DATE: June 3, 2014

An introduction to Number Systems and Operations - Counting and Place Value from Ancient Times - 1

How did we go:

From stones, sticks and caves for a few of us		. to .	smart factories and nice houses for millions of us??
		>>>

SUBJECT: – Mathematics: Grades K-5 Counting and Place Value - from the beginning

INSTRUCTIONAL GOAL: Students should be able to appreciate the value of counting, grouping and place values in number systems to civilization, through history and into the future.

OBJECTIVE:
To help students learn about and appreciate how, from ancient times our ancestors began to use the concepts of units, counting, grouping and, with the smart assistance of objects, symbols, and number systems as tools to measure, record and keep track of quantities in the environment, use them to improve our lives from the cave-dwelling times to the present, and into the future.

CA Common Core K-5 Mathematics Standards: Common CoreMathematics Standards (DOCUMENT) Conceptual understanding: The standards call for conceptual understanding of key concepts, such as Numbers, Place Values and Operations in Base Ten.

Grade K

Work with numbers 11-19 to gain foundations for place value.

CCSS.Math.Content.K.NBT.A.1
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Grade 1

Extend the counting sequence.

CCSS.Math.Content.1.NBT.A.1
Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Understand place value.

CCSS.Math.Content.1.NBT.B.2
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

CCSS.Math.Content.1.NBT.B.2.a
10 can be thought of as a bundle of ten ones — called a "ten."

CCSS.Math.Content.1.NBT.B.2.b
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

CCSS.Math.Content.1.NBT.B.2.c
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

CCSS.Math.Content.1.NBT.B.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Use place value understanding and properties of operations to add and subtract.

CCSS.Math.Content.1.NBT.C.4
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

CCSS.Math.Content.1.NBT.C.5
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

CCSS.Math.Content.1.NBT.C.6
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Grade 2

Understand place value.

CCSS.Math.Content.2.NBT.A.1
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

CCSS.Math.Content.2.NBT.A.1.a
100 can be thought of as a bundle of ten tens — called a "hundred."

CCSS.Math.Content.2.NBT.A.1.b
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

CCSS.Math.Content.2.NBT.A.2
Count within 1000; skip-count by 5s, 10s, and 100s.

CCSS.Math.Content.2.NBT.A.3
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

CCSS.Math.Content.2.NBT.A.4
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Use place value understanding and properties of operations to add and subtract.

CCSS.Math.Content.2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

CCSS.Math.Content.2.NBT.B.6
Add up to four two-digit numbers using strategies based on place value and properties of operations.

CCSS.Math.Content.2.NBT.B.7
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

CCSS.Math.Content.2.NBT.B.8
Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

CCSS.Math.Content.2.NBT.B.9
Explain why addition and subtraction strategies work, using place value and the properties of operations.1

Grade 3

Use place value understanding and properties of operations to perform multi-digit arithmetic.¹

CCSS.Math.Content.3.NBT.A.1
Use place value understanding to round whole numbers to the nearest 10 or 100.

CCSS.Math.Content.3.NBT.A.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

CCSS.Math.Content.3.NBT.A.3
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Grade 4

Generalize place value understanding for multi-digit whole numbers.

CCSS.Math.Content.4.NBT.A.1
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

CCSS.Math.Content.4.NBT.A.2
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

CCSS.Math.Content.4.NBT.A.3
Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

CCSS.Math.Content.4.NBT.B.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 5

Understand the place value system.

CCSS.Math.Content.5.NBT.A.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

CCSS.Math.Content.5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

CCSS.Math.Content.5.NBT.A.3
Read, write, and compare decimals to thousandths.

CCSS.Math.Content.5.NBT.A.3.a
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

CCSS.Math.Content.5.NBT.A.3.b
Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

CCSS.Math.Content.5.NBT.A.4
Use place value understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

CCSS.Math.Content.5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.5.NBT.B.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

MATERIALS AND AIDS (what you will need in order to teach this lesson)

An audiovisual computer classroom allowing each individual student to access the Internet at a reasonable speed for moderate-to-fast multimedia content.
Each station (teacher and student) should have installed the most recent Mozilla Firefox, Google Chrome, or Microsoft Explorer browser, or Apple Safari browser that includes a full-functioning Adobe Flash media player.

PROCEDURE

This lesson uses a "see-think-and-do" approach, supported with focused interactive multimedia mini-lesson activities related to the Common CoreMathematics Standards Number base ten representations and Operations. The content is guided by a narrative webpage Its a Binary world, and the New Generation Science Standards for 3-5 th grades. This lesson is proposed for convenient delivery of context and including sufficient hands-on opportunities for students to experience additional, expanded, focused activity sessions.

INSTRUCTIONAL GOAL Students should be able to begin to appreciate the purpose and value of defining "units" of substance and work with them and integrating (counting) them as parts of larger amounts of the same substance (measurement). Students should also be able to see and experience diverse number systems as smarter ways to represent and manipulate larger amounts of substance integrated in consistent groups (number base grouping of quantity).

PERFORMANCE OBJECTIVE In the end, students should be able to identify and represent correctly various common, small and large numbers in different base systems over 90% of the time . In particular, students should be able to represent and manipulate (add, subtract and multiply) Decimal, base Ten numbers into their corresponding Binary, Base Two equivalents 100% of the time.

RATIONALE An understanding of the nature and form of the Decimal and Binary number representations is helpful in providing a strong foundation and confidence in students for understanding present and future tools of computing, telecommunications and automation we see today and in the future.

LESSON CONTENT: This lesson includes participatory physical and virtual demonstration activities that stimulate the students perceptual and tactile abilities and common sense through real-life objects, coherent animation and computer interactivities. The narrative includes:

PART 1: An initial introduction to ancient and present base number systems

PART 2: Representation of FRACTIONS in different base number systems

PART 3: An introduction to basic manipulation (ADDITION, SUBTRACTION and MULTIPLICATION) for quantities represented in different base number systems. The focus is on the mastering of the human-focused Decimal and the computer-focused Binary, base 2 number systems.

INSTRUCTIONAL PROCEDURES:

SESSION 1 : An introduction to Number Systems Structures - Counting and Place Value, from ancient times.- Students will be presented first with access from their individual computer stations to the main narrative: Its a Binary world (DOCUMENT). The narrative INTRODUCTION presents a series of preliminary ideas and includes links to the most relevant number and base systems in human history:

Ancient Egyptian
Old Babylonian
Chinese
Greek and Roman
Mayan
Hindi-Arabic, Small decimal numbers, Counting Pennies, Counting Weights, Guess the Number, Rounding and Truncating, Rounding Practice Worksheet , Comparing small block counts , Comparing numbers in Decimal , Play skip-count , Addition and subtraction
Binary, Coding numbers in binary , Interactive practice , Worksheet

To conclude their reading, students should fill in the worksheet Multiple representation of numbers.

The INTRODUCTION section includes also a reference to the optimality of the Binary system (Shannons's Information Entropy Theorem), and lowest error probabilities due to minimal ambiguity in the physical definition of its only TWO distinguishable states (0, 1 or OFF, ON) for each of the place value positions. This allows for extraordinary efficiencies and further reliability through data-block integrity checks (Checksum) implemented in current electronic and electromagnetic information technologies .

After completing the review and practice work in the INTRODUCTION, students wil be asked to explore through the REPRESENTATION and FRACTIONS sections of the narrative.

SESSION 2 : The Concept of FRACTIONS, Why? How? - Free Fractioning and Base Fractioning, From Ancient times.- In this session, the students will be invited to access section 4 FRACTIONS of the narrative (Its a Binary world) from their individual computer stations to explore the basic concept and applications of fractions. Students will also be asked to explore the various displays and activities in the section FRACTIONS AND DECIMALS of the Animated Math activities.

SESSION 3 : An introduction to Number Systems Operations - Counting and Place Value, from ancient times.- Students will be requested to explore the ADDITION, SUBTRACTION and MULTIPPLICATION section of the main narrative (Its a Binary world).

ADDITION and SUBTRACTING Decimal: Adding by Counting up, Blocks, Right to Left, Blocks, Left-to-right , Subtracting by Counting Down , Adding with Mechanical Pinions

ADDITION: Binary practice with symbols, Binary addition and subtraction animation, Worksheet. For example, try the following: a) 12 + 8, b) 1250 + 875, and c) 2014 + 100.

MULTIPLICATION- DECIMAL: Basic tables and Division , With symbols , With fractions

MULTILICATION- BINARY: With symbols , Animation

A view of devices and instruments for the various number bases.- Students will be requested to explore the THE PAST, THE PRESENT, AND THE FUTURE section of the main narrative (Its a Binary world), and offer their own thoughts and conclusion about the evolution and usefulness of the various number Bases and devices that helped civilizations though history, and the possible future with the Binary, or possible new number systems. Students write down their thoughts and submit the papers for teacher review.