August-September


 * Number Sense: Place Value, Rounding and Comparing ** ** (August – end of September) **
 * 2.1 The student will**
 * a) read, write, and identify the place value of each digit in a three-digit numeral, using numeration models;**
 * b) round two-digit numbers to the nearest ten; and**
 * c) compare two whole numbers between 0 and 999, using symbols (>, <, or =) and words (greater than, less than, or equal to).**

• The number system is based on a simple pattern of tens where each place has ten times the value of the place to its right. • Opportunities to experience the relationships among hundreds, tens, and ones through hands-on experiences with manipulatives are essential to developing the ten-to-one place-value concept of our number system and to understanding the value of each digit in a three-digit number. Ten-to-one trading activities with manipulatives on place-value mats provide excellent experiences for developing the understanding of the places in the base-10 system. • Models that clearly illustrate the relationships among hundreds, tens, and ones are physically proportional (e.g., the tens piece is ten times larger than the ones piece). • Students need to understand that 10 and 100 are special units of numbers (e.g., 10 is 10 ones, but it is also 1 ten). • Flexibility in thinking about numbers is critical. For example, 123 is 123 ones; or 1 hundred, 2 tens, and 3 ones; or 12 tens and 3 ones. • Rounding is finding the nearest easy-to-use number (e.g., the nearest 10) for the situation at hand. • Number lines are useful tools for developing the concept of rounding to the nearest ten. Students can use the strategy of identifying a number on a number line and finding the multiple of ten that is closest to the identified number. • Once the concept for rounding numbers using a number line is developed, a strategy for rounding numbers to the nearest ten is as follows: – Look one place to the right of the digit you wish to round to. – If the digit is less than 5, leave the digit in the rounding place as it is, and change the digit to the right of the rounding place to zero. – If the digit is 5 or greater, add 1 to the digit in the rounding place and change the digit to the right of the rounding place to zero. • A procedure for comparing two numbers by examining place value may include the following: – Line up the numbers by place value lining up the ones. – Beginning at the left, find the first place value where the digits are different. –Compare the digits in this place value to determine which number is greater (or which is less). – Use the appropriate symbol > or < or words greater than or less than to compare the numbers in the order in which they are presented. – If both numbers are the same, use the symbol = or the words equal to. • Mathematical symbols (>, <) used to compare two unequal numbers are called inequality symbols.
 * UNDERSTANDING THE STANDARD**
 * (Background Information for Instructor Use Only)**

All students should • Understand the ten-to-one relationship of ones, tens, and hundreds (10 ones equals 1ten; 10 tens equals 1 hundred). • Understand that numbers are written to show how many hundreds, tens, and ones are in the number. • Understand that rounding gives a close, easy-to-use number to use when an exact number is not needed for the situation at hand. • Understand that a knowledge of place value is essential when comparing numbers. • Understand the relative magnitude of numbers by comparing numbers.
 * ESSENTIAL UNDERSTANDINGS**

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Demonstrate the understanding of the ten-to-one relationships among ones, tens, and hundreds, using manipulatives (e.g., beans and cups, base-10 blocks, bundles of 10 Popsicle sticks). • Determine the place value of each digit in a three-digit numeral presented as a pictorial representation (e.g., apicture of base-10 blocks) or as a physical representation (e.g., actual base-10 blocks). • Write numerals, using a base-10 model or picture. • Read three-digit numbers when shown a numeral, a base-10 model of the number, or a pictorial representation of the number. • Identify the place value (ones, tens, hundreds) of each digit in a three-digit numeral. • Round two-digit numbers to the nearest ten. • Compare two numbers between 0 and 999 represented pictorially or with concrete objects (e.g., Base-10 blocks), using the words greater than, less than or equal to.
 * ESSENTIAL KNOWLEDGE AND SKILLS**


 * 2.2 The student will**
 * a) identify the ordinal positions first through twentieth, using an ordered set of objects; and**
 * b) write the ordinal numbers.**

• Understanding the cardinal and ordinal meanings of numbers are necessary to quantify, measure, and identify the order of objects. • An ordinal number is a number that names the place or position of an object in a sequence or set (e.g., first, third). Ordered position, ordinal position, and ordinality are terms that refer to the place or position of an object in a sequence or set. • The ordinal position is determined by where one starts in an ordered set of objects or sequence of objects (e.g., left, right, top, bottom). • The ordinal meaning of numbers is developed by identifying and verbalizing the place or position of objects in a set or sequence (e.g., a student’s position in line when students are lined up alphabetically by first name). • Ordinal position can also be emphasized through sequencing events (e.g., months in a year or sequencing in a story). • Cardinality can be compared with ordinality when comparing the results of counting. There is obvious similarity between the ordinal number words third through twentieth and the cardinal number words three through twenty.
 * UNDERSTANDING THE STANDARD**
 * (Background Information for Instructor Use Only)**

All students should • Use ordinal numbers to describe the position of an object in a sequence or set.
 * ESSENTIAL UNDERSTANDINGS**

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Count an ordered set of objects, using the ordinal number words first through twentieth. • Identify the ordinal positions first through twentieth, using an ordered set of objects. • Identify the ordinal positions first through twentieth, using an ordered set of objects presented in lines or rows from – left to right; – right to left; – top to bottom; and – bottom to top. • Write 1st, 2nd, 3rd, through 20th in numerals.
 * ESSENTIAL KNOWLEDGE AND SKILLS**


 * 2.4 The student will**
 * a) count forward by twos, fives, and tens to 100, starting at various multiples of 2, 5, or 10;**
 * b) count backward by tens from 100; and**
 * c) recognize even and odd numbers.**

• The patterns developed as a result of grouping and/or skip counting are precursors for recognizing numeric patterns, functional relationships, and concepts underlying money, time telling, multiplication, and division. Powerful models for developing these concepts include counters, hundred chart, and calculators. • Skip counting by twos supports the development of the concept of even numbers. • Skip counting by fives lays the foundation for reading a clock effectively and telling time to the nearest five minutes, counting money, and developing the multiplication facts for five. • Skip counting by tens is a precursor for use of place value, addition, counting money, and multiplying by multiples of 10. • Calculators can be used to display the numeric patterns resulting from skip counting. Use the constant feature of the four-function calculator to display the numbers in the sequence when skip counting by that constant. • Odd and even numbers can be explored in different ways (e.g., dividing collections of objects into two equal groups or pairing objects).
 * UNDERSTANDING THE STANDARD**
 * (Background Information for Instructor Use Only)**

All students should • Understand that collections of objects can be grouped and skip counting can be used to count the collection. • Describe patterns in skip counting and use those patterns to predict the next number in the counting sequence. • Understand that the starting point for skip counting by 2 does not always begin at 2. • Understand that the starting point for skip counting by 5 does not always begin at 5. • Understand that the starting point for skip counting by 10 does not always begin at 10. • Understand that every counting number is either even or odd.
 * ESSENTIAL UNDERSTANDINGS**

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Determine patterns created by counting by twos, fives, and tens on a hundred chart. • Skip count by twos, fives, and tens to 100, using manipulatives, a hundred chart, mental mathematics, a calculator, and/or paper and pencil. • Skip count by twos, fives, and tens to 100. • Count backward by tens from 100. • Use objects to determine whether a number is odd or even.
 * ESSENTIAL KNOWLEDGE AND SKILLS**