The focus of this session is on the complexities of counting and on the work students do in Investigations around making sense of our base-ten number system and the place value of numbers.

In this session you will:

• read scenarios of students counting and analyze them to see what they do and do not yet understand about counting
• examine activities in which students work on the sequence of numbers and begin to make sense of the structure of the number system
• consider the multiple aspects of place value that students need to understand
• examine some games and activities that highlight the importance of 10 and 100 in our number system

Materials: print Hundredths Grids

Getting Started: Counting

 

Count the butterflies on the poster to the right. You can click on the poster to enlarge it. Double check your first count using a different strategy.

Consider:

How did you keep track of your count? What did you need to know or understand in order to count the number of butterflies?

Read Counting Is More Than 1, 2, 3 which discusses the complexities involved in counting.

 

In this activity, you will look at aspects of counting, analyze counting scenarios and consider what students understand and do not yet understand about counting.

This class is reading Anno's Counting Book, a wordless picture book. Each two-page spread focuses on a particular number and includes the numeral and a tower with that many cubes. (Numbers over ten are shown with two towers, of 10 and some more.) The illustrations represent the quantity in various ways (e.g. 11 trees, 11 birds, 11 houses).

Watch as this group of kindergarteners discusses the 11 page, predicts the number on the next page, and thinks about how to represent it.

Read Counting What's in a Mystery Box in which three students come up with different numbers for the same amount.

What aspects of counting do students engage in and what do they seem to understand about counting in the Anno's Counting Book video and the Counting What's in a Mystery Box reading?

Counting Scenarios

Observing students as they count can give you information about what they understand about counting a set of objects and what they have not yet figured out. This information can be difficult to obtain from only looking at what a student records on paper.

Read Observing First Graders as They Count which describes what you might see as first-graders count.

For each of the three Kindergarten scenarios listed in the Counting Scenarios Forum, share the following information:

• What did the student do?
• What do you think the student understands about counting?
• What -- if anything -- do you think the student does not yet understand about counting?
• What questions might you want to ask the student?

In this activity, you will analyze written counting sequence errors; determine where numbers belong on a 1,000-chart based on the structure of our base-ten number system; and think about the place value of a specific number.

Counting Strips

Throughout Investigations students have many opportunities to work on learning both the oral and written number sequence and the structure of the number system.

• In Kindergarten, students use a number line to practice the rote counting sequence to 100.
• In first and second grade, students look for patterns in the number sequence and work on the structure of the number system up to 1,000.
• In later grades, they continue to examine patterns in the number system and the structure of the number system as they work with very large numbers such as 10,000 or 100,000 and very small numbers such as 0.001.

In first grade and then at the beginning of second grade, students do an activity called Counting Strips. They start at zero and write the numbers vertically in sequence, as high as they can, on a strip of adding machine tape.

Teachers can learn how high their students can count and what students understand and don't yet understand about the written number sequence as they watch their students create counting strips.

Feature - Ongoing Assessment: Observing Students at Work

Observing students as they engage in activities and conversations about their ideas is a primary means of assessing students' learning. Such formative assessment opportunities are built into every Investigations session. One feature is the Ongoing Assessment: Observing Students at Work which offers questions to consider as teachers observe students solving problems, playing math games and working on activities.

Use the Observing Students at Work questions below to help you analyze the mistakes in the following counting strips [Click the image to enlarge it].

View the parts of three strips that illustrate common errors first grade students make as they make the counting strips.

100 Chart

After first grade students have created their own counting strips, the teacher creates a horizontal counting strip of the numbers 1 to 100. The class then cuts the counting strip into rows of 10 and creates a 100 Chart.

A 100 Chart is a tool that is used throughout the grades to learn about the number system and to work on addition, subtraction, and multiplication.

Finding Numbers on a 1,000 Chart

Third grade students create a 1,000 chart using ten blank 100 charts. Students fill in enough numbers on each 100 chart so they can locate any number. Then they find specific numbers and record them on their 1,000 chart.

Use the 1,000 Book to find the numbers in the spaces marked by letters A-E.

Consider the following questions:

• What strategies did you use to determine what number belonged in each square?
• What patterns might students notice as they make counting strips or write numbers on a 1,000 chart that could help them figure out some important aspects of the number system?

Place Value of 163

Understanding the place value structure of our base-ten number system is central to the work students do with numbers and to their ability to efficiently and flexibly solve computation problems. That involves knowing the values of the digits in a number (e.g. in 235, the 2 represents two hundreds, the 3 represents three tens and the 5 represents 5 ones), but there is much more students need to understand about place value of numbers and about the base-ten number system.

Consider these questions:

• What can you say about the number 163?
• What does the 1 stand for in 163?
• Is 163 closer to 200 or 100?
• How many 10s are in 163?

View student responses to the question, "How many 10s are in 163?". Click on the image to enlarge it.

In this activity, you will read and reflect on the different aspects of place value and the number system that students work on; engage in activities that highlight the place value of numbers; and examine the base-ten number system from whole numbers to decimals.

Aspects of Place Value

The following readings focus on the different aspects of place value that students work on throughout Investigations. Choose at least 3 grade levels to read.

Place Value readings by grade level:

• The Foundations of Place Value (Kindergarten)
• Place Value in First Grade
• Place Value in Second Grade
• Place Value in Third Grade

Consider: According to the readings how does the work on place value develop across the grades?

Ten Ones = One Ten

Much of the foundation for work with place value is laid in the early grades. Students engage in activities that highlight the importance of 10 in our number system. One important idea that students are working on in Kindergarten and first grade is that ten ones can also be thought of as one group of ten. This idea is highlighted in many of the activities, particularly when they do activities that involve working with sticks of 10 connecting cubes and when they work with Ten Frames.

A Ten Frame is an important model that students in Kindergarten and first grade use for counting and addition and subtraction; it reinforces the idea that ten ones can also be thought of as a group of ten ones or one ten.

Feature - Math Notes

Each session in the Investigations curriculum units includes sidebars which are a small form of professional development. These sidebars include Math Notes, Teaching Notes, Math Practice Notes, and Professional Development Notes.

Click on the image below to view it at a larger size.

Feature - Games

The games included in Investigations are a central part of the mathematics, not just enrichment activities. Games provide engaging opportunities for students to have repeated practice with important mathematical concepts and skills and to develop and deepen their mathematical understanding and reasoning.

Play a few rounds of the following games:

Consider: What aspects of place value and the composition of numbers are students working on in these games?

Tens and Ones

First and second-grade students move from working just with ones to working with groups of tens and ones using a few different models. This work helps them identify how numbers are composed and ways they can be broken into tens and ones. These models help students understand how two-digit numbers are written: the first digit represents the number of groups of 10 and the second digit designates the number of ones.

Feature - Math Words and Ideas

Math Words and Ideas is a digital resource that provides an overview of a grade level's year of mathematical work, a closer look at the ideas and the kinds of problems students encounter and examples of students' solutions. This component is designed to be used flexibly- as a resource for students to review concepts during class after they have been introduced, as a reference while doing homework, and/or as a reference for families to better understand the work their children are doing in class. It is not meant to be used to teach students new concepts, skills or strategies. In this course, we will use this resource as one means to share how students work on specific concepts in Investigations.

Math Words and Ideas

Watch the following Math Words and Ideas, Tens and Ones, which shows the different models first graders use to think about tens and ones and the important ideas about tens and ones they are working on.

Math Words and Ideas, Tens and Ones

46 Stickers

Second grade students are introduced to a context, a store called Sticker Station, that sells stickers individually (as singles, or ones), in strips of ten or in sheets of 100.

This context helps students think about place value, the principle upon which numbers in our base-ten number system are structured.

Solve the following second grade problem. You may want to record your work.

Imagine you wanted to buy 46 stickers from the Sticker Station.

What are all the different combinations of strips and singles you can buy to make 46 stickers? (You don't have to include strips and singles in each of your combinations.)

• How many combinations are there for this problem?
• How do you know you have all the combinations?

The Number System and Decimal Numbers

When students work with decimal numbers in fourth and fifth grade they learn how our number system extends to include numbers between whole numbers and how the place value structure continues.

Print out the Hundredths Grids. Have one grid of 100 squares equal one whole and use the grids to show 3, 0.3 .03 and 0.003.

• How are 3, 0.3, 0.03 and 0.003 related?
• How do your representations on the grids show this relationship?

Read Extending Place Value to Tenths and Hundredths

Once you have completed the work in this session, go to the Session 1 Discussion Forum.

What struck you about the complexities of counting, and/or learning about place value and the base ten number system as you completed this session? How does this connect to the work you are doing in your classroom?

Note:

• These questions are a stepping off point to start good conversations and therefore the questions are purposefully broad. There are no correct answers to the questions!


• If you start a thread think about what the key point or main focus is and title your post accordingly. This can help focus each conversation.

• Counting is the foundation for understanding the number system and almost all the number work in the primary grades. While counting may seem simple it is actually quite complex and involves the interplay between a number of skills and concepts.
• As students count sets of objects, count orally, write numbers on counting strips, work with number lines and 100 and 1,000 charts, they learn the oral and written counting sequences and make sense of the structure of the base-ten number system.
• Understanding the place value structure of our base-ten number system is central to the work students do with numbers and to their ability to efficiently and flexibly solve computation problems.
• To understand the place value of numbers, students need to understand what each numeral in a number represents; that one 10 is equal to ten 1s; that a number can be expressed as tens and ones (and hundreds and hundredths, etc.) in different ways, and understand a number's relationship to other numbers.