Chapter 12

Developing Strategies for
Multiplication and Division Computation

12.1 recognize how understanding place value and the Properties of operations support the learning of reasoning strategies in multiplication

Direct Modeling: Repeatedly grouping the same number of items through a variety of reprensentations.

Direct modeling of equal groups, counting, and arrays of equal row size.
Adding on and and doubling up.

Decomposition Strategies:

In this strategy, students break apart numbers that show they understand place value. Each factor can be decomposed into groups.
Breaking at least one of the factors into components, making a chain of multiplication calculations with simpler numbers.

Compensation Strategies:

One factor is adjusted to compensate for a change in the other factor. This can help students move numbers to make calculations easier.
The problem is changed to an easier one, and then an adjustment or compensation is made such as halving one factor and then doubling the other.

Number Strings: Also known as cluster problems are helpful in guiding students from easy to more difficult and complex computations.

(Note: All that is required to begin a number string is that your collection of problems eventually leads to a solution. Pearson 2023. Van De Walle, Karp, Bay-Williams)

12.2 Identify a variety of models and recording approaches for developing the multiplication algorithm.

Identifying various models and recording approaches for developing the multiplication algorithm involves using different visual and structured methods, like the array/area model, base-ten grid paper, partial product recording, and lattice multiplication, to help students understand and perform multiplication.

Developing Written Records:

Multiplication Recording Sheets:
- Help students record partial products clearly.
- Teach the method of writing partial products separately to avoid errors.
Recording Partial Products:
- Students can record in any order, reducing mistakes related to regrouping.
- Encourage recording each step to ensure clarity and accuracy.

Alternative Multiplication Strategies:

Array/Area Model:
- Most accurate and popular among students.
- Encourages breaking down multiplication into manageable parts.
- Use base-ten grid paper for visual representation.
- Move from drawing large rectangles to using more precise grid paper.
Cluster Problems:
- Second most accurate and frequently used.
- Develops flexibility in solving problems by breaking them into clusters.

Transition from Models to Written Format:

Partial Products:
- Progress from using physical base-ten materials to drawing rectangles on grid paper.
- Four partial products correspond to sections of the rectangle, connecting to the standard algorithm.
Language and Conceptual Understanding:
- Use base-ten language (e.g., 3 tens x 4 tens = 12 hundreds) to promote understanding of place value.
- Avoid simplifying to single-digit multiplication (e.g., "three times four").

Lattice Multiplication:

Historical and Cultural Method:
- Uses a grid with squares split by diagonal lines.
- Organizes thinking along place-value columns diagonally.

Instructional Tools and Resources:
- Base-Ten Grid Paper:
  - Visual tool for arranging base-ten materials.
- Multiplication Recording Sheets:
  - Support accurate recording of partial products.
- Lattice Multiplication Templates:
  - Provide a structured approach for lattice multiplication.

12.4 Explain the development of standard algorithm for division and ways to record students' thinking. p.279-282 & p.286-292

Below are some exampled of HOW to Record Student Thinking.

Recording Student Thinking

Written Explanations:
- Have students write out their thought process step-by-step while solving division problems. This helps them clarify their understanding and allows teachers to identify misconceptions.
Use of Graphic Organizers:
- Tools like division graphic organizers or step-by-step templates can help students organize their work and make their thinking visible.
Math Journals:
- Encourage students to keep math journals where they reflect on their learning, write about strategies they used, and solve problems. This continuous reflection helps reinforce learning.
Think-Alouds:
- Conduct think-aloud sessions where students verbalize their thought process while solving a problem. This method helps teachers assess students' understanding and provide immediate feedback.
Peer Discussions:
- Encourage students to explain their methods to peers or work in pairs/groups to solve problems. Collaborative learning helps students articulate their thinking and learn from each other.

Multiplication

Standard Algorithm

Conceptual Understanding: Teachers should ensure students comprehend the underlying concepts of multiplication rather than just memorizing procedures. This includes understanding place value and the distributive property, which are fundamental to the standard algorithm.
Connection to Prior Knowledge: Effective multiplication instruction builds on students’ previous knowledge of addition and earlier multiplication strategies. Teachers need to connect the standard algorithm to these foundational skills to help students see the coherence in mathematics.
Error Analysis and Misconceptions: Teachers should be equipped to identify common errors and misconceptions students might have when using the standard algorithm. Understanding these potential pitfalls allows teachers to provide targeted interventions and clarifications.

Division

Standard Algorithm

Understanding Place Value and Estimation: Pre-service teachers need to understand how place value and estimation play crucial roles in the division process. When teaching the standard algorithm, it's important to emphasize how each digit in the dividend and divisor corresponds to specific place values, and how students should estimate the quotient to guide their calculations effectively. This helps students to develop a deeper conceptual understanding rather than just following procedural steps blindly.
Interpreting Remainders: Another important aspect is how to handle remainders in division problems. Pre-service teachers should be adept at explaining the meaning of remainders in different contexts—whether they need to be expressed as fractions, decimals, or left as whole numbers, depending on the problem scenario. This skill is vital for helping students understand the practical applications of division and ensuring they can interpret the results correctly.
Connection to Multiplication and Division Facts: Effective use of the standard algorithm requires strong foundational knowledge of basic multiplication and division facts. Pre-service teachers should ensure that students have fluency in these areas, as it allows for smoother application of the algorithm. This foundational knowledge supports students in checking their work and understanding the relationships between the numbers involved in division problems.

12.5 Identify ways to teach computational estimation for multidigit multiplication and division as a way to develop students' flexibility and ability to recognize reasonable answers. p. 292-297

Estimation skills help students understand an approximation of multiplication and division. Teaching estimation requires the student to show mastery of place value and multiplication/ division concepts.

Division: 12.3

Standard Algorithm: The standard algorithm for long division is a series of steps repeated in this order: divide, multiply, subtract, bring down. With the standard algorithm, we solve division problems one place value at a time.

Compatible Numbers: Students can adjust the divisor or dividend (or both) to close numbers to create division problems that result in whole numbers that are easier to do mentally.

Partial Quotients: Partial quotients, or chunking, uses repeated subtraction to solve problems. This makes the problems easier for students because it pulls from multiplication knowledge that they're familiar with; i.e. 2's, 5's and 10's multiplication facts.

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