Study Guide

Multiplication and Division with Whole Numbers: 1. Represent multiplying and dividing whole numbers with base ten pieces (including the area model and associated partial products for multiplication). 2. Be able to explain how the area model and partial products may be used to help students eventually make sense of the standard multiplication algorithm. 3. Realize that multiple methods exist for multiplying and dividing numbers. These algorithms rely upon sensible ways of composing (grouping together) and decomposing (breaking apart) numbers. 4. Be able to analyze elementary students' alternative algorithms for multiplication and division. 5. Recognize that the Fundamental Counting Principle is another way to represent multiplication, i.e., if an event E can occur in e ways, and after it has occurred, an event F can occur in f ways, then event E followed by event F can occur in e x f ways. Be able to solve such problems using multiplication, a tree diagram, a table, an organized list, and a matching diagram. 6. Be familiar with and able to use the Scaffold Method for Division. Addition and Subtraction with Integers: 7. Represent adding and subtracting integers with two-color counters and the chef story. 8. Know and be able to illustrate that subtracting a negative number has the same result as adding that number. Word Problems: 9. Write, identify, and solve different types of word problems according to the Taxonomy for Multiplication and Division Word Problems. 10. Be able to identify the type of strategy (Direct Modeling, Counting, or Derived Number Fact) used by a child in solving a multiplication or division story problem. 11. Know how to assist children in progressing over time from direct modeling to counting to derived number fact strategies, including considerations of the difficulty of the different types of multiplication and division word problems. Properties of Numbers and Operations: 12. Know that properties such as commutativity, associativity, and distributivity are helpful in mental computations as well as for reducing the number of basic facts that elementary students have to memorize. 13. Be able to state the property that is being used in a computation; this includes the associative and commutative properties for addition and multiplication as well as the distributive property of multiplication over addition. 14. Be able to illustrate with a drawing or with manipulatives the aforementioned properties.