## Reflection: Discourse and Questioning Connecting Multiplication and Division - Section 2: Connecting Multiplication and Division

Over the three sets of problems, students were able to see a pattern connecting the multiplication and division problems.  They were able to see that dividing a number by 2 is the same as multiplying that number by ½.  They were also able to extend this pattern to divisors other than 2.  At the end of this section, I posed a question: 10 divided by 'w' will result in the same answer as 10 x ______ .  Some students immediately reacted to my use of a variable with “I don’t know!” and “W?! How are we supposed to know what w is?!”  I explained that we had been working on finding a connection between multiplication and division, and that I wanted to see if they could use what they’ve learned to demonstrate the connection using variables.  I asked if someone had a number we could substitute in for w.  One student said we could use 5 for w.  10 divided by 5 = 2, so 10 x _____ will also result in 2?  Students participated in a Think Pair Share with their partners.  Students were able to conclude that 10 x 1/5 would also result in 2.  After a bit more discussion, a couple students were able to explain that 10/w is equivalent to 10 x 1/w.  I substituted 2 in for w to demonstrate that it indeed worked.

One of my classes had a little extra time, so I asked them what w divided by w would equal.  Students were able to say that any number divided by itself will result in 1, so w divided by w will equal 1.  Then I asked students what the missing factor would be to also create an answer of 1:  w x ______ = 1.  Students participated in a Think Pair Share.  Some students said w, and I explained that w times w would result in w squared.  I connected this to multiplication, 3 x 3 is equal to 3 squared, or nine.  A student figured it out and I asked her to come to the board to explain her answer.  She said w x 1/w would equal one, because if you multiply the two amounts you would get w/w and any number over itself (like 2/2, 3/3, 4/4, etc) equals one.

Some students continue to struggle to extend patterns using variables and this is okay.  I think it is important to expose students to these questions and have them work through them.  Some students struggle with the idea that a variable could stand for any number.  The more I push students to extend patterns using variables, the more skilled they will become.

Extending Patterns Using Variables
Discourse and Questioning: Extending Patterns Using Variables

# Connecting Multiplication and Division

Unit 4: Fraction Operations
Lesson 12 of 19

## Big Idea: How are multiplication and division connected? Why do 11 divided by 2 and 11 times ½ result in the same answer? Students make connections between multiplication and division and work to develop strategies for dividing whole numbers by unit fractions.

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Standards:
Subject(s):
Math, Number Sense and Operations, division, multiplication, 6th grade, master teacher project, fraction division
60 minutes

### Andrea Palmer

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