Ten
Form of a Number
Proportional Reasoning
Algebraic and Geometric Thinking
Trashketball Stats
Group Name _____________
Team Members: 1._______,(2)_______,(3)________,(4)________
1. Shots taken: Make or Miss:
12345678
Total made/Total taken
_______5/8 Decimal _____ Percentage_____
2. Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Total made/Total taken
_______7/14 Decimal _____ Percentage_____
3. Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10 11
Total made/Total taken
_______7/11 Decimal _____ Percentage_____
4. Shots taken: Make or Miss:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Total made/Total taken
______7/16 Decimal _____ Percentage_____
___5/ 8
___7/ 11
PICTURE IT!
___7/ 14
____7/ 16
Bobcats Stats!
A coach can choose any one player to shoot “technical fouls.”
If you were the assistant coach of the Bobcats and your team
had to shoot technical fouls, who would you suggest to the
head coach to shoot the foul shots and why?
Bobcats Stats
Algebraic
and
Geometric
Thinking
Quantity/
Magnitude
Proportional
Reasoning
Numeration
Language
Form of
Equality
a
Number Base Ten
Components of Number Sense
© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
It is all about the language--“I have heard many People say, ‘Give me
the Ideas. It is no matter what Words you
put them into.’”
To this he replies, “Ideas cannot be Given
but in their minutely Appropriate Words.”
William Blake
(quoted by J. Newman, The World of Mathematics, 1956)

Assignments from Units 3 and 4
• Due Day 3
• Required Reading:
– “Subitizing” Doug Clements
– Continue to read Knowing and Teaching
Elementary Mathematics Liping Ma