1 (3-5) Overview: Introduction to Numbers and Place Value
The teacher will identify and define multiplication and division terms and use them appropriately in context
The teacher will compare and contrast the properties of multiplication and division and explain their relationship as inverse operations
The teacher will identify the connection between multiplication, division, and place value to number sense
The creation of our base-ten system was an important and remarkable jump in mathematics for humanity.
Ten ones compose a new unit called a ten. Ten tens make a unit called a hundred. Because ten units make a unit of the next highest value, only ten digits are needed to represent any quantity in base-ten.
1.1 (3-5) Welcome
Welcome to Session 1: Introduction to Numbers and Place Value! By the end of this module, you will understand the importance of place value with regards to multiplication and division. A deep understanding of place value underlies all computational processes for multiplication and division. Because multiplication and division require students to decompose and compose numbers, students must understand our base ten number system.
In this session, you’ll…
Learn definitions for terms associated with multiplication and division. Multiplication and division play key roles in elementary mathematics. Prior to the CCSSM, instruction primarily focused on helping students develop procedural competencies with basic facts and algorithms. Although procedural fluency with multiplication and division is important, it is essential that students also build a conceptual understanding of multiplication and division through knowledge of place value.
Learn about the inverse relationship between multiplication and division. There is an inverse relationship between multiplication and division. Multiplication is a fundamental operation, with division defined in terms of multiplication. In multiplication, we find the product of two factors, in division we find the missing factor if the other factor and the product are known. Look at the following examples:
If you need a refresher on how multiplication and division are introduced in upper elementary grades, please read all of the following standards below. These standards describe the fundamental ideas of multiplication and division.
Third Grade: Operations and Algebraic Thinking – Standards 1-2Links to an external site.
Third Grade: Operations and Algebraic Thinking – Standard 7
1.2 (3-5) Multiplication and Division Terms
Think back to when you were first taught multiplication and division. The primary focus may have been on rote memorization of facts and applying a step-by-step process to find the correct answer. Do these examples look familiar?
Image: Citro, A. (2015). Unpublished Manuscript. Relay Graduate School of Education.
This type of work demonstrates procedural knowledge. A student’s ability to do multiplication and division calculations and recite multiplication and division facts does not mean that the student understands the concepts. The Common Core State Standards for Mathematics require teachers to take a step back from these approaches and begin teaching multiplication and division by building conceptual understanding.
The following reading synthesizes information from several resources on multiplication and division. This reading will give you precise definitions and examples for terms associated with multiplication and division. In order to build conceptual understanding as you teach, it’s essential to use precise language.
Click here Download here to download the online handout and use it to answer the following questions. You’ll use this handout for Sessions 1 through 4.
Which terms, if any, were familiar to you?
Which terms, if any, were new to you?
What is the difference between asymmetrical and symmetrical multiplication and division problems?
Why is it important to understand and know the distinction between these terms?
1.3 (3-5) Inverse Relationship
Students’ conceptual and procedural understanding of multiplication and division tends to develop slowly. Through exposure to different types of problems and ample practice, students begin to develop multiplicative reasoning—the ability to identify, understand, and use relationships between quantities that are based on multiplication and division.
All multiplicative situations involve three major elements:
groups of equal size
number of groups
Before students can truly understand the inverse relationship between multiplication and division, there are several cognitive shifts that need to take place.
Cognitive Shift #1
Students need to understand numbers as composite units. In an additive situation the number 4 represents 4 discrete, or individual, objects. In a multiplicative situation the 4 represents a unit or a group. Ultimately, students need to understand that a composite group can be counted multiple times and operates as an entity.
Cognitive Shift #2
Students need to understand the units of the quantities within multiplication and division problems. In multiplicative situations, students are dealing with quantities that represent two different units. (In additive situations, students work with quantities that are the same unit). Many multiplication problems are “unit transforming,” meaning that the units of the quantities are not the units of the solution.
In this video you will learn one strategy for supporting students with understanding the inverse relationship between multiplication and division.
As you watch, answer the following guiding questions in your handout:
(Please ignore the reference to the old course code, MATH-101, on the title slide of this video.)
How are multiplication and division related?
What number sense and place value skills do students need in order to understand the relationship between multiplication and division?
How can you support students with understanding the relationship between multiplication and division?
1.4 (3-5) Inverse Practice Analysis
- STUDENT WORK ANALYSIS EXAMPLE
Now that you have had a chance to learn about the relationship between multiplication and division, let’s take a look at some student work examples.
Begin this activity by looking at one annotated example.
Student work example retrieved from KIPP Academy Elementary School (2015). Unpublished manuscript. Used with permission.
MATH-101_3-5_02c_Number.JPG This math problem is a partitive division problem, so students are asked to find the number in each group. We know the original amount (32 glasses) and we know the number of groups (4 groups or rows). We are looking to determine the size of each group or row.
MATH-101_3-5_02_Number2.JPG This student used a skip counting by 4 strategy to determine the number of groups. The student then circled each group of 4 to determine that there were 8 total groups of 4 in the number 32.
MATH-101_3-5_02_Number3.JPG Once the student was able to determine the number of groups of 4 in 32, she wrote two equations—one multiplication and one division. The multiplication problem is labeled to demonstrate her understanding that 4 rows of 8 glasses equals a total of 32 glasses. The student then needed to write the equation to answer the problem’s original question: 32 total glasses are arranged into 4 equal rows. How many glasses in each row? Writing a multiplication and division problem demonstrates multiplicative thinking through recognizing how to determine the unknown factor. Once students have identified the unknown factor, they must then convert the unknown factor into a completed division problem. It is also important for students to recognize that the three numbers in the multiplication problem are the same as the three numbers in the division problem, but arranged differently.
MATH-101_3-5_02_Number4.JPG The student also modeled the multiplication and division problems through creating an array. It appears that this was done last as a way to check her work. An array is typically used as a multiplication strategy. The rows represent the number of groups and the columns represent the number within each group. The student labeled the rows and size of groups in her division equation, which demonstrates her understanding of the relationship between the two operations.
- STUDENT WORK ANALYSIS PRACTICE
Now it is your turn to analyze a student work example. While analyzing the work, you should:
Identify the type of multiplication and division. (Note: You may want to have your multiplication and division reader handy for this activity.)
Identify how the problem illustrates the relationship