## Reflection: Developing a Conceptual Understanding Multiplying with Mathmaster Chef (Day 1 of 2) - Section 2: Warm up

The two things that were most helpful for my students were:

1. Contextualizing the problems with hot and cold cubes.
2. Connecting the idea of equal group in addition and multiplication.

This lesson has a pretty clear focus on the context, but the relationship between addition and multiplication has to be explicitly attended to. The context definitely helps make this relationship clear to students. My students have now had numerous experiences with addition and subtraction and I want their learning of multiplication to build on that. Many of the common misunderstandings and errors I have come to expect in integer multiplication can be confronted and addressed with reference to the work my students have done with addition and subtraction.

Relating multiplication to what my students already know about addition and subtraction prevents them from having to memorize disconnected rules that don't make sense to them. For example:

• 'A negative times a negative is a positive' relates to subtracting negative integers. My students have learned that subtracting negatives is equivalent to adding positives because removing cold cubes and adding hot cubes both result in a temperature increase. Connecting this idea to multiplication requires that students extend their understanding of removing cold cubes to repeatedly removing equal groups/bags of cold cubes.
• 'A negative times a positive is a negative' extends from the same idea of subtracting as equivalent to adding the opposite. Removing positives extends to repeatedly removing equal groups or positives (bags of hot cubes).
• 'A positive times a negative is negative' extends from adding negatives to repeated addition of equal groups of negatives (bags of cold cubes).

In the past my students had little understanding of the origin or mathematical foundations of the integer multiplication rules they had memorized and they often mixed them up or forgot them. They would raise their hand and ask me "is a negative times a negative positive or negative?" and I would have little recourse but to answer their question. Or they would respond that a negative times a positive was negative and I would have nothing to fall back on, but to remind them of the rule. Now when they forget the rules they have a way to figure it out for themselves.  I can ask them questions that relate multiplication to addition and subtraction and to the context of hot and cold cubes rather than just remind them of rules:

• "What does this have to do with/how could we explain this with hot and cold cubes?"
• "How does this relate to addition or subtraction?"
• "What would this look like as addition/subtraction?"

These connections helped my students make sense of rather than recall the rules of integer multiplication.

Developing a Conceptual Understanding: Be explicit about relationship between addition & multiplication

# Multiplying with Mathmaster Chef (Day 1 of 2)

Unit 4: Operations with Integers
Lesson 20 of 24

## Big Idea: Students are introduced to a context for multiplying integers.

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Standards:
Subject(s):
Math, Number Sense and Operations, Operations and Expressions, multiplying integers, providing context, relationship between addition & multiplication
54 minutes

### Erica Burnison

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