##
* *Reflection: Diverse Entry Points
Commutative - Concrete Representations - Section 1: Introduction

I think it's very important for 8 and 9 year old children to have plentiful opportunities to manipulate concrete objects while reasoning their way to an understanding of an abstract concept such as the commutative property. (When I think back on my own experience with math in school, I think I would have been happy to have had more experiences with models and manipulatives even in middle school and high school!) I suspect that for visual learners and kinesthetic learners, this isn't a strategy that we grow out of.

Mathematical Practice 4, Model with Mathematics, speaks to the importance of supporting students in developing their mathematical thinking by using representations, including drawing and using objects. I'm excited that in the Common Core there is more room to address different learning styles instead of the previous exclusive focus on standard algorithms and abstract pencil paper reasoning.

Some students immediately see the connection between the commutative property of addition and the commutative property of multiplication. For others, this is something that has to be seen, felt and experienced many times before they believe it. That's okay! This activity provides an opportunity for the learners who need to see it and feel it to have that time, and in my experience those children who have a more ready grasp of the commutative principle still benefit from and enjoy the process of making the models. It's good for them to have fun with math, and a comment I frequently hear when they are building different models, is, "This is fun!"

Meaningful fun cements learning in a more powerful way than something dogmatic and forced ever will! Isn't it great that we now have the support to go deeper in our mathematical explorations with our students?!

Observing students in focused exploratory settings such as the one in this lesson gives me valuable anecdotal evidence that I can use to communicate with both families and the child about strengths and weaknesses, and notes from situations like this lesson also consistently give me relevant information about how to differentiate for both individual children and the group personality. We all know how different some groups/cohorts can be from the previous years' students, and it's essential that we readjust accordingly.

*Commutative Property of Multiplication- The Importance of Models*

*Diverse Entry Points: Commutative Property of Multiplication- The Importance of Models*

# Commutative - Concrete Representations

Lesson 7 of 11

## Objective: SWBAT make a concrete model (with manipulatives) of a pair of math facts to demonstrate and explain the commutative property. REVISED 9/29/15

## Big Idea: Students are able to illustrate the commutative property and articulate the differences and similarities between two models demonstrating the same pair of facts.

*60 minutes*

#### Introduction

*7 min*

This lesson is straightforward. Students will build models of different multiplication equation pairs that demonstrate the commutative property. By the end of third grade, students are expected to understand the properties of operations and fluently and flexibly apply these rules of how numbers work as strategies to multiply and divide (3.OA.5).

In order to maximize learning time, I have the construction paper work mats out and the cubes/pattern blocks that each student will be using counted out ahead of time. Then playing with the blocks at the start isn’t an issue, but instead can be a reward for children who finish a few minutes ahead of time.

For this lesson I needed 30 cubes and 30 pattern blocks for each child. I intentionally use two different types of manipulatives, because it enables me to do quick visual scans of the room to monitor understanding while I also engaging in one-one or one-two conversations. If I had enough cubes for each child to have a set of 30 in 2 different colors, that would have been an even neater visual.

I start the lesson with a review of the commutative property of addition and asked students how this rule is helpful when they are manipulating numbers. Then I ask them to think about multiplication, *"Do you think the same rule would apply? Why or why not?"*

About half of my students knew that there is a commutative property to multiplication (not the word itself, but the concept) but only a few of them could explain how it works using mathematical terms.

It is important to review procedures for using manipulatives, such as reminders about not building towers (tools, not toys). I also reteach the strategy of moving unneeded blocks to either a side of their desk or in their desk. For example, if the example they were working on is 7 x 4 and 4 x 7, it's imperative that they remove the 2 extra blocks (they started with 30) off their desk. I've found that if I don't explicitly teach students how to do that, at least 1/2 of the class will leave all the blocks on their desk and then that ends up getting them confused. Check the number you are starting with is an important step prior to starting to create the models!

#### Resources

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#### Activity 1

*25 min*

I planned the list of math facts I wanted to use for this activity to ensure a balance of different numerals. I don’t know about you, but when I create equations on the spot, I seem to repeat certain patterns and numerals.

We did the first 3 examples together, and I quickly circulate to help students who needed a better system for getting their blocks on the paper.

Then I projected the remaining set of equations and let kids who were ready go ahead.

##### Resources (5)

#### Resources

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#### Activity 2

*25 min*

The difference between the guided practice portion of this lesson and the independent practice portion is that this section is leveled. If students are working in the middle or far-right column they will need more than 30 cubes. I have a limited number of manipulatives, so I had them work with a partner so that they benefited from both collaboration and sharing of resources.

That took some "on the spot" regrouping of students, based on what I'd observed during the guided practice. This is an example of a time I find a simple list helpful. For my purposes only, I used a checklist and simple symbols. I used a “?” to indicate a student who needed to revisit the simplest example, a smiley face for a student who was ready to make the larger groupings in the middle column, and a star for student who understood the concept well enough to be bored by it and needed more of a challenge.

The examples in the column on the far right are, of course, outside of grade level as 3rd grade content standards only go up to 10 x 10. I think it's okay to have students who are ready, work with more challenging examples, because in doing so they further cement their understanding of, and fluency with, the simpler facts.

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#### Closure

*3 min*

At the conclusion of this lesson, I have students restate in their own words what they learned (the commutative property of multiplication - they don't need to know the name). I have them talk to a partner because that way everyone is participating, and I've gotten very systematic (finally) about rotating around the tables so the squeaky wheels or those in close physical proximity to wherever I'm teaching don't receive a disproportionate amount of my attentions!

After students state what they understand/ learned about the commutative property of multiplication, I then ask, "Why does it matter?" This is very open-ended, and all I look for here is that they're thinking about it. Were I put on the spot, I might be hard-pressed to answer that question, but that doesn't mean it isn't worthwhile.

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- UNIT 1: 1st Week: Getting to Know Each Other Through Graphs
- UNIT 2: Addition and Subtraction
- UNIT 3: Multiplication
- UNIT 4: Introduction to Basic Division
- UNIT 5: Division in Context
- UNIT 6: Time
- UNIT 7: Rounding
- UNIT 8: Place Value Practice
- UNIT 9: Fractions
- UNIT 10: Math and Me: Nutrition, Health and More
- UNIT 11: Geometry in Architecture
- UNIT 12: Time Cycle 2
- UNIT 13: Patterns in Math
- UNIT 14: Area and Perimeter
- UNIT 15: Solving Mult-Step Word Problems Using the Four Operations
- UNIT 16: Musical Fractions
- UNIT 17: Volcanoes (Data Collection, Graphs, Addition & Subtraction)

- LESSON 1: Tubing Tens - Cool Off with an Imaginary River Adventure
- LESSON 2: Tubing Tens - Create a Story
- LESSON 3: African Amphibians - Review Multiplying by Ones & Twos While Learning Fun Facts About Frogs
- LESSON 4: Open Number Lines - Just the Basics
- LESSON 5: Open Number Line - Decomposed Factors
- LESSON 6: Swimming Swamp Monkeys - Using an Open Number Line to Solve Word Problems
- LESSON 7: Commutative - Concrete Representations
- LESSON 8: Fun with Four: Decomposing to Known Factors of 1, 2, and 5
- LESSON 9: Counting Clams - A Tidal Flat Adventure
- LESSON 10: Monster Math
- LESSON 11: The Video Game Model of Adding