1. 1 Overview of Class #10 Comments, questions? Multiplication: concept of multiplication and its main interpretations and models; multiplication algorithm for computing multi-digit numbers Remediating common student errors with multiplication computation The teacher’s role in whole class teaching: enacting high expectations, working on significant mathematics Wrapping up and Assignments

2. 2 Multiplication Write a word problem that can be solved with: 2 x 4 Draw a picture or model that could be used to represent your problem. Compare your problem with the person next to you: How are your problems similar? How are they different?

3. 3 Two Different Interpretations of the Meaning of Multiplication Equal groups: a x b means a “groups” of b “things” each, where what is meant by “groups” can vary Area: a x b gives an area (square units) of width a (linear units) and length b (linear units)

4. 4 Repeated addition as a model for multiplication Counting model Usually used to introduce multiplication as a short cut for adding Convention in U.S.: in a x b, a represents “number of groups” and b represents “size of group”

5. 5 Rectangular array as a model for multiplication Often used along with repeated addition to introduce multiplication Similar to repeated addition, except that items are arranged in an orderly arrangement (“array”) Counting model, even if objects border one another:

6. 6 Cartesian product as a model for multiplication Counting model Each member from the first set is matched with each member from the second set; this mapping produces a set of ordered pairs Often taught without naming in elementary school –(combination problems such as “2 shirts, 4 pairs of pants, how many outfits?”)

7. 7 Area as a model for multiplication Continuous model Convention in U.S.: in a x b, a represents width and b represents length Two kinds of measure: –a and b are lengths (linear units) –a x b is an area (units of area)

8. 8 Models for Multiplication groups area

9. 9 Modeling Multiplication Computation Calculate the answer Show as an area with base ten blocks Explain the correspondence between the area and the written procedure

10. 10 Modeling Multiplication Computation as Area

11. 11 What does NOT work? Repeating the same things over again, slower, more loudly Re-teaching everything Teaching that “coerces” or forces students to get the “right answer” Teacher does the problem for the student Teacher gives students more of the same problems to practice without any intervention Teacher refers the student to easier problems and never comes back to remediate error What DOES work? Identifying carefully where the problem(s) lie(s) Using manipulatives or other representations to focus on and connect the meaning and the procedure Sharing the talk with the student, scaffolding Offering a similar example to try Teacher reduces complexity in some way and then encourages connection of that work with previous problem Remediation

12. 12 Three Common Student Errors What is the error? What produces this error, and what does that show about the student’s understanding? What does the textbook suggest you do to teach this? How adequate are the book’s suggestions? How would you use and also change these suggestions, and why? 123

13. 13 Wrapping Up Assignments (weeks 10 & 11) Questions about Project #2? Collect notebooks

14. 14 Assignments For November 13 (next week): –Part 2 of Project #2 Turn in lesson plan (submit the one you actually used) Explain how you addressed Part 2 considerations For November 20: –Reading (Lampert, Ch. 7) –Representing multi-digit multiplication using the area model –Remediating a common multiplication error