reflection arithmetic challenge - mccarronsitemccarronsite.net/melt-3.pdf · reflection when pride...

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


Example


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


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?


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. 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.


Take-Away Model


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


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.


Comparison Model

Juan has 8 blocks and Susan has 3 blocks. How

many more blocks does Juan have than Susan?

8 − 3 = 5


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

kahoot.it

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


kahoot.it

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


Repeated Addition Model


The Array and Area Models

4 • 5 = 20


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


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.


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.


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.


kahoot.it

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?


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.


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


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

Lattice Base 10 Addition

1578 + 2715

Equal-Additions Base 10 Subtraction

53 16

275 28

144 57

The End of Section 3-4

Gr 3 Wksht “Exercise 9”

Gr 3 Wksht “Exercise 11”

3-4A (p. 142-144) #1b, 3,

7abc, 11abcd

READ #7d

reflection arithmetic challenge - mccarronsitemccarronsite.net/melt-3.pdf · reflection when pride...

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