DEVELOPING WHOLE-NUMBER PLACE-VALUE CONCEPTS

Learning Outcomes

10.1 Pre-Place Value Understandings10.2 Developing Whole Number Place-Value Concepts10.3 Base-Ten Models for Place Value10.4 Activities to Develop Base-Ten Concepts10.5 Reading and Writing Numbers10.6 Place Value Patterns and Relationships 10.7 Numbers Beyond 1000

10.1 Pre-Place Value Understandings

Children know a lot about numbers with two digits (10 to 99) as early as kindergarten. After all, kindergartners learn to count to 100 and count out sets with as many as 20 or more objects. They count students in the room, turn to specific page numbers in their books, and so on. However, initially their understanding is quite different from yours. It is based on a count-by-ones approach to quantity, so the number 18 to them means 18 ones.
Without your help, students may not easily or quickly develop a meaningful use of groups of ten to represent quantities.
A students’ pre-base-ten understanding of number as UNITARY. That is, there are no groupings of ten, even though a two-digit number is associated with the quantity. They initially rely on unitary counts to understand quantities.

10.2 Developing Whole Number Place-Value Concepts

Place-value understanding requires an integration of new and sometimes difficult-to-construct concepts of grouping by tens (the base-ten concept) with procedural knowledge of how groups are recorded in our place-value system and how numbers are written and spoken. Importantly, learners must understand the word grouping.

Integrating Base-Ten Grouping with Counting by Ones

Students move through 3 distinct grouping stages to construct the idea that all of these sets are the same.

Unitary
Base-Ten
Equivalent

● Equivalent Groupings- with fewer than the maximum number of tens. Understanding the equivalence of the base-ten grouping and the equivalent grouping indicates that grouping by tens is not just a rule that is followed, but also that any grouping by tens, including all or some of the singles, can help tell how many. Many computational techniques (e.g., regrouping in addition and subtraction) are based on equivalent representations of numbers.

Integrating Base-Ten Grouping with Words

The way we say a number such as “fifty-three” must also be connected with the grouping-by-tens concept. The counting methods provide a connection. The count by tens and ones results in saying the number of groups and singles separately: “five tens and three.” Saying the number of tens and singles separately in this fashion can be called base-ten language. Students can associate the base-ten language with the standard language: “five tens and three—fifty-three.

10.3 BASE-TEN MODELS FOR PLACE VALUE

When students are learning base-ten concepts, they are integrating multiplicative understanding, Each place value position is ten times greater than the one to its right. This means that moving one place to the left increases the value by a factor of ten, while moving one place to the right decreases it by a factor of ten.
Physical model for base-ten concepts play a key role in helping students develop the idea of a “ten” as both a single entity and as a set of 10 units. The base ten system (decimal system) uses ten digits (0-9). Each digit's position in a number determines its value. The positions are powers of ten (units, tens, hundreds, etc.).
Place value refers to the value of a digit based on its position in the number. For example, in the number 356, the 3 is in the hundreds place, meaning it represents 300.
Base-ten proportional model can be categorized as either groupable or pregroupable.

Groupable models play a crucial role in education by promoting active learning, conceptual understanding, and the development of essential skills across various disciplines. (See Above)

Pregrouped base-ten models, the advantage of this physical model is their ease of use and the efficient way they model large numbers. Pregrouped base-ten models enhance learning by providing clear, efficient, and accurate representations of numerical concepts. (See Above)

Non-proportional models are visual or physical representations that do not preserve proportional relationships between the elements being depicted. The “ten” is not physically ten times larger than the one. They may distort sizes or shapes to emphasize certain characteristics or relationships that are not strictly proportional.

Example: An example of a non-proportional model is a map. Maps often distort the actual sizes and distances of geographic features (such as countries or continents) to fit the entire area onto a flat surface. For instance, Greenland appears much larger on some maps than it actually is in relation to other landmasses, due to the projection method used. Despite this distortion, maps remain valuable tools for navigation and understanding spatial relationships.

10.4 ACTIVITIES TO DEVELOP BASE-TEN MODELS

Grouping activities:

Counting in Groups

Hands-on Learning: Using base-ten blocks allows students to manipulate physical objects, enhancing understanding through tactile and visual learning.

Collaboration: Group work promotes teamwork and communication skills as students work together to achieve a common goal.

Conceptual Understanding: Counting in groups reinforces the concept of place value and the relationship between different place values in numbers

Groups of Ten

Visual Learning: Students see the direct relationship between individual units and groups of ten, reinforcing the concept of place value.

Concrete Understanding: Using manipulatives and base-ten blocks makes abstract concepts tangible and easier to grasp

Engaging students in this activity, they can not only reinforce their base-ten concept but also develop critical thinking skills and a deeper understanding of our numerical system.

Grouping 10 to make 100

Emphasize that grouping into hundreds helps in understanding place value and how larger numbers are composed of sets of hundreds. To reinforce understanding, the teacher needs to ask questions such as how many hundreds are in their number, and how this relates to the place value system.

For example: Instruct the groups to physically group their tens into bundles of ten rods, representing hundreds. For instance, if a group has 70 units represented by rods, they should bundle 7 groups of ten rods to visualize 700 units. Each group count how many full groups of a hundred they have and how many rods are left over.

Equivalent representation

Activities focused on equivalent representations help students understand that numbers can be represented in different ways while maintaining the same value.

For example: the students need to represent the number 53 using the base-ten blocks. They could use 5 rods (representing 50 units) and 3 units (using individual manipulatives) or they could use 4 rods and 13 units, or 3 rods and 23 units.

FIGURE 10.7 Equivalent representations using square-line-dot pictures.

10.5 READING AND WRITING NUMBERS

Helping students connect the oral and written names for numbers with their emerging base-ten concepts of using groups of 10 or 100. It emphasizes that the way numbers are said and written are conventions, not concepts, and must be learned through instruction. Special attention is given to the challenges English Learners (EL) might face, particularly with numbers 11 to 19. In kindergarten and first grade, students should connect base-ten concepts with oral number names. Initially, teachers should use explicit base-ten language (e.g., “4 tens and 7 ones” instead of “forty-seven”) and pair it with standard language. Emphasis is placed on recognizing teens as exceptions that do not fit the base-ten patterns.

“Three-Digit Number Names” explains that teaching these begins by showing students mixed arrangements of base-ten materials (e.g., 4 hundreds, 3 tens, and 8 ones) and having them give the standard name (438). Students should explore different configurations by changing only one type of piece at a time. Teachers should address common misunderstandings and ensure students grasp the relationship between oral and written forms.

A significant challenge with three-digit numbers arises with numbers like 702, where zero-tens need careful explanation. Students often struggle with internal zeros, frequently miswriting numbers such as 7,002 instead of 702. Emphasizing the importance of zero in place value is crucial. Educators are advised to avoid calling zero a “placeholder” and to be mindful of the additional time EL students might need.

Researchers highlight that errors are more frequent with four-digit numbers, underscoring the need to address these comprehensively before expecting students to generalize to larger numbers.

“Written Symbols” discusses using place-value mats to help students understand and organize their base-ten blocks. These mats are divided into sections for ones, tens, and hundreds of pieces. The standard setup places ones on the right, tens in the middle, and hundreds on the left.

The text recommends drawing two ten-frames in the ones section of the place-value mat to clearly display the number of ones. This visual aid helps students see how many additional ones are needed to make the next set of ten, reducing the need to repeatedly count the ones. If modeling two numbers simultaneously, each number can have its own ten-frame.

Students can learn that the left-to-right order of the pieces on the mat corresponds to how numbers are written. Place-value cards, one for each place value, can also be used to show how numbers are constructed.

FIGURE 10.10 Place-value mats with two ten-frames in the ones place promote the concept of groups of ten.

10.5 PLACE VALUE PATTERNS AND RELATIONSHIPS - A FOUNDATION FOR COMPUTATION

The focus will be the relationships of numbers to important special numbers called benchmark numbers and ten-structured thinking-that is, flexibility in using the structure of tens in our number system. These ideas begin to provide a basis for computation as students simultaneously strengthen their understanding of number relationships and place value.

The Hundreds Chart

The Hundreds Chart (see Figure 10.12) deserves special attention in the development of place-value concepts. K–2 classrooms should have a hundreds chart displayed prominently and used often.

Hundreds Chart Activity

Materials:

Hundreds chart with transparent pockets.
100 numeral cards

Activities:

Basic Recognition:

Have students place numeral cards in the correct pockets.

Cover certain numbers and ask students to identify or place them.

Pattern Recognition:

Highlight skip counting by 2s (even numbers), 5s, and 10s.

Have students skip count by 3s and 4s, coloring each number.

In kindergarten and first grade, students recognize two-digit numbers and develop a base-ten understanding by adding multiples of 10. They explore patterns like consistent ending digits in columns, counting by tens down columns, and identifying numbers with identical tens and ones digits along diagonals. Discussions encourage students to describe and discover these patterns.

Key Patterns for Discussion:

Column Consistency: Numbers in a column end with the same digit.

Row Sequence: The tens digit stays the same; the ones digit counts up as you move right.

Column Counting: The tens digit increases by one as you move down a column.

Counting by Tens: The far right-hand column counts by tens.

Diagonal Patterns: Starting at 11, moving diagonally down reveals numbers with identical tens and ones digits (11, 22, 33, etc.).

Students discuss and discover these patterns, enhancing their understanding of place value and number sequences.

RELATIVE MAGNITUDE USING BENCHMARK NUMBERS

Number sense includes understanding the relative magnitude of numbers, which involves comparing sizes to determine if a number is larger, smaller, or similar to another. This understanding is aided by models and benchmark numbers, like multiples of 10 and 100, which serve as reference points.

Key Points:

Hundreds Chart and Ten-Frame Cards: These tools illustrate the distance to the next

multiple of 10, aiding in understanding relative magnitude.

Benchmark Numbers: These are used for informal computation and help students see relationships between numbers. For example, adding up to the next multiple of 10 when finding differences.

Number Line: This tool is essential for exploring numerical relationships and is a strong predictor of future mathematical performance.

Approximate Numbers and Rounding:

Rounding: This is a common form of computational estimation where numbers are changed to simpler, mentally manageable numbers.

Educational Standards:

Third graders should round numbers to the nearest 10 or 100.

Fourth graders should round any multi-digit whole number to any place value.

Fifth graders should round decimals to any place.

Flexibility in Rounding:

Rounding involves selecting compatible numbers, which are easier for mental computation.

Compatible numbers can be close and need not always be multiples of 10 or 100.

Number Line Usage:

Number lines with highlighted benchmark numbers help in selecting compatible numbers. Use number lines to discuss the rounding process, especially the convention of rounding up when the digit is 5.

Connections to Real-World Ideas:

Encourage students to notice and use real-world numbers.

For K-1, focus on numbers up to 100 and 120, respectively.

Second graders should work with numbers up to 1000.

Use real-world examples (e.g., school statistics, measurements) to create engaging activities.

Turn real-world numbers into graphs, stories, and problems to enhance understanding.

Collecting and grouping data in tens, hundreds, or thousands reinforces place value and comparison of quantities.

Connecting classroom activities to real-world contexts is valuable for student learning.

10.6 NUMBERS BEYOND 1000

The importance of helping students grasp the structure and patterns in larger numbers to develop a solid foundation for arithmetic operations and problem-solving. Building on prior knowledge that encouraging students to use their understanding of smaller numbers and place value concepts to interpret and work with larger numbers. Teach the students to recognize the value of digits in larger numbers and to use strategies such as breaking numbers into manageable parts. Have a representation and visualization to students and representations like base-ten blocks, number lines and place values charts to help the students conceptualize large numbers.

Mathematical Language that encourages the use of precise mathematical language to describe numbers and their components example: thousands, ten-thousands and hundred-thousands. Provide students with the opportunities to encounter and work with larger numbers in the real-world context to make learning relevant and meaningful.

Problem-solving activities that the students engaged in activities and exercises that require the application of place value knowledge to solve problems involving large numbers. This chapter aims to deepen student’s number sense and prepare them for more advanced mathematical concepts and operations.

FIGURE 10.15 The triples system for naming large numbers.

With every three places, the shapes repeat. Each cube represents a 1, each long represents 10, and each flat represents a 100.

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