A Model for Multiplication

Developing an Algorithm for Fraction Multiplication

Students develop an algorithm for multiplying improper fractions during the exploration of Problems 3.1 and 3.2. The video focuses on the learning trajectory from first exposure to a model for representing a part to developing an algorithm for the computation.

This video starts with the teacher's comments on her goals for the lesson and what difficulties students might encounter. Most of the discussions occur around Problem 3.1. At the end of Problem 3.2 students are ready to write down an algorithm for multiplying improper fractions. The focus is on student understanding.

Real time is 3 class periods.

As you view the video, consider the following:

• What evidence is there of students' understanding and reasoning? What prior understandings do you see students using or building upon?
• What is the evidence of students' use of the Mathematical Practices from the Common Core Standards for Mathematics (CCSSM)? To what extent are students growing in their sophistication of using the Mathematical Practices?
• Were there any "aha" moments?
• How is the teacher facilitating the development of students' understanding and reasoning?
• As a teacher, how would you use this information from this lesson to plan the next lesson?

• Full Length Video (24:47)
• Problem 2.1
• Problem 3.1
• CMP2 Goals for CMP2 Bits and Pieces II, Problem 3.1
  • Estimate products of fractions
  • Use models to represent the product of two fractions
  • Understand that finding a fraction of a number means multiplication
  • Develop and use strategies and models for multiplying combinations of fractions, whole numbers, and mixed numbers to solve problems
  • Determine when multiplication is an appropriate operation
  • Explore the relationships between two numbers and their product
  • Develop and use an efficient algorithm to solve any fraction multiplication problem
• CMP3 Goals for CMP3 Let's Be Rational, Problem 2.1
  • Numeric Estimation

    • Understand that estimation is a tool used in a variety of situations including checking answers and making decisions, and develop strategies for estimating results of arithmetic operations.

      Use benchmarks and other strategies to estimate results of operations with fractions.Use estimates to check the reasonableness of exact computations Give various reasons to estimate and identify when a situation calls for an overestimate or an underestimate. Use estimates and exact solutions to make decisions.

  • Fraction Operations

    • Revisit and continue to develop meanings for the four arithmetic operations and skill at using algorithms for each.

      Determine when addition, subtraction, multiplication, or division is the appropriate operation to solve a problem.

      Develop ways to model sums, differences, products, and quotients with areas, fraction strips, and number lines.

      Use knowledge of fractions and equivalence of fractions to develop algorithms for adding, subtracting, multiplying, and dividing fractions.

      Write fact families with fractions to show the inverse relationship between addition and subtraction, and between multiplication and division Compare and contrast dividing a whole number by a fraction to dividing a fraction by a whole number. Recognize that when you multiply or divide a fraction, your answer might be less than or more than the numbers you started with Solve real-world problems using arithmetic operations on fractions.

• Video Transcript
• Suggestions for organizing Professional Development
• Student Work
  • Problem 3.1 A
  • Problem 3.1 A
  • Problem 3.1 A
  • Problem 3.1 B
  • Problem 3.1 C1
  • Problem 3.1 C2
  • Problem 3.1 C2
  • Problem 3.1 C3
  • Problem 3.1 C3
  • Problem 3.1 C4
  • Problem 3.1 C4
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