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Reflection
When pride comes, then comes disgrace,
but with humility comes wisdom.
Proverbs 11:2
Arithmetic Challenge
mccarronsite.net/
ArithmeticChallenge.html
In order to keep up in Chapters 3 and 4,
your arithmetic facts have to be
automatic
Goal: complete level 2 for all 4
operations by next week
Warm Up
All Texans wear boots. No one who
wears boots is a wimp. Therefore, no
Texans are wimps.
Today
Warm up
Homework Stumpers
Gifted Hands Chitchat
Shipwreck
Positional Number Systems
Representing Numbers
Writing Numbers
Saying Numbers
Don’t Forget
Pick up today
» Quiz
» HW 2-2A
» Spelling number words
» Mayan Practice
» Practice 3-1
» 3rd Grade worksheets (2)
Turn in today
» HW 2-2B
Gifted Hands Essay
1st Draft (5 points) due 10/20
Final Draft (20 points) due 11/12
300-500 words, double-spaced
Describe the scene or thread you are
referring to
Explain how it will influence you as a
teacher
Homework Stumpers Deconstruction
The Year: 1500 AD
The Yucatan Positional Number System
Each system has a special number called the
“base”
Each position has a place value, determined
by powers of the base
Each digit in a written number has a value,
the digit times the place value.
value = digit place value
Example
Convert Mayan number to base 10:
Example
Convert Mayan number to base 10:
Example
Convert Mayan number to base 10:
Mayan Practice page 1 Mayan Addition
+
Mayan Addition
+
Mayan Practice page 2
The End of Mayan Numbers
Finish Mayan number
worksheet for next time.
Reflection
What causes fights and quarrels among you? Don't they come from your desires that battle within you? You want something but don't get it. You kill and covet, but you cannot have what you want. You quarrel and fight. You do not have, because you do not ask God. When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures.
James 4:1-3
Warm up
Convert the Mayan number to base 10:
Today
Warm up
Mayan Stumpers
Positional Number Systems
Representing Numbers
Writing Numbers
Saying Numbers
Quiz
Don’t Forget
Pick up today
» Spelling number words
» Practice 3-1
Turn in today
» Mayan Practice
Homework Stumpers
Numeration Systems
These systems are testable:
» Base Ten
» Base Five
» Mayan
Other systems may appear in lecture
and homework, not on quizzes or
exams.
Abstract Concrete Semi Concrete
Mathematical Manipulatives
Physical models for abstract concepts
Many, many manipulatives on the
market
At a particular grade level, focus on a
few tools
Abstract Concrete Semi Concrete
Base 10 Blocks
Units (1 cubic cm)
10 Units = 1 Long (aka “rod”)
10 Longs = 1 Flat (aka “tile”)
10 Flats = 1 Block
Three Way Connection
Physical
Model
Numerical
Representation
Words
Make the Connection
Physical
Model
3527
Words
TEKS & Place Value Expansions
Grade 3: Number and operations. The student applies mathematical process standards to represent and compare whole numbers and understand relationships related to place value. The student is expected to:
(A) compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate;
Write in (a) expanded form (b) words
1. 36,713
2. 5098
3. 210,521
4. 2,131,101
Number words
Commas appear in wording where commas appear in numerical representation Five thousand, two hundred twenty-one
Hyphen is used in number words for twenty-one to ninety-nine
“and” is only used for the decimal point or other fractions (follow your text book)
Handout
Spelling!
3-1 Practice #1-2 Examples
Write in base 10:
5 blocks, 7 flats, 3 units
Write the number in words: 764,328,170,915
3-1 Practice #3-4 Write in expanded form
1. 211three
2. 25seven
3. 213five
4. 1101two
3-1 Practice #5 Convert Base 10 to Base 5
23
57
119
348
3-1 Practice #6 The End of Section 3.1
Finish homework from
section 3.1 and skim
section 3.2 for next time.
Reflection
Blessed are the merciful,
for they will be shown mercy.
Matthew 5:7
Warm Up
??? BAn
B A
Today
Warm up
Stumpers
Closure property
Terminology
Cognition and mental models
Mental models for addition
Addition properties
Mental models for subtraction
Don’t Forget
Pick up today
» Practice 3-2 Part 1
» # Lines
Turn in today
» Quiz
» HW 3-1
» Hands Draft
Exam 2 November 5th
Homework Stumpers Closure Property
The sum of two whole numbers is a whole
number.
Generic Closure Property
The set of X is closed under the operation Y if whenever you choose two elements from the set X and combine them using the operation Y , you always get another element of the set X .
X is a set
Y is an operation
Terminology
Every discipline (including mathematics)
has specialized vocabulary
Emerging research indicates students
perform better in mathematics when:
» Their teachers use mathematical
terminology
» The students learn to use the terminology
Terminology
There are some variations. When you
are a classroom teacher, follow your
book
In this course, you will be responsible
for the terminology:
» On the final exam (big matching page)
» On quizzes with warning
Terminology
Addition:
addend + addend = sum
Subtraction:
minuend – subtrahend = difference
A Cognitive Gap
Discrete
Represented by integers
(natural numbers and
whole numbers are
subsets)
There is always “space”
between numbers
Counting things
Continuous
Represented by real
numbers (rational
numbers are a subset)
The set of numbers is
“dense” (you can always
find a number in
between)
Measuring quantities
Discrete or Continuous?
What’s the temperature?
How many people came to the party?
How much water did you drink?
How much does it weigh?
How many chairs will we need?
How many days until Christmas?
What was her time in the 10k?
Mental Models Mental Models
Example: 7+5
Counting On
Set Union
Number Line
3-2 Part 1 Practice #1
1. Illustrate the addition problem on the
number line: 4 + 3
Cuisenaire Rods Remember?
??? BAn
B A
Question
What kind of sets should you choose to
model addition?
3-2 Part 1 Practice #2
2. For which of the following sets is it true that
n(AB) = n(A) + n(B)?
a. A = {x, y, z}, B = {p, d, q}
b. A = {n, u, t}, B = {a, n, t}
Properties of Addition of Whole Numbers
Closure property
Identity property
Commutative property
Associative property
Closed under Addition?
G={4, 8, 12, 16}
H={4, 8, 12, 16, …}
K={2n | nW }
L={0, 1, 2 }
G={n | nW and n>25}
3-2 Part 1 Practice #3
3. Which of the following sets are closed under
addition? If not closed, explain why.
a. M = {3, 6, 9, 12, …}
b. Y = {10, 11, 12}
c. Z = {10, 11, 12, …}
d. K = {5n | n W}
5335
:Example
addition) (of
abba
Commutative Property
235235
:Example
addition) (of
cbacba
Associative Property
505
:Example
0addition) (of
aa
Identity Property
Signal Your Answer…
A. 7 + 0 = 7
B. 3 + 6 = 6 + 3
C. (2+1)+3 = 2+(1+3)
D. (a+b)+c = c+(a+b)
E. (5+2)+0=(5+2)
F. (x+2)+5=(2+x)+5
G. 6 + 203 is a unique
whole number
1. Commutative
Property of Addition
2. Associative Property
of Addition
3. Additive Identity
Property
4. Closure Property of
Addition for Whole
Numbers
3-2 Part 1 Practice #4-7
4. 3 + 7 = 7 + 3 A. Identity Property for Addition
5. (2 + 4) + 6 = 2 + (4 + 6)
6. 0 + 15 = 15
B. Associative Property for Addition
7. (1 + 6) + 8 = (6 + 1) + 8 C. Commutative Property for Addition
The End of Section 3-2 Part 1 Reflection
Live such good lives among the pagans that, though they accuse you of doing wrong, they may see your good deeds and glorify God on the day he visits us.
1 Peter 2:12
Warm Up
Use the Cuisenaire
Rods to find out how
many ways you can
partition 7 into a
sum of three natural
numbers.
(Note: 1+1+5 and
1+5+1 are the same
partition.)
On Your Mobile Device
kahoot.it
Today
Warm up
Stumpers
Subtraction models
Pre-Algebra
Mental models for multiplication
Quiz
Don’t Forget
Pick up today
» Practice 3-2 Part 2
» Quiz
» Grid paper
» Study Questions
Turn in today
» HW 3-2 Part 1
Exam 2 November 5th
Homework Stumpers Subtraction of Whole
Numbers
Subtraction of whole numbers can be modeled in
several different ways:
Take-Away Model – views subtraction as a
second set of objects being taken away from the
original set
Missing Addend Model – an algebraic-type of
reasoning is used where students compute a
difference by determining the value of an
“unknown” addend.
Subtraction of Whole Numbers
Comparison Model – students determine “how
many more” of one quantity exists than another.
Number-Line Model – subtraction is represented
by moving left on the number line a given number
of units.
Subtraction of Whole Numbers
Take-Away Model
Subtraction of Whole Numbers
Missing-Addend Model
This can be thought of as the number of blocks that
must be added to 3 in order to get 8.
8 − 3 =
The number 8 – 3 is the missing addend in the
equation 3 + = 8
5
Subtraction of Whole Numbers
Missing-Addend Model
Definition
Subtraction of Whole Numbers:
For any whole numbers a and b, such that
a ≥ b, a − b is the unique whole number c
such that b + c = a.
Subtraction of Whole Numbers
Comparison Model
Juan has 8 blocks and Susan has 3 blocks. How
many more blocks does Juan have than Susan?
8 − 3 = 5
Subtraction of Whole Numbers
Number-Line (Measurement) Model
5 − 3 = 2
One More
*Counting Down
Properties of Subtraction
It can be shown that if a < b, then a − b is
not meaningful in the set of whole numbers.
Therefore, subtraction is not closed on the
set of whole numbers.
Introductory Algebra Using Whole-Number Addition and Subtraction
Sentences such as 9 + 5 = ☼ and 12 − ◊ = 4
can be true or false depending on the values
of ☼ and ◊.
For example, if ☼ = 10, then 9 + 5 = ☼ is false.
If ◊ = 8, then 12 − ◊ = 4 is true.
If the value that is used makes the equation true, it
is a solution to the equation.
The End of Section 3-2 Part 2 On Your Mobile Device
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Reflection
So when you give to the needy, do not announce it with trumpets, as the hypocrites do in the synagogues and on the streets, to be honored by men. I tell you the truth, they have received their reward in full.
Matthew 6:2
On Your Mobile Device
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Forgot On Study Questions
Convert 234 from base ten to base five.
Answer: 1414five
Today
Warm up
Stumpers
Terminology
Multiplication models
Fact Families
Study Question Stumpers
Don’t Forget
Pick up today
» Practice 3-3
» Quiz
» Grid paper
» Multiplication
Worksheets
Turn in today
» HW 3-2 Parts 1&2
Exam 2 Wednesday
Homework Stumpers
Terminology
factor factor product
4276
Terminology
divisor dividend quotient
11555
Equivalent To
11555
11555
Equivalent To
115/55
11555
Terminology
9728
divisor
dividend
quotient
Equivalent To
9728
divisor
dividend
quotient
7298
3-3 Practice #3 Terminology
8425
divisor
dividend
quotient
2Rremainder
Equivalent To
8425
2R
42285
Practice
910911
10R
Multiplication of Whole Numbers
Repeated Addition Model
If we have 4 groups of 3 brushes, we can use
addition to put the groups together. When we put
equal-sized groups together we can use
multiplication. We can think of this as combining 4
sets of 3 objects into a single set.
3 + 3 + 3 + 3 = 12
four 3’s
Multiplication of Whole Numbers
Repeated Addition Model
Multiplication of Whole Numbers
The Array and Area Models
4 • 5 = 20
Multiplication of Whole Numbers
An Array
4 • 5 = 20
3-3 Practice #1 Reflection
I have much more to say to you, more
than you can now bear.
John 16:12
Today
Properties of Multiplication
Mental Models for Division
Standard Addition Algorithm
Trading Up
Standard Subtraction Algorithm
Trading Down
Nonstandard Algorithms
Engagement Presentations
Standard (TEKS)
Example Assessment Question
Book
Bulletin Board
No More than 5 Minutes
On paper (for doc camera) or PPT sent
an hour before class
Properties of Whole-Number Multiplication
Closure Property of Multiplication
of Whole Numbers If a and b are whole numbers, then a · b is a
unique whole number.
Generic Closure Property
The set of is closed under the operation if whenever you choose two elements from the set and combine them using the operation , you always get another element of the set .
3-3 Practice #2 Multiplication of Whole
Numbers
Multiplication of whole numbers can be modeled in
several different ways:
Repeated-Addition Model – multiplication
can be thought of as repeatedly adding the
multiplicand, multiplier times.
Array and Area Model – multiplication can
be visualized by constructing an array of
crossed sticks or a grid, and counting
intersection points or grid components,
respectively.
That Worksheet from last week That Worksheet from last week
That Worksheet from last week That Worksheet from last week
Properties of Whole-Number Multiplication
Closure Property of Multiplication
of Whole Numbers If a and b are whole numbers, then a · b is a
unique whole number.
Commutative Property of Multiplication
of Whole Numbers
If a and b are any whole numbers, then
a · b = b · a.
Properties of Whole-Number Multiplication
Associative Property of Multiplication
of Whole Numbers If a, b, and c are whole numbers, then
(a · b) · c = a · (b · c).
Identity Property of Multiplication
of Whole Numbers
There is a unique whole number 1 such that for
any whole number a, a · 1 = a = 1 · a.
Properties of Whole-Number Multiplication
Zero Multiplication Property
of Whole Numbers For any whole number a, a · 0 = 0 = 0 · a.
Distributive Property of Multiplication Over Addition for Whole Numbers
For any whole numbers a, b, and c,
Commutative Property of Multiplication
Associative Property of Multiplication
Distributive Property of Multiplication Over Addition
The area of the large rectangle equals the sum of
the areas of the two smaller rectangles.
On Your Mobile Device
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Division by 0 and 1
Let n be any nonzero whole number.
Then,
n ÷ 0 is
0 ÷ n =
0 ÷ 0 is
n ÷ 1 =
undefined.
0.
undefined.
n
Division of Whole Numbers
Set (Partition) Model – the set of elements
representing the dividend is partitioned into
divisor-number of subsets.
Missing-Factor Model – a divided by b is a
unique number c, provided that b • c = a.
Repeated Subtraction Model – the divisor is
continually subtracted from the dividend until
only the remainder is left.
Three models for division
Division of Whole Numbers Set (Partition) Model Suppose we have 18 cookies and want to give an
equal number of cookies to each of three friends:
Bob, Dean, and Charlie. How many should each
person receive?
Division of Whole Numbers
Missing Factor Model If each friend receives c cookies, then the three
friends will receive 3c, or 18 cookies. Therefore,
3c = 18. Since 3 • 6 = 18, we have answered the
division computation by using multiplication.
Division of Whole Numbers
Repeated-Subtraction Model Suppose we have 18 cookies and want to package
them in cookie boxes that hold 6 cookies each.
How many boxes are needed?
If one box is filled, then there are 18 − 6 = 12
cookies left.
If one more box is filled, then there are 12 − 6 = 6
cookies left.
The last 6 cookies will fill another box.
The End of Section 3-3
Final Draft Gifted Hands
Due 11/12
Attach graded first draft
Algorithm
What does it mean?
What is a “standard algorithm”?
Texas standards vs. Common Core
Standard Base 10 Addition
1578 + 2715
Now model with base ten blocks
Testing and Grading
Web Reflection
Defend the cause of the weak and fatherless; maintain the rights of the poor and oppressed. Rescue the weak and needy; deliver them from the hand of the wicked.
Psalm 82:3-4
Today
Stumpers HW & Exam
Engagement
Trading Up
Engagement
Standard Subtraction
Trading Down
Change the game
Nonstandard algorithms
Don’t Forget
Pick up today
» Grade 3 Wksht
“Exercise 9”
» Grade 3 Wksht
“Exercise 11”
Turn in today
» HW 3-3
» Hands final draft
Stumpers Engagement
Game
Trading up
Play a game to 200, rolling 2 dice per turn
Engagement
Standard Base 10 Subtraction
3575 1728
502 236
Game
Trading down
» Play one game down from 40, rolling one die
per turn
» Play a second game down from 150, rolling 2
dice per turn
You probably cheated
How and why
Multi-digit Add or Subract
Trading
up/down
with Base
10 blocks
Standard
Algorithm
Abstract Concrete Semi Concrete
Change the Game
Trading up Base 5
» Play one game to 60ten=220five, rolling one die
per turn
» Play a second game to 200ten=1300five, rolling
2 dice per turn
Demonstrate “standard” algorithm for base
5 addition using paper and supporting with
manipulatives:
134five + 232five
Change the Game
Trading down Base 5
» Play one game down from 40ten=130five,
rolling one die per turn
» Play a second game to 150ten=1100five, rolling
2 dice per turn
Demonstrate “standard” algorithm for base
5 subtraction using paper and supporting
with manipulatives:
231five 132five